Hydrodynamics of disturbed flow and erosion–corrosion. Part II — Two-phase flow study

Erosion–Corrosion in turbulent, two-phase liquid/particle flow with recirculation, after a sudden pipe expansion is studied by the application of a numerical flow model along with two different erosion models. The flow model, which is based on a two-phase flow version of a standard two-equation model of turbulence and a stochastic simulation of particle-fluid turbulence interactions, is capable of successfully predicting local values of time averaged fluid velocities and turbulent fluctuations, as well as predicting particle dispersion and particle-wall interaction. The erosion models used are that of Finnie (1960) and a modified version suggested by Bergevin (1984). The agreement of the predicted and measured hydrodynamic parameters, for flow through a sudden expansion, was satisfactory. Predictions of erosion rates using Bergevin's modified model were in good agreement with the stainless steel erosion measurements for a 2% water/sand slurry flow. The erosion–corrosion model was successful in predicting the overall pattern and rates of metal loss for carbon steel.     

Computation of gas and solid dispersion coefficients in turbulent risers and bubbling beds

A literature review shows that dispersion coefficients in fluidized beds differ by more than five orders of magnitude. To understand the phenomena, two types of hydrodynamics models that compute turbulent and bubbling behavior were used to estimate radial and axial gas and solid dispersion coefficients. The autocorrelation technique was used to compute the dispersion coefficients from the respective computed turbulent gas and particle velocities.The computations show that the gas and the solid dispersion coefficients are close to each other in agreement with measurements. The simulations show that the radial dispersion coefficients in the riser are two to three orders of magnitude lower than the axial dispersion coefficients, but less than an order of magnitude lower for the bubbling bed at atmospheric pressure. The dispersion coefficients for the bubbling bed at 25 atm are much higher than at atmospheric pressure due to the high bed expansion with smaller bubbles.The computed dispersion coefficients are in reasonable agreement with the experimental measurements reported over the last half century.     

Fluid–particle correlated motion and turbulent energy transfer in a two-dimensional particle-laden shear flow

Discrete vortex simulations of a dilute two-dimensional particle-laden shear layer with one-way coupling were performed to study fluid–particle correlated motion and the transfer of turbulent kinetic energy between the phases. The resulting modification of carrier phase turbulence, estimated according to current computational models, was evaluated. Particle Stokes numbers were between 1.0 and 4.5, so that the particles showed considerable temporal concentration fluctuations due to centrifuging by the fluid flow structures, and the mass loading was 12% corresponding to a volume fraction of 6.0×10^?5.Fluid velocities and particle concentration and velocities and their covariances, which appear in a commonly used model equation for carrier phase turbulence modification, were evaluated. Additionally, the probability density functions of fluid velocity fluctuations viewed by the particles are presented and compared with their Eulerian counterparts. It was found that particles view reduced velocity fluctuations due to preferential clustering. The model for carrier-phase turbulence modification predicted turbulence reduction, depending on the particle Stokes number. The mechanism responsible for turbulence reduction was the correlated velocity fluctuations of fluid and particles and this reduction could reach values up to one third of the fluid flow dissipation. Preferential particle concentration together with a relative velocity between the phases could generate turbulent kinetic energy of the gas phase, however this production was nearly an order of magnitude smaller compared to reduction of turbulence due to the correlated motion. The findings were compared with experiments available in the literature and help to clarify the view when turbulence reduction or augmentation occurs.     

Modelling of heavy and buoyant particle dispersion in a two-dimensional turbulent mixing layer

