The Richardson-Zaki Equation Reconsidered: Derivation of the Power Law for Aerated Particle Beds

Analytical derivations of the Richardson-Zaki power law in the case of aeratable fine particle beds where the interparticle forces allow the creation of expanded beds, known also as meta-states, with fixed structures, but not yet fluidized, have been developed. The analysis is based on the concept that the fluid flow through such beds is governed by the porous media equations such as Darcy's and Forchheimer models, upon the assumption of deformable bed structures (driven by the fluid flow) and fluid velocity independent pressure drop.     

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.     

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.     

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 loss and wall shear stress in flow through confined sphere packings

Randomly structured, confined sphere packings with different porosities are generated and the fluid flow within the porous structure is calculated. These locally resolved fluid flow data – instead of integral parameters – are used to investigate the origins of the pressure loss within a packing. First, an analysis and comparison of averaged local velocities is performed to compare the similarity of the simulation approach with empirical relations by means of the void fraction and the velocity distributions. Next, the pressure losses due to mean values of the simulated, locally resolved wall shear stresses are calculated, and these findings are smaller than the results from the integral approach of Kozeny and Carman. This indicates that the pressure drop, even at low Reynolds numbers, is not solely caused by the wall shear stress; the simulated overall pressure drops exceed the Ergun approach, an effect which is caused by the bounded flow within a capillary. To relate the pressure loss due to these secondary pressure losses, the tortuosity of the fluid flow in the porous structure is introduced and this parameter improves the performance of the pressure drop equation.     

Direct numerical simulation and experimental validation of flow resistivity of nonwoven fabric filter

We numerically study and then experimentally validate the flow resistivity of commercial nonwoven fabric filters used for a bag filter system. To represent a realistic flow field inside the filters during simulation, a numerical method that coordinates the filter structure obtained by X-ray computed tomography imaging with computational fluid dynamics using the immersed boundary method is developed. The effects of superficial velocity, porosity of the filter domain, and type of filter on pressure drop are investigated and analyzed based on Darcy's law. The predictions from our numerical method are quantitatively in good agreement with the experimental measurements. We demonstrate that the Kozeny constants of the filters can be estimated by utilizing the solid volume fraction. These results demonstrate that our simulation method can be used to clarify the effects of porosity, fiber arrangement, and fiber shape on the pressure drop. Finally, its application to water droplet permeation is demonstrated.     

Darcy's law for two‐dimensional flows: Singularities at corners and a new class of models

As is known, Darcy's model for fluid flows in isotropic homogeneous porous media gives rise to singularities in the velocity field for essentially two‐dimensional flow configuration, like flows over corners. Considering this problem from the modeling viewpoint, this study aims at removing this singularity, which cannot be regularized via conventional generalizations of the Darcy model, like Brinkman's equation, without sacrificing Darcy's law itself for unidirectional flows where its validity is well established experimentally. The key idea is that as confirmed by a simple analogy, the permeability of a porous matrix with respect to flow is not a constant independent of the flow but a function of the flow field (its scalar invariants), decreasing as the curvature of the streamlines increases. This introduces a completely new class of models where the flow field and the permeability field are linked and, in particular problems, have to be found simultaneously. © 2017 American Institute of Chemical Engineers AIChE J, 2017     

Effect of sample’s length on flow properties of open-cell metal foam and pressure-drop correlations

Many applications require fluid flow through the open pores of metal foam. The foam is usually treated as a porous medium for which the Darcy law and the Hazen-Dupuit-Darcy (or Forchheimer) equation are used to describe the pressure drop, and for obtaining the two important flow properties, i.e., the permeability and the form drag coefficient. Little or no attention is paid to the length (or thickness) of the porous medium in the flow direction. This paper establishes a minimum length necessary for the foam to have length-independent (or bulk) permeability and form drag coefficient. This minimum length is obtained experimentally for various types of open-cell aluminum foam subjected to airflow in the Forchheimer regime. Below this thickness values of the two key flow properties are not constant, and they include entrance/exit effects, which may explain some of the discrepancies in the reported values in the literature. The Forchheimer equation was recast in two different manners, which resulted in new non-dimensional numbers- one representing the form drag and the other the viscous drag. These numbers correlated very well with the thickness of the porous medium. The obtained correlations allow for determining the pressure drop given only the velocity and the thickness of an aluminum foam sample.     

