Progress in Porous Media Research [1 ed.] 9781616683023, 9781606924358

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Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

PROGRESS IN POROUS MEDIA RESEARCH

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

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Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

PROGRESS IN POROUS MEDIA RESEARCH

KONG SHUO TIAN AND

HE-JING SHU

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

EDITORS

Nova Science Publishers, Inc. New York

Copyright © 2009 by Nova Science Publishers, Inc. All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. Any parts of this book based on government reports are so indicated and copyright is claimed for those parts to the extent applicable to compilations of such works. Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS.

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LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA Progress in porous media research / [edited by] Kong Shuo Tian and He-Jing Shu. p. cm. Includes index. ISBN 978-1-61668-302-3 (E-Book) 1. Porous materials. I. Tian, Kong Shuo. II. Shu, He-Jing. TA418.9.P6P76 2009 620.1'16--dc22 2008047770

Published by Nova Science Publishers, Inc. Ô New York

CONTENTS Preface

vii

Research and Review Studies Chapter 1

Chapter 2

Chapter 3

Chapter 4

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Chapter 5

Chapter 6

Chapter 7

Chapter 8

Advances in Synthesis and Applications of Ordered Porous Materials Rajendra Srivastava, Shin-ichiro Fujita and Masahiko Arai Heat and Mass Transfer in Porous Media under Phase Transition Conditions: Freezing of Soils Leonid Bronfenbrener Transient Infinite Element Theory for Simulating Heat Transfer and Mass Transport Problems in Fluid-Saturated Porous Media of Infinite Domains Chongbin Zhao Numerical Simulation of Fluid Flow and Heat Transfer in Porous Media Xiulan Huai, Jun Cai and Weiwei Wang Modeling of Transport Phenomena in Porous Media Using Network Models António A. Martins, Paulo E. Laranjeira , Carlos Henrique Braga and Teresa M. Mata

1 3

55

105

135

165

Advances in Integrated Modeling of Mass Transport and Geomechanics in Coal Seams for CO2 Geo-sequestration F. Y. Wang, Z. H. Zhu, P. Massarotto and V. Rudolph

263

Influence of the Interfacial Drag on Pressure Loss for Two Phase Flow and Coolability in Porous Media Werner Schmidt

341

Microstructure Representations for the Simulation of Physicochemical Processes in Porous Media E.S. Kikkinides

387

vi Chapter 9

Chapter 10

Contents Organic Aerogels as Precursors for the Preparation of Porous Carbons Fernando Pérez-Caballero, Anna-Liisa Peikolainen Mihkel Koel, Robin J. White, Vitaly Budarin and James H. Clark Reducing Polycyclic Aromatic Hydrocarbons by Porous Materials Jian Hua Zhu, Shi Lu Zhou, Ying Wang, Ling Gao, Zhi Yu Yun and Jia Hui Xu

Short Communications A

B

C

D

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Index

Thermal Dispersion and Radiation Effects on MHD Free Convection of a Non-Newtonian Fluid Over a Vertical Cone in a Porous Medium A. M. Rashad, S. M. M. EL-Kabeir and Rama Subba Reddy Gorla Chemical Reaction and MHD Effects on Free Convection Flow Past an Inclined Surface in a Porous Medium M. A. Mansour, N. F. El-Anssary, A. M. Aly and Rama Subba Reddy Gorla Combined Effects Viscous Dissipation and Variable Permeability on Non-Darcy Natural Convection over a Vertical Plate in Porous Medium with Variable Wall Temperature A. M. Rashad and S. M. M. EL-Kabeir Velocity and Turbulence Field Around a Porous Block: Numerical Simulation and Experimental Validation Hsun-Chuan Chan, Yaoxin Zhang, Zhiguo He, Jan-Mou Leu and Tai-Wen Hsu

419

461

485

487

503

525

543

559

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PREFACE A porous material is a solid that is saturated by an interconnected network of pores filled with liquid or gas. It is an inorganic or organic, cross linked or uncross linked, containing pores of all sizes. The pore network is assumed to be continuous, forming two interpenetrating continua such as in a sponge. Examples of porous media range from porous silicon which is porous on the subnanometer scale to limestone caves and underground river systems on the kilometer scale. In this book, fractal structures are generated to model the structures of natural porous media. It also presents the fundamental theory of transient infinate elements, which can be used to effectively and eficiently simulate heat transfer and mass transport problems in fluid-saturated porous media of infinate domains, and addresses recent advances in the intergrated modeling strategies of mass transfer and geo-mechanics in porous media. Chapter 1 – The last decade is the witness for the development of ordered porous materials. Various types of porous materials such as siliceous and non siliceous mesoporous oxides, mesoporous carbon, mesoporous zeolites and layered structure materials are known. These porous materials may be prepared under a wide range of conditions in the presence of cationic, anionic or neutral surfactants. These materials typically have high surface areas and offer unique reaction and adsorption properties. Their synthesis strategies properties are described in this review. Because of their unique flexibility in terms of synthesis conditions, pore size tuning, and framework composition, these materials have been investigated for a number of catalytic and bio-technological applications. Applications of these materials (especially ordered mesoporous ones) for catalysis and drug adsorption/delivery are described. This chapter will cover the state-of-the-art in the synthesis, characterization and application of ordered porous materials. Chapter 2 - Heat and mass transfer processes in porous media occurring under phase transition conditions – in particular freezing of soils – are highly complicated and at the same time very important in many fields of science and engineering. In most studies of the problem, it is assumed that the phase front is a mathematical surface separating the soil region into frozen and unfrozen zones—a two-zone model. Nevertheless, it is known from experimental and theoretical studies on the freezing processes that the front of macroscopic ice formation lags significantly behind the boundary of incipient freezing. This phenomenon may be attributed to the existence of the region of the intensive phase transition in which migration and crystallization of moisture take place simultaneously. This region is characterized by sharp gradients of the equilibrium unfrozen water content, and its width

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depends on the soil properties and freezing conditions. As the experimental investigations show, the freezing zone is formed for the wide group of fine-grained loamy soils, and in this case it is necessary to consider a three-zone model. Thus, in this chapter, on the basis of mass and energy transport laws the system of equations for multiphase media—“soil-water-ice,” is derived. The solution of the moving-boundary problem, related to heat and mass transfer processes in freezing fine-grained porous media under phase transition conditions, is presented. It is assumed that a freezing zone, characterized by a wide temperature range of phase transitions, is formed. Therefore, a three-zone model is developed. The preservation of the terms ∂L ∂t ( L is the ice content) in a system of equations has made it possible to obtain

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the ice distribution within frozen and freezing zones. For loamy soils, the dependence of the freezing process on the characteristic parameters—the Stefan and Lewis numbers—was analyzed. It was found that increasing the enthalpy of phase transition, i.e., decreasing the Stefan ( Ste ) number, resulted in diminution of the frozen zone but, at the same time, the ice content within this zone increased. Intensification of the migration process, i.e., increasing the Lewis number Le , also led to diminution of the frozen zone, in which the ice content and, consequently, the total moisture (including ice) were increased. For large Lewis numbers the freezing zone was observed to decrease with increasing the ice and total moisture content in both frozen and freezing zones. When the water migration process is absent ( Le = 0 ) , the calculations, which were based on the described model, show that in the course of freezing the redistribution takes place only between moisture and ice contents. The total moisture remains constant and equal to the initial water content. The theoretical conceptions and results obtained from the analytical solution are in agreement with experimental findings. The presented model predicts the freezing process in porous media and satisfactorily reflects observed phenomena. Chapter 3 - This chapter presents the fundamental theory of transient infinite elements, which can be used to effectively and efficiently simulate heat transfer and mass transport problems in fluid-saturated porous media of infinite domains. Based on this theory, the infinite domain of a heat transfer and/or mass transport problem is divided into a near field and a far field. Since the near field is simulated using the conventional finite elements, both the complicated geometry and the complex material distribution can be easily taken into account. On the other hand, transient infinite elements are used to simulate the far field, so that the computer efforts can be reduced significantly, without losing the theoretical soundness of simulating an infinite domain. By coupling finite elements and transient infinite elements together, heat transfer and mass transport problems in fluid-saturated porous media of infinite domains can be simulated both effectively and efficiently. To better understand the transient infinite element theory, the details of the related formulation have been derived in this chapter. Meanwhile, the validation and application of the transient infinite element theory have been demonstrated using several examples, for which analytical solutions are available. Chapter 4 - In this chapter, different fractal structures are generated to model the structures of natural porous media. A lattice Boltzman model is developed to simulate fluid flows in fractal porous media under different conditions, and an appropriate mathematical model is established for the heat conduction in fractal porous media. Theoretical investigations were conducted to explore the influence of various factors on fluid flow and heat conduction. The results indicate that the flow field structures of fluid flow exhibits fractal characteristics, and the volume flow rate is proportional to the pressure drop and

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Preface

ix

conforms to an exponential function of porosity. The porosity is the most important factor that determines the effective thermal conductivity of fractal porous media, and the size and spatial distribution of pores, especially the spatial distribution of the bigger pores has substantive influence on heat conduction. Based on the Sierpinski Carpet fractal structure, heat transfer in porous media was first simulated by using Lattice Boltzmann model under the condition of considering the fluid flows. The temperature fields of porous media under the different heat conductivity ratios of fluid and solid were obtained. It was found that the ○ numerical results were in good agreement with the simulation results obtained by FluentR software. This shows that the Lattice Boltzmann model developed by authors can be used to simulate the fluid-solid coupling heat transfer problems of porous media or other materials with complex geometry boundary. Chapter 5 - This article discusses the application of network models to represent the local structure of a packed bed, and their application in the modeling of a fluid flow and mass transport in a porous media. It is divided in two parts. Part A is a critical review of the network models available in literature, with a focus in the main modeling methodologies proposed, its advantages, the main assumptions and limitations. The analysis shows that the local geometrical structure of a porous media is the key factor that controls the observed macroscopic behavior. In Part B, and partly supported by the models described and the conclusions drawn in Part A, a bi-dimensional network model is proposed to describe fluid flow and mass transport in a packed bed and studied in detail. The network itself is made up of two types of elements, the chambers and channels, to better account for the void space variability. A geometrical model is proposed, able to determine the average values of the network elements size distributions. The flow modeling takes into accounting explicitly the relations between the two types of elements. Results show that only for that case it is possible to describe all the possible flow regimens in a porous medium. Good agreement with experimental data is obtained for the packed beds composed by nearly sized particles. The mass transport model was built on network and flow models and it is capable of varying the relative importance of the main transport mechanisms, convection and diffusion, by changing the characteristic geometrical dimensions of the network elements. Nevertheless, results also show that the mass transport can be affected by the flow regimen observed in the network. Chapter 6 - This article addresses recent advances in the integrated modeling strategies of mass transfer and geo-mechanics in porous media with special applications to CO2 sequestration in coal seams. CO2 sequestration in coal seams is a relatively new technique to simultaneously achieve enhanced coal bed methane (ECBM) production and reduced CO2 emission. A methodology is developed in this article that integrates understandings in disparate research fields providing improved insight into the complex nature of the process. Our current overall model, constructed from a number of sub-models, consists of mass transfer in four pore types, namely, fractures, macro-, meso- and micro-pores, all having pore size dependent characteristics. Furthermore, a number of geo-mechanical models represented by matrix equations with different levels of complexity are developed and incorporated into the overall mass transfer model. Effects of adsorption and external stress induced pore size changes on mass transfer operations are analyzed in detail. The proposed modeling strategy is of a multi-scale nature with a variety of time and size scales. The macroscopic level model is validated using a true tri-axial stress coal permeameter (TTSCP), which provides accurate dynamic measurements of systems properties in three orthogonal directions including changes

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Kong Shuo Tian and He-Jing Shu

to the coal matrix volume. The integrated model provides a more complete and flexible representation for the simulation of coupled geo-mechanics and mass transfer in coal seams. Practical model order reduction techniques are proposed to permit fast prediction of process dynamics with acceptable accuracy for particular industrial applications. In particular, a hybrid distributed-lumped parameter model with significantly reduced model complexity has been applied to the depressurization processes for the accelerated recovery of methane residual from coal seams, with good agreement between simulations and experimental data. Finally, three mechanisms are proposed for coal swelling, which is important because it leads to permeability variations during sorption. These are based on molecular dynamics (MD) or quantum mechanics (QM) simulations. Due to the similarity between coals and other porous media, the authors believe the proposed integrated modeling methodology can also be applied to other similar media and processes using porous materials. Chapter 7 - This article investigates the influence of the interfacial drag on the pressure loss of combined liquid/vapour flow through porous media. This is motivated by the coolability of fragmented corium with internal heat sources, which are expected during a severe accident in a nuclear power plant. Due to the decay heat in the particles cooling water is evaporated. To reach steady states the out-flowing steam must be replaced by in-flowing water. The pressure field inside the porous structure determines the water ingression, and in effect the overall coolability. Typically, correlations for the dryout heat flux of porous media are adjusted to measurements where water ingression is from a pool positioned above. In these models the nature of the two-phase flow is included in corrections of the permeability and passability, achieved by simple functions of the void fraction. However, already configurations with possible water ingression from below demonstrate that this treatment is insufficient. The drag between the liquid and the vapour phase supports the co-current water inflow from below, and hinders the counter-current flow from above. Thus, the interfacial drag must be explicitly included in the modeling. This necessity is already seen in the measured pressure loss of simple isothermal air/water flow in porous media as well as for boiling particle beds with water ingression from below. Based on such experiments, two models with explicit consideration of the interfacial drag from the literature are discussed. Different flow patterns for the drag coefficients are included with the most advanced of these models. This article proposes some modifications on this model with respect to the small particle sizes that are expected in the reactor application. Furthermore, modifications to the formulation in the annular flow regime are necessary, as here the original model yields unreliable results. Based on the friction laws referred, the models are applied to typical reactor in-vessel and ex-vessel configurations in two-dimensional geometry. An enhanced overall coolability of the particle bed is already reached by the 2-D nature of the arrangement. Furthermore, the supporting influence of the realistic modeling by explicit consideration of the interfacial drag in the enhanced models is shown. Chapter 8 - The study of physicochemical properties of porous solids is a subject of great importance in the design of the materials for a variety of important industrial applications in gas or liquid separations by adsorption or membranes, catalysis and in the understanding of the mechanisms of various processes involving porous media that include oil and gas production from petroleum bearing reservoirs, transport of contaminants in soils and aquifers and drying processes.

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Preface

xi

The resistance that a fluid encounters as it is transported through the porous matrix is a function of its molecular properties, its interaction with the material that makes up the walls of the pores and the structural architecture in terms of pore connectivity, accessibility and other geometric descriptors. Understanding the relationship between microstructure and transport properties is a general problem in porous media applications. Since the early 50’s, a voluminous number of theoretical and experimental studies have appeared in the literature, concerning transport in porous media and the dependence of the macroscopic transport coefficients on the main structural parameters of the media. However, the direct interpretation of experimental flow or diffusion data from real porous materials in relation to the underlying microstructure is cumbersome due to the complexity of the porous matrix. Although considerable progress has been made in the efficient representation of the internal architecture of porous media using pore networks (see for example and references cited therein), there exists a strong need for a direct quantitative geometric description of the complex microstructure. The relentless advances in computational speed and powerful new methodologies should provide the basis for a reliable determination of macroscopic transport properties by solving the equations that govern the transport phenomena of interest. Studies in the field of petroleum engineering over the last two decades have led to computer based tools for the reconstruction of the actual pore structure by three-dimensional digitized representations and the acquisition of reliable assessments of a number of petrophysical and reservoir engineering properties such as the absolute or relative permeability and electrical conductivity. The developed two-phase structures belong to the general class of random heterogeneous media or simply random media bearing the assumption that any sample of the medium is a realization of a specific stochastic process, or random field. They comprise of direct, process-based, and statistical methods. Attempts have been made for the direct 3D computer reconstruction of detailed pore structure data obtained by serial sectioning of pore casts or computed tomography imaging (e.g., synchrotron X-Ray or Nuclear Magnetic Resonance). Serial sectioning is laborious, time-consuming and operator-depended while tomographic imaging is still limited to micron-range resolution (but getting better), high costs and equipment accessibility. Process based models try to account for the physical processes underlying the formation of certain microstructure. Well known examples include sedimentary rock and sandstone representation by modeling different rockforming geologic processes. In the same context, controlled porous glasses (CPG’s) were generated through a dynamic simulation of the spinodal decomposition process, which is considered to be the main mechanism for the formation of these materials. Process-based reconstruction methods, although more sound from a physical point of view, suffer frequently from severe computational requirements and are limited to the specific material considered. An attractive alternative is offered by the statistical methods in terms of a stochastic simulation of porous media in 3-D utilizing statistical information obtained from relatively few 2-D images of thin sections. The basic underlying principle is that both the real and the model structures should have the same basic statistical content, which is used as input for the creation of the simulated structures under the assumption of statistical homogeneity. Statistical methods have been proposed for the representation of different classes of porous media. In particular, for mesoporous and microporous media tomographic techniques are inadequate to provide 2-D images of the pore space due to the considerably higher (order of nanometer) resolution required. In these cases, information on the aforementioned statistical

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properties of the materials is obtained directly from SEM or TEM images or indirectly by Small Angle Scattering (SAS) data. Chapter 9 - Organic aerogels represent an interesting new class of materials and as the basis for the generation of new carbonaceous materials, both finding application in catalytic and adsorbent technology. These are prepared via the synthesis of an aqueous colloidal gel composed of suitable precursors. The gel state is generated by the formation of new chemical (i.e. polymerisation), and physical (i.e., hydrogen bonding) bonds, as ageing proceeds. The colloidal gel has a well-developed micro- and mesoporosity as well as large surface area. By fine-tuning the synthesis conditions, the pore architectures may be controlled. The solvent is subsequently removed to yield the aerogel, via extraction by supercritical fluids. For that CO2 is a rational choice as its critical point can be easily reached, and offers a wide range of temperatures and pressures, facilitating another dimension in material property control. The most commonly used precursors to generate the gel state are formaldehyde and resorcinol or its derivatives, but may also be prepared from the aqueous gelation and self-assembly of biomass derived polysaccharides. The transformation of organic aerogels via a controlled non-oxidative pyrolysis step, to produce low-density carbons is of increasing interest as the advantages of these materials become more evident. Carbon aerogels are chemically and mechanically more stable than the organic predecessor, provide excellent conductivity, and can be employed at high temperatures and pH. Control of carbon aerogel textural and surface properties is possible by selection of different organic precursors and careful selection of thermal treatment parameters. Interestingly, polysaccharide xerogels and aerogels may also be transformed into like carbons, which are termed Starbons®. Because of close relationship between certain organic and carbon aerogels in this chapter attention will be given to both. A detailed introduction to the synthesis and processing, characterisation and application of organic and carbon aerogels will be presented. The physical and chemical characteristics of these materials will be discussed in detail, with particular regard to density, specific surface area and porosity. Applications of aerogels will also be introduced, with a particularly accent on their use as heterogeneous catalyst supports. Future prospects for this exciting class of materials will also be proposed. Chapter 10 - Adsorption and catalytic cracking of polycyclic aromatic hydrocarbons (PAHs) such as fluoranthene (Flu) and anthracene (Ant) in microporous zeolite NaY, NaZSM-5 and NaA, along with amorphous materials such as alumina, silica and activated carbon, were investigated for the first time to study the selective reduction of PAHs by zeolite in tobacco smoke. Reduction of PAHs in the smoke by zeolite was observed depending on the type of zeolite and PAH; And the elimination mechanism of PAHs was discussed in this article, which resulted in not only the direct adsorbing or cracking the PAHs species, but also the suppressing the formation of PAHs by zeolites in the burning cigarette. Zeolites trapped the intermediates/precursors that were necessary for the formation of PAHs so that the thermal formation of the carcinogenic compounds was thus extinguished, leading to the reduction of PAHs in smoke. Besides, the decrease of PAHs in smoke by zeolites seemed unique because neither tobacco combustion nor the levels of all smoke products were interfered. Moreover, the presence of zeolites additive lowered the mean toxicity of the cigarette smoke as proven by Ames assay and CHO cell assay, which was beneficial for the protection of environment and public health.

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Preface

xiii

Short Comminication A - The present investigation deals with the effects of thermal dispersion, radiation and MHD on free convection boundary layer over a vertical cone in a Non-Newtonian fluid saturated porous medium with variable wall temperature and an exponentially decaying internal heat generation. The coefficient of thermal diffusivity has been assumed to be the sum of molecular and dispersion thermal diffusivity due to mechanical dispersion. Rosseland approximation is used to describe the radiative heat flux in the energy equation. The resulting similarity equations are solved numerically for two cases, one with internal heat generation (IHG) and the other without internal heat generation (WIHG). Numerical results for the dimensionless heat transfer rate in terms of the local Nusselt number are presented in tabular form and in graphical form to illustrate the influence of the various parameters on the temperature profiles with and without heat generation. Short Communication B - A boundary layer analysis has been presented for the free convection flow past an inclined surface in a Newtonian fluid-saturated porous medium. The effects of chemical reaction, MHD, radiation, viscous dissipation and heat generation are included. Rosseland approximation is used to describe the radiative heat flux in the energy equation. Four different cases of flows have been studied, namely, an isothermal surface, a uniform surface heat flux, a plane plume and flow generated from a horizontal line energy source on a vertical adiabatic surface. Numerical results are presented by using the perturbation analysis. The obtained results are compared and a representative set is displayed graphically to illustrate the influences of the flow parameters on the velocity, temperature and concentration. Numerical values for the skin-friction coefficient, Nusselt number and Sherwood number are presented in a tabular form with parameters characterizing the radiation, viscous dissipation, permeability of porous medium, heat generation and chemical reaction. Short Communication C - The effect of viscous dissipation on non-Darcy entire mixed convection regime into natural convection dominated regime in a porous medium form a vertical plate with non-uniform wall temperature incorporating the variation of permeability and thermal conductivity is investigated. The governing equations reduced to local nonsimilarity boundary layer equations using suitable transformation. An appropriate transformation is employed and the transformed equations are solved numerically using the second-level local non-similarity method. Comparisons with previously published work are performed and the results are found to be in excellent agreement. Numerical values of the rate of heat transfer for two cases, uniform permeability (UP) and variable permeability (VP), are presented in tabular form for different values of inertia parameter, Gebhart number, mixed convection parameter and power-law variation of the wall temperature parameter. Velocity and temperature profiles are also calculated at different values of the governing parameters for both UP and VP cases and presented graphically. Short Communication D - In this chapter, the authors present numerical and experimental investigations of turbulent flow characteristics around and within a porous block in an open channel. The turbulent flow field around the porous block was measured using the acoustic Doppler velocimeter (ADV) in a circulating water channel. The porous block model consisted of glass beads in a non-staggered pattern, having geometric porosity equal to 47.5%, was used in this study. A macroscopic model based on the Macroscopic Reynolds-Averaged NavierStokes equations is developed for predicting the flow fields around and within the porous block. The numerical method developed for the present work is based on the finite volume scheme with the semi-implicit method for pressure-linked equation (SIMPLE) method. The

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validity of the numerical method developed in the present work has been evaluated by comparing the numerical predictions with the experimental results. As a result, the recirculation region is shown to be elongated due to the bleed flow passing through the porous block. Turbulent kinetic energies are concentrated in two regions, one just above the structures, and the other behind the structure. The model developed shows a good qualitative and quantitative agreement with the experimental observations. Future applications to the real world problem of flow over a porous block are expected to contribute to the analysis of important engineering.

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RESEARCH AND REVIEW STUDIES

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In: Progress in Porous Media Research Editors: Kong Shuo Tian and He-Jing Shu

ISBN: 978-1-60692-435-8 © 2009 Nova Science Publishers, Inc.

Chapter 1

ADVANCES IN SYNTHESIS AND APPLICATIONS OF ORDERED POROUS MATERIALS Rajendra Srivastava, Shin-ichiro Fujita and Masahiko Arai Division of Chemical Process Engineering, Graduate School of Engineering Hokkaido University, Sapporo 060-8628, Japan

ABSTRACT

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The last decade is the witness for the development of ordered porous materials. Various types of porous materials such as siliceous and non siliceous mesoporous oxides, mesoporous carbon, mesoporous zeolites and layered structure materials are known. These porous materials may be prepared under a wide range of conditions in the presence of cationic, anionic or neutral surfactants. These materials typically have high surface areas and offer unique reaction and adsorption properties. Their synthesis strategies properties are described in this review. Because of their unique flexibility in terms of synthesis conditions, pore size tuning, and framework composition, these materials have been investigated for a number of catalytic and bio-technological applications. Applications of these materials (especially ordered mesoporous ones) for catalysis and drug adsorption/delivery are described. This chapter will cover the state-of-the-art in the synthesis, characterization and application of ordered porous materials.

1. INTRODUCTION Porous materials have attracted considerable attention of chemists and materials scientists because of their commercial applications in chemical separation, heterogeneous catalysis and as host matrices for bio-technological applications. According to the IUPAC definition, porous materials are classified in three categories; microporous (pore size < 2 nm), mesoporous (2–50 nm), and macroporous (> 50 nm) materials [1]. This chapter will review the preparation and application of various porous materials such as microporous zeolites, mesoporous silicas, nanoporous carbons, non-siliceous porous materials, mesoporous zeolites, and layered clay minerals. These materials can show a variety of functions

4

Rajendra Srivastava, Shin-ichiro Fujita and Masahiko Arai

determined by their structural (porosity) and chemical (acid/base) properties, which should be tailored depending on functions to be expected and on structure and size of molecules to be involved. Various attempts have been made so far to design and prepare the porous materials having different structural and chemical properties suitable for the chemical and biochemical applications of interest. Zeolites, crystalline aluminosilicate materials, are widely used as catalysts in oil refining, petrochemistry, and organic synthesis in the production of fine and commodities chemicals [2]. They can function as shape-selective catalysts due to their acidity and uniform micro porosity; however, they are not suitable for chemical reactions of bulky molecules. Larger pores are desired for larger molecules and so mesoporous silica materials such as M41S family, SBA family, KIT-6, and FSM-16 were developed [3]. The acidity and thermal stability of these materials are unsatisfactory in several cases as compared with the zeolites. Various modification procedures have been reported to improve the chemical and structural properties of mesoporous materials [4]. To overcome the above mentioned problem, mesoporous zeolitic materials have also been developed [5]. Unlike oxide molecular sieves, porous carbons are not crystalline materials and thus offer distinct advantages in terms of processibility, tailoring on a macroscopic size scale to form membranes, monoliths, and fibers [6]. Although non-siliceous ordered mesoporous materials have not been investigated well in contrast to siliceous materials, they have the potential for wide-ranging applications [7]. For non-siliceous ordered mesoporous materials, different synthetic strategies may be employed to vary the structural and chemical properties more widely compared with the siliceous ordered mesoporous oxides. A group of ultra-large pore materials consisting of layered structures with interlayer pillars are also of practical importance, which includes smectites, double hydroxides, silicas and metal oxides [8]. This chapter will describe various methods of the above-mentioned porous materials and highlight several successful examples of their applications as catalysts, adsorbents, and carries.

2. ZEOLITES

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Zeolites have been proven to be excellent catalysts for oil refining, petrochemistry, and organic synthesis in the production of fine and specialty chemicals, particularly for molecules having kinetic diameters below 1 nm [9]. Due to following properties, zeolites are most widely used in chemical industries [10]. • • • • •

They have very high surface area and adsorption capacity. The adsorption properties can be controlled from hydrophobic to hydrophilic natures. They have exchangeable cations allowing the introduction of other cations having various catalytic properties. When these cationic sites are exchanged with H+, acid sites can be generated in the framework and their strength and concentration can be tailored. The sizes of their channels and cavities are in the range typical for many molecules of interest (0.5-1.2 nm), and the strong electric fields [11] existing in those micropores together with an electronic confinement of guest molecules [12] are responsible for a preactivation of reactants.

Advances in Synthesis and Applications of Ordered Porous Materials •

5

Their intricate channel structures allow the zeolites to have shape selectivity, which can direct a given catalytic reaction toward the desired product avoiding undesired side reactions.

In addition to these properties, zeolites can be modified to very stable materials resistant to heat, steam and chemical attacks. This also makes zeolites attractive for various types of chemical processes conducted under severe conditions.

2.1. Zeolite structure and composition The properties of zeolites depend on the topology of their framework, the size, shape and accessibility of its free channels, location charge and size of the cations within the framework, the presence of false and occluded materials. Therefore the structural information is extremely important in understanding the adsorptive and catalytic properties of zeolites. The fundamental building unit of zeolite is a tetrahedron of four oxygen anions surrounding one Si or Al ions (Figure 1). These tetrahedra are arranged so that each of the four oxygen anions is shared in turn with another Si or Al tetrahedron, but Al cations do not occupy adjacent tetrahedral sites (Loewenstein’s Rule). The silica and alumina tetrahedra are combined into more complicated secondary building units (SBU) shown in Figure 2. Many zeolite frameworks can be built from several SBUs. The zeolite networks are designated using three-letter codes. Table 1 lists the codes and the zeolite species are classified to each of the codes [13].