Heavy and buoyant particle dispersion in the turbulent mixing layer was investigated numerically using a two-phase flow discrete vortex modelling. It was revealed from the modelling that inclusion of two-way momentum coupling is essential for properly modelling heavy particle dispersive transport in turbulent free shear flows. For heavy particles with small Stokes numbers, the dispersion is predominated by the large-scale vortex structures and they exert small influence on the carrier fluid flow. Heavy particles with large St directionally align along the braid region between the neighbouring vortices. However, the lateral dispersion of particles of large St is smaller than that of particles of small St.For buoyant particles with the density being slightly greater than that of the carrier fluid, numerical simulation revealed that the buoyant particles scatter over the whole vortex core rather than collect along the fringes of the vortex. The Lagrangian statistics calculation of buoyant particle dispersion showed that both the inertial and crossing-trajectory effects affect the particle dispersion behaviour and particle eddy diffusivity. The dispersion behaviour of buoyant particles is highly associated with the particle Stokes number. Large St buoyant particles exhibit a larger dispersion. It was also indicated from the numerical simulation that buoyant particles might disperse larger than the fluid tracers. The correlation between the buoyant particle and fluid tracer velocities was affected by including the coupling effect.     

Axial dispersion characteristics in three phase fluidized beds

Axial dispersion coefficients in three-phase fluidized beds have been measured in a 0.152 m-ID x 1.8 m high column by the two points measuring technique with the axially dispersed plug flow model. The effects of liquid velocity (0.05–0.13 m/s), gas velocity (0.02–0.16 m/s) and particle size (3-8 mm) on the axial dispersion coefficient at the different axial positions (0.06–0.46 m) in the bed have been determined. The axial dispersion coefficient increases with increasing gas velocity but it decreases with an increase in particle size and exhibits a maximum value with an increase in the axial position from the distributor. The axial dispersion coefficients in terms of the Peclet number have been correlated in terms of the ratio of fluid velocities, the ratio of the panicle size to column diameter, and the dimensionless axial position in the bed based on the isotropic turbulence theory.     

Two-phase interactions through turbulent events as described by fluid–particle correlations

An experimental investigation of a vertical upward, two-phase pipe flow was undertaken to measure kinematic parameters of the fluid and solid phases. The kinematic parameters included Reynolds stress distributions based on quadrant analyses that provided insight in understanding the behavior of two-phase kinematic correlation profiles. The data collected was based on a two-color digital particle image velocimetry (DPIV) technique that simultaneously measured the velocity fields of the fluid and solid phases.From quadrant analysis results, differences in Reynolds stress quadrant profiles between the single- and two-phase conditions were observed near the wall in the range 0.8>r/R>0.55, corresponding to wall distances between 35 and 75 viscous lengths (y+). Correlation coefficients between the two phases were then calculated, using the fluctuating velocity components of each phase. The extent of the interaction between the two phases was tracked by the changing correlation values versus distance from the wall. The correlation of the fluid and solid phase velocities was highest in the core region of the pipe (y+∼120), where the effect of turbulent events is reduced; low correlation coefficient values were found at y+<75, where differences of magnitudes, inflection points, etc. of burst and sweep event quadrant analysis profiles were observed.The extent of the influence of wall dynamic turbulent events on the solid phase was observed both by the differences in the relative Reynolds stress quadrant profiles and, more readily, by the changing values of two-phase axial and radial correlation coefficients determined from the simultaneous fluid and solid fluctuating velocities measured by the two-color DPIV methodology. These changing values of the correlation coefficients across the tube reflect the different responses of low inertia (fluid tracers) and high inertia (solid phase glass spheres) particles to the turbulent events. Similar profiles of the axial and radial correlation coefficients were observed, indicating that for the geometry and flow conditions considered, one velocity component of each phase was sufficient to track the spatial extent of turbulent event effects and their interactions with the fluid and solid phases. It is found that the two-color DPIV methodology and two-phase correlation results can give critical insight into the performance of thermal-fluid processes, as burst and sweep events have a large impact on the kinematics and dynamics of particles in the two-phase flow.     

