﻿ 二维方腔热对流系统中纳米颗粒混合及凝并特性的数值模拟
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 力学学报  2015, Vol. 47 Issue (5): 740-750  DOI: 10.6052/0459-1879-15-062 0

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Xu Feibin, Zhou Quan, Lu Zhiming. NUMERICAL SIMULATION OF BROWNIAN COAGULATION AND MIXING OF NANOPARTICLES IN 2D RAYLEIGH-BĖNARD CONVECTION[J]. Chinese Journal of Ship Research, 2015, 47(5): 740-750. DOI: 10.6052/0459-1879-15-062.
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### 文章历史

2015-02-26收稿
2015-06-15录用
2015-07-07网络版发表

1 数学模型 1.1 控制方程

 图 1 物理模型示意图 Fig.1 Schematic of nanoparticle in 2-D Rayleigh-Bènard convection

 $\left. \begin{array}{l} \frac{{\partial u}}{{\partial x}} + \frac{{\partial v}}{{\partial y}} = 0\\ \frac{{{\rm{D}}u}}{{{\rm{D}}t}} = - \frac{{\partial P}}{{\partial x}} + \sqrt {\frac{{Pr}}{{Ra}}} {\nabla ^2}u\\ \frac{{{\rm{D}}v}}{{{\rm{D}}t}} = - \frac{{\partial P}}{{\partial x}} + \sqrt {\frac{{Pr}}{{Ra}}} {\nabla ^2}v + T\\ \frac{{{\rm{D}}T}}{{{\rm{D}}t}} = \frac{1}{{\sqrt {PrRa} }}{\nabla ^2}T \end{array} \right\}$ (1)

 $\dfrac{{\rm D}n}{{\rm D}t} = \dfrac{\partial }{\partial x}\left( {D_{\rm fm} \dfrac{\partial n}{\partial x}} \right) + \dfrac{\partial }{\partial y}\left( {D_{\rm fm} \dfrac{\partial n}{\partial y}} \right) + S_{\rm coag}$ (2)

 ${D_{{\rm{fm}}}} = \frac{{{{(3/4\pi )}^{1/3}}{k_{\rm{b}}}{T_{\rm{g}}}}}{{{\upsilon ^{2/3}}{\rho _{\rm{g}}}c(1 + \pi {r_{\rm{p}}}/8)}}$ (3)

 ${S_{{\rm{coag}}}} = \frac{1}{2}\int_0^\upsilon {\beta \left( {{\upsilon _1},\upsilon - {\upsilon _1},t} \right)} \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} n\left( {{\upsilon _1},x,t} \right) \cdot \;n\left( {\upsilon - {\upsilon _1},x,t} \right){\rm{d}}{\upsilon _1} - \int_0^\infty {\beta \left( {{\upsilon _1},\upsilon ,t} \right)} n\left( {{\upsilon _1},x,t} \right) \cdot \qquad \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} n\left( {\upsilon ,x,t} \right){\rm{d}}{\upsilon _1}$ (4)

 ${\beta _{\rm{B}}^{{\rm{fm}}} = {K_{{\rm{fm}}}}{{\left( {{\upsilon ^{1/3}} + \upsilon _1^{1/3}} \right)}^2}{{\left( {\frac{1}{\upsilon } + \frac{1}{{{\upsilon _1}}}} \right)}^{1/2}}}\\ {{K_{{\rm{fm}}}} = {{\left( {\frac{3}{{4\pi }}} \right)}^{1/6}}{{\left( {\frac{{6{k_{\rm{b}}}{T_{\rm{g}}}}}{{{\rho _{\rm{p}}}}}} \right)}^{1/2}}}$ (5)

