Experimental estimation of the settling velocity and drag coefficient of the hollow cylindrical particles settling in non-Newtonian fluids in an annular channel

The drag coefficient data of particles settling in an annular channel is very much essential for designing different solid–fluid handling equipment, such as the fluidized bed. Experimental settling velocity, wall factor, and drag coefficient data of the hollow-cylinder particle are presented. Carboxymethyl cellulose solution has been used as the working fluid with a flow index of 0.64 ≤ n ≤ 0.91 and a consistency index of 0.31 ≤ K ≤ 1.81. The experimental results covered a wide diameter ratio range (0.14 ≤ d_eq/L ≤ 0.46), hollow cylinder inner to outer diameter ratio (0.2 ≤ d_i/d_o ≤0.8), and Reynolds number (0.05 ≤ Re ≤ 51 and 0.09 ≤ Re_∞ ≤ 55). d_eq, d_i, and d_o are the equivalent inner and outer diameters of the particle, L is the annular gap, and Re and Re_∞ are the Reynolds numbers in the presence and absence of the wall effect, respectively. The wall factor decreased, and the drag coefficient increased with d_eq/L and d_i/d_o ratios. The above parameters declined with the Reynolds number. The hollow cylinder experienced a lesser wall effect than the spherical particles settling in a non-annular channel. In some cases, the wall factor of the hollow cylinder is found to be equal to the spherical particles settling in an annular channel. The developed correlations have successfully predicted the drag coefficients of the hollow cylinder.     

Gas dispersion in air‐lift reactors: Contribution of the turbulent part of the added mass force

Gas dispersion in an airlift reactor focusing on the closure law on turbulent contribution of added mass is presented. A data bank for bubbly flow in an airlift reactor is presented. The liquid velocity is measured by hot film anemometry and gas fraction and velocity are measured with an optical probe. The sensitivity of numerical simulations of gas dispersion to the modeling of turbulent contribution of added mass is shown. Without the turbulent contribution, the bubbles move toward the region where the turbulence is high and the pressure is low. When the turbulent contribution is introduced, the bubble migration towards the low pressure region is counter‐balanced and the void fraction profile is significantly modified. The modeling of the turbulent contribution of added mass is expressed in terms of the turbulent correlations in the gas phase, u_Gi^′u_Gj^′ , that can be related to the Reynolds stress in the liquid phase, u_i^′u_j^′ . © 2011 American Institute of Chemical Engineers AIChE J, 2011     

A spatially‐averaged two‐fluid model for dense large‐scale gas‐solid flows

We present a spatially‐averaged two‐fluid model (SA‐TFM), which is derived from ensemble averaging the kinetic‐theory based TFM equations. The residual correlation for the gas‐solid drag, which appears due to averaging, is derived by employing a series expansion to the microscopic drag coefficient, while the Reynolds‐stress‐like contributions are closed similar to the Boussinesq‐approximation. The subsequent averaging of the linearized drag force reveals that averaged interphase momentum exchange is a function of the turbulent kinetic energies of both, the gas and solid phase, and the variance of the solids volume fraction. Closure models for these quantities are derived from first principles. The results show that these new constitutive relations show fairly good agreement with the fine grid data obtained for a wide range of particle properties. Finally, the SA‐TFM model is applied to the coarse grid simulation of a bubbling fluidized bed revealing excellent agreement with the reference fine grid solution. © 2017 American Institute of Chemical Engineers AIChE J, 63: 3544–3562, 2017     

Estimation of kLa Values in Bench‐Scale Stirred Tank Reactors with Self‐Inducing Impeller by Multiphase CFD Simulations

A multiphase computational fluid dynamics (CFD) simulation methodology is developed and proposed for the estimation of the spatial distribution of k_La values in a bench‐scale reactor equipped with a self‐inducing impeller. The importance of estimating an apparent drag coefficient, which considers the effect of turbulence on the gas bubble rising velocity, is also tackled by applying different correlations available in literature, namely, Brucato, modified Brucato, and Pinelli correlations. The spatial distribution of k_La values in the agitated vessel is found from the CFD results using Danckwert's surface renewal model. An analysis of the gas volume fraction distribution obtained from the simulations is performed in order to choose the most suitable drag model. The modified Brucato correction correlation for the drag force exhibits the best agreement with experimental data.     

