The calculations of dispersion coefficients inside two‐dimensional randomly packed beds of circular particles

The longitudinal (D_L) and transverse (D_T) dispersion coefficients in two‐dimensional (2‐D) randomly packed beds of circular particles in a laminar flow regime are derived. A 2‐D discrete system of particles is divided into cells using modified Voronoi diagrams. The relationship between the variation of the stream function and the averaged vorticity is obtained from computational fluid dynamics (CFD) simulations. The whole flow pattern is then obtained by using the principle of energy dissipation rate minimization. The obtained values of D_L agree well with 3‐D experimental data for all velocities investigated. At very high velocities, D_T in 2‐D appears to be higher than 3‐D experimental data. In addition, the effects of particle‐size distributions, packing structure, and porosity on the D_L and D_T were studied. One result was that an increase in the width of the particle‐size distribution resulted in higher values of D_L and D_T at high velocities. © 2012 American Institute of Chemical Engineers AIChE J, 59: 1002–1011, 2013     

Near-wall porosity characteristics of fixed beds packed with wood chips

Packed beds of fuel wood chips are commonly found in thermal conversion processes such as combustion or gasification. Wood chips in particular are mostly used as fuel for small-scale domestic heating boilers but also for commercial-scale combustion units. The characterization of spatial voidage distribution inside the wood chip beds is of great importance for flow and reactor modelling. This study focuses on the radial porosity variations of cylindrical beds of three different types of commercially available wood chips including chips classified as G30 size class. The conventional technique of consolidating packed beds with a resin was chosen as the experimental procedure. The radial voidage distribution in different cylindrical beds is determined by image analysis of sections of the solidified packings. Additionally, a packing of monosized spheres was investigated in order to assess the selected procedure in comparison with widely available literature data for spheres. The results are discussed and summarized in a mathematical expression correlating the radial voidage distribution depending on average wood chip size, packing core porosity and dimensionless distance from the tube wall.     

Deposition of Brownian particles in packed beds

An analysis is presented to describe the deposition of Brownian particles from suspensions flowing through granular media. The constricted tube porous media model recently proposed by Payatakes et al.[8] was used to characterize the media and the deposition process was assumed to be controlled by convective diffusion as well as the nature of the surface interaction between the media grains and the Brownian particles. Three different tube configurations were considered and the differences among the results were found to be relatively unimportant. Agreement between the analysis and available experimental data was found to be reasonably good for the limiting case when the surface effect is favorable.     

A pore network model for calculating pressure drop in packed beds of arbitrary-shaped particles

A pore network model is built to predict pressure drop in packed beds of arbitrary-shaped particles, using a method that consists of particle packing by the rigid body technique, pore network construction by the maximal sphere algorithm, and numerical calculation of fluid flow. The pore network model is firstly validated by comparing with experiments, Ergun-type equations, and particle-resolved computational fluid dynamics (CFD). The pore network model is as accurate as the particle-resolved CFD, and is remarkably two to three orders of magnitude less computationally intensive. Then, the pore network model is used to calculate the pressure drops in the beds packed with particles of different shapes and sizes, as well as using different flow media. These calculation results prove the versatility of the pore network model. This work provides an accurate yet efficient pore network model for predicting pressure drop, which should be a powerful tool for designing packed beds.     

Effective thermal conductivity of packed beds of silicon-copper particles

The effective thermal conductivity of a bed of silicon-copper particles, as used in the technical synthesis of silicones was measured with a non-stationary method. The particles have an irregular shape and an average diameter of 0·111 mm.From the measured temperature profiles and from literature it was concluded that a mechanism, in which the energy transfer occurs through a series path consisting of an effective solid-path length and gas-path length, plays the main part in the model for heat transfer suggested by Masamune and Smith.The values for the effective thermal conductivity were in the region 0·192–0·440 kcal m^?1 h^?1 deg^?1.     

Radial heat transfer to fixed beds of particles

Radial temperature distributions have been measured when air flows through a fixed bed of large particles heated at the wall. The measurements have been interpreted by two models. In the first, the axial and radial thermal conductivities are assumed constant throughout the bed, and there is a thermal resistance at the wall. In the second, thermal properties have been assumed constant in the bulk, but allowed to vary in a narrow boundary region adjacent to the wall without an additional resistance so that both temperature and temperature gradient are continuous throughout both regions. For small particles and particles of high thermal conductivity, both models interpreted the measurements equally well. For large particles of low thermal conductivity the two-region model was much better.     