O

O M

O

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OM= Si

4+

O

-1

O M

O O-

M= Al3+

Figure 1. Primary building units of zeolite

Si tetrahedra are neutral, because each Si ion has +4 charge balanced by the four tetrahedral oxygen that are -2 charge and shared by two adjacent tetrahedra. Each alumina tetrahedra has a residual charge of -1, since the trivalent Al ion is bounded to four oxygen anions. Therefore each Al tetrahedron requires a +1 charge from a cation in the structure to maintain electrical neutrality. The unit cell formula is usually written as Mn+x/n [(AlO2-)x (SiO2)y]zH2O. Where Mn+ is the cation which balances the negative charge associated with the framework Al ion. These cations are generally alkali metal or alkaline earth metal cations, but they can readily be exchanged with other cations. The tetrahedra are arranged so that the

6

Rajendra Srivastava, Shin-ichiro Fujita and Masahiko Arai

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zeolites have an open framework structure, which define a pore structure with a high surface area. The three dimensional framework consists of channels and interconnected voids or cages. The cations and water molecules occupy these void spaces. The pore structure varies from one zeolite to another. In all zeolites, the pore diameter is determined by the free aperture resulting from 4 to 18 member ring of oxygen atoms. Zeolites can be classified based on this ring size (Table 1). They can be classified in four categories: (a) Small pore zeolite (< 5 Å, 8-member ring), (b) medium pore zeolite (< 7 Å, 10-member ring), (c) large pore zeolites ( 10 , not being possible to define an universal value since this is a 3

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function of the packed bed structure, of the particle characteristic dimensions distributions, flow conditions and even of the fluids properties(Bear, 1972). The fourth zone corresponds to the turbulent flow, despite with characteristics different from the turbulence in tubes due to the irregular structure of the solid matrix and spatial limitations for the development of turbulence (Lage et al., 1997). If the F values as function of Re were represented in logarithmic scales, one would observe a linear dependence in this zone, analogue to the one observed for the fluid flow in rough cylindrical tubes.

Conduit Models One of the first type of models that tried to include, in a very crude way, the tube like nature of the local packing structure are the so called conduit models (Carman, 1937). Although they may be considered as pure phenomenological models, they rely on the definition of an equivalent hydraulic diameter of a tube function of the porous medium characteristics, and therefore can be classified as simple uni-dimensional network models, as the conduits do not cross each other.

Laminar Flow Relating the void volume and the superficial area of the porous with the definition of the hydraulic diameter (Bird, 1960), and assuming laminar flow the following expression for the permeability is obtained

k=

εd h 2T 2 16k0

=

εd h 2

=

16kk

ε3

kk (1 − ε )

2

DP

2

(3)

where k k = k 0 (1 T ) is the Carman-Kozeny constant, T is the porous medium tortuosity 2

that accounts for the irregular structure of the conduits, and k 0 is a constant function of the

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conduits assumed to form the porous medium. For circular capillaries k 0 =2, and its value varies between 2 and 2.5 for other geometries (Happel and Brenner, 1983; Liu et al., 1994). Based on experimental data, (1 T ) ≈ 2.5 , thus the normally called the Carman-Kozeny equation is obtained

k=

ε3

180(1 − ε )

2

Dp

2

(4)

Carman (1937) assumed that k k = 5 was a universal constant and independent of the porous medium characteristics. However, experimental data showed that the value of this constant depends on the porosity and shape of particles (Coulson, 1948; Wyllie and Gregory,

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183

1955). For packed beds made up of spheres, kk ≈ 5 , for porosity values around 0.4 (Fand et al., 1987). In terms of a friction factor, the Blake-Kozeny equation can be written in the form

180 (1 − ε ) Re ε 3

2

fP =

(5)

The previous expression is similar to empirical correlations presented in literature, although they normally have different dependences on the porosity (Rumpf and Gupte, 1971; Agarwal and O’Neill, 1988; Ziólkoskwa and Ziokolwski, 1988; Dullien, 1992). However, those expressions are empiric and strictly valid for the porous media where the experimental data were obtained. Also, the Carman-Kozeny has limited application when the packing is formed by particles with a large size distribution. MacDonald et al. (1991) has proposed the following expression,

1 ε3 k= 180 (1 − ε )2

⎛ X2 ⎞ ⎜⎜ ⎟⎟ X ⎝ 1⎠

2

(6)

where X 2 and X 2 are the first and the second moments of the particle size distribution, and have observed a good agreement for the laminar flow in packed beds. Liu and Masliyah (1996) also considered the basic Blake-Kozeny and tried to extend its validity. Assuming an isotropic medium, and analyzing the dependence of the interstitial velocity with the media local structure, these authors argue that the pressure drop in laminar regimen can be expressed in the form

Δp 36k1 μ (1 − ε ) vT = 2 L D P ε 11 3 2

−

(7)

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where k1 is a constant analogous to the Carman-Kozeny constant, k k . The porosity dependence is different from the one obtained before, but agrees with the dependence inferred by MacDonald et al. (1979) from experimental data.

Turbulent Flow The analysis made for laminar flow can also be extended for turbulent flow (Bird et al., 1960). Equating the pressure drop per unit length with the friction factor f P , the following expression can be written 2

1 1 ⎛ vT ⎞ ΔP ρ⎜ ⎟ 4 fP = LX d h 2 ⎝ ε ⎠

(8)

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Applying the definition of hydraulic diameter, and knowing that the experimental data suggests that 6 f P ≈ 3.5 , the previous expression can be written in the form

ΔP 1 1−ε ρv T 2 3 = 1.75 LX DP ε

(9)

or using a friction factor in the form also known as the Burke-Plummer Equation

f P = 0.875

1− ε

(10)

ε3

Comparing the expressions obtained for laminar and turbulent flow a different porosity dependence is observed. These results are directly obtained from the analogy used, since for turbulent flow regimen the friction factor of the flow in a tube is constant, thus limiting the applicability of this equation to completed developed turbulent flow.

Ergun Equation and Extensions The Blake-Kozeny and Burk-Plummer equations are only valid for the limit regimens of the flow, where either friction or inertial forces are dominant. For the intermediate regimen, Ergun (1952) assumed that the total pressure of the flow through a porous medium is the sum of the values predicted by the two limit expressions, in the form

ΔPT μ (1 − ε ) ρ 1−ε 2 vT + B vT =A 2 3 LX DP ε 3 ε DP

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2

(11)

where A and B are two constants that can be calculated from experimental data. Ergun (1952) obtained initially A = 150 and B = 1.75 . Later, using a larger database, MacDonald et al. (1979) obtained A = 180 , and B = 1.80 for smooth particles and B = 4.00 for rough particles. Even though these constants are considered to be universal, large deviations between predicted and experimental values may occur, especially for packings made up with particles with irregular shapes (Comiti and Renaud, 1989). Even for packings formed by spherical particles with narrow particle size distribution the differences can be relevant. For example the constant values obtained by Kim (1985) and Kececioglu and Jiang (1994) are in agreement with the recommendations of MacDonald et al. (1979), yet the experimental data of Rumpf and Gupte (1971) and Comiti and Renaud (1989) are better described using the values of A and B suggested by Ergun (1952). In many cases the determination of the constants from experimental data is the correct approach. Defining the following dimensionless parameters

F = *

ΔPT D P ε 3

L X ρvT (1 − ε ) 2

(12)

Modeling of Transport Phenomena in Porous Media Using Network Models

Re* =

ρv T D P μ (1 − ε )

(13)

*

*

where F is a equivalent friction factor and Re MacDonal Equation can be written in the form

F* =

185

a generalized Reynolds number, the

A +B Re*

(14)

Experimental data presented in this form in logarithmic coordinates gives two *

asymptotes. One with a slope -1 for low values of Re that corresponds to laminar regimen. *

To high values of Re , where the flow is turbulent a constant value is obtained equal to B . In literature other ways of presenting the experimental data were proposed, based on different definitions of the friction factor (Ergun, 1952), or using other methods to define the characteristic dimension such as the square root of the permeability (Ward, 1969; Kececioglu and Jiang, 1994), among others possibilities (Ahmed and Sunada, 1969; Ziólkoskwa and Ziólkowski, 1988; Venkataraman and Rao, 1998; Trussel and Chang, 1999). The analysis Liu et al. (1994) and Liu and Masliyah (1996, 1999) was also extended to turbulent and the full range of flow regimens, but considering a different porosity dependence and definitions of the relevant dimensionless numbers, the friction factor and the Reynolds number. Comiti and Renaud (1989) tried to extend the validity of the MacDonald equation to packings with particles with shapes not spherical, with the explicit inclusion of the tortuosity, and superficial area of the particles. These authors have proposed the following expression (Mauret and Renaud, 1997) 2 ΔPT f 1 1− ε 2 2 1 (1 − ε ) vT + ρ 3 a vd 3 vT = 2γ M μavd 2 3 LX 2 T T ε ε

where

(15)

γ M is a geometrical factor dependent of the packing local structure, avd is the particle

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specific superficial area and f 2 = 0.0968 . The values of T and aVd are determined from experimental data obtained for packings made up of the particles of interest.

Uni-Dimensional Models The models described in the previous subsection may be loosely considered as network models, because they assume that the local structure of a porous media is a bundle of tubes. However, as they do not try to describe the behavior of the individual elements, but use an analogy with the hydraulic diameter, they were considered separately. The simplest uni-dimensional model is the bundle of parallel capillaries. Assuming that all capillaries have the same diameter for laminar regimen the following expression for the permeability is obtained

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António A. Martins, Paulo E. Laranjeira, Carlos Henrique Braga et al.

k=

εd c 2 32

(16)

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The previous expression assumes that there is only one possible diameter value, and the tubes are straight. However, Scheidegger (1960) showed that the dependence of the permeability on the porosity does not change with the inclusion of a pore size distribution. For describing situations where the flow is uni-dimensional, Scheidegger (1960) suggests to substitute the constant 32 of the denominator by a correction parameter α s , which value equals to 32 and 64 for uni-dimensional and bi-dimensional flows respectively. The previous model does not account for the variations of the local structure of the porous media, equivalent to contractions and expansions that may have a profound impact in the behavior observed at the macroscopic scale. Many of the models and possible geometries are presented in Figure 2. However, a more accurate description of the local structure leads to a more complex description of the flow field, and in many cases the need to use numerical methods. Petersen (1958) and Houpert (1959) were the first authors to consider models with capillaries having a variable section, in particular spatially periodic. Assuming that the flow inside the channels is similar to the flow in an orifice these authors showed that a quadratic equation similar to the Ergun Equation was obtained, but with the advantage that the values of parameters can be obtained directly from the geometrical characteristics of the channels. Blick (1966) and Niven (2002) reached similar conclusions using tubes with orifice constrictions inside the channels. Other authors considered different channel geometries. Azzam and Dullien (1977), Ruth and Ma (1993) and Cao and Kitanidis (1998) considered circular channels with sudden changes in the radius. The computation of the flow field shows that for laminar regimen the pressure drop is a function of the constriction diameters. The transition between linear and non-linear flow regimens is smooth, depending on the geometrical characteristics of the channels, and occurs for Reynolds number much lower than those observed for straight tubes. These results are in qualitative agreement with the behavior observed in porous media, in particular in the transition zone. However, in a real porous medium such as a packed bed, it is not expected to have abrupt variations in the characteristics dimensions of their elements. Thus, several models were proposed in literature where the channels radius varies in a continuous fashion. Pendse et al. (1983) analyzed and compared the relative merits of some of the alternatives available in literature. Payatakes et al. (1973a, 1973b) have considered tubes where their radius varies according to a quadratic function. These authors also developed a geometrical model able to describe the local structure of the porous medium and to obtain the parameters of the quadratic function from experimental data. Results of this model compared well with experimental data for laminar and transition flow regimens. Sáez et al. (1986) considered the same radius variation, but to model the channels in a packed bed having a cubic regular structure. The computational and analytical study leads to the prediction of values for the constant A of the MacDonald Equation in agreement with other theoretical and experimental studies.

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Fedkiw and Newman (1977), Neira and Payatakes (1979), Tilton and Payatakes (1984), Hemmat and Borhan (1995) and Cao and Kitanidis (1998), studied the case in which the radius varies in a sinusoidal form. These authors concluded that the onset on nonlinearities on the flow is due to the formation of recirculation areas in the channels, in particular after the constrictions, depending on the geometrical characteristics of the channels. Also, the results show that for laminar regimen the velocity profile tends to the Poiseuille profile and the constrictions are the aspect controlling the pressure drop in the porous medium. Deiber and Schowalter (1979) and Lahbabi and Chang (1986) studied the transition between flow regimens in channels with sinusoidal walls and showed that the inertial effects are relevant even though not visible at the macroscopic level. The predicted values of the Reynolds number for the transition agrees well with experimental data. Channels that vary according to a hyperbolic function were considered by other authors (Venkatesan and Rajagoplan, 1980; Saeger et al., 1995; Thompson and Fogler, 1997). When compared with other models the predictions are very similar, showing that the form of the channels is not a determining factor. By comparison with the Carman-Kozeny equation, Pendse et al. (1983) concluded that sinusoidal channels are more adequate to describe the behavior of real porous media. For laminar regimen, Sheffield and Metzner (1976) have proposed a different approach to calculate the pressure drop inside the channels, based on the lubrification theory. Assuming parallel and laminar flow, the pressure in an infinitesimal segment of the channel is given by the expression

-

∂P q ∝ 4 ∂x d c

(17)

where q and d c represent respectively the volumetric flow rate and the characteristic dimension of the network element. For a cylindrical tube with constant d c the proportionality constant is equal to 128

π . If d c is a smooth function of the length inside the channel,

integrating the previous equation for a representative section of the channel leads to lp

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ΔP = c p q ∫ 0

dx

d c (x ) 4

(18)

where c p is a proportionality constant dependent on the channel geometrical characteristics. Dias and Payatakes (1986a) used this approach assuming a channel made up of a central circular and sinusoidal on the extremes. Larson and Hidgon (1989) also used it to model the flow in a packed bed of spheres with a cubic structure where the particles are partially fused together. Results showed that the lubrification theory is valid when the porosity value is low, or the resistance to the flow is controlled by the constrictions between the particles. Similar conclusions were obtained by Hemmat and Borhan (1995), having these authors suggested ways of improving the validity of this approach.

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António A. Martins, Paulo E. Laranjeira, Carlos Henrique Braga et al.

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Bi-Dimensional/Tri-Dimensional Models The models described in the previous sections do not consider the interconnected nature of the most real porous media. Bi-dimensional and tri-dimensional network models are used to consider those effects explicitly. However, when modeling the fluid flow inside the network of elements it is necessary to describe not only the behavior of the individual elements but also how they interact with each other. The most relevant changes in relation with the uni-dimensional models is the need to write the mass balance equations to mixing nodes, and possibly expressions to describe the influence on the flow or the interconnection between different network elements. To simplify the description of the flow, many studies assume that the mixing nodes have a negligible influence on the flow, either because it is assumed that they have a negligible volume (Sahimi, 1995), or because their effects are incorporated in the channels (Dias and Payatakes, 1986a and 1986b). However, some examples can be found in literature where the two types of elements were considered explicitly and modeled separately (Koplik, 1982; Thauvin and Mohanty, 1998; Wang et al., 1999b). The first practical application of a network model to describe the fluid flow in a porous media was the work of Fatt (1956) where a bi-dimensional model was used to study biphasic flow in a consolidated porous medium. Since then many more models were proposed in literature, with different strategies to solve the system of balance equations that describes the behavior of the network. The more common is based on the analogy between the flow in the network and the electrical current in a pure resistive circuit, almost always assuming steady state conditions and perfect mixing in the nodes. Considering that the absolute pressure in the nodes and the flow rate in the channels are analogous to the electrical potential and the intensity of the current respectively (Shearer et al., 1967; Palm, 1983), applying the Kirchoff laws to the equivalent electrical network it is possible to use efficient strategies developed for the analysis of circuits (Desoer and Kuh, 1969). Other strategy is based on the Hady-Cross method to determine the flow rates in flow systems (Hampton et al., 1993). This method is iterative and involves the consecutive solution of a linear system of equations, only using the mass balance equations at the network nodes, but requires an initial estimative of the flow rates in the channels that for highly irregular networks may be difficult to obtain. If the network is spatially periodic, Adler and Brenner (1985a and 1985b) have proposed a different methodology to model the fluid flow. These authors showed that the description of the flow can be reduced to the study of a fundamental cell, from which the behavior of a network with any dimensions can be obtained. Both linear and nonlinear flow regimens can be studied for these types of networks, and analytical expressions for the network permeability can be obtained using this method. Whichever methodology is used it is always necessary to characterize the individual behavior of the network individual elements. To simplify the calculations, it is usually assumed that the tubes are cylindrical, although any other shape can be used. The popularity of the analogy with an electrical circuit stems from the fact that if the flow is laminar and the fluid Newtonian the resulting equation systems are linear, symmetric and positive definite, allowing the use of efficient numerical methods to obtain its solution (Dias and Payatakes, 1986a; Kantzas and Chatzis, 1988a; Constantinides and Payatakes, 1989, Suchomel et al. 1998a)

Modeling of Transport Phenomena in Porous Media Using Network Models

189

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However, in some cases, such as high fluid velocities or non-Newtonian flow, the system of equations is non-linear (Thauvin and Mohanty, 1998; Wang et al., 1999b, Tsakiroglou, 2002; Lao, et al, 2004, Balhoff and Thompson, 2006). For these situations iterative algorithms based for example in the Newton-Raphson (Sahimi, 1993b), fixed point methods (Sorbie et al., 1989), or others (Shah and Yortsos, 1995) can be used. As stated before, most of the bi-dimensional and tri-dimensional models do not take into account the influence of the nodes in the flow. However, as in a real porous there are natural variations in the characteristic dimensions of the network elements, and in many cases they represent a significant part of the void space (Berkowitz and Ewing, 1998). Thus, some network models tried to include the influence of the nodes. Dias and Payatakes (1986a) and Constantinides and Payatakes (1989) have considered channels that have a different structure at its ends, to explicitly consider the expansions inside the porous medium. Koplik (1982) considered a bi-dimensional network with two different types of elements, cylindrical pores and spherical nodes, and determined the resistance associated with the connections between the two elements analytically. Ioannidis and Chatzis (1993) used the same strategy assuming that the elements have a rectangular geometry, having determined the correct form of the Koplik correction to that geometry. Thauvin and Mohanty (1998) and Wang et al. (1999b) used tri-dimensional networks with two different elements, and assumed that the resistance due to the connection is equal to the resistance due to the sudden expansions and contractions, and the mixture of fluid in the node. All correction terms are determined from correlations available in literature. Results of this model showed that even for low velocities the inertial effects can be significant. Since the network elements size distributions follow given probability functions, there is the problem of knowing when the results are statistical significant. In general, the larger the number of elements considered, corresponding to a larger sample of elements, the more statistical significant the results are. However, this fact increases the number of equations to solve simultaneously, situation that may limit the size of the network to be studied. Larson and Morrow (1981) studied this problem and concluded that the minimum network dimension that ensures statistical significant results depends on the probability function and the network geometry. As no criteria is available or is easily determined, in many studies the conditions of statistical significance are determined through simulation till the model results, such as, reach a asymptotic limit, or the standard deviation of the average values obtained is below a certain value (Sahimi et al., 1983; Constantinides and Payatakes, 1989).

MASS TRANSPORT MODELING The transport of chemical species inside a porous medium depends ultimately on its local structure, flow field, and the nature of the several solid and/or fluid phases that may be present. Thus, it is essential to have both a good description of the local geometry and flow field within the porous media in order to be able to adequately describe the transport and dispersion of mass. Network models are a natural choice, since they manage to give a simplified yet accurate description of the void space, and from that it is possible to characterize the flow field. In the next subsections the fundamental aspects of mass

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transport/dispersion in porous media and the diverse network models developed to model it are presented and critically discussed.

General Description The phenomena of dispersion is directly related to the way particles of the fluid, or a particular solute, travel through a porous medium. Besides the convective transport of mass, resulting from the movement of the fluid, the transport by diffusion may be relevant in zones of low velocity, like those that exist in the vicinity of the particles or in dead end pores. Both processes take place simultaneously, depending on the relative importance of the flow field characteristics. In a porous medium it is possible to define two more mechanisms directly dependent on the hydrodynamics (Sahimi, 1995). •

•

The first mechanism is kinematic. Due to the irregular nature of the porous media and the flow field, the streamlines will separate and mix together. Thus, the concentration field will change inside the system, controlled by the local structure and the connectivity of the porous medium. The second mechanism is dynamic in nature. Since there is a velocity profile at the local level, the solute molecules that are in different streamlines cross the porous medium in different times, leading to the dispersion of mass.

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In Figure 9 it is presented an example where both mechanisms can be observed (Fried and Combarnous, 1971). It can be seen that due to the presence of particles, the flux lines are naturally curved, and mixing will occur naturally. These processes are dependent on the medium structure, and although the flow field is also relevant, it may be possible to have two porous media with the same permeability value but that behave different when considering mass transport (Bacri et al., 1987).

Figure 9. Different Mechanisms of solute particle dispersion (Fried and Combarnous, 1971).

In most cases, the study of dispersion in a porous medium is done by applying a concentration perturbation at the entrance, and registering the response, also known as breakthrough, of the medium. The treatment of the experimental data gives information about

Modeling of Transport Phenomena in Porous Media Using Network Models

191

the main mass transport mechanics, flow field, among other things. When the underlying transport processes can be considered as linear, the response to a concentration perturbation can be expressed in the form ∞

C S (t ) = ∫ C E (t − t * ) f (t * )dt *

(19)

0

where C E (t ) represents the entrance concentration, and C S (t ) represents the exit concentration. The previous expression is equivalent to the convolution product that can be represented in the following form using the Laplace Transform

C S (s ) = C E (s )G (s )

(20)

where G (s ) is the transfer function of the system. For a impulse perturbation (Dirac),

C E (t ) = δ (t ) , it can be shown that G (s ) is the Laplace Transform of the Residence Time

Distribution, E (t ) . Using the Van der Laan Theorem (Wen and Fan, 1975), the moments of the E (t ) are given by the following expression

⎛ ∂ n G (s ) ⎞ ⎟⎟ n ⎝ ∂s ⎠|s =0

μ n = (− 1)n ⎜⎜

(21)

Methodologies and forms of determining the E (t ) and of analyzing the results are extensively in process and chemical reaction engineering, and excellent descriptions can be found in literature (Levenspiel and Bischoff, 1972).

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Dispersion Model At the local level, and assuming that there is no chemical reaction, the transient mass balance written for the solute is given in general form

∂C + ∇(vC ) = ∇(D M ∇C ) ∂t

(22)

where DM is the solute’s molecular diffusivity, C is the concentration of solute, and v the local velocity. In practice, it is assumed that DM is constant, and the previous equation can be written in the form

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António A. Martins, Paulo E. Laranjeira, Carlos Henrique Braga et al.

∂C + ∇vC = DM ∇ 2 C ∂t

(23)

Even with that simplification, the analytical and/or computational determination of the concentration profiles in time and space is most of times impossible or too much time consuming due to the difficulties in obtaining the flow field and in clearly define the interface between solid and fluid. Assuming that the medium is isotropic and can be considered homogenous, the velocity field can be replaced by an average value, v . Also, the DM may be replaced by a coefficient of effective diffusivity, or dispersion, Deff , that incorporates the effects of the void structure and flow field in the mass transport. Thus, Equation 23 can be written in the form

∂C + v∇C = Deff ∇ 2 C ∂t

(24)

Note that now Deff is not a real diffusion coefficient, but a parameter that models dispersion and takes into account that qualitatively resembles a diffusion process. A similar result was obtained by Taylor (1953, 1954a) and Aris (1956) when modeling the mass transport of solute traveling through a tube at slow velocity. These authors have concluded that sufficient long times, dependent on the flow and fluid characteristics, the concentration is described by an equation similar to Equation 24, being the coefficient Deff function of the molecular diffusivity and the Peclet number. Taylor (1954b) also concluded that for turbulent flow an equation with the same form also holds, but Deff has a different functional form. Nunge and Gill (1969) and Nigam and Saxena (1986) present excellent reviews of extensions of the basic Taylor-Aris model. In practice, Deff is split in two terms: the coefficient of axial dispersion, DL , and the

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coefficient of transversal dispersion, DT , leading to the following equation

∂C + v∇C = DL ∇ 2 C + DT ∇ 2 T C ∂t

(25)

Many times DL is called the axial dispersion coefficient, Dax , and DL the radial dispersion coefficient, D R (Froment and Bischoff, 1990). In many situations, it can be assumed that the transversal dispersion is fast when compared with longitudinal dispersion, or the perturbation imposed in the system is uniform, and the following equation can be written

∂C + v∇C = D L ∇ 2 C ∂t

(26)

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193

This model is widely used in practice and normally is designated by Dispersion Model, DM. In dimensionless form, a Peclet number can be defined as Pe = vL D L , where L is a characteristic dimension. This parameter is a measure of the relative importance of the mass transport by convection and dispersion. Although some authors have questioned its validity, (Sundaresan et al., 1980; Westerterp et al., 1995a, 1995b and 1996) on mathematical and physical grounds, when modeling mass transport in a porous medium it is still the usual choice. The solution of equation requires the definition of an initial condition and a set of boundary conditions. The proper definition of the set of boundary conditions is still an area of intense disagreement, although the set called Danckwerts boundary conditions are normally used (Levenspiel and Bischoff, 1972; Wen and Fan, 1975, Kocabas and Islam, 2000a and 2000b). Although Langmuir (1908) was the first author to propose them, it was Danckwerts in its seminal paper on Residence Time Distribution that gave a theoretical justification and popularized this particular set of boundary conditions. Wen and Fan (1975) using the tanks in series model derived the adequate sets of boundary conditions of the DM for open and closed systems at the entrance and exit (four possible combinations). Wen and Fan (1975) and Barber et al. (1998) have compared the different possible sets of boundary conditions, both theoretically and experimentally, and concluded that only small porous media of low fluids velocities are the results using different boundary condition sets significantly different. The value of DL can be determined experimentally from the response to a concentration perturbation imposed at the entrance. Bischoff and Levenspiel (1962a and 1962b), Levenspiel and Bischoff (1972, and Froment and Bischoff (1990) review some of methods and techniques available. Assuming that the Danckwerts set of boundary conditions is valid, the following expression can be written between the second moments of the E (t ) and the experimental response (Martin, 2000), in the form

σ=

(

)

2 ⎡ 1 ⎤ 1− 1 − e − Pe ⎥ ⎢ Pe ⎣ Pe ⎦

(27)

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Numerous correlations have been proposed in literature to correlate DL as a function of the porous medium properties, in particular for packed beds, and the physical properties of the fluid (Langer et al., 1978; Gunn, 1987; Foumeny et al., 1992). One of the main questions when applying the DM concerns the hypothesis of uniform transversal profile of solute concentrations. Although the results of Oliveros and Smith (1982) showed that even when the wall effect are relevant the presence of the particles makes the concentration profile uniform in the transversal direction of the flow, for small porous media this mixing may not be enough (Johnson and Kapner, 1990; Hackert et al., 1996). The experimental data of Han et al. (1985) showed for packed beds of spheres that for the values of DL are independent of the packed bed length in the main direction of the flow if the following condition is met

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António A. Martins, Paulo E. Laranjeira, Carlos Henrique Braga et al. .

⎛ LX ⎜⎜ ⎝ DP

⎞ 1 ⎟⎟ ⎠ Pe P

⎛1− ε ⎞ ⎜ ⎟ ≥ 0 .3 ⎝ ε ⎠

(28)

where the characteristic dimensional of the Peclet is the average particle size distribution. Many analytical solutions are available and listed in literature for the DM for a wide range of physical systems (Bischoff and Levenspiel, 1962a and 1962b; Brenner, 1962; Levenspiel and Bischoff, 1972; Wen and Fan, 1975, Gill et al., 1975). Some examples include the works of: • • • •

Rasmusson and Neretnieks (1980) that studied dispersion in packed beds or porous particles; Wang and Stewart (1989) that analyzed the case of chemical reaction involving more than one chemical species; Sun et al. (1999a and 1999b) and Clement (2001) that studied the situation where consecutive first order reactions; Aral and Liao (1996), Huang (1996) and Logan (1996) that relaxed the hypothesis of DL constant and analyzed possible spatial and temporal variations of this parameter.

