Numerical calculations of erosion in an abrupt pipe contraction of different contraction ratios

Erosion predictions in a pipe with abrupt contraction of different contraction ratios for the special case of two‐phase (liquid and solid) turbulent flow with low particle concentration are presented. A mathematical model based on the time‐averaged governing equations of 2‐D axi‐symmetric turbulent flow is used for the calculations of the fluid velocity field (continuous phase). The particle‐tracking model of the solid particles is based on the solution of the governing equation of each particle motion taking into consideration the effect of particle rebound behaviour. Models of erosion were used to predict the erosion rate in mg/g. The effect of Reynolds number and flow direction with respect to the gravity was investigated for three contraction geometries considering water flow in a carbon steel pipe. The results show that the influence of the contraction ratio on local erosion is very significant. However, this influence becomes insignificant when the average erosion rates over the sudden contraction area are considered. The results also indicate the significant influence of inlet velocity variations. The influence of buoyancy is significant for the cases of low velocity of the continuous flow. A threshold velocity below which erosion may be neglected was indicated. Copyright © 2004 John Wiley & Sons, Ltd.     

PTV investigation of phase interaction in dispersed liquid–liquid two-phase turbulent swirling flow

An investigation of dispersed liquid–liquid two-phase turbulent swirling flow in a horizontal pipe is conducted using a particle tracking velocimetry (PTV) technique and a shadow image technique (SIT). Silicone oil with a low specific gravity is used as immiscible droplets. A swirling motion is given to the main flow by an impeller installed in the pipe. Fluorescent tracer particles are applied to flow visualization. Red/green/blue components extracted from color images taken with a digital color CCD camera are used to simultaneously estimate the liquid and droplet velocity vectors. Under a relatively low swirl motion, a large number of droplets with low specific gravity tend to accumulate in the central region of the pipe. With increasing droplet volume fraction, the liquid turbulence intensity in the axial direction increases while that in the wall-normal direction decreases in the central region of the pipe. In addition, the turbulence modification in the present flow is strongly dependent on the droplet Reynolds number; however, the interaction of droplet-induced turbulences is significant due to vortex shedding, particularly at high droplet Reynolds numbers and higher droplet volume fraction.     

Influence of large eddies on the suspension of solid particles

The phenomena of solid particles suspensions, in a turbulent flow, can more conveniently be described by stochastic models than by diffusion models, particularly in the case of relatively coarse particles.

The fundamental difficulties of using such models are principally due to the difficulty of performing direct measurements of probabilities, because the number of observations (or tests) necessary to obtain physically representative values is important (theoretically infinite).

We have used such a model to describe the movement of spheres in an inclinable pipe.

To do so, we have identified the movement through a Markov process which permits us to show that we can characterize it by the limit distribution for passage probabilities in a cross section. We have used a special system of close-circuit television to measure it, doing a sufficiently large number of observations for the measurements to be significant.

In the case of a vertical pipe, the phenomena is one-dimensional. By using the model stochastic displacement, we obtain a differential equation which it is possible to integrate by assuming an obviously constant radial dispersion. The interpretation of limit distributions for passage probabilities and visual observations of particles movement in the pipe have caused us to conclude that the mean displacment is due, on one hand, to a radial acceleration bounded to a stochastic rotation of the flow and, on the other hand, to the effect of the mean velocity gradient. The experimental results show that the radial dispersion is a function of the relative dimension of particles with respect to the macroscale of the turbulence.

In the case of an inclined pipe, a two-dimensional stochastic model of the displacement is possible, but the integration of the equation is quite complicated and may be done numerically. We have prefered a two-dimensional simulation model. The results of the simulations permit us to obtain a limit repartition of passage probabilities, the moments of which we have compared with those that we have measured. These comparisons show that the model obviously represents the phenomena when the pipe is horizontal or very slightly inclined but differs in the near vertical case. This is due to the simplicity of the model in which we neglect the radial acceleration we have considered previously and the effect of which is negligible in comparison with gravity when the pipe is inclined.