 $\left. \begin{array}{l} \frac{{{\rm{D}}{M_0}}}{{{\rm{D}}t}} = \sqrt {\frac{{Pr}}{{Ra}}} \frac{1}{{S{c_{\rm{M}}}}}{\nabla ^2}\left( {\frac{{\left( {4 + 5{M_{\rm{C}}}} \right){M_0}{V^{ - 2/3}}}}{9}} \right) + \\ Da\frac{{\left( {65M_{\rm{C}}^2 - 1210{M_{\rm{C}}} - 9223} \right)M_0^2{V^{1/6}}}}{{5184}}\\ \frac{{{\rm{D}}{M_1}}}{{{\rm{D}}t}} = \sqrt {\frac{{Pr}}{{Ra}}} \frac{1}{{S{c_{\rm{M}}}}}{\nabla ^2}\left( {\frac{{\left( {10 - {M_{\rm{C}}}} \right){M_1}{V^{ - 2/3}}}}{9}} \right)\\ \frac{{{\rm{D}}{M_2}}}{{{\rm{D}}t}} = \sqrt {\frac{{Pr}}{{Ra}}} \frac{1}{{S{c_{\rm{M}}}}}{\nabla ^2}\left( {\frac{{\left( {7 + 2{M_{\rm{C}}}} \right){M_2}{V^{ - 2/3}}}}{{9{M_{\rm{C}}}}}} \right) - \\ Da\frac{{\left( {701M_{\rm{C}}^2 - 4210{M_{\rm{C}}} - 6859} \right)M_1^2{V^{1/6}}}}{{2592{M_{{\rm{C0}}}}}} \end{array} \right\}$ (6)

 $\left. \begin{array}{l} S{c_{\rm{M}}} = \nu V_0^{2/3}/{\kappa _{\rm{p}}},\;\;{\kappa _{\rm{p}}} = {D_{{\rm{fm}}}}{\upsilon ^{2/3}} = \\ Da = \frac{H}{U}\frac{{\sqrt 2 {K_{{\rm{fm}}}}{M_{10}}}}{{V_0^{5/6}}}\\ \frac{1}{{\sqrt {RaPr} }}\frac{{{H^2}}}{{{\kappa _{\rm{g}}}}}\frac{{\sqrt 2 {K_{{\rm{fm}}}}{M_{10}}}}{{V_0^{5/6}}} = \\ \frac{1}{{\sqrt {RaPr} }}\frac{{{H^2}}}{{{\kappa _{\rm{g}}}}}\sqrt 2 {K_{{\rm{fm}}}}{M_{00}}V_0^{1/6} \end{array} \right\}$ (7)

1.2 边界条件和初始条件

RB系统四周固壁上流动满足无滑移边界条件. 温度在上下底边满足狄利克雷(Dirichlet)边界条件,左右壁面的梯度为0. 颗粒 群的各阶矩在边界上的梯度为0.

 ${n\left( {\upsilon ,x,y,t} \right) = \frac{N}{{3\sqrt {2\pi } \upsilon \ln {\sigma _0}}}\exp \left( { - \frac{{{{\ln }^2}\left( {\upsilon /{\upsilon _{{\rm{g0}}}}} \right)}}{{18{{\ln }^2}\sigma }}} \right)}\\ {{M_{{\rm{k0}}}} = N\upsilon _{{\rm{g0}}}^{\rm{k}}\exp (9{k^2}{{\ln }^2}{\sigma _0}/2),\;k = 0,1,2}\\ {{\upsilon _{{\rm{g0}}}} = \frac{{M_{10}^2}}{{M_{00}^{3/2}M_{20}^{1/2}}} = \frac{{{V_0}}}{{\sqrt {{M_{{\rm{C0}}}}} }},\;{{\ln }^2}{\sigma _0} = \frac{1}{9}\ln \left( {{M_{{\rm{C0}}}}} \right)}$ (8)

2 计算结果及讨论

2.1 计算工况

2.2 传热参数$Nu$与$Ra$的关系

 图 2 $Nu$与$Ra$关系图 Fig.2 The $Nu$ number vs. $Ra$ number
2.3 流场和颗粒群分布的演化

 图 3 不同时刻下工况I的温度场和速度场(箭头表示速度矢量) Fig.3 The contours of temperature and velocity fields (arrows represent velocity in case I at different times)
 图 4 不同时刻下工况I的颗粒群数浓度云图 Fig.4 The contours of particle number concentration in case I at different times

(1) 扩散阶段(图3(a),3(b),图4(a),图4(b)). 在流动初始阶段,在方腔四个边角附近产生角涡,温度边界层内的等温面发生凹凸,随着时间的发展,凸起的部分渐渐发展成为蘑菇状的羽流结构,如图3(a)所示; 由于方腔中部流体速度很小,流场对颗粒群空间分布的影响非常小,颗粒群向方腔上半区域缓慢扩散,如图4(a)所示. 随着时间的推进,等温面逐渐往中心区域发展,直至整个温度边界层内都产生往外凸起的局部羽流结构,蘑菇状的羽流在浮力或重力的驱动下向上(热羽流)或向下(冷羽流)运动,方腔中部形成4个中尺度的涡结构,如图3(b)所示; 冷热羽流虽然在方腔中部相遇,但其翻转运动仍分别局限于中尺度涡内,上、下半区域间仍然存在界面,远离界面的颗粒群主要跟随羽流做翻转运动,界面附近的颗粒群一部分通过界面扩散至上半区域,一部分跟随羽流运动,如图4(b)所示.

