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由于多孔介质结构的随机性, 很难对其内的胶体粒子输运过程进行建模. Boltzmann输运方程为模拟随机空间中胶体粒子的微观动力学提供了一种可靠的途径. 本文通过Chapman-Enskog(CE)分析, 从胶体粒子的Boltzmann方程导出了宏观输运模型. 该模型具有对流-扩散方程形式, 包括依赖粒子速度分布的扩散项、速度延迟项以及反映微观捕获机制的捕获项. 此外, 还给出了3个输运系数的显式表达. 该宏观模型部分解决了传统胶体输运模型的悖论, 并且在特定条件下与以往模型一致.
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关键词:
- 多孔介质 /
- 胶体输运 /
- 捕获 /
- Boltzmann方程
The structural randomness of porous medium presents significant challenges for accurately simulating colloidal transport. The Boltzmann transport equation (BTE) provides a reliable way for simulating the microscopic dynamics of colloidal particles in random space. By using the Chapman-Enskog (CE) method, a macroscopic advection-diffusion transport model is derived from the BTE. It includes a diffusion term dependent on the particle velocity distribution, a velocity delay term, and a capture term reflecting the microscopic capture mechanism, which tends to preferentially capture high-speed moving particles. These terms explain the delay and capture effects in colloidal transport. Meanwhile, the explicit expressions of the three transport coefficients are presented to provide a quantitative basis for using the model. The model is effective at small mixing filtration coefficients λl. By comparing the outlet concentration profiles of different models, the influences of this mechanism on the advection velocity delay and capture efficiency are elucidated. The model solves some of the paradoxes of traditional colloidal transport models, and under specific conditions, it is consistent with previous models. -
Keywords:
- porous media /
- colloidal transport /
- capture /
- Boltzmann equation
[1] Luna M, Gastone F, Tosco T, Sethi R, Velimirovic M, Gemoets J, Muyshondt R, Sapionet H, Klaas N, Bastiaens L 2015 J. Contam. Hydrol. 181 46
Google Scholar
[2] Tosco T, Gastone F, Sethi R 2014 J. Contam. Hydrol. 166 34
Google Scholar
[3] Fakhreddine S, Prommer H, Gorelick S M, Dadakis J, Fendorf S 2020 Environ. Sci. Technol. 54 8728
Google Scholar
[4] Boccardo G, Sethi R, Marchisio D L 2019 Chem. Eng. Sci. 198 290
Google Scholar
[5] 杨秀清, 胡亦, 张景路, 王艳秋, 裴春梅, 刘飞 2014 物理学报 63 048102
Google Scholar
Yang X Q, Hu Y, Zhang J L, Wang Y Q, Pei C M, Liu F 2014 Acta Phys. Sin. 63 048102
Google Scholar
[6] Salimi S, Ghalambor A 2011 Energies 4 1728
Google Scholar
[7] Winter C L, Tartakovsky D M 2002 Water Resour. Res. 38 23-1
Google Scholar
[8] Russell T, Dinariev O Y, Pessoa Rego L A, Bedrikovetsky P 2021 Water Resour. Res. 57 e2020WR029557
Google Scholar
[9] Zou Z K, Yu L, Li Y L, Niu S Y, Fan L L, Luo W B, Li W 2023 Water 15 2193
Google Scholar
[10] Shapiro A A 2024 Phys. Fluids 36 027118
Google Scholar
[11] Panfilov M, Rasoulzadeh M 2013 Comput. Geosci. 17 269
Google Scholar
[12] Shapiro A A 2022 Chem. Eng. Sci. 248 117247
Google Scholar
[13] Herzig J P, Leclerc D M, Goff P L 1970 Ind. Eng. Chem. 62 8
[14] Bedrikovetsky P 2008 Transp. Porous Med. 75 335
Google Scholar
[15] Bedrikovetsky P, Siqueira A G, de Souza A L S, Altoé J E, Shecaira F 2006 J. Pet. Sci. Eng. 51 68
Google Scholar
[16] Bradford S A, Leij F J 2018 Chem. Eng. Sci. 192 972
Google Scholar
[17] Molnar I L, Pensini E, Asad M A, Mitchell C A, Nitsche L C, Pyrak-Nolte L J, Miño G L, Krol M M 2019 Transp. Porous Med. 130 129
