Viscoelastic flow in packed beds or porous media

An analysis of viscoelastic flow in packed beds or porous media is presented based on a capillary hybrid model of the flow which incorporates a viscous mode and an elongational mode. The development includes modelling of the elongational mode of the flow to obtain the elongational flow contribution to the potential drop for a viscoelastic fluid. A general expression describing viscoelastic flow in porous media is developed which utilizes the viscous response determined by the fluid model equation and an elongational flow response characterized by an elongational viscosity difference for the fluid. The expression applies to all three traditional bed models employing the tortuosity and Kozeny constant. The relationship yielded extensions of Darcy's law applicable to viscoelastic flow in porous media and an expression representing the flow of a viscoelastic fluid in a packed bed or porous core of length L. The relationship of the friction factors and respective Reynolds numbers is also presented.     

The flow of non-Newtonian solutions through packed beds

The rheological behavior of aqueous solutions of Separan AP-30®, polymethylcellulose, and polyvinylpyrrolidone was studied experimentally. These solutions exhibit non-Newtonian flow behavior in simple shear, and are characterized by one of several 2, 3, or 4 parameter rheological equations. The equations used included the power law, the Ellis model, Spriggs equation, the Herschel-Bulkley equation, and Meter's model. The power law model fits the data for each of the solutions over a limited range of shear rates, whereas the other models, which include either a lower shear rate limiting Newtonian viscosity, and/or an upper shear rate limiting Newtonian viscosity, or a yield stress, fit the data well over a wide range of shear rates from 0.00675 to 1076 sec^?1. The pressure drop-flow rate data for the same aqueous solutions flowing through packed beds were correlated well by the Ergun equation using the various rheological models applied in this work to evaluate a modified fluid viscosity. In each case it was found that the rheological model which best fit the viscometric data also gave the best packed bed friction factor correlation, and that no one model, such as the powerlaw, or the Ellis model, is the best one to use in all cases for all solutions. For polyvinylpyrrolidone solutions large deviations between experimental values of friction factor and those from the Ergun equation occurred for modified Reynolds numbers greater than one. A pseudo viscoelastic parameter was used to improve the friction factor correlation empirically at high Reynolds numbers.     

Momentum transfer to Newtonian and non-Newtonian fluids flowing through packed and fluidized beds

This paper deals with the flow behaviour of Newtonian and non-Newtonian fluids flowing through packed and expanded beds. With the help of tube-bundle theory a generalized average shear-stress—shear-rate relationship is derived and found to predict the flow behaviour of power law as well as non-power law fluids. Polyvinyl alcohol solutions in water, a representative of power law fluids, and grease in kerosene, a representative of nonpower law fluids, are studied. The present investigation covers the range of Reynolds number from 10^?3 to 10³. An expression for average shear-stress at minimum fluidization velocity is derived and found to agree with that of our experiment. The generalized frictional Reynolds number is defined and a design chart is also presented for the evaluation of fluidizing velocities.     

Flow of a FENE Fluid in Packed Beds or Porous Media

An analysis describing viscoelastic flow of a FENE fluid in packed beds or porous media is presented based on the capillary hybrid model of the flow. A close similarity is revealed between the functional relationship of the reduced elongational viscosity to and that relating the reduced mean elongational viscosity λ_H ? to the Deborah number which is utilized in the flow expressions. The agreement obtained between the predicted and experimentally determined evaluations of the resistance factor, including the effects of variation of polymer concentration, molecular weight and solvent quality was found to be satisfactory. Onset Reynolds numbers for enhanced flow resistance are also predicted successfully.     

A macroscopic model for shear-thinning flow in packed beds based on network modeling

The flow of non-Newtonian fluids in packed beds and other porous media is important in several applications such as polymer processing, filtration, and enhanced oil recovery. Expressions for flowrate versus pressure gradient are desirable for a-priori prediction and for substitution into continuum models. In this work, physically representative network models are used to model the flow of shear-thinning fluids, including power-law and Ellis fluids. The networks are used to investigate the effects of fluid rheology and bed morphology on flow.A simple macroscopic model is developed for the flow of power-law and Ellis fluids in packed beds using results from the network model. The model has the same general functionality as those developed using the popular bundle-of-tubes approach. The constant β, which appears in these models, is often directly derived from the tortuosity and a simple representation of the porous media. It is shown here that this can lead to incorrect and ambiguous values of the constant. Furthermore, the constant is a weak function of the shear-thinning index, indicating that no single bundle-of-tubes could ever properly model flow for a wide variety of shear-thinning fluids.The macroscopic model is compared to experimental data for shear-thinning fluids available in the literature. The model fits the data well when β is treated as an experimental parameter. The best-fit values of β vary, which is expected because even the constant C in the Blake-Kozeny equation varies depending on the source consulted. Additionally, physical effects, such as adsorption and filtration, as well as rheological effects such as viscoelasticity may affect the value of β. We believe that in the absence of these effects, β equals approximately 1.46 for packed beds of uniform spheres at relatively moderate values of the shear-thinning index (>0.3).     

