On the deposition of particles in liquid metals onto vertical ceramic walls

The deposition of non-metallic particles in liquid-metal flows is a serious industrial problem because the build-up of particles on ceramic walls clogs the flow path and interrupts the production, and this leads to large economic losses. This paper is an effort to extend the current state-of-the-art knowledge of particle deposition in air in order to predict particle deposition rates in liquid-metal flows using an improved Eulerian deposition model and considering Brownian and turbulent diffusion, turbophoresis and thermophoresis as transportation mechanisms. The model was used to predict the rate of deposition of particles in an air flow, and the predictions were compared to published measurements to demonstrate its performance. The model was then modified to take into account the differences in properties between air and liquid metals and thereafter applied to liquid-metal flows. Effects on the deposition rate of parameters such as steel flow rate, particle diameter, particle density, wall roughness and temperature gradient near the wall were investigated. It is shown that the steel flow rate has a very important influence on the rate of deposition of large particles, for which turbophoresis is the main deposition mechanism. For small particles, both wall roughness and thermophoresis have a significant influence on the particle deposition rate. Particle deposition rates under various conditions were successfully predicted.     

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.     

Agglomeration of inertial particles in a random rotating symmetric straining flow

This paper is concerned with the development and validation of a simple Lagrangian model for particle agglomeration in a turbulent flow involving the collision of particles in a sequence of correlated straining and vortical structures which simulate the Kolmogorov small scales of motion of the turbulence responsible for particle pair dispersion and collision. In this particular study we consider the collision rate of monodisperse spherical particles in a symmetric (pure) straining flow which is randomly rotated to create an isotropic flow. The model is similar to the classical model of Saffman and Turner (S&T) (1956) for the collision (agglomeration) of tracer particles suspended in a turbulent flow. However unlike S&T, the straining flow is not frozen in time persisting only for timescales ∼Kolmogorov timescale. Furthermore, we consider the collision of inertial particles as well as tracer particles, and study their behavior not only at the collision boundary but also in its vicinity. In the simulation, particles are injected continuously at the boundaries of the straining flow, the size of the straining region being typical of the Kolmogorov length scale η_K of the turbulence. For steady state conditions, we calculate the flux of particles colliding with a test particle at the centre of the straining flow and consider its dependence on the inertia of the colliding particles (characterized by the particle Stokes number, St). The model replicates the segregation and accumulation observed in DNS and in particular the maximum segregation for St ∼ 1 (where St is the ratio of the particle response time to the Kolmogorov timescale). We also calculate the contributions of the various turbulent forces in the momentum balance equation for satellite particles and show for instance that for small Stokes number, there is a balance between turbulent diffusion and turbophoresis (gradient of kinetic stresses) which in turn is responsible for the build-up of concentration at the collision boundary. As found in previous studies, for the case of inertialess tracer particles, the collision rate turns out to be significantly smaller than the S&T prediction due to a lowering of the concentration at the collision boundary compared to the fully mixed value. The increase in collision rate for St ∼ 0.5 is shown to be a combination of particle segregation (build-up of concentration near the collision boundary) and the decorrelation of the relative velocity between the local fluid and a colliding particle. The difference from the S&T value for the agglomeration kernel is shown to be a consequence of the choice of perfectly absorbing boundary conditions at collision and the influence of the time scale of the turbulence (eddy lifetime). We draw the analogy between turbulent agglomeration and particle deposition in a fully developed turbulent boundary layer.     

Transport and deposition of neutral particles in magnetohydrodynamic turbulent channel flows at low magnetic Reynolds numbers

The effect of Lorentz force on particle transport and deposition is studied by using direct numerical simulation of turbulent channel flow of electrically conducting fluids combined with discrete particle simulation of the trajectories of uncharged, spherical particles. The magnetohydrodynamic equations for fluid flows at low magnetic Reynolds numbers are adopted. The particle motion is determined by the drag, added mass, and pressure gradient forces. Results are obtained for flows with particle ensembles of various densities and diameters in the presence of streamwise, wall-normal or spanwise magnetic fields. It is found that the particle dispersion in the wall-normal and spanwise directions is decreased due to the changes of the underlying fluid turbulence by the Lorentz force, while it is increased in the streamwise direction. The particle accumulation in the near-wall region is diminished in the magnetohydrodynamic flows. In addition, the tendency of small inertia particles to concentrate preferentially in the low-speed streaks near the walls is strengthened with increasing Hartmann number. The particle transport by turbophoretic drift and turbulent diffusion is damped by the magnetic field and, consequently, particle deposition is reduced.     

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.     

