Laminar mixing of shear thinning fluids in a SMX static mixer

Flow and mixing of power-law fluids in a standard SMX static mixer were simulated using computational fluid dynamics (CFD). Results showed that shear thinning reduces the ratio of pressure drop in the static mixer to pressure drop in empty tube as compared to Newtonian fluids. The correlations for pressure drop and friction factor were obtained at Re_MR?100. The friction factor is a function of both Reynolds number and power-law index. A proper apparent strain rate, area-weighted average strain rate on the solid surface in mixing section, was proposed to calculate pressure drop for a non-Newtonian fluid. Particle tracking showed that shear thinning fluids exhibit better mixing quality, lower pressure drop and higher mixing efficiency as compared to a Newtonian fluid in the SMX static mixer.     

CFD investigation of the pipe transport of coarse solids in laminar power law fluids

A numerical parametric study of the laminar pipe transport of coarse particles in non-Newtonian carrier fluids of the power law type has been conducted using an Eulerian-Eulerian computational fluid dynamics (CFD) model. The predicted flow fields have been successfully validated by experimental measurements of particle velocity profiles obtained using a positron emission particle tracking technique, whilst solid-liquid pressure drop has been validated using relevant correlations gleaned from the literature. The study is concerned with nearly-neutrally buoyant particles flowing in a horizontal or vertical pipe. The effects of various parameters on the flow properties of such mixtures have been investigated over a wide range of conditions. The variables studied are: particle diameter (2-9 mm), mean solids concentration (5-40% v/v), mean mixture velocity (25-125 mm s⁻¹), and rheological properties of the carrier fluid (k=0.15-20 Pa sⁿ; n=0.6-0.9). A few additional runs have been conducted for shear thickening fluids, i.e. n>1. Whilst the effects of varying the power law parameters and the mixture flowrate for shear thinning fluids are relatively small over the range of values considered, particle size and solids concentration have a significant bearing on the flow regime, the uniformity of the normalised particle radial distribution and of the normalised velocity profiles of both phases, and the magnitude of the solid-liquid pressure drop. The maximum particle velocity is always significantly less than twice the mean flow velocity for shear thinning fluids, but it can exceed this value in shear thickening fluids. In vertical down-flow, particles are uniformly distributed over the pipe cross-section, and particle diameter and concentration have little effect on the normalised velocity and concentration profiles. Pressure drop, however, is greatly influenced by particle concentration.     

Flow characteristics of surfactant stabilized water-in-oil emulsions

In this paper, an investigation was carried out to study the effect of water fraction and flow conditions on the flow characteristics of surfactant stabilized water-in-oil emulsion. Pressure drop measurements were conducted in 2.54-cm and 1.27-cm horizontal pipes. The influence of water fraction and the flow conditions on emulsion stability, type, conductivity, droplet size distribution, viscosity and pressure drop were reported. The results showed a significant increase in the emulsion stability, viscosity and pressure drop with increasing water fraction up to 70%. In addition, shear thinning behavior was observed for the emulsions especially at high water fractions. Furthermore, pressure drop measurements of high concentrated emulsions showed pipe diameter dependency especially at high Reynolds (Re) numbers. Moreover, drag reduction was observed with decreasing water fraction. The viscosity of surfactant-stabilized water-in-oil emulsions was modeled with a modified fluidity-additivity model.     

Pressure drop during the flow of stokesian fluids through granular beds

The resistance to flow of Stokesian fluids (i.e. time—independent fluids with no yield stress) through granular beds is discussed. A definition of friction factor λ and generalized Reynolds number Re′_BK is proposed for fluids obeying the “power-law” shear stress—shear rate relation.The generalized Ergun equation, derived in this paper, gives the dependence of the friction factor on the generalized Reynolds number and flow behaviour index n. The validity of the generalized Ergun equation was proved experimentally. In the case of Newtonian fluid (for n = 1·0) a more exact form of the classical Ergun equation is obtained.     