Flow of viscoelastic surfactants through porous media

We compare the flow behavior of viscoelastic surfactant (VES) solutions and Newtonian fluids through two different model porous media having similar permeability: (a) a 3D random packed bed and (b) a microchannel with a periodically spaced pillars. The former provides much larger flow resistance at the same apparent shear rate compared to the latter. The flow profile in the 3D packed bed cannot be observed since it is a closed system. However, visualization of the flow profile in the microchannel shows strong spatial and temporal flow instabilities in VES fluids appear above a critical shear rate. The onset of such elastic instabilities correlates to the flow rate where increased flow resistance is observed. The elastic instabilities are attributed to the formation of transient shear induced structures. The experiments provide a detailed insight into the complex interplay between the pore scale geometry and rheology of VES in the creeping flow regime. © 2017 American Institute of Chemical Engineers AIChE J, 64: 773–781, 2018     

Numerical Investigations of Fluid Flow and Lateral Fluid Dispersion in Bounded Granular Beds in a Cylindrical Coordinates System

Results are presented from a numerical study examining the flow of a viscous, incompressible fluid through a random packing of non‐overlapping spheres at moderate Reynolds numbers, spanning a wide range of flow conditions for porous media. By using a laminar model including inertial terms and assuming rough walls, numerical solutions of the Navier‐Stokes equations in three‐dimensional porous packed beds resulted in dimensionless pressure drops in excellent agreement with those reported in a previous study. This observation suggests that no transition to turbulence could occur in the range of the Reynolds number studied. For flows in the Forchheimer regime, numerical results are presented of the lateral dispersivity of solute continuously injected into a three‐dimensional bounded granular bed at moderate Peclet numbers. In addition to numerical calculations, to describe the concentration profile of solute, an approximate solution for the mass transport equation in a bounded granular bed in a cylindrical coordinates system is proposed. Lateral fluid dispersion coefficients are then calculated by fitting the concentration profiles obtained from numerical and analytical methods. Comparing the present numerical results with data available in the literature, no evidence has been found to support the speculations by others for a transition from laminar to turbulent regimes in porous media at a critical Reynolds number.     

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.     

Hydrodynamics of power-law fluids in trickle-flow reactors: Mechanistic model, experimental verification and simulations

A fully predictive one-dimensional mechanistic model was developed for describing the hydrodynamics of power-law fluids in trickle-bed reactors. The model is a generalization of the slit approach to the case of non-Newtonian fluids obeying Ostwald-deWaele rheological behavior. Without recourse to adjustable parameters, the proposed model enabled prediction of the experimental values of (i) total two-phase total pressure drop and total liquid holdup in the trickle flow regime, (ii) frictional pressure drop in single-phase flows through packed beds, and (iii) total liquid holdup in gravity driven liquid downflow and stagnant gas through packed beds. Parametric simulations guided by knowledge of the behavior of highly viscous Newtonian liquids in trickle beds highlighted the capability of the model in the simulation and design of trickle flow operation using power-law fluids.     

Improved parametric characterization of flow geometries

An improved parametric method is presented for characterization of the flow geometry in the rectilinear flow of non-Newtonian fluids in open and closed conduits of arbitrary crosssection and in purely viscous, inelastic flow of non-Newtonian fluids through packed beds and porous media. In the new formulation, an infinite number of geometric parameters characterize the flow geometry. The actual number required in a particular application is shown to be determined by the fluid model equation representing the rheological behavior of the fluid. For Ostwald-de-Waele and Ellis fluids with flow behavior indices s = 1/n and α integers, the number of geometric parameters required to represent the relationship between flow rate and pressure drop is given by s + 1 and α + 1, respectively. The efficacy of the present method is demonstrated by comparisons with available results for various fluid models and flow geometries.     

PRESSURE DROPS OF NON-NEWTONIAN PURELY VISCOUS FLUID FLOW THROUGH SYNTHETIC FOAMS

This work aims at giving a first insight of non-Newtonian fluid flow through synthetic foams.