Although its widespread utilization, some authors argued that the DM is not valid in all situations (Sundaresan, 1980). Thus, extensions of the model were proposed in literature. One of the most interesting is the work of Westerterp and collaborators (Westerterp et al., 1995a, 1995b and 1996). In order to take into account the existence of large gradients in the transversal direction of the flow, situation that occurs immediately after the imposition of concentration perturbation or for fast or highly exothermic reactions, these authors proposed that an additional equation should be considered in the basic DM model, in the form

∂C m ∂C ∂j +v m + =0 ∂t ∂t ∂x

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[1 + τ

w

]

q* (C m , x ,t ) j + τ w

(29)

∂C m ∂j ∂j + τ w (v + v a ) = − DL ∂t ∂x ∂x

(30)

where j represents the additional flux of mass due to the transversal irregularities, and

τw

and v a are the model parameters, function of the flow conditions. The system of equations is hyperbolic and it is similar to wave equations, thus the name given by the authors to this model is Wave Model. The predictions of the model were compared with the values calculated using the full mass balance equations and the Taylor-Aris model (Westerterp et al., 1995b; Benneker et al., 1997), showing the results that the Wave Model are valid for a wider range of conditions when compares with those of the DM model. The same conclusions were obtained by Kronberg et al. (1996) for the description of a laminar flow reactor. Benneker et al (2002) and Iordanidis et al. (2003) have compared experimental data obtained in a packed bed made of

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particles with a narrow particle size distribution for non reactive conditions and reactive conditions, and showed in both cases that the Wave Model is superior to DM, especially for fast or highly exothermic reactions.

Network Models Network models were also used to describe dispersion in porous media even though the number of works that can be found in literature is much smaller than the works available for the modeling of the flow. They can be broadly divided in two families. •

•

Particle tracking methods, where the dispersion characteristics and the values of parameters are determining following in time the evolution of a cloud of particles injected at the entrance of the network. Methods based on the solution of the mass balance equations written for all the network elements, and taking in account their different behavior. From the solution of the resulting system of equations the values of all relevant parameters can be obtained.

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In the following sub-sections both methodologies are reviewed and compared with each other.

Particle Tracking In this modeling strategy it is commonly assumed that the particles travel in the network channels at a velocity that equals the fluid (Sahimi et al., 1983 and 1986). This represents the transport of mass by convection. The nodes behave as mixing nodes without accumulation of particles, representing the dispersion of mass. In some works, the mass transport by diffusion is described assuming that the particles can jump between streamlines, similar to a process of random walk (Sorbie et al., 1991). Saffman (1959a, 1959b and 1960) and Jong (1958) were the first authors to propose models of this type, being the model of Saffman more comprehensive. It is based on a tridimensional network model, where it is assumed that the Darcy law is valid. The movement of the particles inside the medium is described as random walk process with variable distance and duration of each time step. Analysing the different possibilities for the random walk steps, Saffman (1959a) has concluded that the values of DT and D L can be determined from the following expressions

DT =

3 L X vT 16

[

DL n = f Pe(ln Pe ) DT

(31)

]

(32)

The values of the proportionality and power constants depend on the mass transport regimens, and some of the values can be found in Sahimi (1995). This model was extended by

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Chaplain et al. (1998) to the situation where the fluid is non-Newtonian, showing the results that the characteristics of the fluid should also be considered. Sahimi et al. (1983, 1986) used this methodology to describe the dispersion in bidimensional and tri-dimensional regular networks, with nodes of negligible volume. The flow field is modeled using a method similar to those described before, and a time marching algorithm was used to follow the evolution of the cloud of particles. To take into account the possibility of some particles to spend a large time in the network, if only convection is considered and because they are in streamlines close to the channels walls where the fluid velocity is very low, a Random Walk mechanism allows the particles to jump from one streamline to another in the direction normal to the flow. A similar strategy was suggested by Sorbie and Clifford (1991), but in this situation the Random Walk were also considered in the direction of the flow. Results of those models showed that they are capable of qualitatively describing all the mass transport regimens found experimentally (Sahimi et al., 1983 and 1986; Sahimi and Imdakm, 1988). Sahimi et al. (1986) also studied the influence of the mixing characteristics n

of the node showing that DL is proportional to Pe , having n between 1 and 2 and

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dependent of the mixing rules. Similar conclusions were reached also by Weng et al (2004). Many other models were proposed in literature to model mass transport in a porous medium based on particles tracking methods. For the purpose of this article it is not relevant to present all of the different possibilities. Some examples include the works of Moreno and Neretnieks (1993) and Moreno and Tsang (1994) that simulated the mass transport in network of fractures or in porous media with a distribution on the values of the flow conductivities. The results of these authors showed that when the distribution of conductivities is narrow the particle paths are almost parallel, but become ever more tortuous the more disperse are the nom homogeneities of the porous medium. The authors recommend caution when analyzing the response to solute perturbations for highly heterogeneous media, because the values obtained for the dispersion coefficient may not be representative. The influence of the mixing in the nodes for a fracture network was studied analytically by Grubert (2001), showing the results a high dependence on the dispersion coefficient on the mixing degree in the nodes.

Mass Balances Models The other strategy considered in literature to describe the mass in a network model involves writing the mass balance equations for the elements and solving the system of equations. When compared with the Particle Tracking methods, this methodology can be easily extended to include chemical reaction, adsorption and interfacial mass transfer. The breakthrough curve is also simpler to obtain. However, the inclusion of the flow field is not so natural, and the inclusion of the fluid profile in the channels and non ideal mixing effects in the chambers can be difficult to do. Two variants exist: one where the mass balance equations are solved explicitly in the time domain and the other in which the mass balance equations are solved in the Laplace Domain and the dispersion characteristics are determined using the relationship between the transfer function and the moments of the breakthrough curve. The second strategy is valid only for linear systems.

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Time Domain Modeling In this variant it is possible to find models in literature that used results already available, or are based on writing the mass balance equations for the network elements. As an example of the first approach, it is possible to refer the work of Carbonell (1979) that considered a uni-dimensional model of capillaries with a size distribution. Assuming that the Taylo-Aris model is valid in all channels, this author was able to relate the average macroscopic coefficient of longitudinal dispersion with the parameters describing the capillaries diameter distribution, namely its statistical moments. The cases of rectangular tubes and turbulent flow was also analyzed, with similar conclusions. A good agreement was observed between predictions of the model and the experimental data. Based on his results Carbonell (1979) suggested that the changes on the mass transport mechanisms are the result of changes in the flow regimen, in the tubes. In the second variant there is larger variety of models. One of the classical approaches to model mass transport in a porous medium is the Tanks in Series model. In the traditional form, it consists in a sequence of tanks with equal volume. Assuming that the interconnection effects is negligible and no chemical reaction occurs, the transfer function of the model is given by

G (s ) =

where

1 ⎛ τ ⎜⎜1 + T NT ⎝

⎞ s ⎟⎟ ⎠

(33)

NT

τ T is the overall passage time and N T is the number of mixing tanks. The E (t ) can

be easily determined from the previous expression, and equating the second moments of the E (t ) l and the DM, the following relation is obtained between Pe and N T

(

)

1 2 ⎡ 1 ⎤ 1− 1 − e − Pe ⎥ ≈ ⎢ N T Pe ⎣ Pe ⎦

(34)

For high values of Pe , the following expression is obtained

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NT ≈

Pe 1 + , Pe > 2 2 2

(35)

Many of the models and simulation methodologies based in the tanks in series models and its extensions are described elsewhere (Wen and Fan, 1975; Froment and Bischoff, 1990; Sardin et al., 1991). They include the addition of chemical reaction, interfacial mass transport, backflow between tanks, or zones with different hydrodynamic behavior. Although they are mathematically simple, they have the problem that the number of tanks, as well other parameters, has to be determined from experimental data, usually the experimental E (t ) . Nevertheless this type of models is quite popular in practice, due to its simplicity. A better description of the porous media behavior can be done if the tanks are linked with each other, taking into account the interconnect structure of a real porous media. One example

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António A. Martins, Paulo E. Laranjeira, Carlos Henrique Braga et al.

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is the work of Avilés and Levan (1991) that studied dispersion and adsorption in packed using bi and tri-dimensional regular network models, assuming that the channels have no influence on the system behavior. Using different types of isotherms, the authors have showed that the breakthrough curves are strongly influenced by their nature, being more abrupt the more favorable the isotherms are. However, the results show that the influence of the flow field is not significant, even when the wall effect is considered in the model. Bryntesson (2002) also considered the transport of mass in tri-dimensional regular networks, assuming that the mixing nodes are connected by channels with negligible volume. The parameters of the chamber size distribution were determined from porosimetry experimental data, and the results show that transient and steady state effective diffusivities depend on geometrical characteristics of the network. Villermaux and Schweich (1992) and Russel and LeVan (1997) used the same strategy but considering self similar bi-dimensional networks, based on a fundamental cell and a recurrence formula applied in it. The breakthrough curves dependent on the number of recurrence steps considered, but for a sufficiently large number they tend to the limit curves. Depending on the form and the parameters of the recurrence formula, these authors showed that the breakthrough curve is strongly dependent on them, and can be even be multimodal. Deans and Lapidus have proposed a bi-dimensional model formed by mixing tanks to describe the behavior of fixed bed reactors. The thermal effects are considered explicitly, and the behavior of the system can be described solving a system of differential equations under turbulent flow conditions. No attempt was made to compare the model predictions with experimental data. Küfner and Hofmann (1990) also considered a similar model but considering the influence of the channels between the tanks, to better account for the transport of mass by convection. The flow is characterized by a function that describe the transversal variations of velocity in a real packed bed. The comparison between predicted and experimental data showed that this model is better suited than the homogeneous models in the description of packed bed reactors. Other example is the work of Suchomel et al. (1998a), that studied the mass transport with convection and diffusion in the channels using bi and tri-dimensional networks. The system of mass balance equations was solving by an iterative method, starting from the diffusion only solution. The results show that D L is strongly dependent on the channels size distribution parameters, and that the existence of channels with a direction normal to the main direction of the flow leads to occurrence of tails in the breakthrough curve. Suchomel et al. (1998b) extended the model with the inclusion of bacterial growth, modeled using the Monod rate law, in the interface between solid and liquid. Terms for solute adsorption and biofilm erosion were included. A strong dependence on the geometrical characteristics of the network was observed, and good agreement was observed between predicted and experimental data. In other studies the combined influence of the dispersion and diffusion in porous particles was considered. The work of Meyers and collaborators are an example of this (Meyers and Liapis, 1998 and 1999; Liapis et al., 1999, Meyers et al., 2001a). These authors considered tri-dimensional models to describe the behavior of chromatographic columns. Special care was taken to correctly model the behavior of the porous particles, where diffusion dominates, and the behavior in the fluid phase, where the mass transport is controlled by convection. The influence of the porous media connectivity was analyzed, showing the results a strong

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dependence on their characteristics, namely the coordination number and the elements size distributions.

Laplace Transform Models When the system is linear, corresponding to situations where there is no chemical reaction/adsorption or if it occurs is linear, the Laplace Transform can be used to describe mass transport in a network of elements. From the transfer function and using the Van der Laan Theorem it is possible to obtain the moments of the E (t ) , and from there the parameters that characterize the transport of mass in the network. Roux (1986), Arcangelis et al. (1986) and Koplik et al. (1988) were the first to use this strategy, using regular bi-dimensional network with nodes of negligible volume. For each channel it was assumed that the DM is valid, being the longitudinal dispersion coefficient equal to the molecular diffusivity. Considering that the Danckwerts set of boundary conditions is valid, applying the Laplace Transform in each channels the following expressions are obtained

C j ( x , s ) = A j exp(α j x ) + B j exp(β j x )

(36)

2

v j ± v j + 4 DM s

α j ,β j =

2 DM

(37)

where the constants the A j and B j are given by (Koplik et al., 1988)

C j − C j exp(β j ) O

Aj =

E

exp(− β j l j ) − exp(− α j l j ) C j exp(α j ) − C j E

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Bj =

(38)

O

exp(− β j l j ) − exp(− α j l j )

(39)

where the indices E and O refer to the entrance and exit of the channels respectively. To determine the overall system transfer function two different methodologies were proposed. Arcangelis et al. (1986) noted that as the distribution of fluid velocities is available, it is possible to order them in such way that the particle passage time distribution, P (t ) , can be easily determined. That function is related with the particle passage distribution, Pj (t ) , in the Laplace domain by the following expression

P (s ) = ∑ Π Pj (s ) Γ

j∈Γ

(40)

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where P (s ) e Pj (s ) are the Laplace transforms of the functions P (t ) e Pj (t ) respectively, and Γ is variable that takes into account all possible particle paths inside the network. The

efficient calculation of P (s ) is done using a propagation algorithm, where the partial sums

for each node are calculated consecutively starting from the node with higher pressure and continuing through the ordered list till all nodes are considered. After performing the calculation, the numerical inversion of P (s ) gives the response of the network from where

the parameters that characterize the transport of mass in the network are determined. Roux et al. (1986) and Koplik et al. (1988) proposed a different strategy to determine the network response. After writing the mass balance equations for both, channels and chamber, these authors solved the system of equations directly in the Laplace domain. Numerically inverting the solution it is possible to obtain the breakthrough curve. To simplify the determination of the moments of the E (t ) Koplik et al. (1988) showed that using the original system of mass balance equations in the Laplace domain, expanding each term in a Taylor series, it is possible to obtain the moments just by inverting the system coefficient matrix. A similar strategy was used by other authors. An example is the work of Andrade (1993) that studied the dispersion of mass in a packed bed formed by spheres. Alvarado et al. (1997) extended the analysis of Koplik et al. (1988) to the situation where a first order reversible reaction occurs at the channels walls. The results showed that the chemical reaction has a significant effect in the behavior predicted by the model. Also, the authors concluded that the utilization of a homogenous model, like the DM, is adequate only when the spatial distribution of the reaction kinetic constants is homogenous, otherwise is valid only when the chemical reaction is fast.

PART B - NETWORK MODEL PROPOSED

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In the previous part the many types and variants of the network models and how they are used to model and describe transport phenomena in porous media were presented and discussed in detail. Some key aspects can be emphasized from the various models presented above. •

•

•

•

There is no universal network that can be used to describe all porous media that can be found in practical applications or in nature. Also, there are still some problems in obtaining all the data needed to construct the network models, in particular the information directly linked with the local structure of the porous medium. When dealing with fluid flow, there are still some difficulties when considering the non-linear regimens of flow. The models available are valid mostly for linear flow and for simple porous media. When dealing with the transport of mass, network models can be considered under development, as open questions still exist when dealing with mixing at the local level and how the solid and fluid phases may interact with each other. Although the problems faced by network models, currently they represent the best models in terms of the trade-off between the accuracy and the computational

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capacities available. They are very flexible in terms of geometry and possible elements that can be used, making it a good option when dealing with various types of porous media. Therefore, a network model was selected in this work to model fluid flow and the transport in a packed bed, although it can be used for other types of porous media. The model will be hierarchical in the sense that we will start by making a simplified description yet good enough to at least describe qualitatively the local structure of a packed bed. A geometrical model is presented that can be used to determine the network elements size distributions from data readily available. Based on the network model the fluid flow is described taking into account explicitly the relationships between the different elements that constitute the network. Then, using the information obtained of the network and fluid models, the transport of mass is modeled. In the next section each piece of the network model is described in detail, and some results are presented and discussed to assess the capabilities of the model.

NETWORK MODEL The network model used in this work was implemented in a software package that aims to model and describe the transport phenomena in a packed bed. Although the network generator is independent, the interlinks between the several blocks influenced its implementation. The following data is needed to generate the equivalent network of a packed bed: • •

Characteristics of the packed bed from which the analogous will be created. It is required in particular the values of the total porosity and the particle size distribution. Network generation conditions for obtaining networks with the required characteristics. With the imposition of these conditions one intends to simplify the procedures of network generation and of information exchange between the different software blocks that describe different transport phenomena in a packed bed.

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In the next subsections the generation methodology and a geometrical model developed to obtain the parameters elements size distribution are presented and discussed.

Network Generator The network is generated by repetition in both spatial directions of a fundamental cell, constituted by a chamber, assumed to have a spherical geometry, and three channels, assumed to have a cylindrical geometry, as shows in Figure 10. This way, six is the maximum number of channels of a chamber, equivalent to the maximum coordination number of the chamber Ci = 6 . The definition of a fundamental cell allows the simplification of the values attribution and storage to the several network elements during its generation. The attribution of values to

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the network elements depends on the contiguous cells even though the determination process of the network parameters is based on macroscopic parameters, such as the porosity. During the process of values attribution, the morphological characteristics of the elements around the cell are taken into account, but only of the contiguous cells. This way, the network generation is a local process, yet the determination of the network parameters need macroscopic data in order to obtain an adequate analogous of the packed bed under study.

Figure 10. Fundamental cell of the network model.

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The link between fundamental cells in both Cartesian directions, xx and yy , can generate bi-dimensional networks of chambers and channels with different characteristics as shown in Figure 10. In order to define the direction in which the pressure gradient occurs, which is important for the description of the flow through the network of channels and chambers, it was considered that the principal direction of the flow is according to the xx axis, as shown in Figure 11. This way, the channels that were aligned perpendicularly to this axis are designed horizontal channels, being the others the oblique channels. The network may include (Figure 11a and Figure 11b) or not (Figure 11c and Figure 11d) horizontal channels. In the network sides, two different situations may be defined. The first case (Figure 11a and Figure 11c) simulates the existence of periodic boundaries, i.e. the case where the network is considered as a representative part of the infinite structure. The channels in one side of the network are associated to the chamber in the other side of the network, i.e. the channel exiting one side of the network is the entrance channel to the chamber in the other side of the network, in the yy axis direction. The second case (Figure 11b and Figure 11d) simulates the existence of the packed bed walls, being the channels on the boundary, in the network sides, removed or not.

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Figure 11. Types of fundamental networks that can be generated: a) with horizontal channels and with periodic boundaries; b) with horizontal channels and without periodic boundaries; c) without horizontal channels and with periodic boundaries; d) without horizontal channels and without periodic boundaries.

One needs to define for all the network channels and chambers the diameter value of the chamber, and the diameter and length of the channels, as well as the angles of the channels do with the vertical. The network generator should be flexible in order to allow the creation of networks with different characteristics, having however to follow a given set of conditions, which are presented as follows: •

The chamber diameters, Di , and of the channels, d j , are described using two probability density functions, f D (Di ) and f d (d j ) , respectively. The probability density functions f D (Di ) and f d (d j ) are described by the same type functions,

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• •

using different values of the function parameters. Since in many porous media non symmetric pore size distributions with a maximum value are observed, and to ensure that only positive values are generated by the distribution, the Upper Limit Log Normal distribution were used for both functions. A description of this distribution can be found in Mata (2001a). Each channel diameter should be smaller than the associated chamber diameters, i.e. of the chambers linked to that channel. The angle, θ , that the oblique channels make with the xx axis is fixed as shown in Figure 12, and can vary in the interval between 0 and

•

π 2 . This hypothesis implies

a regular network, making it possible to obtain a regular rectangular grid by linking the centers of the network chambers. The variation of θ allows to change the ratio between the two main dimensions of the packed bed and change the network tortuosity, as shown bellow. The distance between two chambers centers is constant either in the vertical direction or in the horizontal one. This fact is a consequence of using a constant value of θ . This characteristic simplifies the network generation, in particular to obtain a distribution function of the channels lengths, f l (l j ) . This is because if the θ and

204

António A. Martins, Paulo E. Laranjeira, Carlos Henrique Braga et al. the distribution function of the chambers length, f D (Di ) , are known, one only needs to know the distance between one type of chambers centers to determine f l (l j ) . In

•

this work it was assumed that the distance between the centers of two chambers linked by an oblique channels is an input to the package. The junction of a chamber (sphere) with a channel (cylinder) results in the formation of a spherical cap, as shown in Figure 12. In the calculation of the total volume of network voids the volume of these spherical caps should be taken into account only one time, in order to obtain the total volume of network voids. The thickness , hij , and corresponding void volume of a spherical cap are given by the following expressions

2⎤ ⎡ ⎛ dj ⎞ ⎥ Di ⎢ h ij = 1 − 1 − ⎜⎜ ⎟⎟ 2 ⎢ ⎝ Di ⎠ ⎥ ⎣ ⎦

VCal =

π

⎞ ⎛D hij ⎜ i + hij ⎟ 3 ⎝ 2 ⎠

(41)

(42)

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Figure 12. Scheme of the spherical cap formed in the junction between a chamber and a channel.

Other conditions can be imposed in the model in order to extend its range of applicability, some of which are described as follows: •

As defined above, the coordination number, Ci , represents the number of channels coupled to a chamber. This parameter value is controlled primarily by the existence or not of horizontal channels and of periodic boundaries. Other form implemented in this work to vary the value of C i and obtain a distribution of values consists in randomly remove channels and/or chambers. In this model its was imposed that the removing process is done to verify an average value of C . The only imposed condition to the model network generator is that each chamber should have at least one entrance channel and one exiting channel, in order to avoid the formation of a line of chambers without channels of entrance and exiting, which would imply a null

Modeling of Transport Phenomena in Porous Media Using Network Models

•

•

205

flow through the network. In the network generator it was implemented the options of removing only horizontal channels or removing any type of channel. Sometimes one needs to determine the network permeability value, for example for comparison with the experimental value of a porous medium under study. To perform this calculation it is needed to know a thickness value of the network, which is not directly used in the network model, because it is bi-dimensional. The determination of the network thickness was done together with the determination of the channels distribution lengths, imposing two conditions. One condition is the equality between the packed bed porosity and the network porosity. The other condition follows directly from the hypothesis of constant distance between the centers of the chambers. One of the conditions already imposed in the model is that of each channel diameter to be smaller than those of the associated chambers. However, if the diameters distribution functions of chambers and channels overlap, the channels may crossover at the chamber effect, and that effect should be taken into account in the volume calculation. Alternatively, it can be imposed the condition that the channels cannot crossover at the chamber entrance. The application of this condition can be very restrictive for situations where there are horizontal channels or where the diameters distribution have very similar average values or when the standard deviation values are large. The application of these conditions for the channels size implies that the network generation is never completely random, making it possible to occur a tendency for the small diameter chambers, when compared to the average value D , to be associated with channels with small diameters. At this stage of development, this situation was not considered in the model.

In another hand, some imposed conditions to the model can be relaxed in a way to simplify the network generation. •

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•

The spherical caps formed by the entrance of the channels in the network chambers can be neglected, especially when the channels and chambers are not very close to each other. Another way to simplify the network creation process is to consider that the chambers and channels diameters have a constant value, similar to having a point distribution. For this distribution, when the boundaries are periodic, the network is named as uniform network, since there are no differences on its local structure.

The last simplification makes it possible, in some limit cases, to analytical model fluid flow and mass transport in the 2D network, as shown in the next sections.

Geometrical Model Description As referred before a model that aims to make a satisfactory description of transport phenomena in a porous media depends largely in its ability of incorporate information about

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its local structure. The failure to do so will limit the validity of the models, that will depend ultimately in parameters fitted to experimental data. Many strategies and experimental techniques were proposed with this intention. The main difficult lies in how the experimental information about the porous media is obtained and it is incorporated in the model. As stated before, the local structure of a packed bed formed by the deposition of particles in a container can be obtained by a number of methods. Based on that information it is even possible to obtain the equivalent network of simple elements (Thompson and Fogler, 1997; Chan and Ng, 1986 and 1988; Balhoff and Thompson, 2006). These methods do not obtain an exact replica of the local structure, but a analogous structure that, if the real packing was random in size and made of a sufficiently large of particles ensures that behavior predicted by the model is close to the real system. However, this methodology is computationally intensive, and problems may arise when defining the elements of the network and how they are interrelated (Chan and Ng, 1986). In this article a simple and faster method is presented, developed to be used for packed beds of spherical shaped particles and with a narrow size distribution. The model is based on readily available or easy to determine data, in particular the porosity and particle size distribution. From all the possible packed beds, those made with particles of the same size with a regular structure are the easiest to characterize. In particular, their behavior can be characterized using a representative cell, reducing enormously the effort needed to characterize them (Hasimoto, 1959; Zick and Homsy, 1982, Sangani and Acrivos, 1982a and 1982b, Edwards et al., 1990). They are six different regular packings, with different geometrical characteristics (Graton and Frasier, 1935; Martin et al., 1951; Pietsch, 1996). In Figure 13 the tri-dimensional structures of each regular packing is presented, and in Table 1 the names of each one as well the corresponding value of porosity are presented (Pietsch, 1996). They encompass a large range of values of porosity, including the values usually encountered in practice, normally around 0.4.

Figure 13. Regular packings of spheres: a) CUB; b) ORT; c)HEX; d) TET; e) RBP; f) RBH.

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Table 1. Porosity values for each of the regular packings (Pietsch, 1996) Type of Packing Cubic Orthorombic Hexagonal Tetragonal Rhomboedral – Pyramidal Rhomboedral – Hexagonal

Code CUB ORT HEX TET RBP RBH

Porosity 476 395 395 302 260 260

Comparing the porosity values of the various packings, it can be seen that there are two pairs with the same porosity value: ORT/HEX and RBP/RBH. Their structures are different, implying that their hydrodynamic behavior must be also different. For examples, albeit the packings ORT and HEX have the same structure, depending only in the way they are observed, depending on the main direction of flow the fluid will cross faces defined by four and three spheres in the ORT and HEX packings respectively. Therefore, both structures will have different hydrodynamic behaviors, as shown experimentally by Martin et al. (1951). An analogous situation occurs for the pair RBP/RBH. Both cases are considered explicitly in the development of the geometrical, being the main assumptions listed below. •

• •

•

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•

As each packing has a regular structure, the values of D , d and l predicted by the model are spatial independent, vary only from packing to packing, and represent only the average values of the network elements size distributions. If the particle size distribution is known, the parameters of the network elements size distribution can be determined directly from it. The values of D , d and l calculate for each regular packing are valid only to that value of porosity. If a packing has a different porosity value, it is assumed that the values of D , d and l can be calculated from correlations based on the values determined for the regular packings. For the pairs of packings that share the same porosity values it is assumed that D , d and l values are an arithmetic average of the values obtained for the two regular packings. As each packing is regular and spatially periodic, the values of D , d and l can be determined directly from the geometrical analysis of the fundamental cells that characterize each regular packing.

The fundamental cells for each packing are presented in Figure 14. Note that each vertex represents a sphere centre, not represented for sake of simplicity. The details of the geometrical manipulations needed to obtain each representative cell starting from the simplest one (cubic packing) are presented elsewhere (Martins, 2006). Analyzing the different fundamental cells, it is possible to define two different types of faces: • •

Square faces, where four spheres touch each other in the same plane, defining the centers of the spheres a square; Triangular faces, where three spheres touch each other in the same plane, defining the centers of the spheres an equilateral triangle.

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António A. Martins, Paulo E. Laranjeira, Carlos Henrique Braga et al.

Figure 14. Fundamental cells for each of the regular packings.

The total number of faces varies from packing to packing, and the values of each are presented in Table 2. Note that as the pairs ORT/HEX and RBP/RBH share the same porosity value, they have similar geometrical structures, and shown in Figure 14, and only of them is considered when determining the number of faces in each regular packing. Note that as the pairs of regular packings that have the same porosity have the representative cell (as expected), they have the same total number of faces, square and triangular faces. The different between them will be highlighted below.

Modeling of Transport Phenomena in Porous Media Using Network Models

209

Table 2. Number of faces for each of the regular packed beds of spheres Packing

Total Number of Faces

Cubic Orthorhombic Tetragonal Rhomboedral - Pyramidal

6 5 5 5

Total Number of Square Faces 6 3 2 1

Total Number of Triangular Faces 2 3 4

The type and number of faces is used directly in the determination of the values of D , d and l . Their determination is based on the hypothesis listed below. • • •

•

The value of D is equal to the diameter of the largest sphere that can be fitted in the regular packing without changing is structure. The values of d and l are determined together to ensure that the porosity value of the representative cell and the regular packing are equal. The number of channels and chambers associated with a representative cell is equal to the number of triangular and square faces respectively. The only exception is the CUB packing, where due to geometrical reasons the number of channels is equal to the number of square faces. The distance between two contiguous layers of spheres in a regular packed bed and the centre of the two chambers in consecutive lines of the network is equal. In Figure 15 an example can be seen for the CUB packing.

From the previous hypothesis it can be concluded that the value of D can be determined independently. For the CUB packing the value of D corresponds to the largest sphere that can be placed inside its representative cell, and equals to

(

)

3 − 1 D P . For the remaining regular

packings, D is the largest sphere that can cross a square face, resulting in D =

(

)

2 − 1 DP .