Hydrodynamic forces on monodisperse assemblies of axisymmetric elongated particles: Orientation and voidage effects

We investigate the average drag, lift, and torque on static assemblies of capsule-like particles of aspect ratio 4. The performed simulations are from Stokes flow to high Reynolds numbers (0.1 ≤ Re ≤ 1,000) at different solids volume fraction (0.1 ≤ ɛ_s ≤ 0.5). Individual particle forces as a function of the incident angle ϕ with respect to the average flow are scattered. However, the average particle force as a function of ϕ is found to be independent of mutual particle orientations for all but the highest volume fractions. On average, a sine-squared scaling of drag and sine-cosine scaling of lift holds for static multiparticle systems of elongated particles. For a packed bed, our findings can be utilized to compute the pressure drop with knowledge of the particle-orientation distribution, and the average particle drag at ϕ = 0° and 90°. We propose closures for average forces to be used in Euler–Lagrange simulations of particles of aspect ratio 4.     

Terminal settling velocity and drag coefficient of biofilm‐coated particles at high Reynolds numbers

The drag force (F_d) on bio‐coated particles taken from two laboratory‐scale liquid–solid circulating fluidized bed bioreactors (LSCFBBR) was studied. The terminal velocities (u_t) and Reynolds numbers (Re_t) of particles observed were higher than reported in the literature. Literature equations for determining u_t were found inadequate to predict drag coefficient (C_d) in Re_t > 130. A new equation for determining F_d as an explicit function of terminal settling velocity was generated based on Archimedes numbers (Ar) of the biofilm‐coated particle. The proposed equation adequately predicted the terminal settling velocity of other literature data at lower Re_t of less than 130, with an accuracy >85%. © 2010 American Institute of Chemical Engineers AIChE J, 2010     

Steady Flow of Power Law Fluids across a Circular Cylinder

The momentum equations describing the steady cross‐flow of power law fluids past an unconfined circular cylinder have been solved numerically using a semi‐implicit finite volume method. The numerical results highlighting the roles of Reynolds number and power law index on the global and detailed flow characteristics have been presented over wide ranges of conditions as 5 ≤ Re ≤ 40 and 0.6 ≤ n ≤ 2. The shear‐thinning behaviour (n < 1) of the fluid decreases the size of recirculation zone and also delays the separation; on the other hand, the shear‐thickening fluids (n > 1) show the opposite behaviour. Furthermore, while the wake size shows non‐monotonous variation with the power law index, but it does not seem to influence the values of drag coefficient. The stagnation pressure coefficient and drag coefficient also show a complex dependence on the power law index and Reynolds number. In addition, the pressure coefficient, vorticity and viscosity distributions on the surface of the cylinder have also been presented to gain further physical insights into the detailed flow kinematics.     

The Voidage Function for the Drag Force Ratio in a Liquid‐Fluidized Bed

Di Felice (1994) has shown that the ratio of the drag coefficient, C_D, on a sphere in a liquid‐fluidized bed of uniform spheres to the drag coefficient, C_DS, on the same sphere in isolation and subjected to the same superficial liquid velocity, u, is given by a function ?^?β, where β was expressed as an empirical function of the particle Reynolds number, Re = duρ/µ. Here it is shown that C_D/C_DS is well approximated by ?^?mm, where the Richardson‐Zaki index n is a function of the terminal free‐settling Reynolds number, Re_t = du_tρ/µ, and m is 2 plus the slope of the standard log C_DS vs. log Re plot at plot at Re = Re_t. The present model, using the best experimentally confirmed equation for n and a new simple equation for and a new simple equation for m, is compared with that of Di Felice in their respective abilities to predict liquid‐fluidized bed expansion.     