Simulation of particles released near the wall in a turbulent boundary layer

A direct numerical simulation was used along with a Lagrangian particle tracking technique to study particle motion in a horizontal, spatially developing turbulent boundary layer along an upper-wall (with terminal velocity directed away from the wall). The objective of the research was to study particle diffusion, dispersion, reflection, and mean velocity in the context of two parametric studies: one investigated the effect of the drift parameter (the ratio of particle terminal velocity to fluid friction velocity) for a fixed and finite particle inertia, and the second varied the drift parameter and particle inertia by the same amount (i.e. for a constant Froude number). A range of drift parameters from 10⁻⁴ to 10⁰ were considered for both cases. The particles were injected into the simulation at a height of four wall units for several evenly distributed points across the span and a perfectly elastic wall collision was specified at one wall unit.Statistics collected along the particle trajectories demonstrated a transition in particle movement from one that is dominated by diffusion to one that is dominated by gravity. For small and intermediate sized particles (i.e. ones with outer Stokes numbers and drift parameters much less than unity) transverse diffusion away from the wall dominated particle motion. However, preferential concentration is seen near the wall for intermediate-sized particles due to inhomogeneous turbulence effects (turbophoresis), consistent with previous channel flow studies. Particle–wall collision statistics indicated that impact velocities tended to increase with increasing terminal velocity for small and moderate inertias, after which initial conditions become important. Finally, high relative velocity fluctuations (compared to terminal velocity) were found as particle inertia increased, and were well described with a quasi-one-dimensional fluctuation model.     

Singularités de la rhéologie de l'air humide saturé et diffusion moléculaire dans les milieux nuageuxSingularities in the rheology of saturated humid air,and molecular diffusion in cloods

Under realistic assumptions, we propose a thermodynamical formalism providing, for the moist-saturated air (cloudy air), a generalized Fick's law. This Fick's law leads to a double diffusive rheology with Dufour effect. The form taken by the energy equation is slightly different from the classical form used in convection problems. We compare the equations with those of the convection in moist unsaturated air (the Dufour effect and all double diffusive effects disappear in this case). As application we demonstrate some consequences of this diffusion in cloudy convection. To cite this article: P.A. Bois, C. R. Mecanique 330 (2002) 627–632.     

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.     

Soret and dufour effects on MHD free convective heat and mass transfer with thermophoresis and chemical reaction over a porous stretching surface: Group theory transformation

The group theoretic method is applied for solving the problem of the combined influence of the thermal diffusion and diffusion thermoeffect on magnetohydrodynamic free convective heat and mass transfer over a porous stretching surface in the presence of thermophoresis particle deposition with variable stream conditions. The application of one-parameter groups reduces the number of independent variables by one; consequently, the system of governing partial differential equations with boundary conditions reduces to a system of ordinary differential equations with appropriate boundary conditions. The equations along with the boundary conditions are solved numerically by using the Runge-Kutta-Gill integration scheme with the shooting technique. The impact of the Soret and Dufour effects in the presence of thermophoresis particle deposition with a chemical reaction plays an important role on the flow field.     

Numerical study of particle deposition in a turbulent channel flow with transverse roughness elements on one wall

A numerical study is presented for the effect of wall roughness on the deposition of solid spherical particles in a fully developed turbulent channel flow based on large eddy simulation combined with a Lagrangian particle-tracking scheme. The interest is focused on particles with response times in wall units in the range of 2.5 ≤ τ_p⁺ ≤ 600 depositing onto a vertical rough surface consisting of two-dimensional transverse square bars separated by a rectangular cavity. Predictions of particle deposition rates are obtained for several values of the cavity width to roughness element height ratio and particle response time. It is shown that the accumulation of particles in the near wall region and their preferential concentration in flow areas of low streamwise fluid velocity that occur in turbulent flows at flat channels are significantly affected by the roughness elements. Particle deposition onto the rough wall is considerably increased, exhibiting a subtle dependence on the particle inertia and the spacing between the bars. The observed augmentation of deposition coefficient can be attributed to the flow modifications induced by the roughness elements and to the inertial impaction of particles onto the frontal deposition area of the protruding square bars.     

