Thermophoretic forces on irregular particles in the free molecular regime

doi:10.11949/0438-1157.20250166

Abstract

Abstract:

In a gas-solid two-phase flow, the particles suspended in the gas are mostly non-spherical, whose geometric shape and orientation have a significant impact on particle transport. Non-spherical particles are usually simplified as sphere particles, but this may cause serious errors. While the specific expressions for non-spherical particles are too complex to be employed for practical applications. In this paper, we derived a general expression for the thermophoretic force of non-spherical particles in the free molecular regime. Then specific expressions for the thermophoretic force on sphere and cylinder can be obtained. Considering the high-speed Brownian rotation of particles in the free molecular region, a uniformly random distribution of the particle orientation can be assumed in the case of weak potential field. Based on this, the expression of the orientation-averaged thermophoretic force on particles of any shape can be derived, which is found to be proportional to the surface area of the particles, and the proportion coefficient is independent of the particle size or shape. This result is verified by direct simulation Monte Carlo, providing a new idea for simplifying the research on particle transport characteristics and its application.

Key words: aerosol, multiphase flow, non-spherical particle, thermophoretic force, kinetic theory, direct simulation Monte Carlo

摘要：

在气固两相流动中，悬浮在气体中的实际颗粒大多为非球形颗粒。基于气体动理论的方法，建立了一种较为简便的计算模型，推导得到了颗粒在自由分子区内所受热泳力的通用表达式，并进一步得到了球体、圆柱体颗粒所受热泳力的具体表达式。考虑到颗粒在自由分子区内的高速布朗旋转，在宏观时间尺度上可以认为颗粒在各个方向上为等概率分布，由此推导得到了任意形状颗粒所受均向热泳力的表达式，结果表明非球形颗粒的均向热泳力正比于颗粒表面积，且比例系数与颗粒形状无关。此结果得到了直接Monte Carlo模拟的验证，为简化颗粒输运特性研究及其应用提供了新思路。

关键词: 气溶胶, 多相流, 非球形颗粒, 热泳力, 动力学理论, 直接Monte Carlo模拟

CLC Number:

O 356

Xinquan CHANG, Kexue ZHANG, Jun WANG, Guodong XIA. Thermophoretic forces on irregular particles in the free molecular regime[J]. CIESC Journal, 2025, 76(8): 3944-3953.

常心泉, 张克学, 王军, 夏国栋. 自由分子区内不规则颗粒的热泳力计算[J]. 化工学报, 2025, 76(8): 3944-3953.

Figures/Tables 13

Fig.1 The coordinate systems for particles in the free molecular regime

Fig.2 The reference systems for spheres

Fig.3 The reference systems for cylinders

Fig.4 The reference systems for spheroids

Fig.5 Direct simulation Monte Carlo model

Fig.6 Temperature distribution profile in the simulation box

Fig.7 Temperature distribution path diagram at Y=0

Fig.8 Thermophoretic force of spherical particle with a radius of 1 μm versus the temperature gradient

Fig.9 Thermophoretic force on spherical particle versus the surface area with varying wall temperature

Fig.10 Influence of the surface area on orientation-averaged forces acting on spheroid particles

Fig.11 The orientation dependence of the thermophoretic force for cylinders

Fig.12 The orientation dependence of the thermophoretic force for spheroids

Fig.13 Influence of the surface area on orientation-averaged forces acting on different shape of particles