Both values are function only of the sphere diameter, and assuming that this relationship is also valid for an irregular packing, D can be expressed in the form D = K D (ε )D P , being

K D a function only of porosity. Considering that K D has a polynomial form, and imposing Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

equal values of K D and a zero value derivative for

ε = 0.395 , the following interpolation

function is obtained

K D (ε ) =

(

2 − 1 + 48.44(ε − 0.395)

)

K D (ε ) =

(

2 −1

)

2

ε > 0.395

(43)

ε ≤ 0.395

(44)

To be able to obtain the values of d and l the other assumptions are used. First, it is needed to relate the distance between the centers of two spheres in consecutives lines of the network with porosity. As seen in Figure 15 for a CUB packing, if lF represents the distance between the centre of two spheres in contiguous layers and LO the distance between the

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António A. Martins, Paulo E. Laranjeira, Carlos Henrique Braga et al.

centre of two chambers connected by an oblique channel, both variables are related by the following expression

l F = LO cosθ

(45)

Figure 15. Relation of the distance between two contiguous lines of spheres and the centres of two chambers in consecutives lines of the network, for a CUB packing.

The values of l can be calculated from LO using l = LO − D + 2h . The values of lF can be determined for each regular packing analyzing how the spheres packed each other. Four different forms of positioning the particles are possible, and for each one lF is function only of DP . Also, the geometrical analysis showed that the pairs ORT/HEX and RBP/RBH have different layer structures, leading to different values of lF .

Thus, as done for D it can be assumed that l F = K F (ε )D P , where K F (ε ) is a

function only of the porosity. The following third degree polynomial correlates well the values obtained for each regular packing.

K F (ε ) = 0.8740 − 0.5859 (ε − 0.26 ) + 13.53(ε − 0.26 ) − 37.57(ε − 0.26 ) 2

3

(46)

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Note that because it is physically impossible for the regular packings considered in this work to have layers distanced more than DP , it is imposed that K F (ε ) = 1 and the first

derivative of the function K F (ε ) is equal to zero for

ε = 0.476 .

The previous results showed that D and lF are proportional to DP , being the proportionality constant only function of the porosity. Assuming that this also true for l , and according to the previous equations the following equation follows

K l (ε ,θ ) = K F (ε ) sec(θ ) − K D (ε ) + 2 K h (ε ) where K h is obtained from the following expression

(47)

Modeling of Transport Phenomena in Porous Media Using Network Models

h = K h (ε )D P , K h (

2 ⎡ ⎛ K d (ε ) ⎞ ⎤⎥ ⎢ ⎟ DP ) = 0.5K D 1 − 1 − ⎜⎜ ⎢ K D (ε ) ⎟⎠ ⎥ ⎝ ⎦ ⎣

211

(48)

The parameter θ can be estimated using the relation between the tortuosity and the geometrical structure of the network, in particular θ . The relation between θ and T is given by the following relation

T=

LX l = F = cos θ LT LO

(49)

where LT is the total distance traveled by a particle of fluid between the network entrances and exit, and L X is the network length in the main direction of flow. The tortuosity of a porous medium depends on the porous medium characteristics, in particular the type of particles and its local structure. Taking into account that the model is based on regular packings of spheres, the correlation of Comiti and Renaud (1989) obtained from experimental data gathered in packings of spheres with a narrow particle size distribution was considered in this work

1 = 1 − 0.49 ln(ε ) T

(50)

θ can be expressed as function of T by the formula θ = ar cos(T ) , and, as T is a function only of the porosity, in this model θ is also only a function of ε . Thus, in The value of

the previous expressions all proportional constants will be only functions of the porosity, simplifying the calculations of the average values of the network elements size distributions. Note that if the previous expression for the tuortosity is valid, then the following equation can be written for K l

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K l (ε ) = K F (ε )[1 − 0.49 ln(ε )] − K D (ε )

(51)

As stated before, the calculation of d and l has to be done simultaneously. According to the condition imposed above that the porosity or the regular cell is equal to the porosity of the regular packing, and taking into account all previous conclusions, the following system of equations can be written to obtain the values of K d and K l , assuming that they are proportional to D P and function only of the porosity

K l (ε ,θ ) = K F (ε ) sec(θ ) − K D (ε ) + 2 K h (ε , θ )

(52)

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António A. Martins, Paulo E. Laranjeira, Carlos Henrique Braga et al.

εK cel

π eq 3 ⎡π eq 2 ⎤ ⎢ 4 N can K d (ε ,θ )K l (ε ,θ ) + 6 N cam K D (ε ) −⎥ =⎢ ⎥ ⎢π N eq K 2 (ε ,θ )[1.5 K (ε ) − K (ε ,θ )] ⎥ cal h D h ⎣⎢ 3 ⎦⎥

(53)

eq

eq

where K cel is a proportional constant function only of the porosity, and N cam , N can and eq N cal are respectively the equivalent number of chambers, channels and spherical caps

associated with a fundamental cell. The values of previous parameters can be determined from the geometrical analysis of the regular packings, and their values are presented in Table 3. With all the previous information it is possible to determine the values of K l and K d for each regular solving a nonlinear system of two equations. The values determined can be represented using the following functions

⎡ (ε − 0.260)⎤ K d = 0.4142 − 0.2522 exp ⎢ − 0.0703 ⎥⎦ ⎣

(54)

K l = 1.072 − 1.892(ε − 0.26 ) + 53.18(ε − 0.26 ) − 237.1(ε − 0.26 ) 2

N eq

N eq

N eq

3

(55)

K

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Table 3. Values of can , cam , cal e Cel of each one fundamental cells of each regular packing. Packing

K Cel

CUB ORT

1

3 4

eq N cam

eq N can

eq N cal

1 1.5

3 1

6 2

TET

3 4

1

1.5

3

RBP

2 6

0.5

2

4

In Figure 16, the values and correlations obtained for K D , K d and K l as a function of the porosity are presented and compared with each other

Modeling of Transport Phenomena in Porous Media Using Network Models

Figure 16. Values and correlations obtained or

KD , Kd

and

Kl

213

as a function of the porosity.

If the distribution of the particle size distributions is known, the geometrical assumes that the chamber and channel are given by the following expressions

( )

f D (Di ) = K D (ε ) f DP D Pj

( )

f d (d j ) = K d (ε ) f DP D Pj

(56) (57)

They follow directly from the results of the model that showed that for the regular packings the average chamber and channel sizes are proportional to the average particle size distribution.

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Limitations and Comparison with Experimental Data One of the main limitations of the geometrical model proposed here is linked to the utilization of regular packings of spheres. Naturally, it is expected that the model will be more adequate for the packings of spheres with a narrow size distribution, homogenous and isotropic. For packing with particles having a shape very different from the spherical or anisotropic, as the model does not include any parameter that takes into account that aspect is not directly applicable in those conditions. For non spherical particles a possibility of extending the model is the utilization of the concept of sphericity or equivalent particle diameter, based on the ratio between the particles volume and superficial area. However, this strategy is always an approximation and does not take into account the influence the particles in the geometrical characteristics of the void space.

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António A. Martins, Paulo E. Laranjeira, Carlos Henrique Braga et al.

Also, the particle size distribution should be uni-modal and have a small value of standard deviation. If there are particles of different sizes, or particle size distribution is wide, there is the possibility of obtaining porosity values lower than 0.260 (Yu et al., 1996), and the resulting of channel diameter distribution will have large values of the standard deviation (Nolan and Kavanagh, 1994; Assouline and Rouault, 1996). This is due to the fact the smaller particles can enter in the voids formed by the smaller ones, as shown in Figure 17.

Figure 17. Cut of a packed bed made up of spheres with a wide range of diameters.

Other limitation is the inability of the model to determine the value of C and corresponding distribution of values, directly from the geometrical analysis of the regular packings. A possible hypothesis was to assume the value of C equal to the number of channels associated with a given fundamental cell. However, due to the geometrical differences between the network model and the packings, it was preferred to maintain C as a free parameter.

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Comparison with Experimental Data Even though the experimental data available in literature is dependent on the structure assumed by the authors for their approximate model of the local structure of the porous media, it is nevertheless instructive to compare the predictions of the model with published data. Kruyer (1958) has performed porosimetry experiments in packed beds made up of nearly equal sized particles. From the extrusion and retraction curves it is possible to conclude that the pore size distribution, equivalent to the channels in our network model, is very narrow and can be accuretaly described using a single average value. Frevel and Kressley (1963) and Mayer and Stowe (1965 and 1966) confirm these experimental findings, and for a value of porosity around 0.90 they predicted values of K d between 0.27 e 0.37, in agreement with the predictions of the geometrical model.

Modeling of Transport Phenomena in Porous Media Using Network Models

215

Nolan and Kavanagh (1994) simulated the deposition of particles with different particle size distributions and analysed the local structure obtained. Their results showed that the chamber and channel size distributions are a function of the particle size distribution, especially its standard deviation. The predicted values of D by both model agree very well for porosities around 0.40. Chan and Ng (1986 e 1988) also studied packed beds formed by the deposition of nearly equal sized spheres in a rectangular container. For porosities around 0.40, the network model constructed by the authors to represent the packing void space gives values of d and of 0.38 and 0.45, in agreement with the geometrical model. Chu and Ng (1989) used the same strategy, but taking into account the influence of container. In this work the channel length is also evaluated. A value of K l ≈ 1.25 was obtained, within the range of values predicted. The previous results show that the model proposed in this work can predict accurately the average sizes of the network elements that represent the packed beds. For porous media formed by particles with a large range of diameters, the data of Payatakes et al. (1980) shows great differences with the geometrical model, revealing its limitations.

HYDRODYNAMIC MODEL Following the general description of the hierarchical model, the hydrodynamic behavior in a porous media is based on the modeling of the local structure of a porous medium, made by the network/geometrical model. The main assumptions made during its development and computational implementation are presented bellow. •

•

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•

•

The flow is incompressible, monophasic, isothermal and in stationary state. The transport properties, in particular the flow viscosity and density, are considered constant. The network elements are saturated by the fluid. It is assumed that the fluid mixture in the chambers is perfect, which ensures that the pressure on its interior is uniform. The influence of the chambers in the flow is being felt through the links with the network channels and by the mixture of the fluid entering the chamber. In the channels there is a plug flow. This hypothesis implies that the velocity profiles in the channels are uniform and uni-dimensional. This way, the pressure drop on a channel can be determined based on a friction factor, function of the flow regimen in the channel. The last hypothesis also implies that the effects of the channels entrances and exits on their velocity profiles are negligible. Similarly, the streamlines at the entrance and exit of the channels are parallel among them, simplifying the modeling of the hydrodynamic behavior in the chambers.

The hypothesis imposed in the flow description implies that the velocity and pressures profiles in the interior of the network are associated to the network channels and branches, respectively. The model implemented to the flow description is described as follows.

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António A. Martins, Paulo E. Laranjeira, Carlos Henrique Braga et al.

Model Description The modeling of the fluid flow through the network chambers and channels are done by writing their mass and momentum conservation equations. When describing the balance equations in the channels, the influence of the links among them and the chambers is taken into consideration. The model formulation is based on the analogy between the hydraulic and electric circuits (Dias and Payatakes, 1986a). That is why it is convenient to associate to the circuit nodes and branches the network chambers and channels, respectively. By applying the analogy with an electric circuit, a system of equations is obtained where the unknowns are absolute pressures in the chambers interior. The nature of the equation system depends on the flow characteristics through the network channels. The primary variables in the flow description are the volumetric flow rate in each channel, q j , and the absolute pressure in each chamber, Pi . The analogous variables are the electric current intensity in a chamber and the electric potential in each node of the electric circuit respectively. In Figure 18 it is presented an example of an electric circuit equivalent to a chamber with horizontal channels and periodic boundaries. As it was assumed that the flow is on steady state, the analogous application allows one to use the methodologies developed in the analysis of electric circuits (Desoer and Kuhn, 1969). The mass balance equations on the network nodes can written in the form

∑q where q j

E j

E

= ∑qj and q j

O

O

(58) represent the volumetric flow rate1 in a chamber entrance and exit

channels, respectively. The behavior in the network branches is described by the following equation, similar to the Ohm law that takes into account the possibility of existence of nonlinear flow terms. s

ΔPj = R j q j + ΔPj − R j q j

s

(59)

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The term ΔPj represents the pressure drop in a network branch, equivalent to the potential difference observed between the nodes and its ends. R j represents the resistance to the fluid flow through the network branch, including this term the influence of the flow friction in the channel walls and of the connections between chambers and channels. The term s

ΔPj represents tension sources, for example when one considers the influence of the flow gravity, capillary pressure due to the superficial tension on a multiphase flow (Payatakes and Dias, 1986a), or physical properties variation, as it is the case of the exothermic reactions s

occurrence in the medium. The term q j represents current sources and can be significant in situation where chemical reactions occur with a variation of the medium moles number.

Modeling of Transport Phenomena in Porous Media Using Network Models

217

Figure 18. Electric analogous used for the flow description: a) electric circuit equivalent to a network with horizontal channels and periodic boundaries; b) analogy for the flow in a channel.

In the hydrodynamic model the flow occurs by imposition of a pressures macroscopic gradient, imposed between the entrance and exit network channels. This term is considered through the definition of a tension source term associated to the fluid entrance in the network channels. For the remaining channels the following equation is valid

ΔPj = R j q j + ΔPj

s

(60)

The mass balance and momentum conservation equations, defined by Equations 58 and 59 respectively, should be solved simultaneously to determine the volumetric flow rates and the absolute pressures in the network channels and chambers respectively. Both types of equations are associated through the term R j that represents the resistance to the flow of a network branch. The nodes and meshes laws can be applied to the network electric analogous to simplify the resolution of the equations system and reduce the number of equation to solve simultaneously (Desoer and Kuhn, 1969). The mass balance equations in the chambers, defined by Equation 58, can be expressed in the following matrix form (nodes law)

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Aq = 0

(61)

where q is a vector of N Can elements having the current intensity values on each branch and

A is the matrix of reduced incidences of N Cam × N Can dimensions. The elements aij of the matrix A can take one of three values as a function of the flow direction in the channel j in relation to the chamber i : -1 if the direction corresponds to flow entrance, +1 if the direction corresponds to flow exit, and 0 if the channel and chamber are not liked among them. Similarly, it is possible to write the meshes law to the electrical analogue. This law says that the different of potential in a closed branch of an electric circuit should equal to zero. The application of this condition results in the following matrix expression

ΔP = A T P

(62)

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António A. Martins, Paulo E. Laranjeira, Carlos Henrique Braga et al.

where ΔP is a vector with N Can elements, containing the differences of potential in each electric circuit branch, P is a vector with N Cam elements, containing the potentials T

(pressures) in the circuit nodes, and A is the transposed incidence matrix. The Equation 61 can be written as follows

(

q j = G j ΔPj − ΔPj

s

)

(63)

where

Gj =

1 Rj

(64)

G j is the branch conductance. The Equation 63 can be written in the matrix form q = G(ΔP − ΔP s )

(65)

where ΔP is a vector with N Can elements, containing the tension sources in each network s

branch and G is a diagonal matrix of dimensions N Can × N Can , containing the conductance values of each equivalent circuit branch. Substituting Equation 65 on Equation 62 and rearranging it, an equation system is obtained where the unknowns are the potentials (pressures) in the network nodes, in the form

YP = q s

(66)

where the matrix Y is defined by the expression

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Y = AGA T

(67) This matrix Y is the admittances matrix, being a square matrix of dimension N Cam × N Cam . The elements yij can be determined from the matrix G , taking into consideration their position in relation to the matrix diagonal, in the form: •

The elements in the matrix diagonal, y ii , are determined by summing the conductance of all network branches associated to node i , representing the admittance of a node i .

•

The elements that do not belong to the diagonal, yij , are symmetric to the conductance value of the branch that links the chambers i and j , representing the

•

mutual admittance between two network nodes. All the remaining elements are nulls.

Modeling of Transport Phenomena in Porous Media Using Network Models

219

According to the structure of the network model implemented, the maximum number of channels associated to a chamber is six. This way, each matrix line Y possess in the maximum seven non null values, since the main diagonal is associated to the nodes. This value is function of the coordination numbers distribution of the network chambers, essentially determined by the existence or not of horizontal channels. According to the second rule, as the elements yij and y ji are equal, Y is a symmetric matrix and positively definite (Suchomel, 1998a and 1998b). This property of Y is important to the selection of the numerical method used for the solution of the system of equations, in particular when there the number of chambers is large. The vector q

s

of dimension N Can represents the influence of the flow and tension

sources, and can be determined using the expression (Desoer and Kuhn, 1969)

q s = AGΔP s

(68)

The equation system defined by the previous equation allows the determination of the potential values in the network nodes, corresponding to the absolute pressures in the s

chambers interior, know as the conductance matrix, G and the vector q . The problem of the flow description through the network chambers and channels is reduced to calculus of the admittance matrix, Y . In the calculus of this matrix one needs to determine the conductance values and of the tension sources associated to each network branch, which is only possible if the geometric characteristics of the network were known, in particular the distributions of the network elements characteristic dimensions.

Modeling of the Network Elements Hydrodynamic In this work it is considered that the network conductance value is the sumo of two terms, representing the action of the viscous forces on the channels walls and the influence of the expansions and contractions between the chambers and channels, in the form F

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Rj = Rj + Rj where R j

F

E

(69)

represents the channel resistance due to the friction on the channel wall and R j

E

represents the expansions and contractions resistance of the channels due to the fluid entrance and exit from the chambers.

Channels Resistance For the channels, the flow resistance is due to the flow friction on the channels wall, and can be written as follows F

ΔPF = R j q j

(70)

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António A. Martins, Paulo E. Laranjeira, Carlos Henrique Braga et al.

where the resistance R j

F

is function of the flow conditions in the channel. The term R j

F

can be written by the following general expression (Lim and Ti, 1998) F

Rj =

8 ρf j l j q j

(71)

π 2d j5

where f j is the friction factor for the flow through the channel. This parameter is function of the Reynolds number in the channels, Re j , defined by the expression

Re j =

ρ v j d j 4 ρq j = μ πμd j

(72)

where v j is the average velocity of the fluid in network channel j . Since the channels are cylindrical, it is possible to define distinct zones for the flow, where it is needed to define different expressions for f j (see for example, Shames, 1982). If Re j < 2300 , the flow is on laminar regimen and f j is given by

fj =

64 Re j

(73)

Substituting the expression above in the Equation 71, one obtain F

Rj =

128μl j

(74)

πd j 4

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Thus, the flow resistance in the network channels for laminar regimen is only function of the geometric characteristics. Other expressions for the calculus of f j on different geometries for laminar regimen can be found in White (1992) and Patzek and Silin (2001). If Re j > 5000 , the flow regimen in the channel is turbulent, being R j

F

a linear

function of q j . Assuming that the channels wall is smooth, the Blasius equation was used

fj =

0.3164 0.25 Re j

(75)

For the transition zone between laminar and turbulent regimens, normally considered to occur for values of Re j between 2300 and 5000, there are no general expressions that allow the

Modeling of Transport Phenomena in Porous Media Using Network Models

221

calculation of f j , due to the large dispersion of available experimental values (Shames, 1982). However, to model the flow in a network of chambers and channels when the value of Re j fall within that range, there is the need for a continuous expression of f j as a function of Re j . Two strategies were considered. In the first one, f j is approximated by linear interpolation between the friction factor values for the superior and inferior extremities of the laminar flow zones, Re j = 2300 and turbulent, Re j = 5000 , respectively. This approximation assures the continuity of f j in the extremes of the transition zone, but not of first derivative of the function f j . Depending on the algorithm used for the equation system resolution, the existence of this type of discontinuities may bring numerical problems, especially when there are many network channels where the flow conditions are close to the limits between the two flow regimens (Bending and Hutchison, 1973). Another way to determine f j was developed in order to guarantee the function continuity and of the first derivative in the limits of the transition zone, using Tchebyshev polynomials (Conte and De Boor, 1984). In this case the limit values of Re j were

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simultaneously determined with the polynomials coefficients. Both approximations are compared in Figure 19. It is observed that both curves are similar, occurring the larger differences in the limits of the definition of the linear fitting. The transition zone defined by the Tchebyshev polynomials is larger than the linear approximation, reducing especially the laminar regimen zone.

Figure 19. Comparison of the

f j curves as a function of Re j in the transition zone for both forms of

fitting proposed.

The hydrodynamic simulator allows the utilization of any approximation, being preferred the second one as the function f j Re j is continuous and has continuous first derivatives in

( )

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António A. Martins, Paulo E. Laranjeira, Carlos Henrique Braga et al.

all the values range of Reynolds numbers. However, the linear approximation simplifies the analytical description of the flow, as one it will be further described in this paper.

Chambers Resistance E

For the chambers, the resistance to the flow, R j , is associated to connections between chambers and channels. For practical questions, this term is associated to the network branches, as shown to Figure 20. Since normally there are two chambers associated to a E

branch, one needs to sum the contributions from each side to determine R j .

Figure 20. Schematic representation of the resistance of the branch and electric analog utilized.

The value of R j E

Rj = Rj

EL

E

+ Rj

The first term, R j term, R j

ET

is the sum of two contributions ET

EL

(76)

, represents the contribution of the viscous strengths and the second

, represents the contribution of the inertial effects.

For the first term, the results Koplik (1982) were used. This author solved analytically the creeping flow problem for a chamber/channel geometry similar to the one used in this work, demonstrating that R j

EL

can be calculated with a small error assuming that the channel j

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has an additional length extension in the chamber interior equal to 2d j

π . Substituting in

Equation 74, and taking into consideration that each branch has two chambers on its extremes, R j

Rj

EL

=

EL

can be expressed in the form

512μ π 2d j 3

The second term, R j

(77)

ET

, represents the inertial effects contribution and their value can be

determined analyzing the chamber flow, in particular in the connections between the chambers and channels. Theoretically it is possible to describe the flow solving the mass and momentum conservation equations written for the network chambers and channels. In this

Modeling of Transport Phenomena in Porous Media Using Network Models

223

work a simpler strategy is considered, based on the calculation procedure normally used to determine the pressure drop due to accidents in pipes and flow systems (Shames, 1982; White, 1992), the K j method. In the R j

ET

determination one needs to take into consideration that there are two

accidents associated to each network branch with distinct hydrodynamic behaviors. Despite the fact that K j is a function of the network flow and geometric characteristics at local level, it is assumed a constant value for the expansions and contractions. This way, R j

ET

is

calculated using the following expression

Rj

ET

=

8 ρq j

π 2d j 4

2

8ρq j

k =1

π 2d j 4

k ∑Kj =

2

∑Kj

(78)

k =1

k

where the parameters K j account for the influence of the flow expansions and contractions. k

Although there are in literature expressions that allows one to estimate K j as a function of the network elements diameters associated among them, these are only valid for cases involving only a chamber and a channel. Thus, these expressions are not applicable to the network structure considered in this work, and ∑ K j is then determined by fitting the predicted values by the hydrodynamic simulation with experimental data obtained in literature, making it possible to directly include in the flow simulation the network structure.

Network Branch Resistance The total resistance of a network branch, R j , is obtained by summing the different terms of flow resistance, obtaining the following general expression as a function of the channel diameter. F

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Rj = Rj + Rj where

∑K j

EL

+ Rj

ET

=

8 ρf j q j l j

π 2d j5

+

512 μ 8 ρq j ∑ K j + π 2d j 3 π 2d j 4

(79)

is the sum of the constant K j associated to a certain network branch. The first

term represents the influence of the flow in the channels and the last two terms the influence of the connections between chambers and channels, composed by two terms in order to distinguish the influence of the viscous and inertial forces. For laminar flow in the channels the previous equation can be expressed in the form F

Rj = Rj + Rj

EL

+ Rj

ET

=

128μ (πl j + 4d j ) + 8ρq j2∑4K j 4 2 π dj π dj

(80)

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António A. Martins, Paulo E. Laranjeira, Carlos Henrique Braga et al.

Assuming that the chambers influence on the flow is negligible, only taking into consideration the friction influence of the flow in the channels wall, the previous expression can be simplified to the following expression F

Rj = Rj =

128μl j

(81)

πd j 4

that is equivalent to the Poiseuille equation for the flow in circular tubes. The model predictions with or without the resistance terms associated to the chambers will be further compared, in order to identify the relevant flow effects.

System Solution The resolution of the equation system defined in the matricial form by the Equation 66 allows one to obtain the pressure values at the network nodes. The most adequate strategy to solve the equation system depends essentially on the nature of the system and on the model selected for the calculus of R j , in particular when the flow conditions are non linear. In order to obtain the solution more efficiently, it is important to take into consideration their properties. Independently of the conditions imposed to the network generator and to the hydrodynamic simulator, the equation system is symmetric and sparse. Each line of the coefficient matrix Y possesses in the maximum seven non zero elements, this situation occurring for a network with horizontal channels and periodic boundaries. Since the characteristic dimensions of the network elements are described by probability density functions, the results obtained by the hydrodynamic simulator are statistically valid if the networks have a large number of elements. As for the networks with a large number of elements almost all the Y elements are zero, the algorithm being implemented should take into consideration this aspect, in order to avoid doing an excessive number of meaningless (Pruess et al., 1992). The nature of the system of equation system being solved depends on the selected model to the calculus of R j and on the conditions imposed to the hydrodynamic simulator, in particular the limit conditions between the several flow regimens. If the R j values are Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

independent from q j , characteristic of the laminar flow regimen, the system of equations is linear. Suchomel et al. (1998a) showed that the system coefficient matrix, behind being symmetric and sparse, also it is positively defined, which makes it possible to use efficient interactive methods for the resolution of the linear equation system. In this case it was used the DSRIS routine from the scientific library ESSL (IBM), that uses the generalized steeptest descent method that takes into consideration the symmetric nature of the coefficient matrix. In the case the inertial terms are significant, the equation system is non linear, being solved using a fixed point method. Since the non-linear resistance terms, R j

ET

, are a function of

q j , it is need to define an initial estimation of the flow rates values in the network channels. Since it is possible in practice to know if the equation system is linear or non-linear from the imposed value of ΔPT , of the network geometric characteristics and of the selected model for

Modeling of Transport Phenomena in Porous Media Using Network Models

225

the R j calculus, the algorithm assumes for the first iteration that the flow is in laminar regimen in all the network elements. This way the R j

ET

values are equal to zero and the

equation system is linear, making it possible to use the usual methods to solve it. The algorithm stops if the flow is laminar in all the network channels, or when the following convergent criterion is verified k k −1 ∑ (q j − q j )

N Can j =1

2

∑ (q j

N Can j =1

)

k 2

< ε tol

where k refers to the iteration number and

(82)

ε tol is the maximum tolerance imposed to the

iterative process, being this value imposed by the user. A similar algorithm was used by Sorbie et al. (1989) for modeling non-Newtonians fluid flow modeling in purely viscous regimen through the porous media, using a bi-dimensional network only composed by channels. In this problem, the non-linear terms also occur in laminar regimen, since the fluid viscosity is a function of the flow characteristics in the channels. As an initial estimative, Sorbie et al. (1989) used the flow profile for Newtonian fluids in laminar regimen, correcting successively the viscosity values according to the flow characteristics in the channels. The performance of the proposed algorithm was satisfactory in all the cases analyzed by these authors. Sahimi (1993) and Wang et al. (1999a) also analyzed the flow of non linear fluid through a porous medium utilizing network models, and solved the non-linear equation system by the Newton-Raphson method. Shah and Yortsos (1995) used the successive over-relaxation, to the flow modeling of non-Newtonian fluids through a porous medium. This methods does not need to have a good estimation as the Newton-Raphson method requires and it is more robust.

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

Special Case of a Regular Network A regular network is a network that possess the following characteristics: • •

the chamber and channeld diameter, and the channel length, take only one value; the coordination number is equal for all network, implying that the boundaries have to be periodic.

In a network with those properties, it can be shown that the hydrodynamic behavior can be characterized analytically (Martins, 2006). The relation between the total pressure, ΔPT , and flow rate, qT , through the network can be expressed in the form N x +1

ΔPT = ∑ Δp j = ( N x + 1)Δp j = ( N x + 1)R j j =1

qT 2N y

(83)

226

António A. Martins, Paulo E. Laranjeira, Carlos Henrique Braga et al. Using the expressions proposed before for R j , its is possible to determine either ΔPT or

qT , depending if qT or ΔPT are known. A list of expressions that can be used to determine either ΔPT or qT can be found in Martins (2006). From those expressions analytical expressions for k and the constants A e B of the MacDonald equation can be also easily obtained. Details of their determination and their analysis are given in Martins (2006). For example, the expressions for the network permeability not considering the volume of spherical caps and assuming that the network is large are the following

k≈

k≈

[(l + D )cosθ ]2 d 2ε

π

16 ⎡ 2 D 3 ⎤ + 2l (sin θ + 1) + (2 sin θ − 1)D ⎥ (πl + 4d ) ⎢ 2 ⎣3 d ⎦

π

[(l + D )cosθ ]2 d 2ε

32 ⎛ 1 D 3 ⎞ ⎜⎜ + l ⎟⎟(πl + 4d ) 2 ⎝3 d ⎠

, C =6

(84)

, C=4

(85)

From the previous expressions it can be concluded that the permeability depends not only on the characteristic dimensions of the network elements, but also on the geometrical structure through θ . The permeability has the expected dependency on the value of d (Dullien, 1992; Guyon, 2005), but, keeping d constant and varying D a distinct behavior can be observed. Considering the geometrical model and assuming C = 4 , the permeability of a packed bed as a function of the permeability is given by the expression

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

k≈

π

[(K l + K D ) cos θ ]2 K d 2ε

32 ⎛ 1 K D ⎞ ⎟(πK l + 4 K D ) ⎜ K + l ⎟ ⎜3 K 2 d ⎠ ⎝ 3

DP

2

(86)

2

The previous expression predicts that permeability is a function of DP , as expected (Dullien, 1992). For a uniform network with C = 6 similar expressions are obtained, and the same dependence of k is observed.

