﻿ 基于三维格子Boltzmann铝点蚀动态数值模拟
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Dynamic numerical simulation of aluminum pitting based on 3D-lattice Boltzmann method
YANG Guangfeng, LU Mengke, XUE Anyuan, LI Hulin, CUI Jing
Airport College, China Civil Aviation University, Tianjin 300300, China
Abstract: As one of the most economical and widely applied materials, aluminum has the advantages of light weight, corrosion resistance, and good thermal and electrical conductivity. However, aluminum is prone to pitting in chloride ion environments. Taking into account the aluminum surface pitting mechanism in neutral solution and liquid flow and phase transition characteristics, this study established a three-dimensional lattice Boltzmann model to examine the pitting phenomenon on aluminum surface in neutral solution, overcoming the deficiency of traditional experimental methods. The whole aluminum pitting process was simulated by the established corrosion model, and the concentration distribution of different components and the relationship between different parameters and corrosion degree were obtained. The results show that the degree of aluminum corrosion increases with the increase of contact time. Furthermore, the higher the initial chloride ion concentration, and the higher the corrosion reaction rate is, the more severe aluminum corrosion becomes. Therefore, the three-dimensional lattice Boltzmann aluminum pitting model provides a new approach to the future numerical study of metal corrosion.
Keywords: lattice Boltzmann method    aluminum pitting    electrochemical reaction    corrosion reaction rate    dynamic numerical simulation    corrosion

1 格子Boltzmann腐蚀模型 1.1 多组分模型

 $\begin{array}{*{20}{c}} {{f_\alpha }(x + c{\mathit{\boldsymbol{e}}_\alpha }\Delta t,t + \Delta t) - {f_\alpha }(x,t) = }\\ { - \frac{1}{{{\tau _v}}}({f_\alpha }(x,t) - f_\alpha ^{{\rm{eq}}}(x,t))} \end{array}$ （1）

D3Q19模型的平衡分布函数为

 $f_\alpha ^{{\rm{eq}}} = {\omega _\alpha }\rho \left[ {1 + \frac{3}{{{c^2}}}({\mathit{\boldsymbol{e}}_\alpha } \cdot \mathit{\boldsymbol{u}}) + \frac{9}{{2{c^4}}}{{({\mathit{\boldsymbol{e}}_\alpha } \cdot \mathit{\boldsymbol{u}})}^2} - \frac{3}{{2{c^2}}}{\mathit{\boldsymbol{u}}^2}} \right]$ （2）

 图 1 D3Q19模型 Fig. 1 D3Q19 model

 ${\rho = \sum {{f_\alpha }} }$ （3）
 ${\rho \mathit{\boldsymbol{u}} = \sum {{f_\alpha }} {\mathit{\boldsymbol{e}}_\alpha }}$ （4）

 ${\mathit{\boldsymbol{F}}_{\sigma \sigma }} = - {G_{\sigma \sigma }}{\psi _\sigma }(x)\sum\limits_{\alpha = 0}^{18} w (|{\mathit{\boldsymbol{e}}_\alpha }{|^2}){\psi _\sigma }(x + {\mathit{\boldsymbol{e}}_\alpha }){\mathit{\boldsymbol{e}}_\alpha }$ （5）

 ${\mathit{\boldsymbol{F}}_{\sigma w}} = - {w_{\sigma w}}{\psi _\sigma }(x)\sum\limits_{a = 0}^{18} w (|{\mathit{\boldsymbol{e}}_\alpha }{|^2})s(x + {\mathit{\boldsymbol{e}}_\alpha }){\mathit{\boldsymbol{e}}_\alpha }$ （6）

 $\mathit{\boldsymbol{u}} = {\mathit{\boldsymbol{u}}^\prime } + \frac{{{\tau _v}{\mathit{\boldsymbol{F}}_\sigma }}}{\rho }$ （7）

