Effects of thermophoresis on Brownian coagulation of spherical particles over the entire particle size regime

In many energy and combustion applications, particles experience large temperature gradients, which can affect the coagulation process due to thermophoresis. This study presents a rigorous theory of thermophoretically modified Brownian coagulation in the entire particle size regime. The theoretical derivations are based on the kinetic theory for the free-molecular regime and the harmonic mean method for the transition regime. The coagulation kernels in different size regimes can be expressed as the basic Brownian coagulation kernel times an enhancement factor. The enhancement factor represents the coagulation rate enhancement induced by thermophoresis and is a function of specific dimensionless numbers. Based on the enhancement factor, the thermophoretic enhancement effects on particle coagulation are further analyzed under a wide range of gas and particle conditions. The results show that thermophoretic enhancement effects are ignorable in the free-molecular regime, but need to be considered in the continuum regime and the transition regime. In addition, the enhancement effects increase significantly with increase of gas temperature and temperature gradient while decrease with increase of gas pressure. The present study can improve understanding of thermophoretic effects on Brownian coagulation in the entire size regime and provide a useful tool to calculate the coagulation rates in presence of thermophoresis.     

An experimental and theoretical study of particle deposition due to thermophoresis and turbulence in an annular flow

The paper describes an experimental and theoretical study of the deposition of small particles from a turbulent annular-flow with cross-stream temperature variation, focusing on the effects of thermophoresis. Various expressions for the thermophoretic force on a spherical particle are critically discussed. The well-known composite formula of Talbot et al. (1980) does not include the ‘second mechanism of thermophoresis’ and it is concluded that the more recent theoretical approach of Beresnev and Chernyak (1995) is probably more reliable. New experimental measurements of particle deposition from a turbulent flow with cross-stream temperature gradients are then presented. The measurement technique is similar to the method of Liu and Agarwal (1974) but in the test section the aerosol flows vertically downwards in an annular gap between two concentric pipes. By heating the outer pipe and cooling the inner it is possible to establish a substantial, near-constant temperature difference between the two walls and hence a thermophoretic force which varies only with radius. Numerical calculations provide a comparison of theory with experiment. The theory is based on the turbulent deposition models of Young and Leeming (1997) and Slater et al. (2003) modified to include thermophoresis and the annular geometry. The theory of Beresnev and Chernyak gives good agreement with the experimental measurements.     

Thermophoresis of nanoparticles hotter/colder than the surrounding dilute gases

Aerosol particles suspended in a diluted gas with non-uniform temperature distribution are expected to experience a thermophoretic force.In theoretical treatmen...     

A stochastic Langevin model of turbulent particle dispersion in the presence of thermophoresis

Direct numerical simulation (DNS) and experimental data have shown that inertial particles exhibit concentration peaks in isothermal turbulent boundary layers, whereas tracer-like particles remain well mixed in the domain. It is therefore expected that the interactions between turbulence and thermophoresis will be strong in particle-laden flows where walls and carrier fluid are at significantly different temperatures. To capture turbulent particle dispersion with active thermophoresis, a coupled CFD-Lagrangian continuous random walk (CRW) model is developed. The model uses 3D mean flow velocities obtained from the Fluent 6.3 CFD code, to which are added turbulent fluid velocities derived from the normalized Langevin equation which accounts for turbulence inhomogeneities. The mean thermophoretic force is included as a body force on the particle following the Talbot formulation. Validation of the model is performed against recent integral thermophoretic deposition data in long pipes as well as the TUBA TT28 test with its detailed local deposition measurements. In all cases, the agreement with the data is very good. In separate parametric studies in a hypothetical cooled channel flow, it is found that turbulence strongly enhances thermophoretic deposition of particles with dimensionless relaxation times τ⁺ of order 1 or more. On the other hand, the thermophoretic deposition of very small inertia particles (τ⁺ < 0.2) in the asymptotic region far from the injection point tends to that which characterizes stagnant flow conditions, in agreement with the DNS results of Thakurta et al.     