Axial dispersion in metal foams and streamwise-periodic porous media

We present a study on the axial dispersion in metal foams and laser sintered reactors. Commercially available metal foams of 20 and 30 ppi are compared to a designed streamwise-periodic structure in terms of axial dispersion coefficients and pressure drops. Therefore tracer pulse experiments were performed and post processed by means of a deconvolution method. The Peclet number Pe_p based on the pore size is ranging from 5×10⁴ to 8×10⁵ which is attributed to the increased velocities due to the high porosity of the material compared to fixed bed reactors. The attained dispersion coefficients ranging from 1.3×10⁻⁴ to demonstrate the trend of packed beds and common packing materials and increase monotone with the Peclet number Pe_p. The pressure drop versus the interstitial bulk velocity follows the Forchheimer equation and can be described by the conventional Ergun model for all investigated porous media. The parameters obtained correspond to values found in literature. The results of this study show the high potential of foam reactors for catalyst driven reactions. They provide the same or even a higher surface area per volume of catalyst bed while inducing a much smaller pressure drop than corresponding fixed beds.     

Water and methane in shale rocks: Flow pattern effects on fluid transport and pore structure

Using molecular dynamics simulations, the two‐phase flow of water and methane through slit‐shaped nanopores carved from muscovite is studied. The simulations are designed to investigate the effect of flow patterns on fluids transport and on pore structure. The results indicate that the Darcy's law, which describes a linear relation between flow rate and pressure drop, can be violated when the flow pattern is altered. This can happen when the driving force, that is, the pressure drop, increases above a pore‐size dependent threshold. Because the system considered here contains two phases, when the fluid structure changes, the movement of methane with respect to that of water changes, leading to the violation of the Darcy's law. Our results illustrate the importance of the capillary force, due to the formation of water bridges across the model pores, not only on the fluid flow, but also on the pore structure, in particular its width. When the water bridges are broken, perhaps because of fast fluid flow, the capillary force vanishes leading to significant pore expansion. Because muscovite is a model for illite, a clay often found in shale rocks, these results advance our understanding regarding the mechanism of water and gas transport in tight shale gas formations. © 2015 American Institute of Chemical Engineers AIChE J, 61: 2993–2999, 2015     

Permeability of porous gelcast scaffolds for bone tissue engineering

The permeability of metallic and ceramic open-cell foams prepared by the gelcasting technique was assessed by fitting of Forchheimer’s equation to experimental pressure drop curves. The ceramic composition was based on pure hydroxyapatite, while the metallic composition was based on titanium metal. Experimental Darcian (k ₁) and non-Darcian (k ₂) permeability constants displayed values in the range 0.40–3.24 × 10^?9 m² and 3.11–175.8 × 10^?6 m respectively. Tortuosity was evaluated by gas diffusion experiments and ranged from 1.67 to 3.60, with porosity between 72 and 81% and average hydraulic pore size between 325 and 473 μm. Such features were compared to data reported in the literature for cancellous bones and synthetic scaffolds for bone graft. A detailed discussion concerning the limitations of Darcy’s law for fitting laboratory data and for predicting fluid flow through scaffolds in real biomedical applications is also performed. Pore size was obtained by image analysis and was also derived from permeation-absorption-diffusion experiments. In both cases, values were within the range expected for porous scaffolds applications.     

Analysis and evaluation of permeability techniques for characterizing fine particles Part I. Diffusion and flow through porous media

The specific surface area and particle size can be deduced with speed and simplicity from appropriate measurements and calculations of fluid flow and diffusion in porous media. The interdependence of these two processes is developed in a series of two articles.In Part I, models are presented for molecular and Knudsen diffusion during flow through aggregates of solid particles at both atmospheric and low pressure permeametric conditions. For a randomly packed bed of granular particles, a cell model is developed that takes into account the tortuosity and variations in the cross-sectional area. A new analytical expression for the Kozeny constant is derived in terms of the bed porosity and particle shape. The effect of porosity on surface area measurements using permeability methods is explained.     

Integral Flow Parameters and Material Characteristics Analysis in Through Air Drying: Part I

The extension of the Darcy law (the Forchheimer flow equation) relating second-order nonlinear pressure drop with flow velocity is studied during fast transient through air drying of sheets of porous biobased materials such as paper. A range of the paper materials with open structure consistent with tissue and towel products (basis weights 25 and 50 g/m²) made using different production processes are analyzed for the factor-specific influences with regard to changes in the fluid resistance from the removal of moisture from the material interstices. A characteristic dimension suitable for the drying process is applied from viscous and inertial momentum transport analysis.     