The interpretation of the measurements by comparison of moments with the two-dimensional model shows that the angular dispersion of solid particles is essentially due to big eddies and that the particle diameters are not essential parameters in this case.

By associating this conclusion with that obtained previously concerning the radial dispersion, it seems that the eddies bigger than the macroscale of turbulence may be of capital importance in the dispersion of solid particles and that it will be of practical interest to characterize them as a function of a mean parameter of the flow.

The study of the movement of sufficiently large particles seems to be a method which is able to give this result.     

垂直湍流液-固流中大颗粒的相对速度

通过量纲分析和实验测量,对于垂直、局部均匀的湍流稀态液一固流中,大颗粒的相对速度,建立了无量纲参数表达式．用分析和实验相结合的方法,确定了表达式中无量纲参数的幂次及有关系数．实验中用激光多普勒分相测量技术,分别测出流体和颗粒的时均速度结果表明,大颗粒相对速度强烈依赖于流体雷诺数,当流体雷诺数较高时,其幂次渐近于1.5。     

Solution of general dynamic equation for nanoparticles in turbulent flow considering fluctuating coagulation

A new averaged general dynamic equation (GDE) for nanoparticles in the turbulent flow is derived by considering the combined effect of convection, Brownian diffusion, turbulent diffusion, turbulent coagulation, and fluctuating coagulation. The equation is solved with the Taylor-series expansion moment method in a turbulent pipe flow. The experiments are performed. The numerical results of particle size distribution correlate well with the experimental data. The results show that, for a turbulent nanoparticulate flow, a fluctuating coagulation term should be included in the averaged particle GDE. The larger the Schmidt number is and the lower the Reynolds number is, the smaller the value of ratio of particle diameter at the outlet to that at the inlet is. At the outlet, the particle number concentration increases from the near-wall region to the near-center region. The larger the Schmidt number is and the higher the Reynolds number is, the larger the difference in particle number concentration between the near-wall region and near-center region is. Particle polydispersity increases from the near-center region to the near-wall region. The particles with a smaller Schmidt number and the flow with a higher Reynolds number show a higher polydispersity. The degree of particle polydispersity is higher considering fluctuating coagulation than that without considering fluctuating coagulation.     

Pipe flow of suspensions

This study shows that fully developed pipe flow of a particulate suspension is defined by four dimensionless parameters of particle-fluid interactions in addition to the Reynolds number. Effects accounted for include the Magnus effect due to fluid shear, electrostatic repulsion due to electric charges on the particles, and Brownian or turbulent diffusion. In the case of a laminar liquid-solid suspension, electrostatic effect is negligible but shear effect is prominent. Solution of the basic equations gives the density distribution of particles with a peak at the center (Einstein, Jeffery) or at other radii between the center and the pipe wall (Segré et al) depending on the magnitudes of the various flow parameters. In the case of a turbulent gas-solid suspension, the Magnus effect is significant only within the thickness of the laminar sublayer. However, charges induced on the particles by the impact of particles at the wall produce a higher density at the wall than at the center of the pipe. The velocity distribution of particles is characterized by a slip velocity at the wall and a lag in velocity in the core from the fluid phase. These results are verified by earlier measurements.     

Pipe flow of suspensions

This study shows that fully developed pipe flow of a particulate suspension is defined by four dimensionless parameters of particle-fluid interactions in addition to the Reynolds number. Effects accounted for include the Magnus effect due to fluid shear, electrostatic repulsion due to electric charges on the particles, and Brownian or turbulent diffusion. In the case of a laminar liquid-solid suspension, electrostatic effect is negligible but shear effect is prominent. Solution of the basic equations gives the density distribution of particles with a peak at the center (Einstein, Jeffery) or at other radii between the center and the pipe wall (Segré et al) depending on the magnitudes of the various flow parameters. In the case of a turbulent gas-solid suspension, the Magnus effect is significant only within the thickness of the laminar sublayer. However, charges induced on the particles by the impact of particles at the wall produce a higher density at the wall than at the center of the pipe. The velocity distribution of particles is characterized by a slip velocity at the wall and a lag in velocity in the core from the fluid phase. These results are verified by earlier measurements.     