(2) 混合阶段(图3(c),图4(c)). 上下两股冷热羽流的碰撞引起了流动的不稳定性,冷热羽流在翻转过程中各自分别从某一侧进入到对方区域,如图3(c) 所示; 冷热羽流分别夹带着数浓度较低和较高的颗粒群从某一侧进入下、上半区域,数目浓度较高和较低的颗粒群之间接触界面变大,两者相互混合加快,如图4(c)所示.

(3)充分混合阶段(图3(d),图4(d)). 当冷(热)羽流到达方腔下(上)底板时,将产生更强的流动不稳定性,在上下底板有更多的羽流结构形成,方腔中部 涡结构在冷热羽流碰撞的影响下形成大尺度涡结构,流动逐渐进入大尺度环流状态,如图4(c); 而对于颗粒群数目浓度,随着混合的不断进行,除局部区域外,颗粒群在方腔中空间分布呈现出均匀状态,见图4(d).

 图 5 $t=52$时颗粒群特征量分布云图(工况I) Fig.5 Contours of moments and mean particle volume at time $t=52$ in case I

 ${C_{{\rm{cor}}}^\Phi = \frac{{\int {\int\limits_A {\left( {\Phi - {{\left\langle \Phi \right\rangle }_A}} \right)\left( {T - {{\left\langle T \right\rangle }_A}} \right){\rm{d}}s} } }}{{\sqrt {\int {\int\limits_A {{{\left( {\Phi - {{\left\langle \Phi \right\rangle }_A}} \right)}^2}{\rm{d}}s\int {\int\limits_A {{{\left( {T - {{\left\langle T \right\rangle }_A}} \right)}^2}{\rm{d}}s} } } } } }}}\\ {RSD = \frac{{\sqrt {\int {\int\limits_A {\left( {\Phi - {{\left\langle \Phi \right\rangle }_A}} \right){\rm{d}}s} } } /A}}{{{{\left\langle \Phi \right\rangle }_A}}}}$ (9)

 图 6 (a)相关系数$C_{\rm cor}^\Phi$, (b)相对标准偏差$RSD$随时间演化 Fig.6 (a) The time evolution of correlation coefficient $C_{\rm cor}^\Phi$ and (b) relative standard deviation $RSD$
2.4 颗粒群特征量长时间演化的渐近行为

 图 7 工况I条件下数值解与渐近解比较 Fig.7 The comparison moments between asymptotic solution and numerical solution under case I

 ${{M_0} = {C_0}K_{{\rm{fm}}}^{ - 6/5}M_1^{ - 1/5}{t^{ - 6/5}}}\\ {{M_2} = {C_2}K_{{\rm{fm}}}^{6/5}M_1^{11/5}{t^{6/5}}}\\ {{M_{\rm{C}}} = 2.20}$ (10)

 $\left. {\begin{array}{*{20}{l}} {{{\left\langle {{M_0}} \right\rangle }_A} = {C_0}{2^{3/5}}\left\langle {{M_1}} \right\rangle _A^{ - 1/5}D{a^{ - 6/5}}{t^{ - 6/5}}}\\ {{{\left\langle {{M_2}} \right\rangle }_A} = {C_2}{2^{ - 3/5}}\left\langle {{M_1}} \right\rangle _A^{11/5}D{a^{6/5}}M_{{\rm{C0}}}^{ - 1}{t^{6/5}}}\\ {{{\left\langle V \right\rangle }_A} = C_0^{ - 1}{2^{ - 3/5}}\left\langle {{M_1}} \right\rangle _A^{6/5}D{a^{6/5}}{t^{6/5}}}\\ {{{\left\langle {{M_{\rm{C}}}} \right\rangle }_A} = 2.20,t \to \infty } \end{array}} \right\}$ (11)

 $\left. {\frac{{{{\left\langle {{M_{\rm{C}}}} \right\rangle }_A} - {M_{Ca}}}}{{{M_{Ca}}}} \le 1\% } \right\}$ (12)