Pressure drop modelling in cellular metallic foams

A theoretical model is presented for the prediction of pressure drop in a Newtonian fluid flowing through highly porous, isotropic metallic foams. The model is based on a rigorous assumption of piece-wise plane Poiseuille flow and a simplistic geometrical model, and shows promise to accurately predict the hydrodynamic conditions in both the Darcy and Forchheimer regimes, without a priori knowledge of the flow behaviour of the particular metallic foam.     

Numerical analysis of reaction kinetic accompanying flow and dispersion in structured ZSM-5@SiC open-cell foam catalyst

Chaotic flow inside porous media accelerates the transport, mixing, and reaction of molecules and particles in widespread natural and factitious processes. Current macroscopic models based on the average pore-scale variations show obvious limitations in the prediction of many chemical processes. In this article, we reconstruct microscopic foam structures using micro-computed tomography to simulate fluid flow in structured ZSM-5@SiC foam catalyst. Moreover, we propose a conceptual model based on the microscopic mean square displacement theory to characterize the effective dispersion inside an open-cell foam. This model will explain the flow characteristics of confined fluid inside the porous media from fluid elements perspective. Particularly, dispersion factor and structure factor, as key parts of this model, perfectly interpret the driving characteristics of pressure drop, velocity different, and reaction in continuous foam media flow. This work also provides a unique means of predicting reaction kinetics of confined fluid in structured foam catalyst.     

The permeability of replicated microcellular structures in the Darcy regime

This study provides an extension to previously published articles on measurements and computational fluid dynamic (CFD) modeling and simulations of airflow across “bottleneck-type” structures in the Forchheimer regime to purely inertial-neglected (Darcy regime) incompressible flowing fluid via tomography datasets. A linear-dependent relationship between the CFD computed unit pressure drop developed across the samples and the superficial fluid inlet velocity was observed for all the samples for creeping fluid flow (0–7 × 10⁻⁰³m·s⁻¹). The permeability of the structures was observed to be dependent on the structural parameters of the porous medium and most importantly, its pore diameter openings. Linear graphical relations between permeability and pore-structure related properties of these materials were observed and these could assist in minimizing the number of design iterations needed for the processing of self-supporting porous metals.     

Permeability of the porous Al2O3 ceramic with bimodal pore size distribution

Porous Al₂O₃ ceramics with bimodal pore size distribution were fabricated by partial sintering with monodispersed PMMA micro balls as pore agent. The porosity of the fabricated porous Al₂O₃ is increased with content of the pore agent increase, the bulk density and bending strength are decreased, accordingly. Relations between pressure drop and flow velocity of the air through the porous Al₂O₃ fit the Forchheimer's equation well for compressible fluid. Due to pores introduced by the pore agent, the Darcy permeability and inertial permeability of the porous Al₂O₃ are increased obviously. For given flow velocity, with increase of the PMMA content, the Forchheimer's number of the fluid through the porous Al₂O₃ is decreased, which results in decrease of the inertial resistance ratio to the total pressure drop. The porous Al₂O₃ ceramics with pores introduced by the monodispersed PMMA micro balls show higher permeability while the filtration selectivity is not deteriorated.     

Wicking and evaporation of liquids in porous wicks: A simple analytical approach to optimization of wick design

Wicking and evaporation of volatile liquids in porous, cylindrical wicks is investigated where the goal is to model, using simple analytical expressions, the effects of variation in geometrical parameters of a wick, such as porosity, height and bead‐size, on the wicking and evaporation processes, and find optimum design conditions. An analytical sharp‐front flow model involving the single‐phase Darcy's law is combined with analytical expressions for the capillary suction pressure and wick permeability to yield a novel analytical approach for optimizing wick parameters. First, the optimum bead‐radius and porosity maximizing the wicking flow‐rate are estimated. Later, after combining the wicking model with evaporation from the wick‐top, the allowable ranges of bead‐radius, height and porosity for ensuring full saturation of the wick are calculated. The analytical results are demonstrated using some highly volatile alkanes in a polycarbonate sintered wick. © 2014 American Institute of Chemical Engineers AIChE J, 60: 1930–1940, 2014