Modifying the inter‐phase drag via solid volume fraction gradient for CFD simulation of fast fluidized beds

The conventional drag model in two‐fluid simulation, which assumes uniform particle distribution in a computational grid, overestimates the drag force, thus failed in capturing the subgrid‐scale strands and resolvable‐scale clusters. This work proposed a new modification to the conventional drag model through considering the heterogeneous distribution of solid volume fraction (SVF), especially, in the inter‐phase boundary (i.e., cluster boundary). The resulting drag model is a function of particle Reynolds number, SVF and the gradient of SVF. This straightforward modification is consistent with the elaborately filtered‐approach‐based modification method in nature. A CFD simulation for a two‐dimensional riser was conducted to validate the new drag model. The outlet solid mass flux, axial and radial time‐averaged voidages from the new drag model agreed well with the experimental measurements, and these results were far better than those from the conventional homogeneous drag models. © 2017 American Institute of Chemical Engineers AIChE J, 63: 2588–2598, 2017     

Direct numerical simulations of dynamic gas‐solid suspensions

Direct numerical simulation results for gas flow through dynamic suspensions of spherical particles is reported. The simulations are performed using an immersed boundary method, with careful correction for the grid resolution effect. The flow systems we have studied vary with mean flow Reynolds number, solids volume fraction, as well as particle/gas density ratio. On the basis of the simulation results, the effect of particle mobility on the gas‐solid drag force is analyzed and introduced into the existing drag correlation that was derived from simulations of stationary particles. This mobility effect is characterized by the granular temperature, which is a result of the particle velocity fluctuation. The modified drag correlation is considered so‐far the most accurate expression for the interphase momentum exchange in computational fluid dynamics models, in which the gas‐solid interactions are not directly resolved. © 2016 American Institute of Chemical Engineers AIChE J, 62: 1958–1969, 2016     

Drop Size Distribution in a Standard Twin‐Impeller Batch Mixer at High Dispersed‐Phase Volume Fraction

The preparation of concentrated aqueous silicone oil emulsions has been investigated with particular attention to the effect of the dispersed‐phase volume fraction ? from 0.01 to 0.5 for a wide range of oil viscosities (50 to 1000 cSt). Oil was added on the top surface of a 6‐L vessel. Drop size distribution and Sauter mean diameter, d₃₂, measurements were carried out over 24 h mixing time. Emulsification was found to be relatively sensitive to the oil phase viscosity, μ_d, for the same ? yielding a narrower drop size distribution for low oil viscosity (50 cSt) and a wider drop size distribution for the highly viscous oil (1000 cSt). For the same ?, increasing μ_d resulted in increasing d₃₂. The equilibrium d₃₂ was found to be well correlated to the viscosity number by for ? = 0.5. For the same oil viscosity, d₃₂ was found to increase with increasing ?. A multiregression of d₃₂ with both ? and V_i for various silicone oil viscosity grades was successfully correlated by with a regression coefficient (R²) of 0.975. This shows a very weak dependence of the equilibrium d₃₂ on ?.     

Verification of filtered two‐fluid models for gas‐particle flows in risers

The effect of solid boundaries on the closure relationships for filtered two‐fluid models for riser flows was probed by filtering the results obtained through highly resolved kinetic theory‐based two‐fluid model simulations. The closures for the filtered drag coefficient and particle phase stress depended not only on particle volume fraction and the filter length but also on the distance from the wall. The wall corrections to the filtered closures are nearly independent of the filter length and particle volume fraction. Simulations of filtered model equations yielded grid length independent solutions when the grid length is ～half the filter length or smaller. Coarse statistical results obtained by solving the filtered models with different filter lengths were the same and corresponded to those from highly resolved simulations of the kinetic theory model, which was used to construct the filtered models, thus verifying the fidelity of the filtered modeling approach. © 2010 American Institute of Chemical Engineers AIChE J, 2011     