循环旋风分离器内气液两相流动数值模拟

采用雷诺应力模型RSM对循环旋风分离器内气液两相流动的情况进行了数值模拟研究,讨论了循环旋风分离器内切向速度、轴向速度、径向速度、压力场、雷诺应力的分布特点以及相同入口速度下分离器内液滴运动轨迹与分离器的分离效率。数值模拟结果表明,循环旋风分离器切向速度呈现明显的驼峰状,轴向速度上行流和下行流明显,径向速度相对较小,压力由轴心向外逐渐升高,雷诺应力分布复杂且无明显规律,分离器对小直径液滴分离效率较低,入口速度对分离效率的影响比较明显。     

Isothermal and nonisothermal,laminar, inelastic,non-Newtonian tube-entrance flow following a contraction

The general equations of motion, including inertial and axial-diffusion terms, and the equation of energy, including axial-diffusion and radial-convection terms, were coupled with a temperature-dependent form of the Powell-Eyring flow equation and then solved simultaneously by numerical methods employing quasilinearization and the method of lines for flow from a larger tube of circular cross section through an abrupt contraction into a smaller tube. The small-tube walls were at a temperature equal to or higher or lower than that of the entering fluid. The Powell-Eyring equation has been demonstrated to represent inelastic flow better than do other simple equations for many simple non-Newtonian fluids, including suspensions of solids in fluids. Results are reported for the case in which the fluid does not adhere to the upstream-tube surface but does adhere to the smaller-tube surface, which approximates flow into the tubes of shell-and-tube heat exchangers and reactors. The computer results presented here included radial and axial velocity profiles; temperature profiles; and lengths of the small tube, L_eq, having a fully-developed-flow pressure loss equivalent to the excess pressure loss attributable to flow contraction and development in the tube. These are presented for a large-tube-small-tube-diameter ratio of 2 as functions of N_Pe of 5–100; N_Re of 0.01–100; z/D for the smaller tube; ψ(H), a temperature factor; and the Powell-Eyring-equation constants. In plots of L_eq against a normalized N_Re, the isothermal-flow non-Newtonian data appear to be close to computed Newtonian functions at higher N_Re. The effects of axial diffusion of momentum and energy and the conditions where they become important are quantatively shown. The results are in good agreement with previous reports in those circumstances where comparisons are appropriate.     

Experimental and numerical characterization of viscous flow and mixing in an impinging jet contactor

The laminar flow in an impinging jet contactor is examined as a first step toward the development of new technology for fast mixing of viscous fluids. The flow, velocity, and stretching fields in an impinging jet contactor are quantified for low Reynolds number flow using three-dimensional numerical simulations and particle image velocimetry measurements. Computational and experimental velocity fields are in close agreement, as quantified by the velocity probability density functions. Two steady-state flow regimes are found to exist: for jet Reynolds numbers (Re_j) < 10, the jets do not impinge and the velocity field scales linearly with Reynolds number; for Re_j > 10, the jets begin to impinge and recirculation regions form above and below the impingement point. The magnitude of the rate-of-strain tensor is calculated as a function of Re_j. While areas of essentially zero stretching occupy most of the flow domain, very high rates of stretching occur at specific locations in the flow. The maximum and average rates of stretching in the contactor increase roughly linearly as a function of Reynolds number. Mixing simulations show that no mixing occurs for the steady flow in a symmetric-jet contactor. However, mixing is improved substantially by a slight modification of the impinging jet geometry that disrupts geometric symmetry.     

Computational Investigation of an Industrial Cyclone Separator with Helical‐Roof Inlet

An industrial cyclone separator with helical‐roof inlet TsN‐11 has been numerically investigated as to pressure and flow field, pressure drop, fractional efficiency, and particle trajectories inside the cyclone. The turbulence was modeled with Reynolds stresses and large eddy simulations (LES) based on three different subgrid‐scales (SGS). The results with the different setups were compared to experimental data from previous studies. For a proper calculation of the flow field, LES combined with a dynamic SGS model was used for predicting cyclone performance. Individual particles were tracked through the unsteady flow field using the Lagrangian approach. The results of the numerical calculations of the tangential and axial velocity, pressure drop, and cut size are in good agreement with experimental measurements.     