Mass transfer between irregular particles and liquid in packed beds

Dissolution of irregular particles has been studied in packed beds for the systems of alabaster-water, marble-aqueous HCl solutions and mossy zinc-aqueous acidic I₂ solutions, where the sphericity of particles varied in a wide range. The observed values of the Sherwood number have been correlated well by

     

Mixing of randomly moving particles in liquid—solid fluidized beds

The deterministic diffusion equation of the Fickian type has been widely employed in modeling solids mixing in fluidized beds. In this work, a stochastic or statistical model has been proposed for the solids mixing in a liquid—solid fluidized bed of relatively large particles. Specifically, the particle diffusivity under the steady fluidized state has been derived by combining the well-known ‘Ornstein-Uhlenbeck’ equation of stochastic processes and the so-called partition function of particle velocities. The values of particle diffusivity calculated from the model have been shown to agree well with the available experimental data. Furthermore, it has been shown that the solids-mixing process in the bed can be related to the time-dependent nature of particle motion.     

Angular porosity distributions in fixed packed beds of low diameter aspect ratio

The angular porosity distribution for fixed packed beds of monosized spheres in cylindrical containers with low diameter aspect ratios (1 ≤ D/d ≤ 2) is determined from the spheres' center coordinates. By dividing the packed bed into a large number of identical wedge-shaped segments, the angular porosity distribution is established from the local angular porosity and the angular positions within the cylindrical container. A correlation which accurately predicts the local angular porosity is determined from the angular porosity distributions. The correlation is a function of D/d and the angular position in the cylindrical container.     

A new pseudo-continuous model for the fluid flow in packed beds

A new pseudo-continuous model, describing the fluid flow through packed bed reactors, is developed by formulating the Navier-Stokes equations for a statistically described domain geometry. A coupling term, occurring in the momentum equation, represents the fluid-solid interaction and is modelled by comparison with the exact Navier-Stokes equations. Static and dynamic, inviscid and viscous contributions are separated, and stringent requirements for the dependence on the local voidage are found which are met by a simple model. Good agreement with correlations and experiments is found for the predictions of the pressure drop and the radial distribution of the axial velocity.     

Gas flow patterns in packed beds: A computational fluid dynamics model for wholly packed domains

This paper describes a general numerical technique for modelling three-dimensional, single-phase gas flow patterns in packed beds. Specifically, it presents a method for implementing a vectorial form of the well-known Ergun equation in a computational fluid dynamics (CFD) package. The approach is validated by comparison with independent experimental results. The general approach can be used to model flow patterns in adsorbers, catalytic reactors, etc., thus forming a useful design tool whereby packed bed configurations can be designed for minimum pressure drop while avoiding gas maldistribution.     

A note on the diffusional deposition of aerosol particles in packed beds

Aerosol penetration through packed beds of spheres under diffusional conditions is given by the empirical equation

$\text{C}{}_{\mathrm{L}}\text{C}{}_{\mathrm{L}}\text{=}\text{exp}\text{?6·39}\text{D}{}^{\mathrm{<mtext>2</mtext><mtext>3</mtext>}}\text{ν}{}^{\mathrm{<mtext>2</mtext><mtext>3</mtext>}}\text{R}{}^{\mathrm{<mtext>5</mtext><mtext>3</mtext>}}\text{L}$

recently proposed by Gebhart et al. (1973). The present note shows that this equation may be derived independently from Wilson and Geankoplis' (1966) correlation for mass transfer coefficients. It therefore follows that standard expressions for such coefficients may be used to predict aerosol collection under diffusional conditions.     

Numerical derivation of dispersion coefficients for flow through three‐dimensional randomly packed beds of monodisperse spheres

The longitudinal (D_L) and transverse (D_T) dispersion coefficients for flow through randomly packed beds of discrete monosized spherical particles are studied. The three‐dimensional (3‐D) porous‐medium model consists of thousands of spherical particles that are divided into cells using Voronoi diagrams. The relationship between the variation of the dual stream function and the vorticity between neighboring particles is derived using Laurent series. The whole flow pattern at low particle Reynolds number is then obtained by minimization of the dissipation rate of energy with respect to the dual stream function. The D_L is obtained by fitting the resulting effluent curve to a 1‐D solution of a continuous model. The D_T is obtained by fitting the numerical concentration profile to an approximate 2‐D solution. The derived D_L and D_T values are in agreement with 3‐D experimental data from the literature enabling a study of the effects of pore structure and porosity on D_L and D_T. © 2013 The Authors AIChE Journal published by Wiley Periodicals, Inc. on behalf of American Institute of Chemical Engineers AIChE J 60: 749–761, 2014     

Flow of generalized newtonian liquids through fixed beds of nonspherical particles

The flow of generalized Newtonian liquids through a random fixed bed of particles has been investigated and a universal method of calculation of the creeping flow was suggested. The usefulness of this method has been verified experimentally for the flow of power law, Ellis and Sutterby liquids through fixed beds of different nonspherical particles.     