1.2 质量传输模型

 $\begin{array}{*{20}{c}} {{g_{k,\alpha }}(x + c{\mathit{\boldsymbol{e}}_\alpha }\Delta t,t + \Delta t) - {g_{k,\alpha }}(x,t) = }\\ { - \frac{1}{{{\tau _{k,g}}}}({g_{k,\alpha }}(x,t) - g_{k,\alpha }^{{\rm{eq}}}(x,t))} \end{array}$ （8）

 $g_{k,\alpha }^{{\rm{eq}}} = {C_k}\left( {{J_{k,\alpha }} + \frac{1}{2}{\mathit{\boldsymbol{e}}_\alpha } \cdot \mathit{\boldsymbol{u}}} \right)$ （9）

 ${J_{k,\alpha }} = \left\{ {\begin{array}{*{20}{l}} 0&{\alpha = 0}\\ {1/6}&{\alpha = 1,2, \cdots ,6} \end{array}} \right.$ （10）

D3Q7模型如图 2所示。

 图 2 D3Q7模型 Fig. 2 D3Q7 model

 ${C_k} = \sum {{g_{k,\alpha }}}$ （11）

 ${D_k} = \frac{1}{2}(1 - {J_{k,0}})({\tau _{k,g}} - 0.5)$ （12）

1.3 电化学反应模型

 图 3 金属铝的点蚀多重反应示意图 Fig. 3 Diagram of multiple reactions for aluminum pitting

1) 钝化膜溶解:

 ${\rm{ ①\ Al}}({\rm{ oxide }}){\rm{OH}} + {{\rm{H}}^ + } \to {\rm{Al}}({\rm{ oxide }}){\rm{OH}}_2^ +$
 $\begin{array}{l} {\rm{②\ Al}}({\rm{ oxide }}){\rm{OH}}_2^ + + n{\rm{C}}{{\rm{l}}^ - } \to \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{Al}}({\rm{ oxide }}){\rm{OH}}_2^ + {\rm{Cl}}_n^{ - n} \end{array}$
 $\begin{array}{l} {\rm{③\ Al(oxide) OH}}_2^ + {\rm{Cl}}_n^{ - n} \to \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{Al}}\left[ {\left( {n{\rm{C}}{{\rm{l}}^ - }} \right)({\rm{oxide }})} \right]{\rm{OH}}_2^ + \end{array}$
 $\begin{array}{l} {\rm{④\ Al}}\left[ {\left( {n{\rm{C}}{{\rm{l}}^ - }} \right)({\rm{oxide }})} \right]{\rm{OH}}_2^ + \to \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{A}}{{\rm{l}}^ + }\left[ {\left( {n{\rm{C}}{{\rm{l}}^ - }} \right)({\rm{oxide }})} \right]{\rm{OH}}_2^ + + {{\rm{e}}^ - } \end{array}$
 $\begin{array}{l} ⑤\ {\rm{A}}{{\rm{l}}^{2 + }}\left[ {\left( {n{\rm{C}}{{\rm{l}}^ - }} \right)({\rm{oxide }})} \right]{\rm{OH}}_2^ + \to \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{A}}{{\rm{l}}^{3 + }}\left[ {\left( {m{\rm{C}}{{\rm{l}}^ - }} \right)({\rm{oxide }})} \right]{\rm{OH}}_2^ + (n - m){\rm{C}}{{\rm{l}}^ - } + {{\rm{e}}^ - } \end{array}$
 $\begin{array}{l} {\rm{⑥}}{\kern 1pt} {\kern 1pt} {\rm{A}}{{\rm{l}}^ + }\left[ {\left( {n{\rm{C}}{{\rm{l}}^ - }} \right)({\rm{oxide }})} \right]{\rm{OH}}_2^ + \to \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{A}}{{\rm{l}}^{2 + }}\left[ {\left( {n{\rm{C}}{{\rm{l}}^ - }} \right)({\rm{oxide}})} \right]{\rm{OH}}_2^ + + {{\rm{e}}^ - } \end{array}$