Measurements of pollen grain dispersal in still air and stationary, near homogeneous, isotropic turbulence

The dispersal of ragweed, pine and corn pollen as well as polystyrene spheres in still air and stationary, near homogeneous, isotropic turbulence (HIT) was investigated using high-speed, digital inline holographic cinematography enabling Lagrangian tracking of the particles. Mean still air settling velocities were similar as reported literature values. Small discrepancies were most likely related to species/size differences and water content of the grains. Near-HIT was generated by loudspeakers mounted on the corners of a 40 cm³ chamber and the turbulent flow field at the center of the chamber was validated using stereoscopic Particle Image Velocimetry (PIV). Results showed near homogeneity and near isotropy with mean velocities 5–10 times smaller than the corresponding rms values of velocity fluctuations. The turbulent kinetic energy dissipation rate was determined from the PIV data sets and used to calculate the Kolmogorov scales and Taylor microscales. Experiments were carried out for two different loudspeaker amplifications corresponding to Taylor microscale Reynolds numbers, R_λ=144 and 162, respectively. The mean settling velocity in turbulent conditions was in all cases higher than the corresponding still air value, the difference becoming smaller as particle Stokes numbers increased. For the present conditions, the still air particle settling velocity was lower than the rms values of air fluctuating velocities. As a result, dispersion was dominated by inertia and for a given R_λ, particle fluctuating velocity autocorrelations fell more rapidly as the particle Stokes number decreased; corresponding particle diffusion coefficients also decreased. Transverse particle diffusion coefficients were lower than those in the direction of gravity in agreement with the continuity effect. Under the present range of experimental parameters, results showed that inertial particles (0.6<St<11) in highly turbulent conditions disperse more effectively than the air.     

Velocity fluctuation interpretation in the near wall region of a dense riser

Tests were conducted in a cold flow circulating fluidized bed to gather computational fluid dynamics (CFD) model validation data. Particle velocity measurements were obtained with an LDV system under various operating conditions at locations near the wall to provide data in terms of a time series of particle velocity values. Time scale criteria were developed to characterize the variance of the velocity fluctuations from LDV measurements as either granular temperature or granular turbulent kinetic energy. By applying these criteria to categorize the variations in the velocities for adjacent particles passing the sample volume, the resulting granular temperatures were found to be much smaller than the granular (particle) turbulent kinetic energy. Average values for the granular temperature in this system ranged between 0.02 to 0.1 m²/s², while the particle turbulent kinetic energy ranged from 0.6 to 0.9 m²/s². Both were dependent upon solids fraction; decreasing with increasing solids fraction. The velocity fluctuation data was also analyzed using the autocorrelation technique providing axial solids dispersion coefficients. These values range from 0.005 to 0.8 m²/s and were found to be a function of both the gas velocity and solids fraction. A method was developed to estimate the local solids fraction with the LDV data.     

Transport property measurements in sheared granular flows

Experiments were performed in a shear cell device under four different solid fractions. The glass spheres with a mean diameter of 3 mm were used as granular materials. The motions of the granular materials were recorded by a high-speed camera. By using image processing technology and a particle tracking method, the average and fluctuation velocities in the streamwise and the transverse directions could be successfully measured and analyzed. Three bi-directional stress gages were used to measure the normal and shear stresses along the upper boundary. The effective viscosity of the granular material flow can be calculated. By tracking the movements of particles continually, the curves of the mean-square diffusive displacements versus time were plotted and were used to determine the self-diffusion coefficients from the slopes of the curves. The fluctuations and the self-diffusion coefficients in the streamwise direction were much higher than those in the transverse direction. The fluctuations were found to increase with the solid fraction, but the diffusion coefficients were greater in a more dilute flow system.     

Influence of interstitial fluid viscosity on transport phenomenon in sheared granular materials

We report experimental measurements of transport properties on sheared granular materials with different interstitial fluids to study the granular flow behaviors. Four kinds of interstitial fluids are used in the experiments. The ensemble velocities, fluctuation velocities, granular temperature and self-diffusion coefficients are successfully measured by PTV method. The results indicate that the interstitial fluid plays an important role in determining the transport properties of the granular flow. The particles motions are more random and interactive collisions are more serious in a dry system (interstitial fluid of air). The values of fluctuations and granular temperatures are smaller as the interstitial fluid is more viscous resulting in the larger viscous force. The thickness of shear band is about three to eight particle diameters and increases with the decreasing interstitial fluid viscosity. The self-diffusion coefficient of granular materials is also discussed in this study. Both the self-diffusion coefficients and the granular temperature increase with the increasing shear rate. The average streamwise self-diffusion coefficient and granular temperature increase with the increase in Stokes number and Bagnold number.     