Direct numerical simulation of particle-laden plane turbulent wall jet and the influence of Stokes number

The distribution and motion of inertial particles in plane turbulent wall jet are investigated using direct numerical simulation, under the assumption of one-way coupling. To our knowledge, this appears to be the first direct numerical simulation of a particle-laden plane turbulent wall jet. It is shown that, in outer part of the wall jet, the behaviour of particles closely resembles that of a free plane jet. Due to the streamwise decay of particle Stokes number, the particle streaks formed in the near wall region of the wall jet are characterized by their intensity variation, which differs significantly from those in the channel flow. The streamwise growth of the particle velocity half-width is approximately equal to that of the fluid velocity half-width and the maximum velocity of particles decays slower than that of fluid due to inertia. The outer scaling can collapse the mean particle velocity in both the inner and outer region for heavier particles. In the buffer region, the particle–fluid velocity difference can be negative or positive depending on the Stokes number since there are two competing effects, namely the memory effect and turbophoresis. In the viscous region, the larger particles are on average faster than fluid and the velocity difference is found to be self-similar depending on outer Stokes number. The near-wall distribution of velocity difference is significantly correlated with the presence of high-momentum particles which are entrained by vortical structures generated in the outer region of the wall jet. These results are useful for environmental and engineering applications.     

Impact of thermophoresis particle deposition and chemical reaction on unsteady non-Darcy mixed convective flow over a porous wedge in the presence of temperature-dependent viscosity

An analysis is presented to investigate the effect of thermophoresis particle deposition and temperature dependent viscosity on unsteady non-Darcy mixed convective heat and mass transfer of a viscous and incompressible fluid past a porous wedge in the presence of chemical reaction. The wall of the wedge is embedded in a uniform non-Darcian porous medium in order to allow for possible fluid wall suction or injection. The governing partial differential equations of the problem, subjected to their boundary conditions, are solved numerically by applying an efficient solution scheme for local nonsimilarity boundary layer analysis. Numerical calculations are carried out for different values of dimensionless parameters arising in the problem. The results are compared with available ones in the literature and excellent agreement is obtained. An analysis of the obtained results shows that the flow field is influenced appreciably by the chemical reaction and thermophoresis particle deposition.     

On model equations for particle dispersion in inhomogeneous turbulence

Comparisons are made between the Advection–Diffusion Equation (ADE) approach for particle transport and the two-fluid model approach based on the PDF method. In principle, the ADE approach offers a much simpler way of calculating the inertial deposition of particles in a turbulent boundary layer than that based on the PDF approach. However the ADE equations that have recently been used are only strictly valid for a simple Gaussian process when particle inertia is small. Using a prescribed, but in general non-Gaussian random particle velocity field, it is shown that the net particle mass flux contains a drift term in addition to that from the mean velocity of the particle velocity field, associated with the compressibility of the velocity field. Furthermore the diffusive flux in general depends not only upon the gradient of the mean concentration (true only for a Gaussian random flow field) but also upon higher order derivatives whose relative contribution depends on diffusion coefficients D_ijk… etc. These coefficients depend upon the statistical moments associated with random displacements and compressibility of the particle flow field along particle trajectories which in turn depend upon particle inertia. In contrast the PDF approach offers the advantage of using a simple gradient (Gaussian) approximation in particle phase space which can lead to a non-Gaussian spatial dispersion process when particle inertia is important. Conditions based on the particle mean free path are derived for which a simple ADE is appropriate. Some of the features of particle transport in an inhomogeneous turbulent flow are illustrated by examining particle dispersion in a random flow field composed of pairs of counter rotating vortices which has an rms velocity which increase linearly from a stagnation point.     

Instationärer Wärmeübergang bei turbulenter Ringspaltströmung

An analysis is made of unsteady convective heat transfer in a hydrodynamically stabilized steady turbulent flow of a viscous incompressible fluid in concentric annuli. The wall heat flux varies with time according to an arbitrary law. The formulation permits the heat transfer coefficient to vary also with time and position. The energy equation is solved using the method of superposition and separating variables by finite integral transform. Use of the generalized results is detailed illustrated by application to the case where the wall heat flux varies exponentially with time. Results are discussed subject to different Reynolds and Prandtl numbers.     

Transportation of heat through Cattaneo-Christov heat flux model in non-Newtonian fluid subject to internal resistance of particles

Thermal conduction which happens in all phases(liquid,solid,and gas) is the transportation of internal energy through minuscule collisions of particles and movement of electrons within a working body.The colliding particles comprise electrons,molecules,and atoms,and transfer disorganized microscopic potential and kinetic energy,mutually known as the internal energy.In engineering sciences,heat transfer comprises the processes of convection,thermal radiation,and sometimes mass transportation.Typically,more than one of these procedures may happen in a given circumstance.We use the Cattaneo-Christov(CC) heat flux model instead of the Fourier law of heat conduction to discuss the behavior of heat transportation.A mathematical model is presented for the Cattaneo-Christov double diffusion(CCDD) in the flow of a non-Newtonian nanofluid(the Jeffrey fluid) towards a stretched surface.The magnetohydrodynamic(MHD) fluid is considered.The behaviors of heat and mass transportation rates are discussed with the CCDD.These models are based on Fourier's and Fick's laws.The convective transportation in nanofluids is discussed,subject to thermophoresis and Brownian diffusions.The nonlinear governing flow expression is first altered into ordinary differential equations via appropriate transformations,and then numerical solutions are obtained through the built-in-shooting method.The impact of sundry flow parameters is discussed on the velocity,the skin friction coefficient,the temperature,and the concentration graphically.It is reported that the velocity of material particles decreases with higher values of the Deborah number and the ratio of the relaxation to retardation time parameter.The temperature distribution enhances when the Brownian motion and thermophoresis parameters increase.The concentration shows contrasting impact versus the Lewis number and the Brownian motion parameter.It is also noticed that the skin friction coefficient decreases when the ratio of the relaxation to retardation time parameter increases.     