Al2+[(nCl-)(oxide)]OH2++e-

2) 阳极反应：

 ${①{\kern 1pt} {\kern 1pt} {\rm{Al}} \to {\rm{Al}}_{{\rm{(ads)}}}^ + + {{\rm{e}}^ - }}$
 ${②{\kern 1pt} {\kern 1pt} {\rm{Al}}_{{\rm{(ads)}}}^ + \to {\rm{Al}}_{{\rm{(ads)}}}^{2 + } + {{\rm{e}}^ - }}$
 ${③{\kern 1pt} {\kern 1pt} {\rm{Al}}_{{\rm{(ads)}}}^{2 + } \to {\rm{Al}}_{{\rm{(ads)}}}^{3 + } + {{\rm{e}}^ - }}$

3) 阴极反应：

 $2{{\rm{H}}_2}{\rm{O}} + {{\rm{O}}_2} + 4{{\rm{e}}^ - } \to 4{\rm{O}}{{\rm{H}}^ - }$

4) 总反应：

 ${\rm{Al}} + 3{{\rm{H}}_2}{\rm{O}} + \frac{3}{2}{{\rm{O}}_2} \to {\rm{A}}{{\rm{l}}^{3 + }} + 6{\rm{O}}{{\rm{H}}^ - }$

5) 沉淀反应：

 ${\rm{A}}{{\rm{l}}^{3 + }} + 3{\rm{O}}{{\rm{H}}^ - } \to {\rm{Al}}{({\rm{OH}})_3}$

6) 水解反应：

 ${\rm{A}}{{\rm{l}}^{3 + }} + {{\rm{H}}_2}{\rm{O}} + {\rm{C}}{{\rm{l}}^ - } \to {{\rm{H}}^ + } + {({\rm{AlOHCl}})^ + }({\rm{eg}})$
 $\begin{array}{l} ①{\kern 1pt} {\kern 1pt} {\rm{A}}{{\rm{l}}^{3 + }} + {{\rm{H}}_2}{\rm{O}} \to {{\rm{H}}^ + } + {({\rm{AlOH}})^{2 + }}\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {K_{1,1}} = 1.07 \times {10^{ - 5}} \end{array}$
 $\begin{array}{l} ②{\kern 1pt} {\kern 1pt} {\rm{A}}{{\rm{l}}^{3 + }} + 2{{\rm{H}}_2}{\rm{O}} \to 2{{\rm{H}}^ + } + {\left[ {{\rm{Al}}{{({\rm{OH}})}_2}} \right]^ + }\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {K_{1,2}} = 3.16 \times {10^{ - 11}} \end{array}$
 $\begin{array}{l} ③{\kern 1pt} {\kern 1pt} {\rm{A}}{{\rm{l}}^{3 + }} + 3{{\rm{H}}_2}{\rm{O}} \to 3{{\rm{H}}^ + } + {\rm{AlO}}{{\rm{H}}_3}\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {K_{1,3}} = 3.16 \times {10^{ - 17}} \end{array}$

7) 自催化反应：

 ${\rm{Al}} + 3{{\rm{H}}^ + } + \to 3/2{{\rm{H}}_2} \uparrow + {\rm{A}}{{\rm{l}}^{3 + }}$
1.4 LB腐蚀边界迁移模型

 ${V_{{\rm{Al}}}}(t + \Delta t) = {V_{{\rm{Al}}}}(t) - A{\kern 1pt} {\kern 1pt} \overline {{V_{{\rm{Al}}}}} {k_1}{C_{{{\rm{O}}_{\rm{2}}}}}\Delta t$ （13）

 $\begin{array}{l} {V_{{\rm{Al(oxide)OH}}}}(t + \Delta t) = {V_{{\rm{Al(oxide)OH}}}}(t) - \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} A{\kern 1pt} {\kern 1pt} \overline {{V_{{\rm{Al( oxide )OH}}}}} {k_2}{C_{{\rm{C}}{{\rm{l}}^ - }}}\Delta t \end{array}$ （14）