References 34

[1]	Zheng F. Thermophoresis of spherical and non-spherical particles: a review of theories and experiments[J]. Advances in Colloid and Interface Science, 2002, 97(1/2/3): 255-278.
[2]	Jiang Q B, Rogez B, Claude J B, et al. Quantifying the role of the surfactant and the thermophoretic force in plasmonic nano-optical trapping[J]. Nano Letters, 2020, 20(12): 8811-8817.
[3]	Krasnikov D V, Marunchenko A A, Koroleva E A, et al. One-step dry deposition technique for aligning single-walled carbon nanotubes[J]. Chemical Engineering Journal, 2024, 498: 155508.
[4]	时国华, 何林珅, 赵玺灵, 等. 余热回收喷淋塔的烟气颗粒物脱除特性研究[J]. 化工学报, 2023, 74(4): 1735-1745.
	Shi G H, He L S, Zhao X L, et al. Study of removal characteristics of particulate matters within flue gas by spray tower for waste-heat recovery[J]. CIESC Journal, 2023, 74(4): 1735-1745.
[5]	Errarte A, Martin-Mayor A, Aginagalde M, et al. Thermophoresis as a technique for separation of nanoparticle species in microfluidic devices[J]. International Journal of Thermal Sciences, 2020, 156: 106435.
[6]	Li Y K, Deng J Q, Han Z W, et al. Molecular identification of tumor-derived extracellular vesicles using thermophoresis-mediated DNA computation[J]. Journal of the American Chemical Society, 2021, 143(3): 1290-1295.
[7]	杨海涛, 郑志坚, 朱家骅, 等. 气液交叉流阵列PM_2.5热泳和扩散泳拟传质模型[J]. 化工学报, 2019, 70(6): 2139-2146.
	Yang H T, Zheng Z J, Zhu J H, et al. Pseudo mass transfer model of PM_2.5 thermophoresis and diffusiophoresis in gas-liquid cross flow array[J]. CIESC Journal, 2019, 70(6): 2139-2146.
[8]	卢慧, 于莹, 凌建亚, 等. 微量热泳动技术在药物靶点发现中的应用进展[J]. 中国药学杂志, 2024, 59(22): 2099-2106.
	Lu H, Yu Y, Ling J Y, et al. Advances in the application of microscale thermophoresis in drug target discovery[J]. Chinese Pharmaceutical Journal, 2024, 59(22): 2099-2106.
[9]	Mustonen K, Laiho P, Kaskela A, et al. Uncovering the ultimate performance of single-walled carbon nanotube films as transparent conductors[J]. Applied Physics Letters, 2015, 107(14): 143113.
[10]	雷舒婷, 张易阳, 李水清. 旋流雾化火焰合成稀土掺杂氧化钇的结构形貌和光电性能调控研究[J]. 工程热物理学报, 2024, 45(2): 569-575.
	Lei S T, Zhang Y Y, Li S Q. Investigation of structural morphology, optical and electrical property optimization of rare earth doped yttrium oxide prepared through swirling spray flame synthesis[J]. Journal of Engineering Thermophysics, 2024, 45(2): 569-575.
[11]	Burelbach J, Zupkauskas M, Lamboll R, et al. Colloidal motion under the action of a thermophoretic force[J]. The Journal of Chemical Physics, 2017, 147(9): 094906.
[12]	Martin K. The Kinetic Theory of Gases: Some Modern Aspects[M]. 3rd ed. London: Methuen&Co., 1950.
[13]	Brock J R. On the theory of thermal forces acting on aerosol particles[J]. Journal of Colloid Science, 1962, 17(8): 768-780.
[14]	Chapman S, Cowling T G. The Mathematical Theory of Non-uniform Gases. An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases[M]. 3rd ed. Cambridge: Cambridge University Press, 1970.
[15]	刘旺旺, 张克学, 王军, 等. 过渡区内纳米颗粒的曳力特性模拟研究[J]. 物理学报, 2024, 73(7): 209-216.
	Liu W W, Zhang K X, Wang J, et al. Simulation study of drag force characteristics of nanoparticles in transition regime[J]. Acta Physica Sinica, 2024, 73(7): 209-216.
[16]	Li L, Loyalka S K, Tamadate T, et al. Measurements of the thermophoretic force on submicrometer particles in gas mixtures[J]. Journal of Aerosol Science, 2024, 178: 106337.
[17]	Waldmann L, Schmitt K H. Thermophoresis and Diffusiophoresis of Aerosols[M]. New York: Aerosol Science: Academic Press, 1966.
[18]	Schadt C F, Cadle R D. Thermal forces on aerosol PARTICLES¹ [J]. The Journal of Physical Chemistry, 1961, 65(10): 1689-1694.
[19]	Beresnev S, Chernyak V. Thermophoresis of a spherical particle in a rarefied gas: numerical analysis based on the model kinetic equations[J]. Physics of Fluids, 1995, 7(7): 1743-1756.
[20]	Wang J, Li Z G. Negative thermophoresis of nanoparticles in the free molecular regime[J]. Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics, 2012, 86(1 Pt 1): 011201.
[21]	Li Z G, Wang H. Thermophoretic force and velocity of nanoparticles in the free molecule regime[J]. Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics, 2004, 70(2 Pt 1): 021205.
[22]	崔杰, 苏俊杰, 王军, 等. 自由分子区内纳米颗粒的热泳力计算[J]. 物理学报, 2021, 70(5): 242-250.
	Cui J, Su J J, Wang J, et al. Thermophoretic force on nanoparticles in free molecule regime[J]. Acta Physica Sinica, 2021, 70(5): 242-250.
[23]	Salmanzadeh M, Zahedi G, Ahmadi G, et al. Computational modeling of effects of thermal plume adjacent to the body on the indoor airflow and particle transport[J]. Journal of Aerosol Science, 2012, 53: 29-39.
[24]	Garcia-Ybarra P, Rosner D E. Thermophoretic properties of nonspherical particles and large molecules[J]. AIChE Journal, 1989, 35(1): 139-147.
[25]	Yu S, Wang J, Xia G D, et al. Thermophoretic force on nonspherical particles in the free-molecule regime[J]. Physical Review. E, 2018, 97(5): 053106.
[26]	Li M D, Mulholland G W, Zachariah M R. The effect of orientation on the mobility and dynamic shape factor of charged axially symmetric particles in an electric field[J]. Aerosol Science and Technology, 2012, 46(9): 1035-1044.
[27]	Li M D, You R, Mulholland G W, et al. Development of a pulsed-field differential mobility analyzer: a method for measuring shape parameters for nonspherical particles[J]. Aerosol Science and Technology, 2014, 48(1): 22-30.
[28]	Li M D, Mulholland G W, Zachariah M R. Understanding the mobility of nonspherical particles in the free molecular regime[J]. Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics, 2014, 89(2): 022112.
[29]	Liu C R, Li Z G, Wang H. Drag force and transport property of a small cylinder in free molecule flow: a gas-kinetic theory analysis[J]. Physical Review E, 2016, 94(2): 023102.
[30]	Clerk Maxwell J. On stresses in rarified gases arising from inequalities of temperature[J]. Philosophical Transactions of the Royal Society of London Series I, 1879, 170: 231-256.
[31]	Liu N, Bogy D B. Forces on a rotating particle in a shear flow of a highly rarefied gas[J]. Physics of Fluids, 2008, 20(10): 107102.
[32]	Liu N, Bogy D B. Forces on a spherical particle with an arbitrary axis of rotation in a weak shear flow of a highly rarefied gas[J]. Physics of Fluids, 2009, 21(4): 047102.
[33]	常心泉, 张克学, 王军, 等. 自由分子区内的非球形颗粒曳力计算[J]. 力学学报, 2024, 56(5): 1251-1260.
	Chang X Q, Zhang K X, Wang J, et al. Drag forces on non-spherical particles in the free molecular regime[J]. Chinese Journal of Theoretical and Applied Mechanics, 2024, 56(5): 1251-1260.
[34]	Bird G A. Recent advances and current challenges for DSMC[J]. Computers & Mathematics with Applications, 1998, 35(1/2): 1-14.