Numerical simulations of transfer and transport properties inside packed beds of spherical particles

In this study, we investigate the transport and transfer properties inside packed beds of spherical particles by means of CFD simulations. Heat and mass transfer properties have been computed in packing configurations of increasing complexity at low to moderate Reynolds numbers (1<R_e<80). Only liquid-phase flows are studied (300<S_c<1000). The problem of contact points between particles, which is inherent to finite-volume numerical methods, is solved by applying a contraction of 2% on all the particles of the bed. We show that this treatment has very little influence on the results when analyzed with dimensionless numbers as N_u=f(R_e, P_r) or S_h=f(R_e, S_c). Finally, a very dense packing of spheres was built using a Discrete Element Method and used to represent the real granular media. Transfer predictions by the model are very realistic. Longitudinal and transverse dispersion coefficients are determined inside geometries containing hundreds of particles. Predictions of dispersion coefficients are consistent with literature, but a correction is applied to improve results, because the bed contraction leads to the underestimation of the transverse dispersion coefficient. The model is found to be very promising to study the “near wall channelling” phenomena inside small packed columns induced by the heterogeneity of the porosity profile close to the wall.     

Experimental investigation of pressure drop in packed beds of irregular shaped wood particles

The knowledge of the pressure drop across a packed bed of irregular shaped wood particles is of great importance for achieving optimal control and maximum efficiency in many applications, such as wood drying, pyrolysis, gasification and combustion. In this work the effect of porosity, average particle size and main particle orientation on the pressure drop in a packed bed is investigated. To this end, particle size distributions and porosities are determined experimentally.Based on the experimental results obtained in this study, the form coefficient C and the permeability K of the Forcheimer equation are calculated for different packed beds. The Ergun equation requires an average equivalent particle diameter that is derived from the measured particle size distribution. This equivalent diameter and the corresponding bed porosity are used in the well known Ergun equation in order to derive adapted shape factors A and B.Since a change in bed porosity and particle size, caused by the degradation of the wood particles and gravity, can be expected in a reacting packed bed, a set of shape factors for use with the Ergun equation is determined that are independent of porosity and particle diameter and fit the experimental data very well.     

Mass transfer and holdup in gas-liquid cocurrent flow packed beds containing water-repellent particles

Water-repellent particles were prepared by spraying polytetrafluoroethylene (PTFE) on activated carbon. Gas-liquid volumetric mass transfer coefficient and holdup were determined in gas-liquid cocurrent upflow and downflow packed beds from the measurements of gas desorption and volume, respectively. As the PTFE loading increased, the gas-liquid volumetric mass transfer coefficients in both upflow and downflow modes were enhanced. Gas holdup in the upflow mode increased with the PTFE loading, while the dynamic liquid holdup in the downflow mode decreased. The enhancement of the mass transfer rate from gas to liquid can be explained by the effect of water-repellency on the surface of activated carbon.     

Computational study of forced convective heat transfer in structured packed beds with spherical or ellipsoidal particles

Randomly packed bed reactors are widely used in chemical process industries, because of their low cost and ease of use compared to other packing methods. However, the pressure drops in such packed beds are usually much higher than those in other packed beds, and the overall heat transfer performances may be greatly lowered. In order to reduce the pressure drops and improve the overall heat transfer performances of packed beds, structured packed beds are considered to be promising choices. In this paper, the flow and heat transfer inside small pores of some novel structured packed beds are numerically studied, where the packed beds with ellipsoidal or non-uniform spherical particles are investigated for the first time and some new transport phenomena are obtained. Three-dimensional Navier–Stokes equations and RNG k–ε turbulence model with scalable wall function are adopted for present computations. The effects of packing form and particle shape are studied in detail and the flow and heat transfer performances in uniform and non-uniform packed beds are also compared with each other. Firstly, it is found that, with proper selection of packing form and particle shape, the pressure drops in structured packed beds can be greatly reduced and the overall heat transfer performances will be improved. The traditional correlations of flow and heat transfer extracted from randomly packings are found to overpredict the pressure drops and Nusselt number for all these structured packings, and new correlations of flow and heat transfer are obtained. Secondly, it is also revealed that, both the effects of packing form and particle shape are significant on the flow and heat transfer in structured packed beds. With the same particle shape (sphere), the overall heat transfer efficiency of simple cubic (SC) packing is the highest. With the same packing form, such as face center cubic (FCC) packing, the overall heat transfer performance of long ellipsoidal particle model is the best. Furthermore, with the same particle shape and packing form, such as body center cubic (BCC) packing with spheres, the overall heat transfer performance of uniform packing model is higher than that of non-uniform packing model. The models and results presented in this paper would be useful for the optimum design of packed bed reactors.     

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).