 图 8 颗粒群达到自保持分布的时间$t_{a}$与各量纲数$Ra$, $Sc_{\rm M}$和$Da$的关系 Fig.8 The relationship between the time to attain a self-preserving distribution and non-dimensional parameters (i.e., $Ra$, $Sc_{\rm M}$, $Da$)

 $\label{eq13} \left.\begin{array}{l} t_a = 6\,870.87 - 320.39\ln Ra \\ t_a = 1\,255.36 - 240.12\ln Sc_{\rm M} \\ t_a = 8\,641.17 + 114.66Da \\ \end{array}\right\}$ (13)

3 结 论

(1) 纳米颗群的演化过程分为3个不同阶段：扩散阶段,混合阶段,充分混合阶段. 颗粒群各特征量与温度的相关系数以及各特征量空间分布的标准偏差在3个不同阶段特征显著不同. 相关系数在扩散阶段缓慢变大,在混合阶段随流场演化而震荡,而在充分混合阶段单调减小; 对空间分布标准偏差,在扩散阶段下降缓慢,混合阶段快速下降,最后在充分混合阶段趋于稳定.

(2) 纳米颗粒分布函数的各阶矩长时间演化存在渐近行为,且与零维情形的渐近解吻合,说明纳米颗粒群分布最后达到了自保持分布.

(3) 该系统中纳米颗粒群达到自保持分布的时间随着$Ra$和$Sc_{\rm M}$呈对数率减小; 而随着$Da$的变大,近似线性增大.