Drag on a sphere in a spherical dispersion containing Carreau fluid

The drag on a rigid sphere in a spherical dispersion containing Carreau fluid is investigated theoretically based on a free surface cell model for Reynolds number in the range [0.1,100], Carreau number in the range [0,10], the power-law index in the range [0.3,1], and the void fraction in the range [0.271,0.999]. The influences of the particle concentration, the nature of the Carreau fluid, and Reynolds number, on the drag coefficient are examined. We show that the drag coefficient declines with the decreasing particle concentration, and the reversal of the flow field in the rear region of a sphere is enhanced by the shear-thinning nature of the fluid. An empirical relation, which correlates the drag coefficient with the void fraction (= 1 − particle concentration), the nature of the Carreau fluid, and Reynolds number, is proposed.     

Effective particle diameters for simulating fluidization of non‐spherical particles: CFD‐DEM models vs. MRI measurements

Computational fluid dynamics—discrete element method (CFD‐DEM) simulations were conducted and compared with magnetic resonance imaging (MRI) measurements (Boyce, Rice, and Ozel et al., Phys Rev Fluids. 2016;1(7):074201) of gas and particle motion in a three‐dimensional cylindrical bubbling fluidized bed. Experimental particles had a kidney‐bean‐like shape, while particles were simulated as being spherical; to account for non‐sphericity, “effective” diameters were introduced to calculate drag and void fraction, such that the void fraction at minimum fluidization (ε_mf) and the minimum fluidization velocity (U_mf) in the simulations matched experimental values. With the use of effective diameters, similar bubbling patterns were seen in experiments and simulations, and the simulation predictions matched measurements of average gas and particle velocity in bubbling and emulsion regions low in the bed. Simulations which did not employ effective diameters were found to produce vastly different bubbling patterns when different drag laws were used. Both MRI results and CFD‐DEM simulations agreed with classic analytical theory for gas flow and bubble motion in bubbling fluidized beds. © 2017 American Institute of Chemical Engineers AIChE J, 63: 2555–2568, 2017     

Comparative analysis of subgrid drag modifications for dense gas‐particle flows in bubbling fluidized beds

Many subgrid drag modifications have been put forth to account for the effect of small unresolved scales on the resolved mesoscales in dense gas‐particle flows. These subgrid drag modifications significantly differ in terms of their dependencies on the void fraction and the particle slip velocity. We, therefore, compare the hydrodynamics of a three‐dimensional bubbling fluidized bed computed on a coarse grid using the drag correlations of the groups of (i) EMMS, (ii) Kuipers, (iii) Sundaresan, (iv) Simonin, and the homogenous drag law of (v) Wen and Yu with fine grid simulations for two different superficial gas velocities. Furthermore, we present an (vi) alternative approach, which distinguishes between resolved and unresolved particle clusters revealing a grid and slip velocity dependent heterogeneity index. Numerical results are analyzed with respect to the time‐averaged solids volume fraction and its standard deviation, gas and solid flow patterns, bubble size, number density, and rise velocities. © 2013 American Institute of Chemical Engineers AIChE J, 59: 4077–4099, 2013     

Combined effects of temperature and Reynolds number on drag‐reducing characteristics of a cationic surfactant solution

The drag‐reducing characteristics in the turbulent channel flow of dilute cationic surfactant solution, cetyltrimethyl ammonium chloride (CTAC)/sodium salicylate (NaSal) aqueous solution, were experimentally investigated in a closed loop fluid flow facility at different temperatures. The mass concentrations of the surfactant solution ranged from 75 to 200 ppm, and the temperatures ranged from 15 to 55°C. The cationic surfactant solution showed a great drag‐reducing ability, which was greatly affected by concentration, temperature, and Reynolds number. It was found that there existed a critical temperature T_c in each solution at different concentrations. Above T_c, drag‐reduction level decreases and reaches the behaviour of water flow without drag‐reducing ability. A new temperature parameters T_f, was proposed, and the difference between T_c and T_f can represent the effective temperature range for the drag reduction at a certain Reynolds number. The variation tendency of T_f and T_c with Reynolds numbers can give the guidance of selecting effective drag reduction range to the practical application in the district heating systems (DHS). It was supposed that temperature and shear stress are two kind of energy applied on the surfactant microstructure, which can be helpful to the surfactant network formation or dissociation depending on their values. © 2011 Canadian Society for Chemical Engineering     