Data Treatment The main results of the network are the pressure and flow field. The values of the permeability can be obtained using the following expression

Modeling of Transport Phenomena in Porous Media Using Network Models

k=

qT μL X q μL = T X AN ΔPT LY E R ΔPT

227

(87)

where qT is the total flow through the network, calculated summing the flow rates in the channels exiting the network. Note that in the previous equation a value of the equivalent thickness is defined, resulting from the fact the permeability is defined as a function of the superficial velocity in the network. The constants A e B can be determined using the *

*

representation of the in terms of F vs Re , and knowing that the MacDonald equation can be represented in the form (see part A for more information)

F* =

A +B Re*

(88) *

When the flow is linear, corresponding to low values of Re , the value of A is given by the *

*

ordinate in the origin of the curve of F as a function of Re , a linear relation in logarithmic coordinates with a slope equal to -1. The value of the constant B is determined in non linear *

flow regimen, for high values of Re , where the curve F constant value equal to B .

*

*

as a function of Re tends a

Sensitivity Analysis Analysis of the Resistance Terms Relative Importance One of the key aspects of fluid model proposed in this work is the explicit calculation of the resistance terms associated with the interconnections between the chambers and channels. Although physically more detailed, it is not clear which terms are relevant for which flow regimen. Thus, in Figure 21 the curves of F connections and without connections, where

*

*

as a function of Re

∑Kj = 0,

thus R

ET j

are given with

= 0 , and R EL = 0. j

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

Also, the MacDonald equation is represented in the figure to determine, from a qualitatively pointo of view, which model better describes qualitatively the behavior observed in practice. *

The results show that for low values of Re , corresponding to linear regimen and dominance of the viscous forces, similar behavior is observed with or without entrance effects. As the ET

value of R j

EL

is small when compared with other remaining terms, and R j

F

and R j predict

*

similar behavior for this range of Re values. The differences observed between the curves EL

F

are due to the inclusion or no of R j , a term small when compared with R j . *

For high values of Re large differences can be observed between the predictions made with or without the entrance effects. In particular, the transition is smooth and starts in the range of modified Reynolds numbers between 10 and 100. The results are also on qualitatively agreement with the MacDonald Equation. When the entrance effects are not considered, the transition occurs for much higher Reynolds numbers and do not show the

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António A. Martins, Paulo E. Laranjeira, Carlos Henrique Braga et al.

smooth behavior of the MacDonald equation. Thus, it can be concluded here that only when all the resistance terms are considered is the model able to describe all the regimens observed for flow in a packed bed, in particular the transition zone.

Figure 21. Plot of and channels.

F*

as a function of

Re* , with and without the entrance effects between chambers

Network Size Effects The network elements characteristic sizes follow given statistical distributions. Thus, it is necessary to determine how large the network have to be to ensure that the results are statistical significant. From a practical point of view, the larger the network the smaller the variance associated with the predicted values, since the larger the samples obtained from the distributions the better. In Figure 22 the values of permeability are presented as a function of N y , for three values of N x . Networks with horizontal channels and periodic boundaries, d D = 0.5 with

D = 0.004 and l 0 = 0.007 fixed (all size dimensions are in meters), and θ = π 4 were Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

considered. Two sets of values of standard deviation are considered:

σ D = σ d = 0.05 and

σ D = σ d = 0.40 . For each set of values of N x and N y twelve simulations were performed varying the seed of the random number generator. In both cases it is observed that if N x ≥ 100 and N y ≥ 100 the values of permeability tend to a limit value. Although the error bars are larger when the values of the standard deviation are high, in both cases they diminish with increasing values of N x and N y . Thus, for networks that verifies the criteria given above it can be assumed that the results obtained are statistical significant.

Modeling of Transport Phenomena in Porous Media Using Network Models

Figure 22. Permeability and a function of

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

and

Ny

for three values of

Nx

and two set of values of

229

σD

σ d : a) σ D = σ d = 0.05 ; and b) σ D = σ d = 0.05 .

Identical behavior was observed independently of the value of C , type of boundary and flow regimen, thus showing that the main factor controlling the statistical significance of the results is the total number of elements in the network. For the remainder of this article, all results dealing with the flow modeling considered 100×100 networks (first the value of N x followed by N y ), d D = 0.5 with D = 0.004 and l = 0.007 . If different values are used in the text they are explicitly stated in the text.

Influence of the Network Characteristics Of the various network parameters and conditions that can be imposed, a qualitatively analysis reach the conclusion that the main factors controlling the network behavior is the distribution of coordination number, C i . The remaining factors, type of boundaries and value of

θ are not significant. For statistical significant networks, varying the type of networks

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António A. Martins, Paulo E. Laranjeira, Carlos Henrique Braga et al.

does not change appreciably the total number of elements, the key factor behind the network behavior, thus not changing significantly the predictions of the model. For the second factor, neither term of resistance is function of θ , thus, no change in the flow behavior is observed. Two forms of varying the coordination number were considered: by the inclusion or no horizontal, or by randomly removing channels. In this last form, two variants were considered: removing only horizontal channels or removing horizontal and oblique channels regardless or their characteristics. Both processes have different impacts when analyzing the local structure that results from the removal process. In particular, removing only horizontal channels maintains the structure of the oblique channels, the elements controlling the flow field, situation not observed in the other form of removal. To better assess the relative impact of both forms, the best way is to compare the values of permeability predicted by the model for networks with the same value of C but obtained using the two forms. To show the different effects of the removal process, the values of the ratio kC k6 as a function of C are presented in Figure 23, for σ D = σ d = 0.40 . In any case, a network with C = 6 and periodic network boundaries was the starting point for the removal process. The comparison between the results shows that the removal process have a profound impact in the behavior predicted by the model. In particular, removing only horizontal channels leads to an increasing in k , whereas the removal of any kind of channel reduces k . When any kind of channel can be removed, including oblique channels, the number of paths available to the fluid will be reduced. Hence, the global flow resistance will increase, leading to lower values of the permeability. When only horizontal channels are removed, the structure of the oblique channels is maintained. Thus, the main effect controlling the permeability value is the variation of void volume, necessary to calculate the value of the equivalent network thickness. For a uniform network the flow rate in the channels is null, and it can be shown that the ratio k C k 6 can be expressed in the form (see Martins, 2006, for more details) 6

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

k C VV = ≈ k 6 VV C

2 D 3 + 6ld 2 + 3l h d 2 ⎞ ⎛C 2 D 3 + 6ld 2 + 3⎜ − 2 ⎟l h d 2 ⎠ ⎝2

(89)

where l h is length of a horizontal channel in a uniform network. The previous expression predicts that there is a linear relation between k and C in a uniform network, in agreement with the results presented in Figure 23.

Modeling of Transport Phenomena in Porous Media Using Network Models

231

Figure 23. Permeability ratio for the cases of removal of channels or only horizontal channels.

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Influence of the Elements Characteristics The network elements are characterized by their characteristic size distribution. Since the resistance to the flow is a function of the channel diameter, it can be predict that it will be that parameter that will have a more profound impact on the hydrodynamic model predictions. To show that this is correct in Figure 24 the values of permeability are presented as a function of the ratio d D for the cases where D or d is kept constant. Note that due to the restrictions imposed in the network generator, this ratio an vary only from 0 to 1. Two sets of values of standard deviation are considered: σ D = σ d = 0.05 and σ D = σ d = 0.40 . The values obtained assuming that the network is uniform are also present, to assess if that approximation is a valid one. The results show good agreement between the permeability values obtained for the uniform network and using low values of the standard deviation. Thus, it can be concluded that the results are adequate for network with narrow distributions of the network elements characteristic dimensions. For larger distributions, the deviations increase the larger the value of d is. This is a direct result of the restrictions imposed between the values of D and d at the local level, that when d ≈ D will lead to a move of the distribution of values of d to lower values, resulting in a higher resistance to the flow than expected. In Figure 25 the values of permeability as a function of d are presented for

d D = 0.25 or D constant and equal to 0.004. Networks with the same geometrical characteristics as those used in the previous figures were also considered here. The predictions of the uniform network are also presented, confirming the results the preceding conclusions that this model is valid if the values of σ D and σ d are small.

232

António A. Martins, Paulo E. Laranjeira, Carlos Henrique Braga et al. 10

d cons tan t

k 1

D cons tan t 0.1

Uniform

σ d = σ D = 0.05

0.01

σ d = σ D = 0.40 0.001 0

0.2

0.4

Figure 24. Permeability values as a function of

0.6

d D

keeping

0.8

D

or

1

d D

d

constant.

When d D is kept constant, the permeability is a function of d

2

for all range of values.

However, when D is constant, it is possible to observe two different zones. For low values of

d , the permeability is proportional to d 3 , difference from the dependence on d 2 observed for high values of d . 10

σ d = σ D = 0.05

k

σ d = σ D = 0.40

1

Uniform 0.1

d D = 0.25

0.01

D cons tan t

0.001 Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

0.0001

Figure 25. Permeability values as a function of

d 0.01

0.001

d

for

d D = 0.25

or

D

constant.

Comparison with Experimental Data The validity of the flow model is ultimately assessed comparing its predictions against experimental data. Because the geometrical model will be used simultaneously, the main restrictions of that model have to be considered. Thus only data obtained for packed beds made up by nearly sized spheres should be used.

Modeling of Transport Phenomena in Porous Media Using Network Models

233

The experimental data of Kim (1985) was selected because it verifys the conditions of applicability of the geometrical and includes the linear and non linear regimens of flow. In Figure 26 the predictions of the hydrodynamic model are compared with experimental data, varying C and ∑ K j . The results show that only when C = 4 is the model capable of describing the behavior of a real packed bed. This fact aggress with the experimental results of Sederman et al (1997, 1998) and Baldwin et al. (1996), that obtained experimentally using NMR values of C between 4 and 5. Using other experimental techniques, Yanuka et al. (1986) obtained similar values. For C = 4 , the agreement is good for all values of considered, being better when

∑Kj

∑ K j ≈ 3 . For the data presented, the best fitting occurs for

∑ K j = 3.48 . 100

F* C = 6, ∑ K j = 1.2

10

KIM 2

∑Kj =4 C =4 ∑Kj =3

∑Kj =2

1 1

10

100

Re* 1000

Figure 26. Comparison between the predictions of the hydrodynamic model and the experimental data

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

of Kim (1985), as a function of

C

and

∑Kj .

In Martins (2006) more comparisons between experimental data and the predictions of the model can be found, all confirming the previous conclusions.

MASS TRANSPORT According to the model proposed in this work, the transport of mass in the network is based on the description of the local structure, performed by the network of elements coupled with the geometrical model, and of flow field inside the network. Although in some situations the determination of the flow field and the dispersion of mass are linked together, as for example when a perturbation with a high concentration of solute is imposed at the fluid entrance or a exothermic reaction occurs in the media, here its is assumed that the flow and

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António A. Martins, Paulo E. Laranjeira, Carlos Henrique Braga et al.

the transport of mass do not interfere with each other, allowing the decoupling the system of balance equations. This condition is valid in the tracer limit, where the solute concentrations are low everywhere in the porous medium and do not change appreciably the physical properties of the fluid, or when the solutes particles are indistinguishable from the fluid particles. In the next sub-sections it will described and analyzed in detail how the mass transport in the network elements can be modeled, the algorithm implemented to solve the mass balance equations, and how the network geometrical and flow field characteristics influence the dispersion of mass. At this stage only a qualitatively analysis will be performed. In future publication the predictions of the model will be compared with experimental data.

Model Description As stated before, the transport of mass inside the network elements will rely on the results obtained by the network/geometrical and flow models describe above in this article. Following the main assumption of this model, the simulation of the dispersion of mass can be done independently, though it must be done after the determination of the flow field. In brief, the model that will be described aims to determine the behavior of the network model, and consequently of a porous medium, from the response obtained to a perturbation in the concentration fed to the network.

General Model Using the hypothesis made before in the flow modeling, it can be concluded that the chambers behave as perfectly mixers and the channels as plug flow units. Thus, the channels will be responsible for the mixing of the solute, corresponding to dispersion, and the channels will be responsible for the transport of mass, corresponding to convection. So, different mechanisms of mass are responsible by different mass transport mechanisms. Base on the behavior imposed for the network elements, and assuming no chemical reaction or interfacial mass transfer, the dimensionless mass balance equations for the chamber i and channel j can be written in the form

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

τi

τj

df i (t ) + f i (t ) = ∑ α ij E f j (1, t ) dt j ∂f j ( z , t ) ∂t

+

∂f j (z , t ) ∂z

=0

(90)

(91)

where t represents time; z is the axial coordinate in the channels normalized by the channel length l j ; f i and f j are the dimensionless concentrations inside the channels and chambers respectively;

α ij E is the total fraction of flow that enters chamber i through channel j , and

Modeling of Transport Phenomena in Porous Media Using Network Models

235

τ i and τ j are the passage times through the chambers and channels, respectively. The dimensionless concentrations and parameters are defined by the following expressions

f i (t ) =

C i (t ) < C 0 ( y ,t ) >

C 0 ( y, t ) f 0 ( y, t ) = < C0 ( y, t ) > where

∑ qs

f j (t ) = lj V τi = i , τ j = vj ∑ qs

C j ( z ,t )

< C 0 ( y ,t ) >

α ij = E

qj

(92)

E

∑q

(93) s

is the sum of the flow rates exiting the chamber, determined from the global

mass balance; Vi is the volume of the chambers; v j is the fluid velocity inside the channels;

qj

E

is the flow rate in the chamber i and channel j ; and < C 0 ( y , t ) > is a reference

concentration that renders the chambers and channels concentrations dimensionless. The correct value of < C 0 ( y , t ) > depends on the characteristics of the perturbation imposed. In some cases its value its value is self evident, as for example in a spatial uniform step, where < C 0 ( y , t ) > = C 0 , being C 0 the concentration limit value. In other cases, < C 0 ( y , t ) > may be equal to an average of the solute concentration entering the network, although the correct definition may vary depending on the situation. Considering the overall mass balance written for a chamber it can be shown that E E ∑ α ij = 1 . The distribution of values of α ij

is a measure of the influence of the local

flow field in the dispersion of mass. The wider the distribution the more important the dispersion of mass will be. The following set of initial and boundary conditions must be used to solve the system of mass balance equations

t = 0, z > 0 ⇒ f i (t ) = 0 , f j ( z , t ) = 0 t > 0 ⇒ f j (0, t ) = f 0 ( y , t )

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E

(94) (95)

where y is the spatial direction normal to the main direction of the flow. The first equation implies that when the concentration perturbation is applied the network has no solute. The second condition is the definition of the dimensionless concentration perturbation that it is imposed at the network entrance. The function f 0 ( y , t ) can be a function of time, space, or both, and can be defined in many different forms, ranging from the simple, such as step or pulse uniform perturbations, or more complex such as spatially non uniform or random perturbations.

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António A. Martins, Paulo E. Laranjeira, Carlos Henrique Braga et al.

Algorithm Implemented to Solve the System of Mass Balance Equations The solution of the general system of equations can only be accomplished if their mathematical properties are accounted for explicitly, both when solving them as well in the definition of the convergence criteria. A key aspect that has to be considered when solving the system of differential equations is the need to keep the concentration for previous times. From a practical point of view, the concentration that enters a given chamber at time t through channel j is the concentration that was observed in the chamber or outside the network, depending on the relative position of the chamber on the network, but for a time t − τ j . In other words, the channels act as pure delays, and incorporating it on the mass transport equations the following general expression is obtained

τi

df i (t ) E + f i (t ) = ∑ α ij f ij (t − τ j ) dt j

(

(96)

)

where f ij t − τ j represent the concentration in the chamber i on the inlet of channel j for

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a time t − τ j . The system of equations now corresponds to a system of delay differential equations, where the solution depends on the all solution history and not only on the initial and boundary conditions. (Haier et al., 1987). They arise and are important to describe the behavior of systems and processes which behavior depends on their evolution in time. Examples include the study of population dynamics (Bocahrov and Rihan, 2000), epidemiology (Nelson and Pereelson, 2002), control systems (Ramirez and Puebla, 1999), among others. Reviews of applications and numerical methods available to solve this type of differential equations can be found in the works of Baker and co-workers (1994 and 1995). The solution of the system of modified mass balance equations was done using a RungeKutta method of fourth order. Some key aspects have to be taken into account when solving it. First a time increment Δt must be defined. Because our goal is to describe as accurate as possible the behavior of all network elements, Δt is defined as a percentage of the minimum values of the passage time distributions of both channels and chambers. These distributions can be obtained directly from the geometrical and flow field characteristics of the network. This procedure ensures that the behavior of all the network elements is properly considered. Second, when solving the mass balance equation for chamber i , it is necessary to know how f i (t ) changes in time. From the general mass balance, Equation 96, it is clear that it is

[

only need to know the history in the interval t , t − τ

max j

], where τ

max j

represents the

maximum value of the flow time passage in the channels j that enter the chamber i . In

practice

this

is

done

dividing

the

channels

in

N INC j

points,

where

N INC = Int (τ j Δt ) . In each time step, the values of f i (t ) are calculated first, based on the j

values of f i (t − Δt ) . After, the values of f j ( z , t ) are updated, starting from the entrance to

the exit of the fluid in the channel j . This updating process is analogous to the traveling of

Modeling of Transport Phenomena in Porous Media Using Network Models

237

the concentration wave between two spatial positions inside the channels, and is akin to a particle tracking method, here applied instead to the concentration of solute. Thus the classification of this methodology as a mix between particle tracking and mass balance models. Instead of following the evolution of a cloud of particles, one follows the spatial and time evolution of the local concentration field, after imposing a concentration perturbation at the fluid entrance. Other key aspect is the convergence criteria. In contrast with the flow modeling, here it is not possible to define unambiguous stop criteria. In this work a combination of criteria was considered, depending the possible combinations on the characteristics of the perturbation imposed at the network entrance. Whenever one of the criteria is met, the simulation stops. The general stop criteria, independent of the perturbation properties, is to define a maximum simulation time, proportional to the network passage time, τ G . This ensures that the simulations came to an end, although an adequate time depends on the geometrical and flow characteristics of the network. For perturbations that tend to a constant value for long times (t >> τ G ) , such as uniform step, the simulation stops if the overall mass balance is verified within a small error. In this work this criterion was implemented only for the uniform step and pulse perturbations. Also for those types of perturbations, a third stop criteria was defined based in the expected limit criteria. For example, for a uniform step it is expected that the limit exit concentration will approach the step concentration. When the entrance and exit by a sufficiently small amount, it can be considered that the simulation reached the steady state and can be stopped. For all simulations performed a combination of criteria was always used.

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Strategies to Reduce the Effort Needed to Simulate Mass Transport The general algorithm presented in the previous sub-section is capable of describing the transport of mass inside a network of elements regardless its geometrical characteristics. However, in some situations the effort needed to solve the equations and obtaining the concentration profile can be prohibitive, thus the interest in developing strategies to reduce the overall effort. Three forms were considered in this work. The first considers network with horizontal channels or wide distributions of the network elements size. In these networks, the velocity of the fluid in the channels will also have a wide distribution, situation that will lead to prohibitively small values of Δt . Noting that channels with low values of v j correspond low values of

α ij E , from Equation 96 it can be

concluded that the influence is significant only if t >> τ G . Also, as it was concluded in the flow modeling that the best agreement was obtained for C = 4 (networks without normal channels), a similar situation was considered to be valid for the modeling of the mass transport in the network. Therefore, only network without horizontal channels were considered in this work. The second strategy can be applied for perturbations that tend to a constant value for long times, and only to the chambers. It stems directly from the characteristics of those kind og perturbations, where f i (t ) will always tend to the limit concentration defined by the perturbation given enough time. When this situation is reached, solving the mass balance

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António A. Martins, Paulo E. Laranjeira, Carlos Henrique Braga et al.

equation for chamber i is meaningless. Thus, if the difference between the values of f i (t ) in consecutive times is smaller than a given tolerance, the chamber is considered inactive and the concentration in it equal to the limit value defined by the perturbation. This can reduce significantly the number of differential equations that have to be solved simultaneously. On the other hand, in the starting moments of the simulation it is not need to solve all mass balance equations, simply because the perturbation did not reach them. So, using the passage time distributions, it is possible to know at each time step the maximum possible distance traveled by the perturbation, and solve only the chambers that were already reached by it.

Treatment of the Results As stated before, a run of the mass transport simulator is equivalent as simulating a tracer experiment. The breakthrough curve that is obtained should be processed to obtain the values of the relevant parameters. In this study three main parameters were defined: the Peclet number, Pe , the longitudinal dispersion coefficient, D L , and the normalized breakthrough time, Θ B . The parameter Θ B is the ratio between the minimum time necessary for the perturbation to reach the network exit and the overall passage time of network. The value of this parameter can be obtained directly from the transit time distributions of the channels, dependent only of the geometry and flow fields inside the network. Θ B values vary in the range between 0 and 1. Values close to 1 correspond to the situation where convection is dominant, values close to 0 correspond to a dominance of dispersion. The value of Pe is obtained matching the second statistical moment of the network response with the expression predicted by the DM for closed-closed boundary conditions. In this work this was done using the normalized residence time distribution, E (Θ ) , where Θ is the normalized time defined by Θ = t

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F (t ) =

q js f js (t )

∑ q js

E (t ) =

τ G , and using the following expressions

1 dF E (Θ ) = E (t ) dt τG

(97)

where F (t ) is the dimensionless response to a step perturbation; and f js (t ) and q js are the outlet concentration and flow rates, respectively, in the fluid exit channels of the network. The value of Pe is calculated using the next expression Θ*

σ = ∫ (Θ − 1)2 E (Θ )dt σ 2 = 2

0

(

)

v L 2 1 − 1 − e − Pe Pe = T x 2 DL Pe Pe

(98)

Modeling of Transport Phenomena in Porous Media Using Network Models

239

Special Case of a Regular Network In the flow modeling it was shown that a regular network is a special case where an analytical solution of the flow field is possible. The same situation occurs in the modeling of mass transport. For a uniform network, the flow rate in the normal channels is equal to zero and equal for all oblique channels. Thus, the time to reach a chamber in a line of the network is independent of the particle path through the network, and depends only in the conditions imposed the network entrance. Thus, to be able to obtain an analytical solution it must be also imposed that the concentration imposed has to be spatially uniform. If those conditions are met, the response of the network model and a model composed by channels and chambers in series is equivalent. To the analogy be correct it is necessary to define a correct passage time for the chambers,

τ i* , that takes into account that in the network

two channels goes to a given chamber, in the analogue only one. Thus,

τ i* is defined by the

expression

τ i* =

Vi 2q j

(99)

where Vi is the volume of the chamber and q j is the flow rate in the channels of the uniform network. Assuming that there is no chemical reaction and mass transfer between phases, the system is linear and the G (s ) can be expressed in the form Nx

⎛ 1 ⎞ ⎟ exp − ( N x + 1)τ j s G (s ) = ⎜⎜ * ⎟ 1 τ + s i ⎠ ⎝

[

]

(100)

The residence time distribution E (t ) is obtained inverting G (s ) . Using the previous

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equation the expression for E (Θ ) can be written in the form

⎧0 Θ < ΘB ⎪ N x −1 E (Θ) = ⎨ N x [Θ − Θ B ] exp[-β (Θ − Θ B )] Θ ≥ Θ B ⎪β (N x −1)! ⎩ where

(101)

β = τ G τ i* and Θ B are the normalized breakthrough times.

The values of the main parameters can be determined easily for the simplified model. For

Θ B the following expressions can be written

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António A. Martins, Paulo E. Laranjeira, Carlos Henrique Braga et al.

ΘB = 1 −

1 N +1τ j 1+ x N x τ i*

The ratio τ

j

(102)

τ i* in a uniform network is equivalent to a volume ration between channels

and chambers. Expressing that ratio as a function of the geometrical characteristics of the network elements, the following general expression for Θ B is obtained

(103)

1

ΘB = 1− 1+

⎧ 2 Nx ⎡ N x + 1 ⎪ 3d 2 l ⎛ d ⎞ ⎤⎥ ⎢ − − − 1 1 ⎜ ⎟ ⎨ Nx +1⎢ N x ⎪ D3 ⎝ D ⎠ ⎥⎦ ⎣ ⎩

In most cases the expression

2

2 ⎫ ⎡ d ⎤ ⎢1 + 0.5 1 − ⎛⎜ ⎞⎟ ⎥ ⎪⎬ ⎢ ⎝ D ⎠ ⎥⎦ ⎪ ⎣ ⎭

τ j τ i* ≈ 3d 2 l D 3 is a good approximation to the time

passage ratio, resulting in the following approximation to Θ B

ΘB ≈ 1 −

1 N + 1 3d 2l 1+ x N x D3

(104)

From the previous results an dimensionless group,

γ , can be defined to characterize the

relative importance of the transport of mass by convection and dispersion in a uniform network, in the form

γ=

Nx +1τ j N x τ i*

(105)

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From the definition of this dimensionless group, it can be concluded that when the value of is large when convection dominates the transport of mass. For low values of the controlling mechanism. Also, as

γ , dispersion is

γ is comparable to the ratio d D , the same conclusions

also hold for that parameter. The Pe value can be estimated solving the following non-linear equation

1 Nx

⎛ 1 ⎜⎜ ⎝1+ γ

γ

2

⎞ 2 2 ⎟⎟ = − 2 [1 − exp(− Pe )] Pe Pe ⎠

(106)

Modeling of Transport Phenomena in Porous Media Using Network Models

241

If Pe > 5 , the exponential term is not significant and an explicit expression for Pe can be obtained in the form

⎡ Pe ≈ N x (1 + γ )⎢1 + γ + ⎣

(1 + γ )2 −

1 ⎤ ⎥ Nx ⎦

(107)

For high values of N x the previous expression can be simplified to Pe ≈ 2 N x (1 + γ ) . This 2

expression is similar to the relation predicted using the tanks in series model, but includes an

additional factor equal to (1 + γ ) , that accounts for the existence of two different types of 2

elements in the equivalent network. The previous expression also shows that a linear dependence exists between Pe and the length of the network in the main direction, showing that the model should be applied mainly when convection is the dominant mass transport mechanism. Figure 27 presents the curves of E (Θ ) predicted for an uniform network as a function of

γ , for two values of N x = 10 and N x = 30 .

Figure 27. Curves of

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b)

E (Θ ) predicted for an uniform network as a function of γ

, for: a)

N x = 10 ;

N x = 30 . The results show that the larger the value of

γ the more pronounced is the Gaussian

character of the response, in the sense that they become increasingly more symmetric around Θ = 1 , and the dispersion of values is lower. Thus, it is possible to conclude from here that an increase in the value of γ leads to an increase in the relative importance of convection. For equal values of

γ , the increase of N x also leads to the same situation. From the

expressions obtained above for an uniform, this also corresponds to larger values of Pe , in agreement with the conclusions of the tanks in series model. A uniform network is also a limit case regarding the influence of the flow field in the dispersion of mass. In particular, due to the regularity of the flow field in a uniform network, when compared with a real network, it will have the lowest value of dispersion and

242

António A. Martins, Paulo E. Laranjeira, Carlos Henrique Braga et al.

consequently of D L . Therefore, the ratio between the values of Pe and Θ B obtained by the mass transport simulator and predicted using a uniform network, Pe and Θ B , is a measure *

*

of the influence of the flow field on dispersion, as it will be shown below.