EVALUATION OF LAGRANGIAN PARTICLE DISPERSION MODELS IN TURBULENT FLOWS

This work evaluates the performance of Lagrangian turbulent particle dispersion models based on the Langevin equation. A family of Langevin models, extensively reported in the open literature, decompose the fluctuating fluid velocity seen by the particle in two components, one correlated with the previous time step and a second one randomly sampled from a Wiener process, i.e., the closure is at the level of the fluid velocity seen by the particle. We will call those models generically the “standard model.” On the other hand, the model proposed by Minier and Peirano (2001) is considered; this approach is based on the probability density function (PDF) and performs the closure at the level of the acceleration of the fluid seen by the particle. The formulation of a Langevin equation model for the increments of fluid velocity seen by the particle allows capturing some underlying physics of particle dispersion in general turbulent flows while keeping simple the mathematical manipulation of the stochastic model, avoiding some pitfalls, and simplifying the derivation of macroscopic relations. The performance of the previous dispersion models is evaluated in the configurations of grid-generated turbulence (Snyder and Lumley, 1971; Wells and Stock, 1983), simple shear flow (Hyland et al., 1999), and confined axisymmetric jet flow laden with solids (Hishida and Maeda, 1987).     

稀疏气固两相湍流边界层拟序结构变动特性

应用PIV两相同时测量方法,对壁面Reynolds数为430的水平槽道稀疏气固两相湍流边界层拟序结构变动特性进行了研究。选取质量载荷为10^-4~10^-3的110 μm聚乙烯颗粒作为离散相。结果表明,低载荷颗粒仍能显著改变湍流拟序结构,进而影响宏观湍流属性。颗粒重力沉降形成的粗糙壁面增强了壁面附近湍流猝发行为,导致黏性底层中的气相法向脉动速度和雷诺剪切应力显著增大。颗粒与壁面的碰撞加强了低速流体上抛、削弱了高速流体下扫,同时增强了轨道交叉效应,从而抑制了湍流拟序结构发展,显著减小了黏性底层以上区域的法向脉动速度和雷诺剪切应力。此外,颗粒惯性还减小了黏性底层厚度、增大了流向速度梯度,导致气相流向脉动速度峰值增大,且其对应位置也更加靠近壁面。     

Axial dispersion characteristics of three (liquid-liquid-solid) phase fluidized beds

The effects of the continuous and dispersed phase velocity and particle size on the axial dispersion of the continuous phase have been determined in two (liquid-liquid) and three (liquid-liquid-solid) phase fluidized beds. In a cocurrent liquid-liquid flow system, the axial dispersion coefficient increases with both the dispersed and continuous phase velocities. In three phase fluidized beds, the coefficient increases with dispersed phase velocity but it decreases with the particle size. Also the coefficient exhibits a maximum value with an increase in the continuous phase velocity at the lower dispersed phase velocities, but it increases with the continuous phase velocity at higher dispersed phase velocities. The axial dispersion coefficients in terms of Peclet number have been correlated in terms of the ratio of fluid velocities and the ratio of the particle size to column diameter, based on the isotropic turbulence theory.     

Prediction of turbulent gas‐solid flow in a duct with a 90° bend using an Eulerian‐Lagrangian approach

A dilute, particle‐laden turbulent flow in a square cross‐sectioned duct with a 90° bend is modeled using a three‐dimensional Eulerian‐Lagrangian approach. Predictions are based on a second‐moment turbulence closure, with particles simulated using a Lagrangian particle tracking technique, coupled to a particle‐wall interaction algorithm and a random Fourier series method used to model particle dispersion. The performance of the model is tested for a gas‐solid flow in a horizontal‐to‐vertical duct, with predictions showing good agreement with experimental data. In particular, the consistent use of anisotropic and fully three‐dimensional approaches throughout yields predictions that result in fluctuating particle velocities in acceptable agreement with data. © 2011 American Institute of Chemical Engineers AIChE J, 2012     