Thermophoresis particle deposition effects in a free convective doubly stratified medium over a vertical plate

An analysis is performed to study the thermophoresis effects in a transient free convective flow of a viscous, incompressible fluid past an isothermal vertical plate in a doubly stratified medium. The governing boundary layer equations are solved numerically using an implicit finite difference scheme of Crank-Nicolson type. The influence of thermophoresis on particle deposition velocity and particle concentration in a doubly stratified medium are analyzed and illustrated graphically. As well the influence of thermal and mass stratification on velocity, temperature and concentration are also investigated and presented. The influence of the parameters on local as well as average skin-friction, the rate of heat and mass transfer are presented graphically and discussed. The results are compared with particular solutions available in the literature and are found to be in good agreement.     

湍流边界层中固体颗粒运动的数值模拟

根据Ｌａｇｒａｎｇｅ颗粒运动微分方程及不可压缩湍流边界层中流体的壁面速度分布规律，数值求解了颗粒在湍流边界层中的运动，考虑了Ｓａｆｆｍａｎ升为对颗粒运动的影响，壁面对运动阻力的影响，给出了固体颗粒沉积边壁，在边界层外缘上所需的最小速度和最小入射角，计算结果还表明边界层对固体颗粒撞击边壁的速度和入射角有较大影响，从数值结果可可以发现一个重要现象。     

Mass and heat transfer during solid sphere dissolution in a non-uniform fluid flow

We studied a nonisothermal dissolution of a solvable solid spherical particle in an axisymmetric non-uniform fluid flow when the concentration level of the solute in the solvent is finite (finite dilution of solute approximation). It is shown that simultaneous heat and mass transfer during solid sphere dissolution in a uniform fluid flow, axisymmetric shear flow, shear-translational flow and flow with a parabolic velocity profile can be described by a system of generalized equations of convective diffusion and energy. Solutions of diffusion and energy equations are obtained in an exact analytical form. Using a general solution the asymptotic solutions for heat and mass transfer problem during spherical solid particle dissolution in a uniform fluid flow, axisymmetric shear flow, shear-translational flow and flow with parabolic velocity profile are derived. Theoretical results are in compliance with the available experimental data on falling urea particles dissolution in water and for solid sphere dissolution in a shear flow.     

Longitudinal diffusion of solid particle admixture in a flow through a tube

The propagation of solid particle admixture in a flow through a flat channel is studied.The processes of diffusion and convective transfer as well as solid particle deposition due to gravity result in varying admixture concentration both in depth and longtitudinally.The study of admixture longitudinal distribution is of great interest in a lot of applications, therefore this paper gives the derivation of longitudinal diffusion equation for a mean cross-section admixture concentration.The equation contains three effective parameters; i.e. convective tranfer velocity, longitudinal diffusion coefficient and particle deposition time. These parameters integrally reflect local processes of matter transfer as well as momentum.The proposed model is specific and differs from Taylor equation for longitudinal diffusion, since the fact of particle deposition and adhesion is taken into account. As a result of particle deposition a sediment layer is formed on the channel bottom which increases in thickness with time. To describe this process balance conditions for the whole flow mass and admixture mass on sediment sediment surface are formulated and a condition for matter movement towards the channel bottom is derived that is different from zero due to particle adhesion.     

Flow and migration of nanoparticle in a single channel

A numerical simulation based on a combined Euler and Lagrange method is investigated in this work to simulate the flow and migration of nanoparticles in a single channel. The motion of discrete nanoparticles is determined by the Lagrangian trajectory method based on the Newton’s second law that includes the influence of the body force, various hydrodynamic forces, the Brownian motion and the thermophoresis force. The coupling of discrete particles with continuous flow is realized through the modification of the source term of the continuous equation. The results reveal the two-phase flow nature of nanoparticle suspensions and their implications to the convective heat transfer of nanofluids.