Google Scholar
[18] Zhang H, Malgaresi G V C, Bedrikovetsky P 2018 Int. J. Non - Linear Mech. 105 27
Google Scholar
[19] Arns C H 2004 Physica A 339 159
Google Scholar
[20] Arns C H, Knackstedt M A, Martys N S 2005 Phys. Rev. E 72 046304
Google Scholar
[21] Russell T, Bedrikovetsky P 2021 Phys. Fluids 33 053306
Google Scholar
[22] Russell T, Bedrikovetsky P 2023 J. Comput. Appl. Math. 422 114896
Google Scholar
[23] Bhatnagar P L, Gross E P, Krook M 1954 Phys. Rev. 94 511
Google Scholar
[24] Grad H 1963 Phys. Fluids 6 147
Google Scholar
[25] Bradford S A, Yates S R, Bettahar M, Simunek J 2002 Water Resour. Res. 38 63-1
Google Scholar
[26] Tufenkji N, Elimelech M 2004 Environ. Sci. Technol. 38 529
Google Scholar
[27] Andrade J S, Araújo A D, Vasconcelos T F, Herrmann H J 2008 Eur. Phys. J. B 64 433
Google Scholar
[28] Wang H Q, Lacroix M, Masséi N, Dupont J P 2000 C. R. Acad. Sci. - Ser. IIA - Earth Planet. Sci. 331 97
Google Scholar
[29] Yang Y, Bedrikovetsky P 2017 Transp. Porous Med. 119 351
Google Scholar
[30] Malgaresi G, Collins B, Alvaro P, Bedrikovetsky P 2019 Chem. Eng. J. 375 121984
Google Scholar
[31] Hashemi A, Nguyen C, Loi G, Khazali N, Yutong Y, Dang-Le B, Russell T, Bedrikovetsky P 2023 Chem. Eng. J. 474 145436
Google Scholar
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图 1 孔隙尺度下多孔介质中粒子所受作用力和捕获机制
Fig. 1. Forces and capture mechanisms on particles in porous media at the pore scale.
图 2 孔隙尺度下粒子的速度分布示意图
Fig. 2. Schematic diagram of the velocity distribution of particles at the pore scale.
图 3 多孔介质中粒子输运和捕获示意图
Fig. 3. Schematic diagram of particle transport and capture in porous media.
图 4 混合矩$ {\bar R_{ij}} $与$ {C_v} $的关系
Fig. 4. Relationship between mixing moments $ {\bar R_{ij}} $ and $ {C_v} $
图 5 本模型和R-平均模型(average model)的$ 1/P{e^*} $比较 (a) $ {C_v} = 0.1\left( {\alpha = 1} \right) $; (b) $ {C_v} = 1.19\left( {\alpha = 0.8} \right) $
Fig. 5. Comparison of the parameter $ 1/P{e^*} $ between the present model and the average model: (a) $ {C_v} = 0.1\left( {\alpha = 1} \right) $; (b) $ {C_v} = $$ 1.19\left( {\alpha = 0.8} \right) $.
图 6 本模型和R-平均模型(average model)的延迟数θ (a) $ {C_v} = 0.1\left( {\alpha = 1} \right) $; (b) $ {C_v} = 1.19\left( {\alpha = 0.8} \right) $
Fig. 6. Comparison of the parameter θ between the present model and the average model: (a) $ {C_v} = 0.1\left( {\alpha = 1} \right) $; (b) $ {C_v} = $$ 1.19\left( {\alpha = 0.8} \right) $.
图 7 本模型和R-平均模型(average model)的参数$ \varPsi = \varOmega /\left( {\lambda L} \right) $比较 (a) $ {C_v} = 0.1\left( {\alpha = 1} \right) $; (b) $ {C_v} = 1.19\left( {\alpha = 0.8} \right) $
Fig. 7. Comparison of the parameter $ \varPsi = \varOmega /\left( {\lambda L} \right) $ between the present model and the average model: (a) $ {C_v} = 0.1\left( {\alpha = 1} \right) $; (b) $ {C_v} = 1.19\left( {\alpha = 0.8} \right) $.
图 8 本文模型的宏观系数θ对$ {C_v} $和$ \lambda l $的敏感性分析 (a) $ {C_v} \in (0, 0.1],\; \lambda l \in [0, 200] $; (b) $ {C_v} \in (0, 50],\; \lambda l \in [0, 0.1] $
Fig. 8. Sensitivity analysis of the macroscopic coefficient θ of the present model to $ {C_v} $ and $ \lambda l $: (a) $ {C_v} \in (0, 0.1], \;\lambda l \in [0, 200] $; (b) $ {C_v} \in (0, 50], \;\lambda l \in [0, 0.1] $.