Two-dimensional unsteady flow of power-law fluids over a cylinder

The unsteady flow of incompressible power-law fluids over an unconfined circular cylinder in cross-flow arrangement has been studied numerically. The two-dimensional (2-D) field equations have been solved using a finite volume method based solver (FLUENT 6.3). In particular, the effects of the power-law index (0.4?n?1.8) and Reynolds number (40?Re?140) on the detailed kinematics of the flow (streamline, surface pressure and vorticity patterns) and on the macroscopic parameters (drag and lift coefficients, Strouhal number) are presented in detail. The periodic vortex shedding and the evolution of detailed kinematics with time are also presented to provide insights into the nature of flow. The two-dimensional flow transits from steady to unsteady behaviour at a critical value of the Reynolds number Re∼(40-50) and the von-Karman vortex street is observed beyond the critical Reynolds number (Re). Obviously, both the lift coefficient and Strouhal number values are zero for the steady flow, but their values increase with the increasing Reynolds number (Re) in the unsteady flow regime. For highly shear-thickening fluids (n=1.8), the flow becomes unsteady at Re=40 and unsteadiness in the flow appears at Re=50 for all values of power-law index (n). As expected, the evolution of the kinematics and vortex shedding show a complex dependence on the flow parameters near the transition in the flow. For a fixed value of the Reynolds number (Re), the drag coefficient increases and lift coefficient decreases with increasing value of the power-law index (n). For a fixed value of the power-law index (n), the drag coefficient gradually increases with the Reynolds number (Re). Similar to the drag coefficient, lift coefficient also shows a complex dependence on the power-law index (n) near the transition zone. The value of the Strouhal number (St) decreases with the increasing value of the power-law index (n) at a fixed value of the Reynolds number (Re).     

Gas displacing liquids from tubes: high capillary number flow of a power law liquid including inertia effects

Computer simulation has been used as a virtual experimental tool to investigate the displacement of a shear thinning Power Law liquid from a cylindrical tube by a gas, in the limit of high capillary number and in the absence of gravity effects. Two scenarios have been considered. In the first, gas enters at a steady rate, and the gas penetration velocity and residual wall layer thickness attain steady values. In the second, a constant gas pressure is applied at the inlet, and the gas penetration rate accelerates as the column of liquid ahead of it becomes shorter. The first set of experiments confirm that the developed wall layer thickness falls with increased degrees of shear thinning and with increased Reynolds Number, and quantifies the latter effect for the first time. The relationship is summarized by a correlation formula for dimensionless layer thickness as a function of Power Law index, n, and an appropriately defined Reynolds group in the range 0.1?n?1.0,0.001?Re?100. The flow pattern ahead of the gas bubble throughout the range of these experiments was always of the ‘by-pass’ type, consistent with a generalized criterion for the transition between by-pass and re-circulating flow which is derived for a Power Law liquid. In the second set of experiments, where a constant gas inlet pressure is applied, giving accelerating gas penetration, a comparison of layer thickness values at various axial positions, with those obtained at corresponding Reynolds number in steady flow, showed close agreement, though a small discrepancy for the highest Reynolds numbers could indicate some influence of inertia in the accelerating liquid column. At higher Reynolds number, in both steady and accelerating flow, the gas bubble near to the inlet shows a concave region on the axis, with re-circulation in the liquid ahead of it. As the bubble moves down the tube, the radius of this concavity decreases and a steady convex profile is eventually attained, with reversion of the flow to by-pass type. We show that the origin of this is inertial.The results have applications in a number of technologies, including gas-assisted injection moulding of plastics and certain gas liquid reactors.     

Pipe flow of a dense emulsion: Homogeneous shear‐thinning or shear‐induced migration?

The flow field of a 70% concentrated noncolloidal o/w emulsion in a pipe has been investigated by means of Particle Image Velocimetry in a matched refractive index medium. At steady state and in laminar regime, the shape of axial velocity profiles is not parabolic and exhibits a shear‐thinning behavior of the dense emulsion, with a flow index of 0.5 and a negligible yield stress (less than 1 Pa). However, instead of a square root law, the pressure drop increases linearly with U_m. To explain this apparent inconsistency, two mechanisms of different nature are considered. The first originates from a possible relation between the consistency factor and the drop mean diameter. The second mechanism is shear‐induced migration and leads to the development of a concentration gradient in the pipe cross section. Both mechanisms considered reconcile the experimental data, the apparent local shear‐thinning behavior and the linear evolution of the pressure drop with the flow rate. © 2017 American Institute of Chemical Engineers AIChE J, 2017     