 $\begin{array}{l} {V_{{\rm{Al(OH}}{{\rm{)}}_{\rm{3}}}}}(t + \Delta t) = {V_{{\rm{Al(OH}}{{\rm{)}}_{\rm{3}}}}}(t) + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} A{\kern 1pt} \overline {{V_{{\rm{Al(OH}}{{\rm{)}}_{\rm{3}}}}}} {k_3}{C_{{\rm{Al(OH}}{{\rm{)}}_{\rm{3}}}}}\Delta t \end{array}$ （15）

 ${{D_{{{\rm{O}}_{\rm{2}}}{\rm{(aq)}}}}\frac{{\partial {C_{{{\rm{O}}_{\rm{2}}}{\rm{(aq)}}}}}}{{\partial \mathit{\boldsymbol{n}}}} = {k_1}{C_{{{\rm{O}}_{\rm{2}}}{\rm{(aq)}}}}}$ （16）
 ${{D_{{\rm{C}}{{\rm{l}}^ - }({\rm{aq}})}}\frac{{\partial {C_{{\rm{C}}{{\rm{l}}^ - }({\rm{aq}})}}}}{{\partial \mathit{\boldsymbol{n}}}} = {k_2}{C_{{\rm{C}}{{\rm{l}}^ - }({\rm{aq}})}}}$ （17）
 ${{D_{{\rm{A}}{{\rm{l}}^{3 + }}({\rm{aq}})}}\frac{{\partial {C_{{\rm{A}}{{\rm{l}}^{3 + }}({\rm{aq}})}}}}{{\partial \mathit{\boldsymbol{n}}}} = - {k_1}{C_{{\rm{A}}{{\rm{l}}^{3 + }}({\rm{aq}})}}}$ （18）

1.5 结点钝化模型

 $\varepsilon = \left\{ {\begin{array}{*{20}{l}} 0&{R > P}\\ 1&{R \le P} \end{array}\quad P = P({C_{{{\rm{O}}_{\rm{2}}}}})} \right.$ （19）

 ${V_{{\rm{Al}}}}(t + \Delta t) = {V_{{\rm{Al}}}}(t) - \varepsilon A{\kern 1pt} {\kern 1pt} \overline {{V_{{\rm{Al}}}}} {k_1}{C_{{{\rm{O}}_{\rm{2}}}}}\Delta t$ （20）

2 数值模拟 2.1 点蚀演化过程分析

 变量 真实物理量 真实物理单位 格子数值 氯离子初始浓度CCl- 1.00 mol/m3 1 000 氯离子扩散系数DCl- 2.86×10-8 m2/s 0.017 5 氧气扩散系数DO2 4.50×10-8 m2/s 0.017 5 液体黏度v 1.00×10-6 m2/s 0.17 氧气初始浓度CO2 1.00 mol/m3 1 134.7 Al的摩尔质量$\overline {{V_{{\rm{Al}}}}}$ 7.09×10-3 m3/mol 2.93×10-3 Al(OH)3摩尔质量$\overline {{V_{{\rm{Al}}{{({\rm{OH}})}_3}}}}$ 3.053×10-2 m3/mol 0.019 1 腐蚀反应速率k1 2.44×10-3 m/s 8.14×10-5 钝化膜溶解速率k2 1.21×10-3 m/s 4.07×10-5 腐蚀产物沉淀速率k3 0.94×10-3 m/s 5.09×10-6 腐蚀产物饱和浓度Cs 1.00 mol/m3 1 000

 图 4 在x=20处点蚀坑形貌随时间变化的剖面图 Fig. 4 Profile of pitting morphology with time at x=20

 图 5 点蚀过程中氯离子浓度随时间分布的截面图 Fig. 5 Sectional view of concentration distribution of chloride ions over time during pitting

 图 6 点蚀过程中铝离子浓度随时间分布的截面图 Fig. 6 Sectional view of concentration distribution of aluminum ions over time during pitting

 图 7 点蚀过程中含氧腐蚀溶液里氧气浓度随时间分布的截面图 Fig. 7 Sectional view of concentration distribution of oxygen over time in corrosive oxygenated solution during pitting

 图 8 点蚀过程中水解产生的氢离子浓度随时间分布的截面图 Fig. 8 Sectional view of concentration distribution of hydrogen ion concentration by hydrolysis over time during pitting