Application of particle image velocimetry to combusting flows: design considerations and uncertainty assessment

Methodological aspects concerning the application of the PIV technique to the study of turbulent flames are discussed in this paper. The physical features of the flow, which have implications for the experimental set-up, image processing and measurement accuracy are identified. Design considerations are developed focusing on several factors: spatial resolution, particle performance, seeding technique, image formation and recording, and image post-processing for the evaluation of the displacement. Relevant uncertainty concerns are related to the effect of the thermophoretic force, acting on a seeding particle while crossing the flame front, and to the non-homogeneity and time-dependence of the refractive index field. The uncertainty due to thermophoresis is assessed by numerically studying the motion of a particle crossing a reference temperature profile. The effect of the refractive index variation is evaluated by means of theoretical analysis of light propagation and image formation, supported by experimental tests designed for this special purpose. Received: 25 November 1999/Accepted: 31 March 2000     

The onset of convection in a horizontal nanofluid layer of finite depth

This paper presents a linear stability analysis for the onset of natural convection in a horizontal nanofluid layer. The employed model incorporates the effects of Brownian motion and thermophoresis. Both monotonic and oscillatory convection for free–free, rigid–rigid, and rigid–free boundaries are investigated. The oscillatory instability is possible when nanoparticles concentrate near the bottom of the layer, so that the density gradient caused by such a bottom-heavy nanoparticle distribution competes with the density variation caused by heating from the bottom. It is established that the instability is almost purely a phenomenon due to buoyancy coupled with the conservation of nanoparticles. It is independent of the contributions of Brownian motion and thermophoresis to the thermal energy equation. Rather, the Brownian motion and thermophoresis enter to produce their effects directly into the equation expressing the conservation of nanoparticles so that the temperature and the particle density are coupled in a particular way, and that results in the thermal and concentration buoyancy effects being coupled in the same way.     

Thermophoretic motion of large heated aerosol spherical particles

The stationary motion of a large spherical aerosol particle in the external field of a temperature gradient in zero gravity is theoretically described using the Stokes approximation and the assumption that the average temperature of the particle surface differs considerably from the temperature of the surrounding gaseous medium. The gas dynamics equations are solved taking into account the power-law temperature dependence of the molecular transport coefficients (viscosity, thermal conductivity) and the density of the gaseous medium. Numerical estimates show that the dependence of the thermophoretic force and velocity on the average temperature of the particle surface is nonlinear.     

Thermophoretic augmentation of particle deposition in natural convection flow through a parallel plate channel

Present paper deals with temperature driven mass deposition rate of particles known as thermophoretic wall flux when a hot flue gas in natural convection flow through a cooled isothermal vertical parallel plate channel. Present study finds application in particle filters used to trap soot particles from post combustion gases issuing out of small furnaces with low technical implications. Governing equations are solved using finite difference marching technique with channel inlet values as initial values. Channel heights required to regain hydrostatic pressure at the exit are estimated for various entry velocities. Effect of temperature ratio between wall and gas on thermophoretic wall flux is analysed and wall flux found to increase with decrease in temperature ratio. Results are compared with published works wherever possible and can be used to predict particle deposition rate as well as the conditions favourable for maximum particle deposition rate.     

A generalized mass transfer law unifying various particle transport mechanisms in dilute dispersions

A generalized mass transfer law for dilute dispersion of particles (or droplets) of any sizes suspended in a fluid has been described, which can be applied to turbulent or laminar flow. The generalized law reduces to the Fick’s law of diffusion in the limit of very small particles. Thus the study shows how the well-known and much-used Fick’s law of diffusion fits into the broader context of particle transport. The general expression for particle flux comprises a diffusive flux due to Brownian motion and turbulent fluctuation, a diffusive flux due to temperature gradient (thermophoresis plus stressphoresis) and a convective flux that arises primarily due to the interaction of particle inertia and the inhomogeneity of the fluid turbulence field (turbophoresis). Shear-induced lift force, electrical force, gravity, etc. also contribute to the convective flux. The present study includes the effects of surface roughness, and the calculations show that the presence of small surface roughness even in the hydraulically smooth regime significantly enhances deposition especially of small particles. Thermophoresis can have equally strong effects, even with a modest temperature difference between the wall and the bulk fluid. For particles of the intermediate size range, turbophoresis, thermophoresis and roughness are all important contributors to the overall deposition rate. The paper includes a parametric study of the effects of electrostatic forces due to mirror charging. The present work provides a unified framework to determine the combined effect of various particle transport mechanisms on mass transfer rate and the inclusion of other mechanisms not considered in this paper is possible.     