Sensitivity Analysis

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As in the sensitivity analysis performed for hydrodynamic model, a similar procedure will be performed for the mass transport simulator. However, there are some important differences between the two situations that have to be considered explicitly. One of the most important is related to the network dimensions that are used in some simulations, in particular when the number of elements in the network is large. Although the results may not be statistical significant, since the nature of the response depends on the size of network in the main direction of the flow, as shown in the analysis of an uniform network, in some case there is a need to use small networks. Other relevant deals with the type of perturbation that is more adequate. From a practical point of view and because is also easier to implement, uniform step perturbations were used to obtain most of the results presented in this work. While the results of a impulse perturbation are easier to process, its definition at the network entrance is not easy and not instantaneous, since the model does not permit to imposed perturbation with a duration smaller than a time step. Also, following the previous conclusions on the relative importance of oblique and horizontal channels on the transport of mass, only networks without horizontal channels were used in the simulations. This restriction agrees also with the results of the flow model, that showed that the best agreement with agreement occurs for C ≈ 4 . For the solution algorithm, whenever possible the strategies implemented to reduce the total effort needed to obtain the response of the network, ensuring that the results are meaningful. Some representative results are presented in the next sub-sections of the capabilities and main results already obtained by the model. In future publications more results will be presented.

Algorithm Validation Two main aspects need to be considered when assessing if the algorithm is valid, at least qualitatively, to model mass transport in the network: the ability to model different types of networks, and what are the algorithm parameters, in particular the value Δt that ensures that the results are valid. As an example of the types of perturbations that be dealt by the model, in Figure 28 four snapshots of the spatial and temporal evolution of the concentration field for a punctual step perturbation applied at the network entrance. A network 30×10 without horizontal channels and periodic boundaries, and σ D = σ d = 0.20 and d D = 0.5 was considered. It can be observed that the width of the solute plume increase with the distance traveled by the fluid in the network. Also, the concentration values decreases, as a result of the mixing of the fluid with and without fluid. For long times, the concentration profile reaches a steady state

Modeling of Transport Phenomena in Porous Media Using Network Models

243

situation, and does not change in time. This behavior is similar to what is observed experimentally (Yun et al., 1998; Ganganis et al., 2005). The simulator was tested using other types of pertubations (Martins, 2006). In each case, the behaviour predicted and the results are the expected, showing that the model implemented is quite flexible and can deal with many different types of perturbations easily.

Figure 28. Snapshots of the spatial and temporal evolution of the concenCurves of an uniform network as a function of

γ

E (Θ ) predicted for

, for: a) N x = 10 ; b) N x = 30 .

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Concerning the parameter Δt , it is important here the definition of a criteria that ensures that the results are significant. Qualitatively, the Δt the smaller the overall error is. However, the computational effort increases, and a trade-off must be reached between accuracy and computational effort. In Figure 29 the error in the overall mass balance are presented for networks 20×30, without horizontal channels and periodic boundaries, considering σ D = σ d = 0.05 . Three values of d D used to assess if the relative importance of convection or dispersion has an influence on Δt . The results show that all curves have the same qualitatively, showing that the value of d D is not important when selecting Δt . For all cases, if Δt is equal and lower than 2% of the minimum value of the chambers and channels transit times, the error made on the mass balance is always lower that 1%. This criteria is independent of the geometrical and the flow characteristics of the network. Yet, in many situations a larger value of Δt can be used, reducing the computational effort.

244

António A. Martins, Paulo E. Laranjeira, Carlos Henrique Braga et al. 0.11 d D = 0.25

Erro (%)

0.08

d D = 0.50

0.05 d D = 0.75

0.02

-0.01 0

0.2

0.4

Figure 29. Error on the material balance for as a function of

0.6

0.8

Δt

for three values of

1 Δt

1

d D.

Influence of Network Characteristics Since no simulations were performed for networks with horizontal channels or with the removal of channels and/or chambers, according to the results obtained for the uniform network it is N x the only parameter that influences significantly the transport of mass. Figure 30 presents the curves E (Θ ) obtained by the simulator for different values of

N x , for networks with N y = 30 , d D = 0.5 and σ D = σ d = 0.20 . It is observed that the

larger the value of N x , the more Gaussian like are the curves of E (Θ ) , in accordance with the conclusions obtained for a uniform network. The deviations observed for small values of N x correspond to networks where the lateral mixing of solute is still significant and the concentration is still developing, thus the more importance of the dispersion of mass. To gauge the importance of the lateral mixing and the development of the concentration *

profile, in Figure 31 the values of Pe as a function of N x are presented for several values Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

of N x , with periodic boundaries and using

σ D = σ d = 0.20 . When the value of Pe* tends

to a fixed value, it can considered that the relative importance of dispersion and convection reached a limit value. Note that a uniform network is a limit case where the dispersion of mass is the smallest possible for a given network. For all cases it can be concluded that if N x ≥ 100 a limit value is reached. This result also implies that a limit value of D L is also reached is that criteria is met, in qualitatively agreement with experimental data available in literature (Han et al., 1985). When N x is low, meaning that the network is small in the main *

direction of flow, Pe can be larger than 1, confirming the previous conclusions that dispersion and the lateral mixing of solute are very significan in the first part of the network.

Modeling of Transport Phenomena in Porous Media Using Network Models

245

10.0 E (Θ ) 8.0

Nx 150 100

6.0

50 30

4.0

20 10

2.0 0.0 0.7

Figure 30. Curves of and

0.8

0.9

1.0

1.1

E (Θ ) for different values of N x

1.2

1.3

for networks with

1.4

Θ

1.5

N y = 30 , d D = 0.5

σ D = σ d = 0.20 . 1.3 Ny=10

Pe*

Ny=20 Ny=30

1.1

Ny=50

0.9

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0.7 0 Figure 30. Values of

Pe *

50 as a function of

Nx

100 for various values of

Nx

150

Ny .

The influence of N y is only relevant for small networks. Comparing the results obtained for different N y the same behavior is observed, this confirming that the main factor is the network dimension in the main direction of flow.

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António A. Martins, Paulo E. Laranjeira, Carlos Henrique Braga et al.

Influence of Flow Field The results of the hydrodynamic model have shown that depending on the flow regimen the behavior observed is different. So, it is convenient to judge if for the mass transport the characteristics of the flow field also have an impact on the predictions of the mass transport model. Figure 31 presents the predicted values of Pe e Θ B as a function Re for networks *

with 20×20, d D = 0.5 ,

*

*

σ D = σ d = 0.20 , and with periodic boundaries. To determine if

any possible changes are the result of changes in the flow regimen, the values of FV (defined as FV = F * Re ) as a function of Re are also presented. The results show two limit zones *

*

where Pe e Θ B are constant, that correspond to the zones of low and high Re . In the *

*

*

transition regimen, both Pe e Θ B increase, showing that changes in flow field, in this case *

*

the transition between linear to fully developed non linear flow have an impact on the transport of mass. The increase of Pe e Θ B indicates that the velocity distribution becomes *

*

more narrow, leading to less dispersion and an increase importance of the mass transport by convection. 1.0

10000

*

Pe

Θ*B

Θ*B 0.9

FV

1000

0.8

0.7

Pe* 0.6 Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

0.1

Figure 31. Values of

1.0

10.0

100.0

1000.0

Re *

100 10000.0

FV , Pe* e Θ*B em função de Re* .

Influence of the Elements Characteristics The main parameters associated with the network elements that influence the mass transport are the ratio d D and the values of the standard deviation. Concerning d D , as concluded above, the higher its value the more significant is the importance of convection. Also, increasing the values of either σ D or σ d will lead to a wider distribution of fluid velocities, due to the larger distribution of flow resistance in the network elements, leading to more dispersion.

Modeling of Transport Phenomena in Porous Media Using Network Models

247

To confirm these conclusions, in Figure 32 the values of Pe e Θ B as a function d D are compared with each other. Networks 100×30 with periodic boundary were used, keeping d = 0.002 in all cases, and σ D = σ d = 0.05 and σ D = σ d = 0.20 . As expected, the values of Pe e Θ B predicted for high values of the standard deviation are smaller, as concluded before. In the figure it can be also observed that Θ B tends to a limit for high values of d D . Therefore, even when d D will tend to one, situation where the channels and the chambers are indistinguishable, there will be dispersion due to the mixing in the nodes. Although the same behavior is not observed for the Pe values, a similar situation should occur for sufficiently large values of d D .

100000

1 ΘB

σD =σd 0.05

Pe

0.20

0.8

Pe 10000

ΘB

0.6 0.4

1000

0.2 0

100 0.1

0.3

0.5

Figure 32. Comparison between the values of

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of

σD

and

Pe

and

ΘB

0.7

d D

as a function of

d D

0.9 to two set of values

σd .

CONCLUSION Because this article is divided in two main parts, the same division will be considered in this section of conclusions for sake of simplicity. In part A it was given a thorough description and analysis of the various network models proposed in literature to describe the geometrical structure of real porous medium, the hydrodynamic behavior and the transport of mass. The focus was not in practical applications, but on the different methodologies and strategies available. From the analysis of the various works it can be concluded that network models are a good option when describing and

248

António A. Martins, Paulo E. Laranjeira, Carlos Henrique Braga et al.

modeling transport phenomena in a porous media. Although for most cases they are a simplified approximation of a porous medium local structure, their bottom-up approach (from the microscopic to macroscopic scales) produces good results and especially insight of the key aspects controlling transport phenomena in porous media. In part B a bi-dimensional model consisting of two types of elements interconnected with each other is presented that was designed to be used for packed beds. A geometrical model was presented relating the main parameters that characterize a packing, in particular the porosity and the average particle diameter, the network elements size distributions. The prediction of the mode agrees well with experimental data for packed beds formed by spheres with a narrow size distribution. The network model as used to model the fluid flow based on a analogy with a purely resistive electrical circuit. To be able to model all possible flow regimens in single-phase flow, from laminar to turbulent, the effects of the interconnections between the chambers and channels have to be taken into account explicitly. The model results show that the key factors controlling the hydrodynamic behavior of the network are the channels size distribution and the spatial distributions of the oblique channels. For a uniform network it was shown that an analytical solution of the flow is possible, that gives good predictions of the main flow parameters, such as the permeability, when the network elements size distributions are narrow. The comparison between predicted and experimental data was adequate for C = 4 and K > 1.5 . Based on the network/geometrical and flow models the transport of mass was modeled in the network. The behavior of the network is described by a system of delay differential equations, solved by a algorithm similar to a Particle Tracking method. Special care was taken to reduce the computational effort and to ensure that the results are physically significant. The model can handle a wide variety of concentration perturbations. The results show that relative importance of dispersion and convection is a function of d D , and the

D L tends to asymptotic value, function of the total distance traveled by the fluid inside the

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network (Han et al., 1985). Also, an influence of the flow regimen was observed in the predicted values of Pe and θ B , an indication of changes in the flow field when passing from linear to non linear flow regimens. Further work includes the optimization of the calculation algorithm, specially the way in which the time delays due to the channels are considered, the study of the effect of different types of networks, namely networks with different connectivities. The influence of chemical reaction will be also considered.

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In: Progress in Porous Media Research Editors: Kong Shuo Tian and He-Jing Shu

ISBN: 978-1-60692-435-8 © 2009 Nova Science Publishers, Inc.

Chapter 6

ADVANCES IN INTEGRATED MODELING OF MASS TRANSPORT AND GEO-MECHANICS IN COAL SEAMS FOR CO2 GEO-SEQUESTRATION F. Y. Wang, Z. H. Zhu, P. Massarotto and V. Rudolph Division of Chemical Engineering, School of Engineering The University of Queensland, Brisbane, Qld. 4072, Australia

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ABSTRACT This article addresses recent advances in the integrated modeling strategies of mass transfer and geo-mechanics in porous media with special applications to CO2 sequestration in coal seams. CO2 sequestration in coal seams is a relatively new technique to simultaneously achieve enhanced coal bed methane (ECBM) production and reduced CO2 emission. A methodology is developed in this article that integrates understandings in disparate research fields providing improved insight into the complex nature of the process. Our current overall model, constructed from a number of sub-models, consists of mass transfer in four pore types, namely, fractures, macro-, meso- and micro-pores, all having pore size dependent characteristics. Furthermore, a number of geo-mechanical models represented by matrix equations with different levels of complexity are developed and incorporated into the overall mass transfer model. Effects of adsorption and external stress induced pore size changes on mass transfer operations are analyzed in detail. The proposed modeling strategy is of a multi-scale nature with a variety of time and size scales. The macroscopic level model is validated using a true tri-axial stress coal permeameter (TTSCP), which provides accurate dynamic measurements of systems properties in three orthogonal directions including changes to the coal matrix volume. The integrated model provides a more complete and flexible representation for the simulation of coupled geo-mechanics and mass transfer in coal seams. Practical model order reduction techniques are proposed to permit fast prediction of process dynamics with acceptable accuracy for particular industrial applications. In particular, a hybrid distributed-lumped parameter model with significantly reduced model complexity has been applied to the depressurization processes for the accelerated recovery of methane residual from coal seams, with good agreement between simulations and experimental data. Finally, three mechanisms are proposed for coal swelling, which is important because it leads to permeability variations during sorption. These are based on molecular

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F. Y. Wang, Z. H. Zhu, P. Massarotto and V. Rudolph dynamics (MD) or quantum mechanics (QM) simulations. Due to the similarity between coals and other porous media, we believe the proposed integrated modeling methodology can also be applied to other similar media and processes using porous materials.

NOTATION a

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a1-an a to i A A b bi-bn B B Ci Cp C D D e E E f fr g h I IB Jμ_i Jμ K kads kdes L mi m M M Ni N NC pb

aperture of slit fractures, m; ratio of activation energy to adsorption energy, dimensionless parameters in ARX model empirical parameters in Equation (4) cross section area of coal specimen, m2 collocation matrix, dimensionless parameter in Langmuir or Langmuir-Freundlich isotherm, m3/mol parameters in ARX model macro- and meso-pore permeability, m2 Stefan-Maxwell matrix, s/m2 gas phase concentration for species i, mol/m3 compressibility factor. 1/MPa vector of gas phase concentration, mol/m3 diffusivity, m2/s diffusivity matrix, m2/s error in ARX model adsorption energy, J/mol Young’s modulus, MPa probability density function fractional uptake, dimensionless gravitational acceleration, m/s2 cleat spacing, m identity matrix inverse of matrix B, m2/s molar flux in micro-pores for species i, mol/(m2 s) vector of molar flux in micro-pores, mol/(m2 s) permeability, m2 bulk modulus, MPa adsorption rate constant, s-1/(N/m2) desorption rate constant, s-1/(N/m2) length of coal specimen, m block-to-fracture mass cross flow rate for species i, mol/(m2 s) vector of block-to-fracture mass cross flow rate, mol/(m2 s) molecular weight, kg/mol mass matrix for matrix representation of differential algebraic equations molar flux for species i, mol/(m2 s) vector of molar flux, mol/(m2 s) number of components, dimensionless bulk pressure, N/m2

Advances in Integrated Modeling of Mass Transport and Geo-mechanics… solvation pressure, N/m2 total pressure, N/m2 partial pressure for species i, N/m2 vector of partial pressure, N/m2 volumetric gas flow rate, m3/s function defined in Equation (18), m/s ideal gas constant, J/(K mol) change of adsorbed mass, kg/m3 time variable, s temperature, K input variable in ARX model relative volume change per unit volume of specimen, m3/m3 gas velocity, m/s half-width of slit pore, m spatial coordinates, m mole fraction, dimensionless; output variable in ARX model spatial coordinates, m compression factor, dimensionless

ps P Pi P Q Qi R ΔS t T u ΔU V w x y z Z

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Greek Symbols α β γ ε ζ η λ Λ μ ν Π ρ τ τq ψ

parameter in Archie’s law, dimensionless; volumetric swelling coefficient, m3/kg adsorption affinity of adsorbent, (K kg/mol)1/2/ (mol/m3) ratio of surface diffusivity to self diffusivity at high temperature, dimensionless porosity, dimensionless dimensionless spatial variable defined in [0, 1] parameter in Langmuir-Freundlich isotherm, dimensionless half distance between two fractures, m diagonal matrix defined by Equation (23), s/m2 viscosity, (N s)/m2 Poisson’s ratio, dimensionless pressure acting between micro-pore walls, N/m2 gas density, kg/m3 tortuosity, dimensionless time constant in Equation (88), s-1 correction parameter in Darcy’s law, dimensionless

Subscripts B c d

butt cleat coal matrix; convection; critical value diffusion

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F. Y. Wang, Z. H. Zhu, P. Massarotto and V. Rudolph e f F hy i ij K m ma me r t V μ μs

effective fracture face cleat hypothetic species i binary pair i-j Knudsen diffusion mean value; gas mixture; coal matrix macro-pore meso-pore reduced property total value vertical direction micro-pore adsorption saturation

Superscript E-F effective medium theory with the smooth field approximation s geometric factor

Mathematical Symbols 〈•〉 arithmetic average 〈•〉e effective average

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INTRODUCTION A porous material is one which contains pores or voids embedded in the solid matrix. Fluids may or may not be able to penetrate a particular porous material. In this article, only permeable porous materials which allow fluids to pass through are considered. A wide variety of porous materials are encountered in all areas of nature, including living and inanimate systems, and over a broad range of scales. We confine our attention here to natural geological porous media, and most pertinently, to coal. The important processes that occur in porous media operations even from this limited perspective include complex fluid flow, adsorption/desorption, molecular and surface diffusions, material deformation, and pore structure evolution. Studies on porous media involve a very broad range of scientific areas including material synthesis and characterization, geo- and fracture- mechanics, fluid flow and transport phenomena, thermodynamics of porous media, mathematical modelling and simulation, process control and system optimization. As with most other complex systems, research methodology can be divided into two interconnected categories, namely fundamental studies for the exploration of physical insights based on rigorous theoretical analysis, and

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practical approaches for the development of empirical correlations using experimental data. In practice, these two methods are normally combined together to develop the so-called “greybox models”. Many working models used in industries can be classified as grey-box models. In this article, we address recent advances in integrated modeling of mass transfer and geo-mechanics in porous media, with special focus on CO2 sequestration in coal seams. Most materials on the modeling of mass transfer processes presented in this article are based on recent research outcomes published by the authors in international journals, but with significant modifications to incorporate the most recent advances in the field. The geomechanical models in matrix representations together with the integration of mass transfer with geo-mechanics present new research results.

Mass Transfer in Coal Seams for CO2 Sequestration

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CO2 sequestration in coal seams is a relatively new technique to simultaneously achieve enhanced coal bed methane (ECBM) production and reduced CO2 emission. In this process, shown schematically in figure 1, industrially generated CO2 is injected into relatively deep coals to replace adsorbed methane (CH4). Coals generally display a much higher adsorption affinity for CO2 than CH4. The significance and current status of this research area have been comprehensively reviewed in two recent review papers (Tsotsis et al., 2004; White et al., 2005). In any particular practical case, the feasibility, economics and risk evaluation require and depend upon the determination of a number of operational conditions. Both theoretical analysis and experimental investigation are essential to the development of effective ECBM strategies. A multidisciplinary approach is required, covering a number of challenging scientific areas such as multi-component transport in porous media, coal characterization, geophysics and geochemistry. In spite of an extensive literature in relevant areas in recent years, significant gaps in knowledge can still be identified due to the complexity and broad scientific coverage required of the field. These are identified as follows. 1. The multi-scale nature, which ranges from molecular interactions in the length scale of nanometers and time scale of micro seconds all the way to coal seam operations with lengths of kilometers and performance analyzed in hours, days, and even weeks. Significant research advances have been achieved in both microscopic level using molecular simulations, and macroscopic level studies through commercialized software packages, for coal bed methane (CBM) simulations. However, coupling these two scales remains elusive as evidenced by the fact that the recent conceptual advances achieved from fundamental studies are scarcely incorporated into the large scale simulators. For example, it can be shown that neglecting micro-pore size distributions in CBM models leads to considerable errors. It is necessary to develop adequate transition procedures to bridge the micro- and macro-scale formulations. 2. A process can be described by many models with different complexity and accuracy. We adopt the position of Box and Draper (1987) that: “Essentially, all models are wrong, but some are useful.” They also emphasized that: “… all models are wrong; the practical question is how wrong do they have to be to not be useful.” A model could be useful for a particular application, but may fail for other purposes. A study

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F. Y. Wang, Z. H. Zhu, P. Massarotto and V. Rudolph on model hierarchy could be an effective strategy for model form selections based on applications. Unfortunately, no systematic study on the hierarchical modeling of ECBM processes has been reported in the literature. Guidelines for model form selections for specified end-uses are yet to be developed. 3. A significant number of modeling papers relied on the experimental data reported by other researchers for model validation and parameter estimation. Because of the complexity of coal characteristics, and possible inconsistency between experimental and simulation conditions, the validation and estimation results may not be reliable. It is highly desirable to integrate the theoretical study, numerical simulations, and experimental investigations into a unified research program. 4. Numerical values of important coal parameters reported in some papers have lost their original physical significance even though outcomes with acceptable accuracies were obtained, e.g., Clarkson and Bustin, 1999; Shi and Durucan, 2003. For example, both pore and surface diffusivities estimated in these papers by back-fitting, are many orders smaller than values computed from known thermodynamic relationships. There are two main reasons for the dramatic deviations between different estimation methods: (1) model structures cannot adequately represent the real physical systems, leading to error accumulation in the estimated parameters; and (2) estimation methods based on macroscopic measurements are insufficient for the unique determination of multiple parameters associated with different mechanisms. The effective pore and surface diffusivities estimated using the overall measurement data have been called “pseudo diffusivities” by Yang (1997), since they are many orders of magnitude different from thermodynamically derived diffusivities. Modified modeling strategies should incorporate computations based on the fundamental theory where appropriate, and be able to provide convincing physical explanations about discrepancies. 5. Although adsorption induced dimensional changes (swelling and shrinkage) have been investigated extensively (Jakubov and Mainwaring, 2002; Ustinov and Do, 2006), the dynamic evolutions of coal structures in CO2 sequestration processes have not been fully studied. An important consequence of coal swelling/shrinkage during sorption processes is the permeability change, which has attracted considerable research attention. However, models developed for the prediction of this phenomenon normally assume constant geophysical properties, such as constant volumetric swelling coefficient and Young’s modulus, and negligible effects on adsorption and diffusion processes. There is strong evidence to show that geophysical properties of coal change dynamically and nonlinearly. For example, the overall volumetric swelling coefficient for a core block, including matrix and fractures, does not change linearly with the amount of gas adsorbed. As more gas is adsorbed, the strain seems to reduce. Also the Young’s modulus of the fractures (Ef) goes larger as more gas adsorbed due to the fracture porosity reduction. Consequently, some key parameters must be identified dynamically. This involves dynamic optimization, which has not been reported previously.

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Figure 1. Schematic Diagram of CO2 Sequestration in Coal Seams.

There are two popular dynamic modelling strategies for the representation of transport phenomena in porous media, namely, the pore network approach and the particulate approach. In the particulate approach, the solid material is represented as small uniformly sized spheres. These are aggregated into uniform macrospheres, which are in turn assembled into a unit cell for the diffusion-adsorption analysis. Since the system provides dual porosity and diffusivity within and between the macro-spheres, the model is named the bi-disperse model. This technique was originally developed for spherical catalysts and adsorbents with quite restrictive mathematical assumptions, including negligible pressure gradient and isotropic properties of the adsorbents (Yang, 1997). It has been extended to coal analysis in recent years (Clarkson and Busting, 1999; Shi and Durucan, 2003; Cui et al., 2004). Because of the limitations of the bi-disperse model, described by Yang (1997), and the complexity of coal structures, its extension to coal has resulted in the loss of physical significance of several key parameters, such as pore and surface diffusivities. The pore network approach, reviewed by Tsotsis et al. (2004), involves fluid transport through the macro-pore and cleat fracture networks, diffusion through the meso-pore and micro-pore regions of the coal matrix, adsorption of CO2 onto and desorption of CH4 from the surface of the micro-pores. The fractures (that is, macro-pores, fractures and cleats) and the coal matrix itself (ie meso- and micro-pores and solid material) represent separate classes of transport paths in the sense that each has its own distinct characteristics. In spite of scientific advances and the practical significance of the two-path pore network approach (Gilman and Beckie, 2000), it can be demonstrated through numerical simulations that the mass transfer mechanisms in each path are much more complicated than that incorporated in the conventional models. For example, there is no justification for the assumptions used in the development of the models regarding negligible diffusive mass transfer in macro-pores, or no convective mass transfer in meso-pores. Furthermore, there are four experimentally distinguishable pore types in coals that have a role in dynamic structural changes associated with the pressurized adsorption/desorption processes. The combination of macro-pores with cleats leads to significant error. These issues should be adequately addressed in modelling studies.

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F. Y. Wang, Z. H. Zhu, P. Massarotto and V. Rudolph

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There are advantages and disadvantages associated with both strategies. A comparative study between two approaches will certainly lead to valuable insights. Methods to relax the restrictive assumptions imposed on the particulate approach in order to broaden its application areas is also an important research topic. These may include the development of three dimensional particulate models and studies on multi-sized particle packing. The processes of CO2 sequestration in coal seams can be depicted by a spectrum of representations ranging between a very simple black-box model based on input-output data only, and a highly complex mechanistic model consisting of a large scale partial differential equation (PDE) system with parameters estimated through molecular simulations. Intermediate models, denoted as grey-box models, may be developed between these two extremes to form a hierarchical tree. The selection of model forms depends on the application requirements. It is impractical at this time to develop a pure mechanistic model due to incomplete knowledge in a variety of relevant fields. Attempts have been made by the authors to fill in part some of the identified gaps in knowledge through the development of an integrated modeling approach to the dynamics of mass transfer in porous media (Wang et al., 2007a), followed by model simplification for fast estimation of complex dynamic processes (Wang et al., 2007b). That is, emphasis is first placed on the development of a mechanistic model based on some physical understanding of the process, followed by model order reduction for industrial applications. Our modeling strategy covers the following areas. 1. Develop an overall model structure incorporating: convective flow in cleats associated with aperture and permeability computations, convective and diffusive flows in macro- and meso-pores, adsorption/desorption and surface diffusion in micro-pores. Micro-pore size distribution and pore network correlations are incorporated into the overall model. 2. Estimate key parameters using thermodynamic relationships, molecular simulations, and reliable correlations. On-line identification techniques using measurement data are only used for the estimation of a minimum number of parameters, which cannot be directly or independently measured, or computed from theory. 3. Validate the macroscopic level model using a true tri-axial stress coal permeameter (TTSCP), which provides accurate dynamic measurements of gas flow-rates, compositions, dimensional variations of specimens, temperatures and pressures in three orthogonal directions. 4. Study the mechanisms of coal swelling leading to dynamic permeability changes during adsorption through molecular simulations, which quantitatively show the adsorption induced bond length and pore aperture extensions. 5. Investigate the effects of pore size distributions on fluid flow and adsorption behavior. This is particularly important for the mass transfer in micro-pores. 6. Reduce model complexities using proper mathematical approximations to provide simplified but fast estimation of process dynamics, for industrial applications. In the integrated overall model, the Stefan-Maxwell analysis (Krishna and Wesselingh, 1997) and computations of viscous flow (Jackson, 1977) are applied to the fluid phase inside meso and macro-pores (Wang and Bhatia, 2001) within the dusty-gas modelling framework. In recent years, studies on Stefan-Maxwell equations for the adsorption and diffusion of pure components and binary mixtures in zeolites and carbon nanotubes have been carried out using

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molecular dynamics (MD) simulations with promising results (Krishna and van Baten, 2005; 2006). The limitations of the reported applications of Stefan-Maxwell analysis to the micropore mass transfer are identified as: (1) The materials under study consist of well defined micro-pore structures with thoroughly characterised surface roughness; and (2) Studies have been restricted to pure components and binary mixtures. Because of the complexity of the coal structures, the first limitation becomes a real hurdle for the extension of the reported MD simulations to coal research. This implies that the validity of Stefan-Maxwell analysis applied to multi-component surface diffusion in micro-pores within the coal matrix is yet to be justified. We adopt the technique suggested by Wang and Do (1999) based on the concept of hypothetical concentration to address the surface diffusion for multi-component systems. The pore networks for meso-pores are characterised by using the effective medium theory with the smooth field approximation (EMT-SFA) procedure. The pore size distributions in micro- and meso-pores are incorporated into the mathematical models and computer algorithms to replace the conventionally used methods based on mean pore sizes. The adsorption and desorption kinetics are investigated using experimental data obtained in our laboratories with new representations in the matrix format ready for the development of compact computer codes based on matrix operations. The physical properties are computed dynamically within the integrated computational algorithm. Compared with the models reported in the literature, the newly developed model possesses the following advantages.

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1. The integrated model, consisting of four pore types with six fluxes along each axis, is more complete than conventional dual or triple porosity models typically dealing with three fluxes. 2. The model is more general due to the elimination of several restrictive assumptions used in the development of current conventional models. The eliminated assumptions include those requiring negligible pressure drop, isotropic and homogeneous conditions for the development of bi-disperse models (Yang, 1997), single mass flux and unique pore size in each pore type for the development of both bi-disperse and pore network models (Tsotsis et al., 2004; Yang, 1997). 3. The model is more flexible due to dynamic computation of key model parameters using validated methods. These include gas phase diffusivities and viscosities, and the permeability of cleat, macro- and meso-pores. The remaining shortcomings of the integrated model developed in this work can be identified as follows. 1. It is difficult to measure directly the required physical properties: surface diffusivities, micro-pore size distribution, pore-size dependent adsorption energies, and concentrations in the coal matrix, with consistent results. 2. The overall algorithm involves the solution of partial differential-integral equations with symbolic differentiation operations, which is more complicated than the numerical schemes for solving the conventional models. Furthermore, numerical oscillations induced by step changes in boundary conditions, known as the Gibbs phenomenon, need to be accommodated.