The flow of non-Newtonian slurries through fixed and fluidised beds

Measurements have been taken of the flow rate, pressure drop and bed height characteristics when non-Newtonian slurries flow through fixed and fluidised beds of uniformly sized spherical particles.In the case of fixed beds, the pressure drop-flow rate data has been interpreted using the capillary model of a porous medium together with rheological data for the slurries obtained from a tubular viscometer. The resulting friction factor-Reynolds number relationship is

This correlation was used to satisfactorily predict the minimum fluidisation velocity for a given solid/liquid system by equating the pressure drop to the net weight per unit area of particles in the bed. However, the correlation was not adequate for the prediction of bed expansion in the fluidised state. For systems which have a Reynolds number at minimum fluidisation, Re′_mf′ less than 40 an effect of particle diameter to bed diameter was observed. For systems having Re′_mf >40 the velocity, υ, and voidage, ?, were related to their values at minimum fluidisation by

It is therefore clear that, in the fluidised state, the capillary model does not present an adequate basis for the prediction of bed expansion.     

Pressure drop in a split‐and‐recombine caterpillar micromixer in case of newtonian and non‐newtonian fluids

Microreactors are very promising tools for the design of future chemical processes. For example, emulsions of very narrow size distribution are obtained at much lower energy consumption than the one spent with usual processes. Micromixers play thereby an eminent role. The goal of this study is to better understand the hydrodynamic properties of a split‐and‐recombine Caterpillar micromixer (CPMM) specially with regard to handling viscoelastic fluids, a topic hardly addressed so far in the context of micromixers in general, although industrial fluids like detergent, cosmetic, or food emulsions are non‐Newtonian. Friction factor was measured in a CPMM for both Newtonian and non‐Newtonian fluids. For Newtonian fluids, the friction factor in the laminar regime is f/2 = 24/Re. The laminar regime exists up to Reynolds numbers of 15. For shear‐thinning fluids like Carbopol 940 or viscoelastic fluids like Poly Acryl Amide (PAAm) aqueous solutions, the friction factor scales identically within statistical errors up to a generalized Reynolds number of 10 and 0.01, respectively. Above that limit, there is an excess pressure drop for the viscoelastic PAAm solution. This excess pressure drop multiplies the friction factor by more than a decade over a decade of Reynolds numbers. The origin of this excess pressure drop is the high elongational flow present in the Caterpillar static mixer applied to a highly viscoelastic fluid. This result can be extended to almost all static mixers, because their flows are generally highly elongational. © 2013 American Institute of Chemical Engineers AIChE J, 59: 2679–2685, 2013     

Performance of Kenics static mixer over a wide range of Reynolds number

The present study deals with the numerical simulation of flow patterns and mixing behaviour in Kenics static mixer over a wide range of Reynolds number. Three different sets of Kenics mixer (aspect ratio = 1.5) comprised of 3, 9 and 25 elements each have been characterized. The Reynolds number was varied in the range of 1 to 25,000 (i.e., from laminar to turbulent flow regime). The numerical approach takes into account the aspects of the fluid flow at higher Reynolds number values including circumferential velocity profiles at different cross-sections within the Kenics mixer, which were neglected in previous studies. It was observed that cross-sectional mixing in the turbulent flow regime takes place up to 30% of each element length at element-to-element transition; beyond that velocity profiles were uniform. The experiments were also carried out to measure the circumferential and axial velocity profiles and pressure drop in three different Kenics Mixers using air as fluid. The pressure drop per unit element (ΔP/η) was found to be independent of the number of Kenics mixing elements used in the system. The total pressure drop across Kenics mixer obtained by CFD simulations were compared with the experimental pressure drop values and correlations available in the literature. The numerical results were found in good agreement with the experimental as well as the results reported in the literature. A new pressure drop correlation in the Kenics static mixer has been developed.     