INSTANTANEOUS VELOCITY PROFILE MEASUREMENTS IN A TURBULENT BOUNDARY LAYER

Instantaneous wall shear stress and streamwise velocities have been measured simultaneously in a flat-plate, turbulent boundary layer at moderate Reynolds number in an effort to provide experimental support for large eddy simulations. Data were obtained using a buried-wire, wall shear gage and a hot-wire rake positioned in the log region of the flow. Fluctuations of the instantaneous U ⁺ versus Y ⁺ profiles about a mean law of the wall are shown to be significant and complex. Peak cross-correlation values between wall shear stress and the velocities are high, and reflect the passage of a large structure inclined at a small angle to the wall. Estimates of this angle are consistent with those made by other investigators. Conditional sampling techniques were used to detect the passage of various sizes and types of flow disturbances (events), and to estimate their mean frequency of occurrence. Events characterized by large and sudden streamwise accelerations were found to be highly coherent throughout the log region and were strongly correlated with large fluctuations in wall shear stress. Phase randomness between the near-wall quantities and the outer velocities was small. The results suggest that the flow events detected by conditional sampling applied to velocities in the log region may be related to the bursting process.     

Dispersion and slip effects in hydraulic hoisting of solids

Concentration changes which occur in cyclical hydraulic hoisting operations can be modelled with a form of the convection-diffusion equation. In order to solve this equation it is necessary to specify the axial dispersion coefficient and the effective slip of the particles relative to the mean velocity. The effective slip depends upon drag and particle migration effects. Experimental observations of these concentration changes are interpreted with numerical solutions of the appropriate equation. Fine particles develop a near-Gaussian concentration distribution while coarse particles produce a saw-tooth pulse. Axial dispersion coefficients are similar to those observed with solutes in turbulent slurry flows and slip velocities are shown to be concentration dependent.     

Mean and fluctuating gas phase velocities inside a circulating fluidized bed

Gas phase velocities is an area in circulating fluidized beds (CFB) that has traditionally received little attention. The dynamics and motion of particles or clusters inside the bed has been the main focus of research. This is because particles dominate the fluid mechanics and heat transfer inside a CFB. However, gas phase motions also effect particle motion. Gas eddies or fluctuations can play an important role in transporting particles to and from the wall. They also help in providing a uniform temperature throughout the bed by promoting mixing. This paper deals with how particles effect the mean and fluctuating gas velocities throughout the cross-section of a riser.Gas velocities were measured inside a cold scale model CFB using a shielded hot wire anemometer. At the centerline, typical mean gas velocities were measured which were approximately twice the superficial gas velocity. These high velocities are likely caused by the negligible net gas upflow in the annulus region. The presence of many dense, downward flowing clusters in the annulus makes this a reasonable assumption.Previous work on gas phase turbulence in two phase flows has typically used either laser measurement techniques in very small diameter risers or in larger risers with very low particle concentration. The general results have shown that smaller particles, on the same order of magnitude as those typically used in CFB and FCC reactors, tend to damp out the gas phase fluctuations. This implies that gas phase motion behaves close to a laminar fashion. This present research measures gas phase fluctuations with typical particle concentrations inside a CFB (∼1-5%). The results indicate that at larger particle concentrations where clusters are formed, the gas phase fluctuations increase dramatically. This suggests that length scales based on cluster size, as opposed to particle size, should be used in estimating the increased levels of gas fluctuations caused by the solid phase. Hence, models which ignore the effect of clusters on the gas or which treat the gas phase as laminar like flow, yield a misleading picture of the flow dynamics inside a CFB riser.     