At first, a review of experimental pressure drops measured with Newtonian fluids through various foams is proposed as well as a recall of a capillary-type flow model used to determine structural parameters. In this particular case of Newtonian fluid flow, a single equation is shown to correlate experimental data whatever the grade of the foam. Results of an image analysis study are also given; they allow to give a physical sense to the value of the equivalent diameter of pore given by the flow model.

In a second part, results of pressure drops measured with a non-Newtonian fluid are reported. A model allowing the determination of the pressure drops for non-Newtonian purely viscous fluid flow through packed beds of particles, based on the same capillary representation of porous media, is tested in the case of fluid flow through synthetic foams. The model predictions are acceptable for foams of high grades, but a discrepancy is observed for foams of low grades. An attempt is made to explain the underevaluation of pressure drops with the aid of results of image analysis.     

Quantitative Estimation of Permeability with Lattice Boltzmann Simulations: Representative Porous Media from Composite Processing

Understanding of the flow of fluids through beds of fibrous media is extremely important for composites processing. In this work, we have investigated a steady flow of a Newtonian fluid through two‐dimensional porous media using lattice Boltzmann methods. The porous domains studied in this work represent different types of porous media encountered in composite processing. Initially, the methodology was validated with a simulation of flow through random porous media. Flow through porous media with circular and elliptical inclusions was simulated with different geometric arrangements. Simulations were also carried out with anisotropic porous media. The permeability was estimated as a function of porosity, geometric arrangements and the degree of anisotropy. The simulation results agree well with those from analytical, empirical and experimental studies. The results demonstrate that such a method will be very useful in simulating composite processing.     

An experimental study of non-newtonian fluid flow through fixed and fluidized beds of non-spherical particles

Extensive measurements of pressure drop in fixed beds, minimum fluidization velocity and expansion characteristics for beds of non-spherical particles are reported in the following ranges of conditions: 10^-3 ≤ Re ≤ 20; 0.66 ≤ n ≤ 1 and 0.41 ≤ ? ≤ 0.75. Based on an analysis of these results, it is illustrated that the existing frameworks originally developed for Newtonian fluid flow through beds of spherical particles are also satisfactory for power law fluid flow through beds of non-spherical particles, provided a volume equivalent diameter modified by a sphericity factor and a modified Reynolds number are used instead of their usual definitions.     

SINGLE FLUID FLOW IN POROUS MEDIA

A comprehensive study on single fluid flow in porous media is carried out. The volume averaging technique is applied to derive the governing flow equations. Additional terms appear in the averaged governed equations related to porosity ε, tortuosity τ, shear factor F and hydraulic dispersivity D _h. These four parameters are uniquely contained in the volume averaged Navier-Stokes equation and not all of them are independent. The tortuosity can be related to porosity through the Brudgemann equation, for example, for unconsolidated porous media.

The shear factor models are reviewed and some new results are obtained concerning high porosity cases and for turbulent flows. It is known that there are four regions of flow in porous media: pre-Darcy's flow, Darcy's flow, Forchheimer flow and turbulent flow. The transitions between these regions arc smooth. The first region, the pre-Darcy's flow region represents the surface-interactive flows and hence is strongly dependent on the porous media and the flowing fluid. The other flow regions are governed by the flow strength of inertia. For Darcy's flow, the pressure gradient is found to be proportional to the flow rate. The Forchheimer flow, however, is identified by a strong inertia! effects and the pressure gradient is a parabolic function of flow rate. Turbulent flow is unstable and unsteady flow characterized by chaotic flow patterns. The pressure drop is slightly lower than that predicted using the laminar flow equation.

The hydraulic dispersivity is a property of the porous media. It may be considered as the connectivity of the pores in a porous medium. It characterizes the dispersion of mementum, heat and mass transfer. In this paper, only the dispersion of momentum is studied.

Single fluid flow through cylindrical beds of fibrous mats and spherical particles has been used to show how to solve the single fluid flow problems in porous media utilizing the knowledge developed in this communication. Both the pressure drop and axial flow velocity profiles are computed using the developed shear factor and hydraulic dispersion models. Both the predicted velocity profile and pressure drop compare fairly well with the published experimental data.     

生物膜填料床内含有生化反应的多相传输模型

废气处理生物膜滴滤塔的多孔填料床内是带有气液两相流动、有机污染物扩散、生物膜内生化反应的复杂生化反应体系.在平行平板理论模型基础上,建立了生物膜多孔填料床内含生化反应的多元多相流动及传输特性的多相混合模型,获得了废气处理生物膜滴滤塔净化效率的理论计算方法.模型的理论预测值与生物膜滴滤塔净化低浓度甲苯废气的实验结果基本吻合.     