水平液─固流中颗粒抑制湍流的行为和条件

用激光多普勒分相测量方法，考察了水平含颗粒水流的主流区，亚毫米颗粒对水流湍流有抑制行为，湍流削弱程度与颗粒浓度、流动发展长度、水流雷诺数等因素有关．按照不出现尾涡脱落和至少能响应一部分大尺度涡的运动这两个基本条件，分析了湍流抑制与有关参量的依赖性．结果表明，流动尺度和颗粒尺寸之比，对于颗粒消耗而不增生湍流起着最主要的作用．在一定的水流雷诺数下，只要这一比值很大，亚毫米颗粒就会表现出对湍流的抑制作用     

Extremely large-amplitude oscillation of soft pipes conveying fluid under gravity

In this work, the nonlinear behaviors of soft cantilevered pipes containing internal fluid flow are studied based on a geometrically exact model, with particular focus on the mechanism of large-amplitude oscillations of the pipe under gravity. Four key parameters, including the flow velocity, the mass ratio, the gravity parameter, and the inclination angle between the pipe length and the gravity direction, are considered to affect the static and dynamic behaviors of the soft pipe. The stability analyses show that, provided that the inclination angle is not equal to π, the soft pipe is stable at a low flow velocity and becomes unstable via flutter once the flow velocity is beyond a critical value. As the inclination angle is equal to π, the pipe experiences, in turn,buckling instability, regaining stability, and flutter instability with the increase in the flow velocity. Interestingly, the stability of the pipe can be either enhanced or weakened by varying the gravity parameter, mainly dependent on the value of the inclination angle.In the nonlinear dynamic analysis, it is demonstrated that the post-flutter amplitude of the soft pipe can be extremely large in the form of limit-cycle oscillations. Besides,the oscillating shapes for various inclination angles are provided to display interesting dynamical behaviors of the inclined soft pipe conveying fluid.     

Numerical investigation of the influence of gravity on flutter of cantilevered pipes conveying fluid

We have carried out a numerical investigation of the three dimensional nonlinear dynamics of a cantilevered pipe conveying fluid in the presence of gravity. The pipe may be misaligned at the clamped end with respect to gravity, and the effects of this misalignment are the main objects of the present investigation. The problem has been formulated using the Cosserat rod model. First, we have computed the equilibrium solutions and used them to experimentally validate both the Cosserat model and the constitutive law. Then, we have analyzed the occurrence of flutter, via Hopf bifurcation, for critical values of the relevant parameters of the problem, such as fluid to total mass ratio, dimensionless flow rate, dimensionless gravity and misalignment angle. The influence of the equilibrium solution on flutter has been explored, and the results of the linear stability analysis show that the stabilizing or destabilizing effect of fluid flow, either in or out of the plane of the pipe, depend crucially on the misalignment. We have also computed the non-linear periodic behavior after flutter instability by two different methods: the first one is by solving the full nonlinear equations by direct integration in time and space, while the second one is by assuming the time dependence given by an appropriate ansatz. Circular periodic orbits have then been studied and found that its loss of stability via Hopf bifurcation gives rise to stable planar periodic orbits. Finally, we have also computed the multiply periodic and chaotic behaviors which take place for sufficiently large values of the flow rate.     

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.     