A numerical study of particle wall-deposition in a turbulent square duct flow

The deposition of dense solid particles in a downward, fully developed turbulent square duct flow at Re_τ = 360, based on the mean friction velocity and the duct width, is studied using large eddy simulations of the fluid flow. The fluid and the particulate phases are treated using Eulerian and Lagrangian approaches, respectively. A finite-volume based, second-order accurate fractional step scheme is used to integrate the incompressible form of the unsteady, three-dimensional, filtered Navier-Stokes equations on an 80 × 80 × 128 grid. A dynamic subgrid kinetic energy model is used to account for the unresolved scales. The Lagrangian particle equation of motion includes the drag, lift, and gravity forces and is integrated using the fourth-order accurate Runge-Kutta scheme. Two values of particle to fluid density ratio (ρ_p/ρ_f = 1000 and 8900) and five values of dimensionless particle diameter (d_p/δ × 10⁶ = 100, 250, 500, 1000 and 2000, δ is the duct width) are studied. Two particle number densities, consisting of 10⁵ and 1.5 × 10⁶ particles initially in the domain, are examined.Variations in the probability distribution function (PDF) of the particle deposition location with dimensionless particle response time, i.e. Stokes number, are presented. The deposition is seen to occur with greater probability near the center of the duct walls, than at the corners. The average streamwise and wall-normal deposition velocities of the particles increase with Stokes number, with their maxima occurring near the center of the duct wall. The computed deposition rates are compared to previously reported results for a circular pipe flow. It is observed that the deposition rates in a square duct are greater than those in a pipe flow, especially for the low Stokes number particles. Also, wall-deposition of the low Stokes number particles increases significantly by including the subgrid velocity fluctuations in computing the fluid forces on the particles. Two-way coupling and, to a greater extent, four-way coupling are seen to increase the deposition rates.     

INSTANTANEOUS VELOCITY PROFILE MEASUREMENTS IN A TURBULENT BOUNDARY LAYER

Instantaneous wall shear stress and streamwise velocities have been measured simultaneously in a flat-plate, turbulent boundary layer at moderate Reynolds number in an effort to provide experimental support for large eddy simulations. Data were obtained using a buried-wire, wall shear gage and a hot-wire rake positioned in the log region of the flow. Fluctuations of the instantaneous U⁺ versus Y⁺ profiles about a mean law of the wall are shown to be significant and complex. Peak cross-correlation values between wall shear stress and the velocities are high, and reflect the passage of a large structure inclined at a small angle to the wall. Estimates of this angle are consistent with those made by other investigators. Conditional sampling techniques were used to detect the passage of various sizes and types of flow disturbances (events), and to estimate their mean frequency of occurrence. Events characterized by large and sudden streamwise accelerations were found to be highly coherent throughout the log region and were strongly correlated with large fluctuations in wall shear stress. Phase randomness between the near-wall quantities and the outer velocities was small. The results suggest that the flow events detected by conditional sampling applied to velocities in the log region may be related to the bursting process.     

A Stochastic Model for Particle Deposition and Bounceoff

A new model for particle deposition and bounceoff that combines current knowledge of turbulent bursts with the stochastic properties of turbulent fluctuations is presented. The model predictions for deposition velocities agree with experimental results in the literature for dimensionless particle relaxation time τ_p ⁺ > 2. For τ_p ⁺ > 10, most of the particles delivered to the edge of the viscous sublayer are able to deposit onto the surface due to their inertia; the deposition velocity approaches an asymptotic value because the process becomes limited by the rate of turbulent delivery to the viscous sublayer. Because of the penetration of turbulent fluctuations into the viscous sublayer, the minimum values of vertical velocities needed for particles to deposit onto the surface are smaller than those predicted by the free flight model. Most of the deposition occurs from those turbulent fluctuations at the upper tail of the distribution of the vertical component of air velocity.

In addition to the deposition velocity, the model is able to provide the distribution of particle velocities on reaching the surface which is used to compute the fraction of particle bounceoff. The model predictions for the fractions of rebound agree reasonably with the measured results from a wind tunnel experiment for τ_p ⁺ > 2. However, both the deposition velocity and the fraction of rebound are underestimated by the model for τ_p ⁺ < 2. Other mechanisms such as Brownian diffusion must be included in further revisions to this model in order to obtain satisfactory predictions for smaller values of τ_p ⁺.