Effects of walls temperature variation on double diffusive natural convection of Al2O3–water nanofluid in an enclosure

The effect of wall temperature variations on double diffusive natural convection of Al₂O₃–water nanofluid in a differentially heated square enclosure with constant temperature hot and cold vertical walls is studied numerically. Transport mechanisms of nanoparticles including Brownian diffusion and thermophoresis that cause heterogeneity are considered in non-homogeneous model. The hot and cold wall temperatures are varied, but the temperature difference between them is always maintained 5 °C. The thermophysical properties such as thermal conductivity, viscosity and density and thermophoresis diffusion and Brownian motion coefficients are considered variable with temperature and volume fraction of nanoparticles. The governing equations are discretized using the control volume method. The results show that nanoparticle transport mechanisms affect buoyancy force and cause formation of small vortexes near the top and bottom walls of the cavity and reduce the heat transfer. By increasing the temperature of the walls the effect of transport mechanisms decreases and due to enhanced convection the heat transfer rate increases.     

Thermophoresis and photophoresis in a rarefied gas

The present article discusses the problem of the force acting on a spherical particle in a heated rarefied gas (thermophoresis) and the problem of the force acting on such a particle in an isothermal rarefied gas, heated by an external heat flux (photophoresis). Both problems are solved in a linear statement, i.e., under the assumption of the smallness of the temperature gradient of the gas and of the external heat flux, respectively. The rising interest in these problems is due to problems of atmospheric contamination, the physics of clouds, etc.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 178–182, September–October, 1976.     

Influence of induced magnetic field on the peristaltic flow of nanofluid

This article investigates the effects of an induced magnetic field on the mixed convection peristaltic motion of nanofluid in a vertical channel. Transport equations involve the combined effects of Brownian motion and thermophoretic diffusion of nanoparticles. Analysis has been addressed subject to long wavelength and low Reynolds number assumptions. Explicit expressions of stream function, magnetic force function, temperature and nanoparticles concentration are developed. Analytic expressions are validated with the obtained numerical solutions. Peristaltic pumping rate is found to increase upon increasing the strengths of electric and magnetic fields and the buoyancy force due to temperature gradient. Moreover temperature rises and nanoparticles concentration decreases with an intensification in the Brownian motion effect.     

Numerische Berechnung des Partikeltransportes zu einer laminar angeströmten Kreisscheibe

Increasingly process steps become important, in which particles as product particles or contaminants are deposited on substrates out of the gas phase. In this paper the particles transport processes are investigated close to the surface of a circular plate surrounded by a laminar flow. The analogy between the governing equations of momentum, energy and mass is applied to the extended diffusion equation. In the nondimensional form the results of the numerical calculations give informations about velocity, temperature and particle concentration boundary layer thickness as well as their distributions. Especially the impact of external forces on particle concentration boundary layer thickness and profile is discussed. The transport of submicron particles to the surface due to convection, diffusion, gravity and thermophoretic forces acting independently is investigated. In the used normalized form the different forces are acting as one resulting force independently of their origin. Their resulting effect in comparison to the effect due to convective diffusive transport is important for particle deposition.     

纳米颗粒在人类鼻腔中沉降率的研究

借助欧拉和拉格朗日方法数值模拟了纳米颗粒在人类鼻腔中的输运和沉降. 在采用有限体积法以及k-w湍流模型求解流场的基础上, 通过单向耦合的拉格朗日方法得出了水动力、热泳力和布朗力等综合作用下的纳米颗粒的运动轨迹以及沉降率. 研究发现, 非常微小的纳米颗粒在鼻腔内的沉降率非常高; 粒径在1~10nm的纳米颗粒在鼻腔中的沉降率从80%降至18%; 粒径在10sim150nm之间的纳米颗粒在鼻腔中的沉降率变化很小,且其值介于15%~18%之间.      

Lie group analysis for thermophoretic and radiative augmentation of heat and mass transfer in a Brinkman-Darcy flow over a flat surface with heat generation

A boundary layer analysis is presented to investigate numerically the effects of radiation,thermophoresis and the dimensionless heat generation or absorption on hydromagnetic flow with heat and mass transfer over a flat surface in a porous medium.The boundary layer equations are transformed to non-linear ordinary differential equations using scaling group of transformations and they are solved numerically by using the fourth order Runge-Kutta method with shooting technique for some values of physical parameters.Comparisons with previously published work are performed and the results are found to be in very good agreement.Many results are obtained and a representative set is displayed graphically to illustrate the influence of the various parameters on the dimensionless velocity,temperature and concentration profiles as well as the local skin-friction coefficient,wall heat transfer,particle deposition rate and wall thermophoretic deposition velocity.The results show that the magnetic field induces acceleration of the flow,rather than deceleration(as in classical magnetohydrodynamics(MHD) boundary layer flow) but to reduce temperature and increase concentration of particles in boundary layer.Also,there is a strong dependency of the concentration in the boundary layer on both the Schmidt number and mass transfer parameter.     