 图 9 点蚀过程中氢氧根离子随时间的浓度分布 Fig. 9 Concentration distribution of hydroxide ions over time during pitting

 图 10 铝腐蚀蚀坑最大深度随时间变化关系 Fig. 10 Relationship of maximum depths of aluminum corrosion pits over time
 $h(t) = A{t^b}$ （21）

 图 11 蚀坑1中心(16，20，30)和蚀坑2中心(16，20，30)竖直方向不同点的浓度分布 Fig. 11 Concentration distribution at different points in vertical direction of center of pit 1 (16, 20, 30) and that of pit 2 (16, 20, 30)
2.2 影响因素分析 2.2.1 氯离子初始浓度

 模拟工况 腐蚀反应速率k1 Cl-初始浓度 Cl-扩散系数 1 1.0 2 1.1 3 5.08×10-3 1.2 0.017 5 4 1.3 5 1.4
 图 12 氯离子初始浓度不同条件下最终钝化膜xoz平面的形貌 Fig. 12 Morphology of xoz plane of final passivation film at different initial chlorine ion concentrations

 图 13 氯离子初始浓度不同条件下的蚀坑内氢离子分布情况 Fig. 13 Distribution of hydrogen ions in etch pit under different initial chlorineion concentrations

 图 14 不同初始氯离子浓度与蚀坑平均最大深度关系 Fig. 14 Relationship of different initial chlorine ion concentrations and average maximum depths of pit
2.2.2 腐蚀反应速率

 模拟工况 腐蚀反应速率k1 Cl-初始浓度 Cl-扩散系数 1 6.73×10-3 2 5.81×10-3 3 5.08×10-3 1.2 0.017 5 4 4.52×10-3 5 4.07×10-3
 图 15 不同腐蚀反应速率下的蚀坑xoz平面形貌 Fig. 15 Etch pit morphology of xoz plane with different corrosion reaction rates

 图 16 在不同腐蚀反应速率下蚀坑内氢离子浓度的分布 Fig. 16 Hydrogen ion concentrations in corrosion pit under different corrosion reaction rates

 图 17 腐蚀反应速率与蚀坑平均最大深度的关系 Fig. 17 Relationship of corrosion reaction rates and mean maximum depths of pit
3 结论

1) 使用格子Boltzmann方法建立了不同腐蚀条件下的流场以及不同组分的浓度场数学模型，可模拟真实的液体组分流动扩散；以电化学反应中半反应的方式为创新，对腐蚀体系中阴阳极的作用进行了描述；将金属钝化概率引入结点体积法之中，对金属阳极腐蚀速率和蚀坑形貌进行修正处理，并使用结点体积法对腐蚀产物的形态变化进行了研究，结点体积法可快速寻找到腐蚀边界。

2) 纯铝点蚀的全过程中，点蚀逐步经过了点蚀成核、亚稳态点蚀过程以及稳态点蚀过程；并且随着时间的增加，金属铝被腐蚀的体积越来越大。

3) 相同的腐蚀时间下，氯离子初始浓度的升高使得吸氧腐蚀电池更早地形成，铝点蚀坑内pH降低，因此金属铝被腐蚀的程度加深；在相同的腐蚀时间下，随着腐蚀电池中铝阳极反应速率的增加，蚀坑内生成大量铝离子，使得铝离子水解向正向移动，加速了蚀坑内的自催化反应，铝点蚀的程度加深。

4) 将模拟中铝点蚀蚀坑最大深度与时间进行拟合，得到的模拟结果与实验经验公式h(t)=Atb吻合良好。

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http://dx.doi.org/10.7527/S1000-6893.2019.23582

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#### 文章信息

YANG Guangfeng, LU Mengke, XUE Anyuan, LI Hulin, CUI Jing

Dynamic numerical simulation of aluminum pitting based on 3D-lattice Boltzmann method

Acta Aeronautica et Astronautica Sinica, 2020, 41(10): 423582.
http://dx.doi.org/10.7527/S1000-6893.2019.23582