Power‐Law Fluid Flow Over a Sphere: Average Shear Rate and Drag Coefficient

Based on the consideration of the rate of mechanical energy dissipation, an expression for the average shear rate for a sphere falling in a power‐law fluid in the creeping flow regime has been deduced. The average shear rate in a power‐law fluid (n<1) appears to be higher than that in an equivalent Newtonian fluid. This in turn has been combined with the numerical predictions of drag coefficient (up to Reynolds number of 100) of a sphere to develop a generalized drag correlation for power‐law liquids encompassing both n > 1 and n < 1 which appears to apply up to much higher values of the Reynolds number. The available experimental data have been used to demonstrate the reliability and accuracy of the new correlation for shearthinning liquids. Also, in the limit of n = 1, this expression reproduces the standard drag curve with a very high accuracy.     

Minimum pickup velocity: The transition between nano‐scale and micro‐scale

The transport of nano‐scale particles has become increasingly important, but the knowledge base available is limited. This study aims to bridge the knowledge gap between the nano‐ and micro‐scales for pneumatic conveying. A key parameter is the minimum pickup velocity (U_pu), which is the minimum fluid velocity required to initiate motion in a particle originally at rest. The U_pu values of nine alumina particles with particle diameters (d_p) ranging from 5 to 110,000 nm were determined using the weight loss method, then compared against the established pickup Zones (analogous to the Geldart Groups). Results indicated that: (1) U_pu varied non‐monotonically with increasing d_p, thus revealing the missing link between the nano‐ and micro‐scales; (2) the intermediate particle diameters surprisingly did not agree with any pickup Zone; (3) Zone III (analogous to Geldart Group C) is inadequate for all the nano‐scale particles, so new boundaries and a new Zone are proposed. © 2016 American Institute of Chemical Engineers AIChE J, 63: 1512–1519, 2017     

Effects on the melt rheology of systems of Nd‐Fe‐B particles suspended in a poly(phenylene sulfide) and liquid crystalline polymer blend

The melt rheology of blends of a liquid crystalline polymer (LCP) and poly(phenylene sulfide) (PPS) and their composites with ferromagnetic Nd‐Fe‐B particles (MQP) was studied. We investigated the effects of LCP concentration, Nd‐Fe‐B particle volume fraction and size, distribution, and shear rate on the rheological properties of these composites. Enthalpy of fusion changes that were observed resulted from the addition of the LCP and Nd‐Fe‐B particles to the polymer blends/composites. The shear rate and frequency dependencies of the materials revealed a viscosity reduction at low (1–3 wt%) and moderate (10–15 wt%) LCP concentrations, and strong effects on the shear‐thinning characteristics of the melt. The suspensions of polydispersed Nd‐Fe‐B particle configurations in PPS that were of lower size ratios gave better processability, which is contradictory to previously reported behavior of suspensions containing spherical particles. Specifically, the compositions with unimodal and a bimodal distribution of Nd‐Fe‐B particles gave the lowest viscosities. The experimental data were correlated with semi‐empirical viscosity model equations of Maron‐Pierce, Krieger‐Dougherty, Eilers, and Thomas and were found to be consistent with the data. The maximum packing fraction, ϕ_m, of the MQP particles was estimated to be within the range of 0.78 ϕ ≤_m ≤ 1.0 through graphical and parametric evaluation methods.