Particle deposition due to thermal force in a tube

Particle deposition in a tube with laminar flow is investigated. An analytical procedure is developed for predicting the particle deposition efficiency by incorporating the velocity of thermophoresis in the equation of conservation of particles. Effects of important parameters, such as temperature difference between the inlet gas and the tube wall, particle size and the Lewis number, on the particle precipitation efficiency are examined. Also considered in this work is the assumption of constant temperature gradient as a limiting case. It is found that particle precipitation efficiency predicted by using constant temperature gradient is much optimistic.     

Effect of thermophoresis and other parameters on the particle deposition on a tilted surface

In this study a modified version of v2-f turbulence model (φ-α), is applied to simulate a non-isothermal air-flow. The φ-α model and a two-phase Eulerian approach complement each other to predict the rate of particle deposition on a tilted surface. The φ-α model can accurately calculate the normal fluctuations, which mainly represent the non-isotropic nature of turbulence regime near the wall. The Eulerian model was modified considering the most important mechanism in the particle deposition rate when compared to the experimental data. The model performance is examined by comparing the rate of particle deposition on a vertical surface with the experimental data in a turbulent channel flow available in the literature. The effects of lift force, turbophoretic force, thermophoreric force, electrostatic force, gravitational force and Brownian/turbulent diffusion were examined on the particle deposition rate. The results show that, using the φ-α model predicts the rate of deposition with reasonable accuracy. The results of modified particle model are in good agreement with the experimental data. This study highlights the paramount effect of thermophoretic force on the particle deposition rate and clearly shows that when the temperature difference exceeds a certain limit, the electrostatic force has insignificant effect on the particle deposition rate. Furthermore, it is indicated that even at small temperature differences, the effect of tilt angle on the particle deposition rate for intermediate-size particles is negligible.     

Mechanics of thermophoretic and thermally induced edge forces in carbon nanotube nanodevices

A double walled carbon nanotube thermal actuator consisting of a short outer tube sliding along a long inner tube under a temperature gradient is used as a model system to investigate the mechanics of thermophoretic and thermally induced edge forces in nanoscale contact based on the theory of lattice dynamics. It is shown that the total thermophoretic force has two components: a gradient force due to the change in van der Waals energy in the direction of temperature gradient and an unbalanced edge force due to the temperature difference between the two tube ends. Closed-form analytical expressions are derived for the gradient and unbalanced edge forces, with results in excellent agreement with molecular dynamics simulations. This study represents a first analytical study of thermophoretic and thermally induced edge forces between two solid bodies, and may have far reaching implications on thermomechanical nanodevices and nanoscale contact.     

Analytical and numerical solutions for axisymmetric flow of nanofluid due to non-linearly stretching sheet

This article reports the laminar axisymmetric flow of nanofluid over a non-linearly stretching sheet. The model used for nanofluid contains the simultaneous effects of Brownian motion and thermophoretic diffusion of nanoparticles. The recently proposed boundary condition is considered which requires the mass flux of nanoparticles at the wall to be zero. Analytic solutions of the arising boundary value problem are obtained by optimal homotopy analysis method. Moreover the numerical solutions are computed by Keller–Box method. Both the solutions are found in excellent agreement. The behavior of Brownian motion on the fluid temperature and wall heat transfer rate is insignificant. Further the nanoparticle volume fraction distribution is found to be negative near the vicinity of the stretching sheet.     

On the effect of the fugacity on aerosol thermophoresis

The problem of thermophoresis in gases is studied with allowance for phase and (or) other physico-chemical conversions undergone by the particle medium. It is found that these conversions influence the thermophoresis in two ways: firstly, the temperature distribution changes and, secondly, the ambient space is saturated with the vapors of the particle medium, which results in the thermodiffusion of the mixture and uncompensated momentum transfer by the gas molecules. The first effect is more pronounced for low-thermal-conductivity bodies, while the second is especially important for high-thermal-conductivity bodies. The fact that the experimental thermophoresis rate is significantly greater than the theoretical value obtained from the Epstein formula cannot be explained by the phase transition effect.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 181–186, September–October, 1995.