Reynolds number effects in a simple planetary mixer

Planetary mixers are widely used in a diverse range of industrial applications. This paper presents an experimental investigation of mixing in a planetary mixer, and a comparison with numerical simulations based on a simple mathematical model of the flow. The model allows an exact expression for the velocity field in the Stokes flow regime, apparently the first for a mixer with genuinely moving parts, which permits accurate numerical tracking of material interfaces. Experiments performed at low Reynolds number (Re?1) show good agreement with corresponding numerical simulations, but as the Reynolds number is increased, the agreement between experiments and Stokes-flow numerics worsens, in a manner that reflects improving experimental mixing quality. Specifically, we find that islands of poor mixing shrink as Re increases. Our results suggest that, while numerical simulations in the Stokes flow regime may be used as a ‘sieve’ to select good mixing protocols at small Re, experiments or computational fluid dynamics simulations are required properly to evaluate mixing protocols operated at finite Reynolds numbers.     

Relationship of Annular and Parallel-Plate Poiseuille Flows for Power-Law Fluids

This paper deals with both the axial and the tangential annular flows of power-law fluids in concentric annuli, and it specifically focuses on the relation volumetric flow rate Q against pressure drop P. Initially the mutual compatibility of the hitherto obtained results is analyzed, and from this, new results concerning behavior of a parameter λ [a location of maximum axial (tangential) velocity component] are derived. Then, approximative expressions for relations Q against P (independent of A.) are proposed in a sufficiently large region (K,n) where K represents annular aspect ratio and n the flow behavior index n. The deviations of these simple relations from the exact ones are negligible.     

Non-Newtonian thin films: Theory and experiment

The flow of thin films of Newtonian and non-Newtonian, Power law fluids down a vertical plate was studied in the laminar and wavy flow regimes. A theoretical development based on the existence of a steady, periodic solution to the equations describing film flow was used to predict the flow characteristics of the film. Double logarithmic plots of film thickness versus Reynolds number (Re_{P. L.}) were linear, as predicted, up to a Re_{P. L.} = 100. The ratio of surface to average velocity was found to be approximately independent of the Reynolds number at values predicted by a laminar velocity profile. The wavelength of stable waves was found to be independent of the Reynolds number for a given fluid.     

Wall effect in laminar flow of non-Newtonian fluid through a packed bed

Experiments were conducted on laminar flow of non-Newtonian fluid through a packed bed of low column to packing particle diameter ratio (3.8) to elucidate the wall effect on pressure drop and mass flux. Carboxy methyl cellulose (CMC) at different concentrations was passed through the packed bed and the pressure drop was measured at different CMC concentrations and flow rates. It was found that the pressure drop increases with the increase in CMC flow rate. The pressure drop also increases with the increase in CMC concentration for a given flow rate. The friction factor is plotted against Reynolds number and the data for different CMC concentration are found to be scattered around a line expressed as f=1.03/Re^0.87. The tri-regional model of Cohen and Metzner [1] predicted correctly the mass flux in the packed bed at different pressure drop values and CMC concentrations with parameters K₀ (related to pore geometry) value of 1.5 and L_e/L (related to effective path length) value of 1.2, respectively.     

A model for gas-solid flow in a horizontal duct with a smooth merge of rapid-intermediate-dense flows

Granular flow in the rapid flow regime is dominated by particle-particle collisions and the constitutive relations for the solid stress are obtained from the classic kinetic theory of granular flow. In the dense flow regime, on the other hand, particles interact via enduring contacts and the solid stress can be deduced from soil mechanics theories. In this paper, constitutive equations, recently proposed by Tardos et al. [2003. Slow and intermediate flow of frictional bulk powder in the Couette geometry. Powder Technology 131, 23-39.] has been incorporated in the simulation of gas-solid flow in a horizontal duct. These equations smoothly merge the rapid granular flow solution with the so-called “intermediate” regime (where both kinetic/collisional and frictional contributions might play a role) and reduce to Coulomb yield condition for slow frictional flow (shear rate → 0). The results of this new modelling approach have shown good qualitative agreement with the reported experimental observation on wide range of gas-solid flow conditions. In this study, we also present the definition of boundaries between rapid-intermediate-dense flow regimes based on the dimensionless shear rate (λ), and a modified Reynolds number (Re). We have shown that the intermediate flow regime can be classified at approximately 0.1<λ<1.0 and 100<Re<3000.