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F. Y. Wang, Z. H. Zhu, P. Massarotto and V. Rudolph

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3. Since deep coal seams are often below the water table, the single phase fluid flow addressed in this paper needs to be extended to include water flow to provide a more general result for ECBM operations. Because of the complexity of ECBM operations, a staged development of a model is inevitable in order to make the problem tractable. The model and framework outlined here permits incorporation of scientific advances or additional complexity, as these become available. The integrated model developed by the authors provides certain physical insights and process details. However, some model outputs may not be necessary for particular applications. The model complexity and long computing time also restrict its applications. In many cases, simplified models are sufficient to the purpose. In this article, a simplified dynamic model for accelerated methane residual recovery from coals developed by the authors previously is described to demonstrate model order reduction strategies. Effective recovery of residual methane from coal seams after major extraction operations is an important but difficult task due to the small driving forces and low mass transfer coefficients. An obvious option to achieve nearly complete methane recovery is the depressurization technique. We develop here a model for the prediction of mass transfer dynamics between coal matrix and fractures under depressurization conditions. The fluid flow in fractures is modeled using Darcy’s law described by partial differential equations, whereas the variations of gas and adsorbed phase concentrations within the coal matrix are represented by a lumped parameter model represented by ordinary differential equations, which are originally described by partial differential equations in the integrated model. Due to the replacement of the distributed parameter model by a lumped parameter model for mass transfer within coal, significant model order reduction is achieved. Although this simplification leads to certain loss of physical insights, it is reasonably accurate for industrial applications and has the advantage of significantly reduced solution time. In particular, the relative importance of the convective and diffusive flows can be readily quantified through numerical simulations using the simplified model, which is essential to the design of depressurization operations. It is our contention that richer physical insights are provided by the complicated model, providing a basis for conceptual advances, whereas the faster estimation using the simplified model is more suitable for industrial applications

Anisotropic Geo-Mechanical Properties of Coals for Permeability Computations In order to study fluid flow through coal seams, it is essential to determine the permeability, which is a function of pore size, geometry, connectivity, and tortuosity. These characteristics depend on geo-mechanical properties and vary with external stress, pore pressure and adsorption/desorption processes. Coal is highly compressible and the petrophysical structures of coals are decidedly anisotropic. A systematic study of three dimensional (3D) geo-mechanical properties of coal specimens is necessary for accurate predictions of the performance of coal bed methane (CBM) reservoirs. We adopt the 3D representations in matrix formats widely used in rock mechanics (Goodman, 1989) as our model structure for geo-mechanical studies. As part of the CO2 sequestration project, an

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experimental investigation has also been carried out at The University of Queensland using a number of 80 mm cube specimens tested in the TTSCP, which provides dynamic measurements of system properties in three mutually-orthogonal directions (generally aligned with the bedding, butt and face cleat orientations), including external stresses and bulk coal strain measured using strain gages on each face. Based on the measured stress-strain data, the principal geo-mechanical properties along each orthogonal direction (Young’s modulus, Poisson’s ratio and bulk modulus) can be determined using parameter identification techniques. Experiments were carried out in a broad stress range varying from 0 to 75 MPa. In addition to variations of stress magnitudes, coal specimens were also tested under a variety of other dynamic operational conditions, such as one dimensional stress changes only, and isotropic 3D stress changes. There is considerable difference between the anisotropic 3D data and the uni-axial and isotropic 3D data. Both experimental and computational results have confirmed the severe anisotropic characteristics of coal specimens under study. These characteristics can be readily incorporated into 3rd generation (3G) permeability prediction models for calculating gas and water flow rates. One of the research topics in our current permeability investigations requires the evaluation of Young’s moduli for coal fractures/cleats and matrices, separately. Thus, for predicting permeability changes with changing stresses, we apply the cleat modulus, whereas for predicting permeability changes with desorption-induced strain (which happens in the matrix), we use of a matrix modulus. Previously, these two separate responses were not distinguished and the physical significance of the coal response in terms of the different behaviours of the cleats and the matrix, was poorly recognized as a lumped average. We have derived an expression to link this overall bulk modulus to the moduli representing the different material components via geometric averaging. We report a strategy using the bulk (average of 3D) compressibility-isotropic stress curves to estimate the cleat modulus under conditions of low stress and the matrix modulus when the stress is high. This method can be extended to 3D anisotropic systems by replacing the bulk modulus and volume compressibility in 1D systems by directional Young’s moduli and the three orthogonal linear strains in 3D systems, respectively. Since all of the primary geo-mechanical properties are determined independently, there are no model parameters that need to be extracted from measured permeability. Consequently, measured permeability data can be applied to and reserved for model validation and fine tuning of predictions. Our procedure for geo-mechanics model development can then be summarized as follows. 1. Development of 3D geo-mechanical models based on well established principles in rock mechanics (Goodman, 1987), but with major modifications to incorporate the adsorption/desorption induced dimensional changes. 2. Experimental measurement of 3D stress-strain data and adsorption/desorption induced dimensional changes for the estimation of model parameters including directional Young’s modulus, Poisson’s ratios, and volumetric swelling coefficient. 3. Integration of the mass transfer and geo-mechanical models.

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Model Order Reduction The mechanistically based, rigorous models provide physical insights with high accuracy. However, simulations using these models may currently be viewed as too time consuming for practical applications. In many cases, it is beneficial to simplify the models to balance model complexity and accuracy (Hangos and Cameron, 2001). A number of model order reduction strategies are explained in this article. These include the following strategies. 1. Incorporation of the effective medium theory with the smooth field approximation (EMT-SFA) procedure, to compute the mass transfer in meso-pores with broad poresize distribution. 2. Approximation of a two dimensional (2D) system by two 1D systems. 3. Computation of the overall effective diffusivity to replace individual diffusivities in various pore types. 4. Development of the hybrid distributed-lumped parameter models to replace the pure distributed parameter models. 5. Formulation of black-box models based on experimentally measured data. Illustrative examples related to CO2 sequestration in coal seams are provided for each model order reduction strategy.

EXPERIMENTAL EQUIPMENT AND PROCEDURE FOR CO2 GEO-SEQUESTRATION

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Experimental Equipment A high pressure, true tri-axial stress coal permeameter (TTSCP) is installed at The University of Queensland. The routine maximum operating pressure in the TTSCP is 14 MPa, although much higher pressures can be applied in the equipment. Fourteen MPa replicates underground conditions at about 1400 m depth, well in excess of the critical values of CO2 (Tc = 304.1 K (31oC) and Pc = 7.38 MPa). At 14 MPa, pure CO2 is in the supercritical condition. However, in most practical processes, there are impurities in the CO2 which significantly increase of the critical pressure. Furthermore, many prospective coal seams suitable for CO2 sequestration are relatively shallow and the underground conditions in terms of both temperature and pressure will be in the gas part of the phase diagram. As a prelude to further studies on mixed phase and supercritical CO2 flows, we report here normal single phase gas flow under pressures of 400-5000 kPa. The extension of the method to more general models dealing with multiphase and/or supercritical fluid flows is proceeding. The generalized process flow diagram of the TTSCP is schematically shown in figure 2. A variety of fluids may be made to flow through a cubic specimen which is held under controlled pressure and temperature conditions. The specimen is subject to three independently-set, mutually-orthogonal external compressive stresses. The size range of coal specimens, which can be accommodated by the TTSCP is 40-200 mm side length cubes. The coal specimen size most frequently used by the authors is 80 mm cube cut in general

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alignment with the bedding plane and the cleat directions. Cubic samples have the significant advantage that they can be rotated, permitting the permeability to be measured in the three main directions. The three dimensional strain changes under operational conditions are measured using strain gauges imbedded in the coal specimen. The gas phase composition is accurately measured by employing twin gas chromatographs (GC) operating in tandem. This arrangement provides fast response times (~20s) for gas composition. Detailed design features and measurement techniques are described by Massarotto (2002).

Figure 2. Schematic Diagram of the TTSCP.

Experimental Procedure for CO2 Sequestration A typical experimental set consists of three steps, namely setting up a standard initial condition, in which He flows through the sample for an extended flushing period, replacement of He by CH4, and CO2 flush to replace the CH4. The complete procedure is

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schematically depicted in figure 3. Upon returning to Step 1 after Step 3, the coal specimen can be reused for another set of experiments (eg at different stress conditions).

Figure 3. Experimental Procedure.

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The transient data include flow-rates, pressures and temperatures for inlet and outlet gas streams, outlet gas concentrations, three dimensional external stresses on the coal specimen, and the three dimensional strains. Exit gas concentrations are measured using a GC system. Three strain gauges are embedded in the coal specimen for the determination of strains in three orthogonal directions. The dynamic data are recorded every two minutes by the computer data acquisition system for offline analysis.

COAL SPECIMEN PREPARATION AND SURFACE CHARACTERISATION Preparation includes coal specimen extraction from the coal seam, surface characterisation, mercury porosimetry and helium pycnometry analyses on small subsamples, sorption isotherms, specimen cutting to the desired sizes, and surface polishing. These procedures are described in detail by Massarotto (2002). The most frequently tested specimen size in our laboratory is 80mm (side length) cube, which is also used as the basis for the simulation studies. This 80mm specimen size provides a convenient compromise between

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being large enough to be representative of the coal and cleating properties and the practicalities of sample availability. Typically, 40mm cubes can be cut from standard HQ cores and 80mm cubes can be cut from large cores specially retrieved for our experiments (requiring special drilling, coring and retrieval treatments). We have also tested some 200mm cubes, but these need to be individually mined, a significant logistical problem in underground retrieval, handling, transport and cutting. For a 200mm cube, cut in a specific orientation, a very much larger original block is required. Smaller size samples, e.g. 40mm, are easier to handle, but show a much higher degree of heterogeneity between samples. As an illustrative study on coal permeability, the permeability of an 80 mm cube was first measured in the TTSCP with the result of 5.3 mD. This was then cut into eight 40 mm daughters and the permeability of each of these measured. The surface features and the measured permeability of these sub-cubes are shown in figures 3a and 3b. The following ten cleat characteristics can be obtained from the sub-specimens shown in figures 3a and 3b:

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1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

Cleat frequency (density); Cleat spacing distribution; Cleat aperture distribution; Cleat orientation/truncation; Cleat length and height; Cleat continuity; Cleat connectivity; Cleat filling (with minerals and clay); Cleat tortuosity; Cleat damage in sample preparation processes.

The significance of these characteristics can be analysed as follows using figure 4. The sub-cube F2 with the highest permeability (5.0 mD) had high cleat density, connectivity, continuity, large cleat apertures and low tortuosity. Although the cleat density for the subcube F3 with the second highest permeability (2.6 mD) is low, the connectivity and continuity for both face and butt cleats are high, leading to the effective use of all cleat spaces. One main cleat in sub-cube F4 was filled with silica because a part of the sample fell off in the polishing process and was then glued back. Small sections of the corners or edges on some sub-cubes required repair after cutting, indicated in figure 3. The continuity and connectivity of the remaining cleats in this sub-cube are very low and this leads to the lowest permeability (0.4 mD). The effects of cleat orientation (truncation) on the coal permeability can be clearly demonstrated using sub-cubes R1 and R2 with relatively low permeability (0.8 and 0.5 mD respectively). A number of angular cleats in these two sub-cubes became unavailable for vertical fluid flow because they dead-end against the sides of the specimen holder. The bottleneck effect can be seen in sub-cube R3 with low permeability (0.6 mD), in which several cleats were contracted into a single point on the surface. Sub-cube R4 should possess higher permeability than the measured value (1.0 mD) due to large cleat apertures, low tortuosity, and reasonable values of cleat density and connectivity. We believe this lower than expected permeability was caused by technical problems in the data acquisition system, providing a not entirely reliable data set. Because the larger sample has significantly less cleat damage, truncation induced cleat discontinuity and disconnections, and boundary effects, the mother specimen should possess higher permeability than the individual sub-specimens. Although

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seven out of eight sub-cubes possess much lower permeability than the overall permeability, it can be semi-quantitatively shown that if the effects of cleat damage are removed, and the original continuity and connectivity are recovered, it is reasonable that the permeability of the assembly of the eight sub-cubes reaches the value of 5.3 mD.

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Figure 4a. Front four elements in the 80 mm cube coal specimen.

Figure 4b. Rear four elements in the 80 mm cube coal specimen.

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ADSORPTION ENERGY AND PORE CHARACTERISATION Adsorption Energy Adsorption energies for gases adsorbed on various porous media are normally computed using the Lennard-Jones potential energy equation or its modified formats, such as the Steele potential equation (Steele, 1977). The Steele 10-4-3 potential equation developed for slitshape pores is described as: 10 σ sg10 ⎤ ⎡ σ sg4 σ sg4 ⎤ 5 ⎧⎪ 2 ⎡ σ sg Φ ( z , w) = Φ 0 ⎨ ⎢ + + ⎥−⎢ ⎥ 3 ⎪⎩ 5 ⎣ (w + z )10 (w − z )10 ⎦ ⎣ (w + z )4 (w − z )4 ⎦

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⎤ ⎫⎪ ⎡ σ sg4 σ sg4 −⎢ + 3 3 ⎥⎬ ⎣ 3Δ (0.61Δ + w + z ) 3Δ (0.61Δ + w − z ) ⎦ ⎪⎭ 6 Φ 0 = πρ sε sgσ sg2 Δ 5

(1)

In Equation (1), z is the distance from the central plane of the pore, w is the half width of slit-shaped pores, Φ0 is the minimum interaction energy between a gas molecule and a single lattice layer of the adsorbent, ρs is the number of lattice molecules per unit volume, Δ is the spacing of lattice layers, εsg and σsg are the cross potential well depth and the effective diameter for the adsorbate-adsorbent molecule atoms. These cross parameters are calculated using the Lorentz-Berthelot rules as follows: εsg = (εsεg)1/2, and σsg = (σs+σg)/2. The pair (εs, σs) and (εg, σg) are the Lennard-Jones parameters for a surface atom and a gas molecule, respectively. Figure 5 shows the adsorption energy in slit pores with different widths using the Steele 10-4-3 potential energy equation given by Equation (1). The significance of adsorption energy computations for coal specimens includes the determination of the parameters in adsorption isotherms and the estimation of surface diffusivities which will be explained later. The adsorption energies of N2 and CH4 depicted in figure 5 are very close to that reported by Cui et al. (2004), except for the CO2 profile, which we believe they underestimated. Importantly, for micro-pores in the range of 0.36 < w < 0.46 nm the adsorption energy of CO2 is always much larger than that of CH4. Our laboratory measurements of the pore size distribution using density functional theory (DFT) indicate that a significant amount of micropores do lie in this range.

Pore Size Distributions Porosity in coal is composed of micro-pores (5 Å to 20 Å), meso-pores (20 Å to 500 Å, macro-pores (500 Å - 0.1 μm), and cleats (0.1 μm – 2 mm). The size range of cleats is very broad. The mean cleat aperture in the unstressed coal specimen shown in figure 4 is estimated as 110 μm, which is slightly higher than the most frequently measured cleat size range of 0.1

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– 50 μm reported by Gamson et al. (1993) (suspected stressed samples). Since the Young’s modulus of the cleat is much smaller than that of coal matrix, the cleat aperture reduces rapidly with increased external stress. Consequently, the coal specimen used in our study represents the general situation reasonably well.

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Figure 5. Adsorption Energy in Slit Shape Micro-Pores.

A comprehensive literature review carried out by Massarotto (2002) presents some commonly accepted coal data from the open literature. Generally, the data reported by Levine (1993) and Gamson et al. (1993) are well regarded by other researchers. The relevant points are summarized as follows. The total porosity of a coal is a strong and nonlinear function of the rank of coal measured by carbon content. It may vary from 4% to 18%. For high rank coals, the total porosity could be in the range of 4% - 8%. We adopt the following ratios for various pore types: cleats: 1 – 5%, macro-pores: 10 – 15%, meso-pores: 5%, and micro-pores: 70 – 80% of total pore volume. Pore size distributions for meso-pores in a number of coal specimens have been measured in our laboratory. The distributions vary significantly for different coal specimens. However, the curve shapes of the distribution densities match lognormal distributions well. Consequently, we use lognormal distribution for meso-pores (Wang et al., 2007a). The general representation of the lognormal distribution is given by:

f ( w) =

⎧ − [log(w / m)]2 ⎫ 1 exp ⎨ ⎬ wσ ( 2π )1/ 2 2σ 2 ⎩ ⎭

(2)

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where f is the probability density function, also known as the distribution function, the parameters m and σ are median and standard deviation, respectively. The computed distribution function with σ = 0.6 and m = 65 Å is very close to the measured one for mesopores in a number of coal specimens. The adsorption and surface diffusion characteristics are affected strongly by micro-pore size distributions (Wang and Do, 1999; Wang et al., 2007). Unfortunately, there are no commonly accepted techniques for the accurate determination of micro-pore size distributions. A number of methods have been tested in our laboratory for micro-pore characterisations with a low repeatability due to the complexity of coal specimens. Because of the diversity of the experimental results, the micro-pore size distributions can be represented by lognormal, Γ, or other appropriate distributions based on the available measurement data. The general representation of a Γ-distribution is described as:

f ( w) =

(w / b )c −1 [exp (− w / b )] bΓ ( c )

(3)

∞

Γ( c ) = ∫ exp (− u )u c −1du 0

where b and c are denoted as the scale and shape parameters, respectively. The lognormal and Γ distributions for micro-pores in coal specimens with σ = 0.6, m = 4.5 Å, c = 2 and b = 1 are reasonably close to a number of the measured distributions. Based on the most recent measurement carried out by Bae and Bhatia (2006) for the same coal under our studies, we proposed the following probability density function for micro-pore size distribution with the multi-modal pattern in our simulations (Wang et al., 2007b):

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⎛ ( x − b) 2 ⎞ ⎛ ( x − e) 2 ⎞ ⎛ ( x − h) 2 ⎞ ⎟ ⎜ ⎟ ⎜⎜ − ⎟ f ( x ) = a exp⎜⎜ − d g + − + exp exp ⎜ 2i 2 ⎟⎠ 2 f 2 ⎟⎠ 2c 2 ⎟⎠ ⎝ ⎝ ⎝

(4)

where a-i are fitting parameters. The measured data and the computational result are shown in Figure 6. Both Equations (3) and (4) were used to compute the size dependent parameters in our recent work depending on the available data. It must be pointed out that the development of reliable experimental techniques for the determination of micro-pore size distributions is a very important, yet underdeveloped research area, requiring urgent research attention. For coal, conventional modelling strategies using averaged pore sizes could be acceptable for macro-pores, but lead to significant errors in meso- and micro-pores. The necessity for the study of size effects on masstransport in meso- and micro-pores can be analysed as follows. Since Knudsen diffusion, which is strongly affected by the pore size, is the dominant mass transfer mechanism in meso-pores, accurate computational results cannot be obtained without taking size distribution into account. For mass transport in micro-pores, Figure 5 shows that adsorption energies change considerably with minor variations of pore widths in the half-pore size range from 3 to 5 Å. However, a significant amount of micro-pores are in this sensitive size range as estimated by Equations (2)-(4) and depicted by figure 6, signifying the pore-size effects on mass transport in micro-pores. The importance of micro-pore size distribution will

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be further justified in the Results and Discussion section. Macro-pores and cleats, where adsorption and Knudsen diffusion are negligible, can be characterised in the model development using the averaged pore sizes with acceptably small error.

Figure 6. Micro-pore Size Distribution.

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Pore Network Models In this work, we adopt the pore network approach to the dynamics of fluid transport in coal specimens. There are different definitions for the porosities. For example, the porosity can be defined either based on the total sample volume or based on the remaining volume, which is the total sample volume minus the summation of larger pore volumes. The first definition is popular in the coal literature, whereas the second definition could be better suited to pore network computations. In order to make the definitions consistent with the majority of the publications in the porous media literature, we adopt the following strategies: a. Separate cleats from other three pore types; b. Combine micro-pores with solid materials (Do and Wang, 1998), and c. Exclude larger pore volumes in the determination of the porosity of smaller pores. Consequently, the total porosity does not include the cleat- and micro-porosity, leading to the following definition:

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ε t = ε ma + ε me _ c = ε ma + (1 − ε ma ) ε me

ε me _ c = (1 − ε ma ) ε me

283

(5)

where εt, εma and εme stand for total, macro-, and meso-pore porosities, respectively with the total porosity consisting of macro- and meso-porosities only, and εme_c is the alternative definition of meso-pore porosity adopted in the coal literature. The advantage to combine micro-pores with solid materials is as follows. Three key parameters for micro-pores, namely porosity εmi, tortuosity τmi and surface diffusivity Dpμ, can neither be computed rigorously nor measured directly, and therefore require online identification. If the micro-pores and solid materials are treated together, these three parameters are combined together defining a composite surface diffusivity Dμ represented by Dμ = (εmi Dpμ)/τmi, significantly simplifying the parameter identification problem. Probably, as measurement techniques advance, micropores and solids can be treated separately in the future providing improved accuracy and physical understanding. Archie’s law is used to estimate the tortuosities of both macro- and meso-pores (Sahimi, 1995):

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ε = εα τ

(6)

where α is the parameter to be determined experimentally. The range of α estimated by Sahimi (1995) is from 1.3 to 4. For an activated carbon sample with the porosity εma = εme = 0.31 and measured tortuosity (Do and Do, 2000) τma = 4.70, the computed α is 2.32. Due to the small coal porosities, this value of α leads to very large totuosity values for macro- and meso-pores. It seems that α for coals is in the lower part of the value range suggested by Sahimi (1995). For example, if we assume that the range of α is from 1.3 to 2, the range of macro-pore tortuosity is from 4 to 100, which seems plausible. A nominal value of α = 1.4 is used for both macro- and meso-pores as the starting point of our simulations. It must be pointed out that although micro-pores are combined with solid materials, the estimation of the micro-pore tortuosity cannot be avoided for accurate computations due to the adsorption/desorption induced connectivity changes. The detailed modelling strategy will be described in later sections. The Effective Medium Theory coupled with the Smooth Field Approximation (EMT-SFA procedure) is employed to address mass transfer in meso-pore networks (Wang et al., 2007a).

DEVELOPMENT OF MASS TRANSFER MODELS A simplified diagram of a computational element for a coal specimen is shown in figure 7. There are seven mass fluxes in the coal specimens. These are: convective fluid flow through cleats (Flux 1), bulk diffusions in macro and meso-pores (Fluxes 2 and 3), viscous flows in macro and meso-pores (Fluxes 4 and 5), surface diffusion within micro-pores (Flux 6), and a potential gradient induced flow (Flux 7) in micro-pores. Flux 7 is normally ignored in most mathematical models for porous media, but it could be very significant. We

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incorporate the first six fluxes in the integrated model for CO2 transport in coal specimens in the current section, and address Flux 7 separately in another article for clarity of presentation. A multiple one-dimensional modelling strategy is developed in this work for notational and computational simplicity. It can be easily extended to multi-dimensional systems. We use the orthogonal collocation on finite element (OCFE) method to solve the model equations numerically. In the model development, we mainly demonstrate the derivations of the governing equations in the z-direction, and use averaged properties along the x-axis. However, in the study of mass transfer between coal matrix and face cleats, concentration gradients in the x-direction are taken into account. It will be shown that the strategy of the combination of multiple one-dimensional models is more accurate than the pure onedimensional models but much simpler than multi-dimensional models.

Figure 7. Schematic Diagram of an Element in Coal Specimen.

The following eight assumptions are made for the model development: 1. The fluid phase and the adsorbed phase are in local equilibrium with each other at any time in the coal matrix, however, fluid phase in cleats is not in equilibrium with the adsorbed phase. 2. The Effective Medium Theory coupled with the Smooth Field Approximation (EMTSFA procedure) is an acceptable approximation for the investigation of transport

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3. 4.

5.

6. 7. 8.

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phenomena in pore networks for meso-pores (Sotirchos, 1989; Sotirchos and Burganos, 1988). The real gas law using the compression factor is applicable to the fluid phase. The slit pore structure is assumed for all the pore types. However, certain analogies between slit and cylindrical pores are also assumed in order to use the wellestablished pore network models. The pore size distribution for meso-pores is assumed as lognormal, and that for micro-pores is assumed as either Γ-distribution or multi-modal distribution described by Equation (4). A uniform pore length is assumed in the EMT-SFA procedure. The system is isothermal, but extendable to non-isothermal operations. Water flow is neglected at this stage.

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It should be pointed out that some of the assumptions may be relatively easily relaxed. For example, Assumption (1) can be eliminated by the computation of the mass transfer rate between the fluid phase and the adsorbed phase (Wang and Do, 1999); Assumption (2) can be replaced by more advanced characterisation techniques described by Sahimi (1995) and Adler (1992) for the determination of Knudson diffusivities and permeability parameters in macroand meso-pores; and Assumption (5) can be modified by using real measurement data for individual specimens. In fact, the generalised algorithm for numerical simulations is developed in such a way that the users are able to select pre-documented functions based on particular applications. Because of the difficulties encountered in the determination of some parameters and functions, the more advanced techniques have not been widely applied to coal research. These parameters and functions include the mass transfer coefficients between the fluid phase and the adsorbed phase, pore network geometry and connectivity, pore length distribution, and micro-pore size distribution. It can be seen that at this stage, the bottleneck for the relaxation of the listed assumptions is the lack of reliable measurement techniques to provide the necessary detail regarding physical properties, rather than mathematical difficulties. The assumption of negligible water flow leads to significant errors for the prediction of mass transfer in coal seams below the water table. However, it makes the problem more tractable and forms a theoretical and computational foundation for further studies on multiphase fluid flow.