图 9 本文模型的宏观系数$ \varPsi = \varOmega /\left( {\lambda L} \right) $对$ {C_v} $和$ \lambda l $的敏感性分析 (a) $ {C_v} \in (0, 0.1],\; \lambda l \in [0, 200] $; (b) $ {C_v} \in (0, 50], \;\lambda l \in [0, 0.1] $
Fig. 9. Sensitivity analysis of the macroscopic coefficient $ \varPsi = \varOmega /\left( {\lambda L} \right) $ of the present model to $ {C_v} $ and $ \lambda l $: (a) $ {C_v} \in $$ (0, 0.1],\; \lambda l \in [0, 200] $; (b) $ {C_v} \in (0, 50], \;\lambda l \in [0, 0.1] $.
图 10 从微观参数对$ \left( {\lambda , l, {C_v}} \right) $计算宏观参数对$ \left( {P{e^{ - 1}}, \theta , \varOmega } \right) $的结果 (a) $ \alpha = 1 $; (b) $ \alpha < 1 $
Fig. 10. Results of calculating the macroscopic parameter for $ \left( {P{e^{ - 1}}, \theta , \varOmega } \right) $ from the microscopic parameter for $ \left( {\lambda , l, {C_v}} \right) $: (a) $ \alpha = 1 $; (b) $ \alpha < 1 $.
图 11 三个微尺度过滤系数模型(平均速度的正负代表粒子运动方向)
Fig. 11. Three microscopic filtration coefficient models (the positive and negative values of the average velocity represent the direction of particle motion).
图 12 三种捕获机制下的宏观系数关于平均速度u的灵敏度研究($ s = {10^{ - 2}},\, l = {10^{ - 6}}, \,\alpha = 0.5—1 $) (a) $ P{e^{ - 1}} $; (b) $ \theta $; (c) $ \varOmega $
Fig. 12. Sensitivity analysis of the macroscopic coefficient under three capture mechanisms to the average velocity u $( s = {10^{ - 2}}, \;l = {10^{ - 6}}, \;\alpha = 0.5-1 $): (a) $ P{e^{ - 1}} $; (b) θ; (c) Ω.
图 13 粒子速度为正值时本文模型与传统模型在$ \lambda l $很小时的系数比较 (a) $ P{e^{ - 1}} $; (b) $ \theta $; (c) $ \varOmega /\left( {\lambda L} \right) $
Fig. 13. Comparison of the coefficients between the present model and traditional models at very small $ \lambda l $ when the particle velocities are positive: (a) $ P{e^{ - 1}} $; (b) $ \theta $; (c) $ \varOmega /\left( {\lambda L} \right) $.
图 14 不同模型的出口浓度曲线比较 (a) $ \lambda L = 5 $; (b) $ \lambda L = 10 $, 其他参数为$ {C_v} = 0.1, \;l/L = {10^{ - 2}} $
Fig. 14. Comparison of outlet concentration profiles for different models: (a) $ \lambda L = 5 $; (b) $ \lambda L = 10 $, other parameters are $ {C_v} = 0.1,\; l/L = {10^{ - 2}} $.
图 15 不同模型的出口浓度曲线比较 (a) $ \lambda L = 5 $; (b) $ \lambda L = 10 $, 其他参数为$ {C_v} = 1.19,\; l/L = {10^{ - 2}} $
Fig. 15. Comparison of outlet concentration profiles for different models: (a) $ \lambda L = 5 $; (b) $ \lambda L = 10 $, other parameters are $ {C_v} = 1.19,\;l/L = {10^{ - 2}} $.
表 1 宏观系数的显式表达
Table 1. Explicit expression of macroscopic coefficients.