RHEOLOGICAL CHARACTERIZATION OF NON-NEWTONIAN FLUIDS WITH A NOVEL VISCOMETER

A hydrostatic head viscometer and its novel viscosity equation were developed to determine flow characteristics of Newtonian and non-Newtonian fluids. The objective of this research is to test capabilities of the hydrostatic head viscometer and its novel non-Newtonian viscosity equation by characterizing rheological behaviors of well-known polyethylene oxide aqueous solutions as non-Newtonian fluids with 60 wt.% sucrose aqueous solution as a reference/calibration fluid. Non-Newtonian characteristics of 0.3–0.7 wt.% polyethylene oxide aqueous solutions were extensively investigated with the hydrostatic head viscometer and its non-Newtonian viscosity equation over a 294–306 K temperature range, a 0.14–40 Reynolds number range, and a 55–784 s^?1 shear rate range at atmospheric pressure. Dynamic viscosity values of 60 wt.% sucrose aqueous solution were determined with the calibrated hydrostatic head viscometer and its Newtonian viscosity equation over a 3–5 Reynolds number range at 299.15 K and atmospheric pressure and compared with the literature dynamic viscosity value.     

Onset of pulsing in trickle beds with non-Newtonian liquids at elevated temperature and pressure—Modeling and experimental verification

The onset of pulse flow in trickle-bed reactors involving gas-non-Newtonian liquid systems was predicted from a stability analysis of the solutions around equilibrium steady-state trickle flow of a transient two-fluid model based on the volume-average mass and momentum balance equations. The model was developed for the versatile Herschel-Bulkley constitutive rheological equation from which special solutions for plastic Bingham fluids, power-law shear-thinning and thickening fluids, as well as Newtonian fluids were derived. The impact of yield stress, consistency and power-law indices, and temperature and reactor pressure on the trickle-to-pulse flow transition was analyzed theoretically. Model predictions of the trickle-to-pulse transition for gas-non-Newtonian liquid systems were confronted with elevated temperature and pressure experimental transition data obtained for air-0.25 and 0.5 mass(carboxymethylcellulose) CMC solution systems measured by means of an electrical conductivity technique. In addition the model version offspring corresponding to the Newton case (n=1,k=μ_?,τ₀=0), confronted with measured high temperature/pressure-transition data from this work and high-pressure transition data from Wammes et al. [1990. The transition between trickle flow and pulse flow in a cocurrent gas-liquid trickle-bed reactor at elevated pressure. Chemical Engineering Science 45, 3149; 1991. Hydrodynamics in a cocurrent gas-liquid trickle bed at elevated pressures. A.I.Ch.E. J. 37, 1849] and Burghardt et al. [2002. Hydrodynamics of a tree-phase fixed-bed reactor operating in the pulsing flow regime at an elevated pressure. Chemical Engineering Science 57, 4855] proved equally successful.     

Lattice‐Boltzmann simulation of sago (Metroxylon sagu) starch solutions in porous media

From rheological experiments in gelatinized sago starch solution already reported in the literature and a Lattice‐Boltzmann simulation, we provide some insight into the understanding of the non‐Newtonian fluid dynamics of sago‐starch‐type solutions in porous media. In this paper, permeability and wall shear stress in arbitrarily generated and randomly generated porous media are predicted in the range of the modified Darcy's law. Additional results on flow paths, velocity, shear‐stress tensor, and pressure fields are provided. We prove that our LBE model for sago starch solutions reproduces Blake‐Kozeny and Ergun laws. The model presented in this paper is intended to be used for simulating packed beds.     

ONE- AND TWO-PHASE FLOW IN NETWORK MODELS OF POROUS MEDIA

We discuss the low Reynolds number flow of one or two immiscible Newtonian fluids in network models of microscopically random porous media. For the case of a single fluid, we reduce the flow problem to an analog random electrical resistor problem and use an 'effective medium theory' to express the permeability of such networks in terms of the pore space geometry. For the flow of two fluids we use the Washburn approximation to incorporate capillary pressure differences, and show that this problem may also be formulated as a random electrical network. In this case, the capillary menisci correspond to moving batteries, and we follow the motion of the fluid-fluid interface (the ensemble of analog batteries) by a time-step procedure. We study the time evolution of the interface and the dynamics of blobs of one fluid contained in the other, as a function of the network geometry.’