Simultaneous measurements of velocity and deformation in flows through compliant diaphragms

Flow through a circular orifice in a deformable diaphragm mounted in a pipe was studied experimentally as a simple yet suitable case for validating numerical fluid/structure interaction (FSI) codes including structures with significant deformation and strain. The flow was characterized using pressure taps, particle image velocimetry (PIV), and hot-film anemometry while deformation of the compliant diaphragm was determined directly from PIV images. The diaphragm material properties were measured independently by a uniaxial tensile testing machine. The diaphragm material modulus, orifice diameter, and pipe Reynolds number were varied over ranges appropriate for simulations of flows through heart valves. Pipe Reynolds numbers ranged from 600 (laminar upstream condition) to 8800 (turbulent upstream condition). The pressure drop across the diaphragm resulted in a concave deformation for all cases studied. For the range of Reynolds number tested, the Euler number decreased with increasing Reynolds number as a result of orifice expansion. The flow immediately downstream of compliant diaphragms was jet-like with strong inward radial velocity components and vena contracta. Laminar low Reynolds number flow (Re=600) through both rigid and compliant diaphragms yielded early and regular roll up of coherent vortex rings at a fixed frequency in contrast to turbulent higher Reynolds number flow (Re=3900), which yielded a broad range of vortex passage frequencies. Expansion of the compliant orifice for Re=3900 resulted in an initially broader slower jet with delayed shear layer development compared with the equivalent rigid case.     

Dynamics of disk-like particles in turbulent vertical channel flow

The dynamical behavior of inertial disk-like particles in turbulent vertical channel flow is investigated by an Eulerian–Lagrangian point-particle approach. Gravity effects on distribution, translation, rotation and orientation statistics of non-spherical particles modeled as oblate spheroids have been studied both in an upward and a downward flow and compared with results obtained in the absence of gravity. Altogether 12 different particle classes have been studied, with inertia and shape parameterized by means of Stokes number St and aspect ratio λ ≤ 1. The St = 5 disk-like particles distribute more evenly across the channel in upward than in downward flow. The gravity effect on the particle concentration diminishes with large inertia and the spheroid shape has only a modest influence. Although the gravity significantly affects the streamwise and wall-normal mean slip velocities with increasing inertia, the particle shape rarely has any impact on the translational motion, except for the mean wall-normal velocity. The fluctuations of the velocity of disk-like particles are mainly ascribed to inertia, whereas the gravity and shape only have marginal effects. The presence of gravity is moreover found to have a negligible effect on the particles’ orientation and rotation, in spite of the striking effect of λ on the orientation and rotation seen in the near-wall region. The tendency of the disks to align their symmetry axis orthogonal to the fluid vorticity in the channel center is stronger for particles with modest inertia. In the near-wall region, however, oblate spheroids preferentially align with the fluid vorticity for St >> 1. The observed behavior is believed to be caused by the influence of the gravity force on the turbophoresis; i.e. that inertial particles move towards low-turbulence regions.     

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.     

Wave regimes of ascending flow of a thin layer of a viscous liquid in contact with a gas

The flow of a liquid in thin layers is one of the hydrodynamic problems of chemistry and heat engineering. The large surface area of films and their small thickness make it possible to accelerate thermal, diffusive, and chemical processes at the gas-liquid boundary.Theoretical studies of liquid flow in a vertical descending thin layer are presented in [1–4]. In this paper we study ascending wave flows of a liquid in a thin vertical layer in contact with a gas, i.e., flows in the direction opposite the action of the force due to gravity, with account for the action of the gas on the liquid surface. Such motions are encountered when oil is extracted from strata that are saturated with gas. At some distance from the stratum the oil and gas separate: the gas travels at high velocity inside the pipe, occupying a considerable portion of the pipe, and the liquid is displaced toward the pipe walls, forming a thin film. In certain cases a wave-like interface develops between the oil and gas that travels with a velocity greater than that of the liquid but less than the average gas velocity. Similar phenomena are observed in high velocity mass exchangers.We examine the effect of the gas for both laminar and turbulent flow.Studies that neglect the effect of the gas flow on the liquid show that for waves on the film surface whose lengths are considerably longer than the average thickness of the layer, the liquid motion in the film is described by boundary layer equations in which account is taken of the mass force, i.e., the force due to gravity. With some approximation, we can assume that in accounting for the effect of the gas on the liquid the liquid flow is described by these same equations.     