α = 1 α < 1 $ {{Pe}}^{-1} $ $ {({l}{/}{L}){{C}}_{{v}}^{2}}_{}^{} $ $ ({l}{/}{L}){{C}}_{{v}}^{2} $ θ $ 2{ \lambda l}{{C}}_{{v}}^{2} $ $ 2{ \lambda l}\left\{\left(1+{{C}}_{{v}}^{2}\right)\left[2\varPhi\left(\displaystyle \frac{1}{{{C}}_{{v}}}\right)-1\right]+\displaystyle \frac{2{{C}}_{{v}}}{\sqrt{{2\pi}}}\text{exp}\left(-\frac{1}{2{{C}}_{{v}}^{2}}\right)\right\} $ Ω $ \lambda{ L}{(1}-{ \lambda l}{{C}}_{{v}}^{2}) $ $\lambda L \left[ \left[ 2\varPhi\left( \dfrac{1}{C_v} \right) - 1 \right] + \dfrac{2C_v}{\sqrt{2\pi}} \exp\left( -\dfrac{1}{2C_v^2} \right) - \lambda l \left( C_v^2 + 1 - \left\{ \left[ 2\varPhi\left( \dfrac{1}{C_v} \right) - 1 \right] + \dfrac{2C_v}{\sqrt{2\pi}} \exp\left( -\dfrac{1}{2C_v^2} \right) \right\}^2 \right) \right] $ 注:$ \varPhi \left( x \right) = \displaystyle \int_{ - \infty }^x {\frac{{\exp \left( { - {y^2}/2} \right)}}{{\sqrt {2{\pi{}}} }}} {\text{d}}y. $ 
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[1] Luna M, Gastone F, Tosco T, Sethi R, Velimirovic M, Gemoets J, Muyshondt R, Sapionet H, Klaas N, Bastiaens L 2015 J. Contam. Hydrol. 181 46
Google Scholar
[2] Tosco T, Gastone F, Sethi R 2014 J. Contam. Hydrol. 166 34
Google Scholar
[3] Fakhreddine S, Prommer H, Gorelick S M, Dadakis J, Fendorf S 2020 Environ. Sci. Technol. 54 8728
Google Scholar
[4] Boccardo G, Sethi R, Marchisio D L 2019 Chem. Eng. Sci. 198 290
Google Scholar
[5] 杨秀清, 胡亦, 张景路, 王艳秋, 裴春梅, 刘飞 2014 物理学报 63 048102
Google Scholar
Yang X Q, Hu Y, Zhang J L, Wang Y Q, Pei C M, Liu F 2014 Acta Phys. Sin. 63 048102
Google Scholar
[6] Salimi S, Ghalambor A 2011 Energies 4 1728
Google Scholar
[7] Winter C L, Tartakovsky D M 2002 Water Resour. Res. 38 23-1
Google Scholar
[8] Russell T, Dinariev O Y, Pessoa Rego L A, Bedrikovetsky P 2021 Water Resour. Res. 57 e2020WR029557
Google Scholar
[9] Zou Z K, Yu L, Li Y L, Niu S Y, Fan L L, Luo W B, Li W 2023 Water 15 2193
Google Scholar
[10] Shapiro A A 2024 Phys. Fluids 36 027118
Google Scholar
[11] Panfilov M, Rasoulzadeh M 2013 Comput. Geosci. 17 269
Google Scholar
[12] Shapiro A A 2022 Chem. Eng. Sci. 248 117247
Google Scholar
[13] Herzig J P, Leclerc D M, Goff P L 1970 Ind. Eng. Chem. 62 8
[14] Bedrikovetsky P 2008 Transp. Porous Med. 75 335
Google Scholar
[15] Bedrikovetsky P, Siqueira A G, de Souza A L S, Altoé J E, Shecaira F 2006 J. Pet. Sci. Eng. 51 68
Google Scholar
[16] Bradford S A, Leij F J 2018 Chem. Eng. Sci. 192 972
Google Scholar
[17] Molnar I L, Pensini E, Asad M A, Mitchell C A, Nitsche L C, Pyrak-Nolte L J, Miño G L, Krol M M 2019 Transp. Porous Med. 130 129
Google Scholar
[18] Zhang H, Malgaresi G V C, Bedrikovetsky P 2018 Int. J. Non - Linear Mech. 105 27
Google Scholar
[19] Arns C H 2004 Physica A 339 159
Google Scholar
[20] Arns C H, Knackstedt M A, Martys N S 2005 Phys. Rev. E 72 046304
Google Scholar
[21] Russell T, Bedrikovetsky P 2021 Phys. Fluids 33 053306
Google Scholar
[22] Russell T, Bedrikovetsky P 2023 J. Comput. Appl. Math. 422 114896
Google Scholar
[23] Bhatnagar P L, Gross E P, Krook M 1954 Phys. Rev. 94 511
Google Scholar
[24] Grad H 1963 Phys. Fluids 6 147
Google Scholar
[25] Bradford S A, Yates S R, Bettahar M, Simunek J 2002 Water Resour. Res. 38 63-1
Google Scholar
[26] Tufenkji N, Elimelech M 2004 Environ. Sci. Technol. 38 529
Google Scholar
[27] Andrade J S, Araújo A D, Vasconcelos T F, Herrmann H J 2008 Eur. Phys. J. B 64 433
Google Scholar
[28] Wang H Q, Lacroix M, Masséi N, Dupont J P 2000 C. R. Acad. Sci. - Ser. IIA - Earth Planet. Sci. 331 97
Google Scholar
[29] Yang Y, Bedrikovetsky P 2017 Transp. Porous Med. 119 351
Google Scholar
[30] Malgaresi G, Collins B, Alvaro P, Bedrikovetsky P 2019 Chem. Eng. J. 375 121984
Google Scholar
[31] Hashemi A, Nguyen C, Loi G, Khazali N, Yutong Y, Dang-Le B, Russell T, Bedrikovetsky P 2023 Chem. Eng. J. 474 145436
Google Scholar
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