 [1] 赵海波, 郑楚光. 离散系统动力学演变过程的颗粒群平衡模拟. 北京: 科学出版社, 2008, 1-30 (Zhao Haibo, Zheng Chuguang. Population Balance Modeling of Dynamic Evolution in Dispersed System. Beijing: Science Press, 2008, 1-30 (in Chinese)) [2] Metzger S, Lelieveld J. Reformulating atmospheric aerosol thermodynamics and hygroscopic growth into fog, haze and clouds. Atmospheric Chemistry and Physics, 2007, 7: 3163-3193 [3] Hanna V, Ilona R. Thermodynamics and kinetics of atmospheric aerosol particle formation and growth. Chemical Society Reviews, 2012, 41: 5160-5173 [4] Gao Y, Zhang M, Liu Z, et al. Modeling the feedback between aerosol and meteorological variables in the atmospheric boundary layer during a severe fog-haze event over the North China Plain. Atmospheric Chemistry and Physics, 2015, 15: 1093-1130 [5] 方传波, 夏智勋, 胡建新等. 强迫对流下硼颗粒燃烧特性影响因素研究. 航空学报, 2014, 35: 1-9 (Fang Chuanbo, Xia Zhixun, Hu Jianxin, et al. Influence factors of combustion characteristics of boron particle in forced convective flow. Acta Aeronautica et Astronautica Sinica, 2014, 35: 1-9 (in Chinese)) [6] Golman B, Julklang W. Simulation of exhaust gas heat recovery from a spray dryer. Applied Thermal Engineering, 2014, 73(1): 899-913 [7] Shi D, El-Farra NH, Li MH, et al. Predictive control of particle size distribution in particulate processes. Chemical Engineering Science, 2006, 61(1): 261-281 [8] Lee KW. Change of particle size distribution during Brownian coagulation. Journal of Colloid and Interface Science, 1983, 92(2): 315-325 [9] Gelbard F, Tambour Y, Seinfeld JH. Simulation of multicomponent aerosol dynamics. Journal of Colloid and Interface Science, 1980, 76(2): 541-556 [10] Tandon P, Rosner DE. Monte Carlo simulation of particle aggregation and simultaneous restructuring. Journal of Colloid and Interface Science, 1999, 213(2): 273-286 [11] McGraw R. Description of aerosol dynamic by the quadrature method of moments. Aerosol Science and Technology, 1997, 27(2): 255-265 [12] 于明州, 江影, 张凯. 湍动剪切微米尺度粒子凝并TEMOM模型研究. 力学学报, 2011, 43(3): 447-452 (Yu Mingzhou, Jiang Ying, Zhang Kai. The study on micro-scale particle coagulation due to turbulent shear mechanism using TEMOM model. Chinese Journal of Theoretical and Applied Mechanics, 2011, 43(3): 447-452 (in Chinese)) [13] Yu MZ, Lin JZ, Chan TL. A new moment method for solving the coagulation equation for particles in Brownian motion. Aerosol Science and Technology, 2008, 42(9): 705-713 [14] Chen ZL, Lin JZ, Yu MZ. Direct expansion method of moments for nanoparticle Brownian coagulation in the entire size regime. Journal of Aerosol Science, 2014, 67: 28-37 [15] Suha SM, Zachariaha MR, Girshicka SL. Numerical modeling of silicon oxide particle formation and transport in a one-dimensional low-pressure chemical vapor deposition reactor. Journal of Aerosol Science, 2002, 33(6): 943-959 [16] Settumba N, Garrick SC. Direct numerical simulation of nanoparticle coagulation in a temporal mixing layer via a moment method. Journal of Aerosol Science, 2003, 34(2): 149-167 [17] Xie ML, Yu MZ, Wang LP. TEMOM model to simulate nanoparticle growth in the temporal mixing layer due to Brownian coagulation. Journal of Aerosol Science, 2012, 54: 32-48 [18] Yu MZ, Lin JZ, Chan T. Numerical simulation of nanoparticle synthesis in diffusion flame reactor. Powder Technology, 2008 181(1): 9-20 [19] Zhou K, Attili A, Alshaarawi A, et al. Simulation of aerosol nucleation and growth in a turbulent mixing layer. Physics of Fluids, 2014, 26: 065106 [20] 周全, 夏克青. Rayleigh--B′enard 湍流热对流研究的进展、现状 及展望. 力学进展, 2012, 42(3): 231-251 (Zhou Quan, Xia Keqing. Advances and outlook in turbulent Rayleigh--B′enard convection. Advances in Mechanics, 2012, 42(3): 231-251 (in Chinese)) [21] Friedlander SK. Smoke, Dust and Haze: Fundamentals of Aerosol Dynamics, 2nd ed. New York, Oxford University Press, 2000: 14- 210 [22] Shishkina O, Stevens RJAM, Grossmann S, et al. Boundary layer structure in turbulent thermal convection and its consequences for the required numerical resolution. New Journal of Physics, 2010, 12: 075022 [23] Ahlers1 G, Grossmann S, Lohse D. Heat transfer & large-scale dynamics in turbulent Rayleigh-B′enard convection. Reviews of Modern Physics, 2009, 81(2): 503-537 [24] 徐炜, 包芸. 利用FFT 高效求解二维瑞利--贝纳德热对流. 力学学 报, 2013, 45(4): 666-671 (Xu Wei, Bao Yun. An effcient solution for 2-D Rayleigh-B′enard convection using FFT. Chinese Journal of Theoretical and Applied Mechanics, 2013, 45(4): 666-671 (in Chinese)) [25] Xie ML,Wang LP. Asymptotic solution of population balance equation based on TEMOM model. Chemical Engineering Science, 2013, 94: 79-83 [26] Spicer PT, Pratsinis SE, Trennepohl MD, et al. Coagulation and fragmentation: the variation of shear rate and the time lag for attainment of steady state. Industrial & Engineering Chemistry Research, 1996, 35(9): 3074-3080
NUMERICAL SIMULATION OF BROWNIAN COAGULATION AND MIXING OF NANOPARTICLES IN 2D RAYLEIGH-BĖNARD CONVECTION
Xu Feibin, Zhou Quan, Lu Zhiming
Shanghai Institute of Applied Mathematics and Mechanics, Shanghai 200072, China
Fund: The project was supported by the National Natural Science Foundation of China (11272196, 11332006).
Abstract: In the present simulations, the first three moments of the particle size distribution of nanoparticles in a two dimensional Rayleigh-Bénard convection system are calculated with the combination of SIMPLE algorithm and the Tayler-series expansion method of moments (TEMOM) to probe into Brownian coagulation and mixing of nanoparticles. Driven by Brownian coagulation, diffusion and thermal convection, the number concentration of nanoparticles decreases, while the average volume increase generally as time goes on. The temporal evolution of nanoparticles can be divided into three stages, named the diffusion stage, the mixing stage and the fully mixing stage respectively. The correlation coefficients between moments of nanoparticles and the temperature, and relative standard deviation of moments experience distinct characteristics in three stages. The long-time behavior for moments of nanoparticles is obtained and is in good agreement with the asymptotic solution. Finally, the time to attain such an asymptotic solution is investigated and its dependence on Ra, ScM and Da is also determined numerically. The results show that the time decreases logarithmically when Ra and ScM increase, while it increases linearly when Da increases.
Key words: TEMOM    nanoparticle    Rayleigh-B′enard convection    mixing    coagulation    numerical simulation