Transient thermal entrance heat transfer in laminar pipe flows with step change in pumping pressure

Analysis is made for the transient heat transfer phenomena in the thermal entrance region of laminar pipe flows. The transient results from both the change in flow field, a step change in pressure gradient from zero to a fixed value, and the change in thermal field, a step change in the inlet temperature. An exponential scheme has been employed to solve the energy equation with the presence of axial heat conduction in the fluid. In order to demonstrate the results more clearly, a modified Nusselt number is introduced. The unsteady axial variations of conventional Nusselt number, modified Nusselt number, bulk fluid temperature and pipe wall temperature are presented for water and air over a wide range of outside heat transfer coefficients. It is observed that the outside heat transfer coefficient has a significant influences on the transient heat transfer processes. The results can be comprehensively interpreted by the interactions among the axial convection, axial diffusion, and radial diffusion.     

含颗粒污染物的油液均质流的数值分析

含颗粒污染物的油液是浓度很稀、粒径极小的伪均质流,为了掌握颗粒污染物在输送过程中的浓度分布,利用一维扩散方程构建了污染油液的数学模型;通过特征线法数值求解,获得了污染油液中各相的动态特征。结果表明,油液压力和速度沿管长呈脉动规律运动,且随着时间的延长逐渐衰减;颗粒污染物对油液速度具有极好的跟随特性;颗粒污染物的浓度分布也随着油液流速的变化而呈现规律性的变化;在不同运行时间内油液压力沿管长的衰减趋势不同,油液速度沿管长的变化趋势与压力的趋势相反;颗粒污染物速度和浓度分布沿管长与油液速度具有紧密的联系。     

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.     

CFD-DEM investigation into flow characteristics in mixed pulsed fluidized bed under electrostatic effects

Static electricity has an important effect on gas–solid fluidized bed reactor fluidization performance. In the process of fluidization, electrostatic interaction between particles will obviously accelerate particle agglomerate formation, which consequently reduces the fluidization performance. Pulsed gas flow injection is an efficient method to enhance particle mixing, thereby weakening the occurrence of particle agglomerate. In this study, the two-dimensional hybrid pulsed fluidized bed is established. The flow characteristics are studied by using the coupled CFD-DEM numerical simulation model considering electrostatic effects. Influences of different pulsed frequencies and gas flow ratios on fluidized bed fluidization performance are investigated to obtain the optimal pulsed gas flow condition. Results show that in the presence of static electricity, the bubble generation position is lower, which is conducive to the particle flow. Pulsed gas flow can increase the particle velocity and improve the diffusion ability. The bubble generation time is different at different frequencies, and the frequency of 2.5 Hz has the most obvious effect on the flow characteristics. Different gas flow ratios have significant impacts on the particle movement amplitude. When the pulse gas flow accounts for a large ratio, the particle agglomerate tends to be larger. Therefore, in order to improve the fluidization effect, the ratio of pulsed gas flow to stable gas flow should be appropriately reduced to 0.5 or less.     

Thermo-mechanical modeling of turbulent heat transfer in gas–solid flows including particle collisions

A thermo-mechanical turbulence model is developed and used for predicting heat transfer in a gas–solid flow through a vertical pipe with constant wall heat flux. The new four-way interaction model makes use of the thermal k_θ–τ_θ equations, in addition to the hydrodynamic k–τ transport, and accounts for the particle–particle and particle–wall collisions through a Eulerian/Lagrangian formulation. The simulation results indicate that the level of thermal turbulence intensity and the heat transfer are strongly affected by the particle collisions. Inter-particle collisions attenuate the thermal turbulence intensity near the wall but somewhat amplify the temperature fluctuations in the pipe core region. The hydrodynamic-to-thermal times-scale ratio and the turbulent Prandtl number in the region near the wall increase due to the inter-particle collisions. The results also show that the use of a constant or the single-phase gas turbulent Prandtl number produces error in the thermal eddy diffusivity and thermal turbulent intensity fields. Simulation results also indicate that the inter-particle contact heat conduction during collision has no significant effect in the range of Reynolds number and particle diameter studied.