Fluid Flow in Cleats (Flux 1) Gilman and Beckie (2000) developed a mass balance model for flow in cleats by imposing five assumptions. Two out of five assumptions are impractical for CO2 sequestration processes. They are: [A1] methane is the only moving substance in the coal seam (single component system is assumed); and [A4] methane behaves like an ideal gas and its viscosity does not depend on pressure. These two assumptions are removed in this article. The multi-component fluid flow equation in matrix form is given by:

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F. Y. Wang, Z. H. Zhu, P. Massarotto and V. Rudolph

∂ (ε f C f ) ∂t

∂ (ε f VC f )

=−

∂z ⎡ C f _1 ⎤ ⎥ ⎢ Cf = ⎢ # ⎥ ⎢⎣C f _ NC ⎥⎦ ⎡ m1 ⎤ m=⎢ # ⎥ ⎢ ⎥ ⎢⎣m NC ⎥⎦

+ m( z, t )

(7)

where εf is the fracture (cleat) porosity, Cf the vector of concentrations, Cf_i the concentration of the i-th component, i = 1, … , NC, NC the total number of components, m the vector of the blocks-to-fracture mass cross flow rate, mi the blocks-to-fracture mass cross flow rate for the i-th component, which could be either positive or negative, V the gas velocity. Equation (7) can be converted into the partial pressure representation as:

∂ (ε f P f ) ∂t

=−

∂ (ε f VP f )

∂z ⎡ Pf _ 1 ⎤ ⎢ ⎥ Pf = ⎢ # ⎥ ⎢⎣ Pf _ NC ⎥⎦

+ Z m RTm( z, t ) (8)

where Pf is the vector of partial pressures in fractures, Pf_i is the partial pressure for the species i, and Zm is the compression factor of the gas mixture. The total pressure is the summation of the partial pressures given by:

∂ (ε f Pf ∂t

) = − ∂ (ε

f

VPf

∂z

)+ Z

NC

m RT ∑ mi ( z , t ) i =1

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NC

(9)

Pf = ∑ Pf _ i i =1

For the coordinate system defined by figure (9) with gases passing through specimens from the bottom to top, gas velocity in fractures is governed by Darcy’s law as follows:

ε fV = −

⎤ ⎤ K o ⎡ ∂Pf K ⎡ ∂Pf g ρ = − + gρ m ⎥ + m ⎢ ⎢ ⎥ ψμm ⎣ ∂z μ m ⎣ ∂z ⎦ ⎦

(10)

where μm and ρm are the viscosity and density for the gas mixture respectively, Ko is the overall permeability on the basis of the total cross section area of the specimen, K the

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287

permeability of cleats, V the gas velocity , g the gravitational acceleration, and ψ > 1 the correction parameter. The necessity for the incorporation of the correction parameter ψ is that the Darcy flows occur not only in cleats but also in macro- and meso-pores. Without accounting for the contributions of other pore types to the permeability in addition to cleats, the velocity in cleats will be over estimated. The overall permeability in Equation (10) is measured in our laboratory using the following approximation:

Ko =

Q m Lμ m AΔP

(11)

where Ko is the overall permeability, Qm the volumetric gas flow-rate, L and A the height and cross section area of coal specimens, respectively, and ΔP the total pressure drop. Substitute Equation (10) into Equation (7) with the assumption of constant pressure gradient ∂Pf/∂z, we obtain:

∂ (ε f C f ) ∂t

=

⎞ ∂C f K ⎛ ∂Pf ⎜⎜ + m(z, t ) + gρ m ⎟⎟ μm ⎝ ∂z ⎠ ∂z

(12)

The permeability can be estimated by using the following correlation for slit shape fractures:

K=

a2 ε a 3 1 w2 ε = = 12 τ 12h τ 3 τ

(13)

where a and w are the width and half width of slit fractures respectively, h is the distance between fractures. The physical insight provided by Equation (13) will be analyzed in the section titled “Geo-mechanical Properties and Permeability of Coal Specimens”. With the application of the Archie’s law, Equation (13) becomes:

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K=

w2 3

⎛ 2w ⎞ ⎜ ⎟ ⎝ h ⎠

α

(14)

Since the tortuosity τ of coal fractures is low, the parameter α is at the lower end of the value range as described previously. We use α = 1.4 as a representative starting point. The apertures of the fractures change dynamically due to adsorption induced swelling and the variations of external stresses. Consequently, the permeability also changes dynamically during CO2 sequestration operations. This issue will be addressed in later sections. A technique was proposed by Gilman and Beckie (2000) to estimate single phase mass cross-flow rate with diffusion as the only mass transfer mechanism based on the concept of effective diffusivity. This estimation method is easy to understand and simple to use. However, original authors (2000) could not explain the tremendous differences between the experimentally measured effective diffusivities and the Knudsen diffusivity. We provide a

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F. Y. Wang, Z. H. Zhu, P. Massarotto and V. Rudolph

more rigorous estimation method for the multi-component systems with both of the diffusive and convective mass transfer fluxes as follows:

∂C ε D ⎛ε D ⎞ 1 mi = ⎜⎜ ma i _ ma + me i _ me + (1 − ε t ) μ ,i Dμ ,i ⎟⎟ 2 (Ci − C f ,i ) ∂Ci τ me ⎝ τ ma ⎠λ Kc (Pc − Pf )Ci for Pf < Pc + 2

μ cm λ

∂C μ ,i ⎛ ε ma Di _ ma ε me Di _ me ⎞ 1 + + (1 − ε t ) mi = ⎜⎜ Dμ ,i ⎟⎟ 2 (Ci − C f ,i ) ∂Ci τ me ⎝ τ ma ⎠λ Kc (Pc − Pf )C f ,i for Pf > P + 2

μ cm λ

−1

⎛ ⎞ n y ⎟ ⎜ 1 j +∑ Di = ⎜ ⎟ ; i = 1, " , NC ⎜ Di _ K j =1 Dij ⎟ j ≠i ⎝ ⎠ 2 2 2 w ε w ε w 2 α ma wme α me K c = ma ma + me me = ma ε ma + ε me 3 τ ma 3 τ me 3 3

α ma = α me = 1.4

(15)

In Equation (15), Di is the combined Knudsen and molecular diffusivity, Di_K is the

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Knudsen diffusivity, Dij is the binary diffusivity, Dμ ,i is the surface diffusivity in micro-pores, Ci and Cf_i are the gas phase concentration in the coal matrix and fracture, respectively, Cμ,i is the adsorbed concentration, Pf_i is the partial pressure in the fracture, the subscript i = 1, …, NC is the index for identifying the species, Kc is the permeability of coal matrix in xdirection, Pc and Pf are the total pressures in the coal matrix and fractures, respectively, μcm is the viscosity of gas mixture in the coal matrix, λ is the half distance between fractures, the subscripts ma and me stand for the macro- and meso-pores, respectively. Because of the simplicity, the methods based on the concept of the effective diffusivity are widely used. However, an empirical estimation of the effective diffusivity may lead to unexplainable discrepancies, as observed by Gilman and Beckie (2000). A rigorous mathematical analysis will be carried out in the section titled “Model Order Reduction” to determine the accurate representation of the effective diffusivity.

Fluid Phase Diffusion in Porous Media (Fluxes 2 and 3) The fluid phase diffusions in macro and meso-pores consist of molecular diffusion and Knudsen diffusion. The molar fluxes within the macro and meso-pores in the Dusty-Gas model format with the porosities defined in Equation (5) are represented as:

Advances in Integrated Modeling of Mass Transport and Geo-mechanics…

ε ma ⎛ −1 ∂C ⎞ ⎟ ⎜ B ma τ ma ⎝ ∂z ⎠ (1 − ε ma ) ε me ⎛ −1 ∂C ⎞ = − ⎟ ⎜ B me τ me ∂z ⎠ ⎝

289

N dma = − N dme

(16)

The general form for Stefan-Maxwell matrix B is given by:

B (i, i ) =

D

B (i, j ) = −

yn ; i = 1, " , nc, DinE,−eF

nc

1 E−F ki ,e

+∑ n =1 n ≠i

yi DijE,−e F

(17)

, i, j = 1, " , nc; i ≠ j.

The Effective Medium Theory coupled with the Smooth Field Approximation (EMTSFA) is employed to address diffusion in meso-pore networks. In order to simplify the computations, we assume a uniform pore length. This is because the real pore length distribution cannot be measured, and a non-uniform pore length distribution can be accounted for by adjusting the ε/τ value in the parameter identification procedures. The proposed methods can be extended to include pore length distribution if known. For a network of uniform length pores, the EMT-FSA Knudsen diffusivity for slit shape pores is given by:

D

E −S Ki ,e

=

1

τ

ε Qi

w3

e

w2

(18)

1 8 RT 2 πM i × 10 −3

Qi =

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The EMT-FSA effective binary diffusivity in slit pores takes the following form:

E −S ij ,e

D

=

1

τ

ε Dij

w2 w

e

2

(19)

In Equations (18) and (19), the effective average is defined as:

qa

e

=∫

qmax

qmin

q a f (q ) dq

(20)

where f(q) is an appropriate distribution function, which could be the lognormal-, Γ- and multi-modal-distribution functions given by Equations (2)-(4), respectively, qmin and qmax are

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F. Y. Wang, Z. H. Zhu, P. Massarotto and V. Rudolph

the minimum and maximum values of q, respectively. The arithmetic average is represented as follows:

qa =

(qmax

qmax 1 q a dq ∫ q min − q min )

(21)

For the coal specimens under study, we adopt average pore size for macro-pores, lognormal distribution for meso-pores, Γ- or multi-modal-distribution for micro-pores.

Viscous Flows in Porous Media (Fluxes 4 and 5) The molar fluxes of viscous flows within the macro and meso-pores are given by:

ε ma Bma 0 Z m RT ∂C −1 B ma Λ ma C τ ma μm ∂z (1 − ε ma )ε me Bme 0 Z m RT ∂C −1 B me Λ me C =− τ me μm ∂z

N vma = − N vme

(22)

The generalised form of the diagonal matrix Λ is given by:

Λ (i, i ) =

1 DkiE,−eF

, i, j = 1,", nc,

(23)

Λ (i, j ) = 0; i ≠ j.

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Equation (23) is applicable to both macro and meso-pores with respective Knudsen diffusivities. The permeability parameters Bma0 and Bme0 in Equation (22) are computed using:

Bma 0

4 1 wma = 2 3 wma

e

Bme 0

4 1 wme = 2 3 wme

e

(24)

Surface Diffusion Model (Flux 6) We separate cleats from other pores and combine the micro-pores with the solid phase. Consequently, the total porosity is only contributed by macro- and meso-pores defined by Equation (1). We also embed the tortuosity of micro-pores into the surface diffusivity to make

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291

it consistent with the majority of publications in the adsorption literature. The local surface flux of each species can be written respectively in terms of the hypothetical and the adsorption concentration as (Wang and Do, 1999):

J μ ,i ( w) = − Dμ ,i ( w)

C μ ,i ( w) C hy ,i

NC

∂Chy ,i

j =1

μ, j

∑ ∂C

( w)

∇C μ , j ( w )

(25)

where Chy,i and Cμ,i are the hypothetic bulk concentration and adsorbed phase concentration for component i, respectively. The representation of Chy,i was developed by Wang and Do (1999) based on the extended Langmuir isotherm. However, since the coal surfaces are not uniform, it may be better to use the extended hybrid Langmuir-Freundlich equation explained by Yang (1997) in adsorption studies for coals given by:

C μ ,i ( w) = C μs ,i

bi ( w)Ciηi NC

1 + ∑ b j ( w )C j

(26)

ηj

j =1

where ηi, i = 1, …, NC, are additional parameters in the extended hybrid LangmuirFreundlich equation, and Cμs,i is the adsorption capacity for component i with a temperature dependent property estimated by:

C μs ,i = C μs 0,i exp[δ i (T − T0 )]

(27)

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in which Cμs0,i and δi represent adsorbed capacity at reference temperature (273 K) and temperature dependent coefficient of adsorption capacity, respectively, for species i. Based on the extended hybrid Langmuir-Freundlich equation, the adsorption and desorption rates ra,i and rd,i are originally proposed by Wang and Do (1999) are modified as follows through the incorporation of the parameter ηi: NC ⎡ ⎤ ra ,i = k ads ,i Ciηi ⎢C μs ,i − ∑ C μ , j ⎥ j =1 ⎣ ⎦ rd ,i = k des ,i C μ ,i

(28)

where kads and kdes are adsorption and desorption rate constants, respectively. By setting ra,i = rd,i at the equilibrium point, we directly obtain the mathematical representation for the hypothetic bulk concentration as follows:

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F. Y. Wang, Z. H. Zhu, P. Massarotto and V. Rudolph

C hy ,i

⎡ ⎤ ⎢ 1 ⎥ C μ ,i ( w) ⎥ =⎢ NC ⎢ bi ( w) ⎥ C μs ,i − ∑ C μ , j ( w) ⎥ ⎢ j =1 ⎣ ⎦

1 / ηi

(29)

In Equation (29), the pore size-dependent parameter bi can be estimated by using the following approximation:

bi ( w) =

β

⎡ E (i , w ) ⎤ exp ⎢ M iT ⎣ RT ⎥⎦

(30)

In Equation (30), and parameter β is assumed to be solid specific and independent with respect to adsorbate. The surface diffusion model is then formulated as:

∂C μ ,i ( w) ∂t

=

⎞ C μ ,i ( w) NC ∂C hy ,i 1 ∂ ⎛⎜ s ⎟ ∇ z D ( w ) C ( w ) ∑ j i , , μ μ ⎟ z s ∂z ⎜⎝ C hy ,i j =1 ∂C μ , j ( w) ⎠

(31)

where s is defined as the geometric factor with values of s = 0, 1, 2 for slab, cylinder and sphere, respectively; and the adsorbed phase diffusivity Dμi(w) is given by:

⎡ a E ( w) ⎤ Dμ ,i ( w) = Dμ0i ,∞ exp ⎢ − i i RT ⎥⎦ ⎣

(32)

In Equation (32), a is the ratio of activation energy for surface diffusion to adsorption energy (Wang and Do, 1999), Dμ,∞0 is the surface diffusivity at zero loading and infinite temperature, and E is the adsorption energy. Both a and Dμ,∞0 are normally treated as fitting parameters. It has been suggested by the authors in their recent paper to use self diffusivity at 0

high temperature as a starting point for the estimation of Dμ ,∞ (Wang et al., 2007a) as Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

follows:

Dμ0 ,∞ = γ

ε mi Ds ,∞ τ mi

where Ds,∞ is the self diffusivity at high temperature, and γ is the ratio of surface diffusivity to self diffusivity at high temperature. Under certain circumstances, the ratio γ can be estimated without relying on online measurement data. For example, γ = 2 / 3 for diffusion in a sinusoidal potential field (Karger and Ruthven, 1992). Applicable methods for the computation of the self diffusivity include that developed by Mathur and Thodos (1965), and Zhu et al. (2002). An effective technique for the incorporation of pore size dependent

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293

properties into the mathematical model and numerical algorithm has also been proposed by Wang et al. (2007a). We have recently developed rigorous models for the computation of the surface diffusivity (Want et al., 2008) without empirical parameters. Equation (25) can be rewritten in the matrix model format as:

J μ ( w) = − (1 − ε t ) D μ ( w)

∂C hy ∂Cμ

T

( w)∇Cμ ( w)

(33)

where the total porosity εt is defined in Equation (5) without counting the micro-porosity, and Dμ(w) is a diagonal matrix defined as:

Cμ ,1 ( w) ⎤ ⎡ ( ) 0 0 D w , 1 μ ⎥ ⎢ Chy ,1 ⎥ ⎢ D μ ( w) = ⎢ % 0 0 ⎥ C μ ,NC ( w) ⎥ ⎢ 0 0 Dμ ,NC ( w) ⎢ Chy ,NC ⎥ ⎦ ⎣

(34)

and the vector differentiation is given by:

∂C hy ∂Cμ

T

∂Chy ,1 ⎤ ⎡ ∂Chy ,1 ⎥ ⎢ ∂C ( w) " ∂C μ , NC ( w) ⎥ ⎢ μ ,1 # % # ( w) = ⎢ ⎥ ∂Chy ,NC ⎥ ⎢ ∂Chy ,NC ⎥ ⎢ ∂C ( w) " ∂C μ , NC ( w) ⎦ ⎣ μ ,1

(35)

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Overall Mass Transfer Model In each coordinate direction, there are six fluxes in coal matrices (blocks), and one flux in cleats. Since the computation of the potential gradient induced convective flow within micropores requires detailed knowledge on micro-pore structures, this additional complexity will be addressed in a separate section. We temporarily incorporate the convective mass transfer stream into surface diffusion flux through adjusting the surface diffusivity. Consequently, the mass transfer model within coal blocks consisting of five mass fluxes is developed as:

∂ Cμ ∂C e + (1 − ε t ) ∂t ∂t 1 ∂ s =− s z N dma + N vma + N dme + N vme + J μ z ∂z

εt

{[

e

]}− m

(36)

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F. Y. Wang, Z. H. Zhu, P. Massarotto and V. Rudolph

where vector m is defined in Equation (7), which can be roughly estimated by using Equation (15). The overall mass transfer model consists of the sub-model for fluid flow (Flux 1) in cleats described by Equation (7) and the mass transfer sub-model within coal blocks represented by Equation (36). The five mass fluxes in coal blocks (Fluxes 2-6) are defined in Equations (16), (22) and (34) respectively. The major advantage to combine the mass transfer in micro-, meso- and macro-pores together is the elimination of the computation of the mass transfer between the micro-pores and other two pore types. The time derivative term of Cμ in Equation (36) is given by:

∂ Cμ

∂C μ

=

e

∂t

∂C

T e

∂C ∂t

(37)

where

∂Cμ ∂CT

= e

∂Cμ ,1 ( w) ⎤ ⎡ ∂C μ ,1 ( w) " ⎢ ∂C ∂Cnc ⎥ 1 ⎥ ⎢ # % # ⎥ ⎢ ⎢ ∂Cμ ,NC ( w) " ∂C μ ,NC ( w) ⎥ ⎢ ∂C1 ∂C NC ⎥ ⎦ ⎣

(38)

e

Consequently, the overall mass transfer model described by Equation (36) becomes:

⎧⎪ ∂C μ ⎫⎪ ∂C ⎨ε t I + (1 − ε t ) ⎬ ∂CT e ⎪⎭ ∂t ⎪⎩ 1 ∂ s =− s z N dma + N vma + N dme + N vme + J μ z ∂z

{[

e

]}− m

(39)

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where I represents an identity matrix. The effective average is defined by Equation (20).

GEO-MECHANICAL PROPERTIES AND PERMEABILITY OF COAL SPECIMENS In order to solve the overall mass transfer model represented by Equations (12) and (36), it is necessary to compute the permeability values for cleats, macro- and meso-pores. A number of permeability models have been reported in the literature, such as the models described by Equations (13) and (14) for cleats, and model represented by Equation (24) for macro- and meso-pores as functions of cleat or pore sizes. However, most conventional models assume constant structures and sizes of cleats and pores, ignoring dynamic variations of these parameters, which leads to significant errors. There are two influences leading to pore structure and size changes, namely external stress variations and adsorption/desorption

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295

induced coal swelling and shrinkage. A detailed knowledge of the geo-mechanical properties of coals, so far unavailable, is essential for the computation of permeability variations. Coal is highly compressible and the petrophysical structures of coals are highly anisotropic. A systematic study of three dimensional (3D) geo-mechanical properties of coal specimens is necessary for accurate predictions of the performance of coal bed methane (CBM) reservoirs. In this section, we develop 3D mathematical models with different model complexities followed by experimental investigations using a number of 80 mm cube specimens tested in the TTSCP described in the previous section.

Adsorption Induced Dimensional Changes It is well established that coal properties change during adsorption/desorption processes, leading to significant permeability variations. Selected coal swelling data measured in our laboratory using the experimental rig shown in figure 2 are listed in table 1. The parameter α in table 1 is defined as the volumetric swelling coefficient for the computation of the volume change of coal specimens using the following equation (Van Krevelen, 1993):

ΔU = αΔS

(40)

where ΔU and ΔS are changes of the specimen volume and adsorbed mass, respectively for constant external stresses. It can be seen that the dimensional changes take place in all of the three directions. Consequently, the condition of negligible lateral strain assumed by Gilman and Beckie (2000) cannot be applied to our coal specimens.

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Table 1. Adsorption Induced Coal Specimen Swelling & Permeability Change Property Mean Pressure Pm (kPa) x-Direction Strain (ΔLx/L) y-Direction Strain (ΔLy/L) z-Direction Strain (ΔLz/L) Gas Adsorbed (kg/m3) Permeability (mD) Average α (m3/kg gas ads.) Literature Range of α (Van Krevelen, 1993) (m3/kg gas ads.)

CH4 Replacing He 400 2000 0.0012 0.0010 0.0013 4.6 14.1 3.1-3.2 2.5×10-4 -4 1.0~3.0×10

CO2 Replacing CH4 400 2000 0.0019 0.0017 0.0020 39.6 70.5 1.9-2.1 0.8 1.3×10-4

Matrix Representations for Isotropic Geo-Mechanical Systems The development of matrix models for geo-mechanical properties follows the general technique described by Goodman (1989) with modifications to incorporate adsorption/desorption induced dimensional changes. We first consider the geo-mechanical properties of isotropic systems with unique Young’s modulus E and Poisson’s ratio ν in all directions. A two parameter model can be developed as follows:

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F. Y. Wang, Z. H. Zhu, P. Massarotto and V. Rudolph

⎡ 1 ⎡ε B ⎤ ⎢⎢ E ⎢ε ⎥ = − ν ⎢ F⎥ ⎢ E ⎢⎣ε V ⎥⎦ ⎢⎢ ν − ⎢⎣ E

−

ν

E 1 E

−

ν

E

−

ν⎤

E ⎥ ⎡σ B ⎤ ν⎥ − ⎥ ⎢σ F ⎥ E⎥ ⎢ ⎥ 1 ⎥ ⎢⎣σ V ⎥⎦ E ⎦⎥

(41)

where ε and σ stand for strain and stress, respectively, the subscripts B, F and V indicate three directions, namely the butt cleat, face cleat, and vertical directions, respectively. Equation (41) can be represented in the following equivalent form:

⎡ 1 ⎡σ B ⎤ ⎢⎢ E ⎢σ ⎥ = − ν ⎢ F⎥ ⎢ E ⎢⎣σ V ⎥⎦ ⎢ ν ⎢− ⎢⎣ E

−

ν

E 1 E

−

ν

E

−

ν⎤

−1

E ⎥ ⎡ε B ⎤ ν ν ⎤ ⎡ε B ⎤ ⎡1 − ν E ν⎥ ⎢ ⎥ ⎢ 1 −ν − ⎥ εF = ν ν ⎥ ⎢ε F ⎥ (42) ⎥⎢ ⎥ E ⎥ ⎢ ⎥ (1 + ν )(1 − 2ν ) ⎢ 1 − ν ν ν ε ⎢ ⎥⎦ ⎢⎣ε V ⎥⎦ ⎢ ⎥ 1 ⎥ ⎣ V⎦ ⎣ E ⎥⎦

Through the incorporation of adsorption/desorption induced dimensional changes, Equation (42) is extended to:

⎡1 − ν ⎡σ B ⎤ E ⎢ ν ⎢σ ⎥ = ⎢ F ⎥ (1 + ν )(1 − 2ν ) ⎢ ⎣⎢ ν ⎣⎢σ V ⎦⎥ ⎡1 − ν E ⎢ ν = (1 + ν )(1 − 2ν ) ⎢ ⎢⎣ ν

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ε SB = ε SF = ε SV =

α 3

ν

ν ⎤ ⎧ ⎡ε B ⎤ ⎡ε SB ⎤ ⎫ ⎪ ⎪ ν ⎥ ⎨ ⎢ε F ⎥ + ⎢ε SF ⎥ ⎬ 1 −ν ⎥ ⎢ ⎥ ⎢ ⎥ ν 1 − ν ⎦⎥ ⎪⎩ ⎣⎢ε V ⎦⎥ ⎣⎢ε SV ⎦⎥ ⎪⎭ ν

ν ⎤ ⎡ε B ⎤ ⎡1⎤ α ⎥ ⎢ ⎥ ν ε + 1 −ν EΔS ⎢1⎥ ⎥ ⎢ F ⎥ 3(1 − 2ν ) ⎢⎥ ν 1 − ν ⎥⎦ ⎢⎣ε V ⎥⎦ ⎢⎣1⎥⎦

(43)

ΔS

where the subscripts SB, SF and SV to ε indicate adsorption/desorption induced strains in B, F and V directions, respectively, α is the volumetric swelling coefficient, and ΔS is the increment of the adsorbed mass. In order to simplify Equation (43), the following assumptions are made:

εB = εF = 0 σ V = − ΔPf

(44)

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297

where ΔPf stands for the pressure variation in the fluid phase. The first assumption in Equation (44) implies σB = σF. Under the conditions described by Equation (44), Equation (43) can be simplified to:

σB =σF = −

ν 1 −ν

ΔPf +

α

3(1 − ν )

EΔS

(45)

Equation (45) is structurally similar, but not exactly the same as the model developed by Gilman and Beckie (2000), which is described by:

σB = σF = −

ν 1 −ν

α

ΔPf +

(1 − ν )

EΔS

(46)

It can be shown through mathematical derivations and numerical simulations (Wang et al, 2007) that Equation (45) is correct, and Equation (46) is erroneous with an over predicted effect from adsorption induced dimension changes.

Matrix Representations for Anisotropic Geo-Mechanical Systems For anisotropic systems, the stress-strain relationship can be represented by a nine parameter model as:

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⎡ 1 ⎢ ⎡σ B ⎤ ⎢ E B ⎢σ ⎥ = ⎢ − ν FB ⎢ F ⎥ ⎢ EF ⎢⎣σ V ⎥⎦ ⎢ ν ⎢ − VB ⎣⎢ EV

−

−

ν BF

−1

E B ⎥⎥ ν − FV ⎥ EF ⎥ 1 ⎥ ⎥ EV ⎦⎥

EB 1 EF

ν VF EV

⎡ε SB ⎤ ⎡γ B ⎤ ⎢ε ⎥ = αΔS ⎢γ ⎥; γ ≈ ⎢ SF ⎥ ⎢ F⎥ i ⎣⎢ε SV ⎦⎥ ⎣⎢γ V ⎦⎥ Ei =

−

ν BV ⎤

(1 / Ei ) 3

∑ (1 / E ) i =1

⎧ ⎡ε B ⎤ ⎡ε SB ⎤ ⎫ ⎪⎢ ⎥ ⎢ ⎥ ⎪ ⎨ ⎢ε F ⎥ + ⎢ε SF ⎥ ⎬ ⎪ ⎢ε ⎥ ⎢ε ⎥ ⎪ ⎩ ⎣ V ⎦ ⎣ SV ⎦ ⎭

(47)

i

∂ε ∂σ i ; ν ij = j ; i, j = B, F , V ∂ε i ∂ε i

The nine parameters are EB, EF, EV, νBF, νBV, νFB, νFV, νVB and νVF. The nine parameter model described by Equation (47) can be reduced to six parameter model by introducing the following assumptions:

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F. Y. Wang, Z. H. Zhu, P. Massarotto and V. Rudolph

ν BF EB

=

ν FB ν BV EF

;

EB

=

ν VB ν FV EV

;

EF

=

ν VF EV

(48)

Equation (47) is then simplified to:

⎡ 1 ⎢ ⎡σ B ⎤ ⎢ E B ⎢σ ⎥ = ⎢ − ν FB ⎢ F ⎥ ⎢ EF ⎣⎢σ V ⎦⎥ ⎢ ν VB ⎢− ⎢⎣ EV

−

−

ν FB EF 1 EF

ν VF EV

−

ν VB ⎤

−1

EV ⎥⎥ ⎧ ⎡ε ⎤ ⎡γ B ⎤ ⎫ B ν VF ⎥ ⎪ ⎢ ⎥ ⎪ − ε F + αΔS ⎢γ F ⎥ ⎬ ⎨ ⎢ ⎥ EV ⎥ ⎪ ⎢ ⎥ ⎪ ε ⎢ ⎥ ⎥ V ⎣ ⎦ ⎣⎢γ V ⎦⎥ ⎭ ⎩ 1 ⎥ EV ⎥⎦

(49)

Cleat Permeability for Isotropic Specimens We consider the isotropic systems first followed by the extension to anisotropic systems. Since the cleats are nearly vertical, we have:

dσ B = dσ F = − E f

da a

(50)

where a is the mean fracture aperture and Ef is the Young’s modulus of the fracture. The following proportionalities can be intuitively observed for the slit-shaped cleats:

K ∝ abφ ; b ∝ a; φ ∝

a h

(51)

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where K is the vertical direction permeability, b the mean fracture length in face cleat direction, h the cleat spacing, and φ the vertical direction porosity. Consequently, the permeability can be represented as:

K=

a3 cτ f h

(52)

where c is a constant and a function of cleat geometry and surface roughness, and τf is the fracture tortuosity. Since the structure of Equation (52) is the same as Equation (13), the physical insights and mathematical conditions for the validity of Equation (13) are clarified. Through calculus and algebraic operations on Equation (52), we obtain:

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dK da 3a 2 cτ f h da = 3 = 3 K cτ f h a a

299

(53)

Substitute Equation (50) to Equation (53), we have:

dK dσ F = −3 K Ef

(54)

A combination of Equation (45) and (54) followed by integration operations leads to:

⎤ ⎡ S ⎤ ⎡ Pf 3ν αE K = K 0 exp ⎢ ∫ dPf ⎥ exp ⎢ ∫ − dS ⎥ P S ⎥⎦ ⎣⎢ 0 (1 − ν )E f ⎦⎥ ⎣⎢ f 0 (1 − ν )E f

(55)

Equation (55) is similar to the permeability model developed by Gilman and Beckie (2000) described as:

⎡ 3ν ΔPf K = K 0 exp ⎢ ⎢⎣ (1 − ν ) E f

⎡ 3αE ΔS ⎤ ⎤ ⎥ ⎥ exp ⎢ − ⎢⎣ (1 − ν ) E f ⎥⎦ ⎥⎦

(56)

It should be noted that there are following two important differences between Equations (55) and (56): (1) Two coefficients in the second exponential term in Equation (56) are unsupported; and (2) Equation (56) is only applicable to the systems with constant parameters ν, E and Ef. Consequently, Equation (54) presented in this article is more general with meaningful model parameters.

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Cleat Permeability for Anisotropic Specimens We assume that the structure of anisotropic coal specimens can be represented as bundled match sticks (BMS). In this case, the contributions from face cleats and butt cleats can be treated independently. The vertical permeability consists of contributions from both butt and face cleats represented as:

dK BV dσ B dK FV dσ F = −3 ; = −3 K BV E Bf K FV E Ff Equation (57) leads to:

(57)

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F. Y. Wang, Z. H. Zhu, P. Massarotto and V. Rudolph

⎛ ⎞ σB 1 K BV = K BV 0 exp⎜ − 3∫ dσ B ⎟ ⎜ ⎟ σ B0 E Bf ⎝ ⎠ ⎛ ⎞ σF 1 K FV = K FV 0 exp⎜ − 3∫ dσ F ⎟ ⎜ ⎟ σF0 E Ff ⎝ ⎠ KV = K BV + K FV

(58)

where the stress variations σB and σF can be determined through solving Equation (47) with additional conditions. For example, if conditions described by Equation (44) are imposed, Equation (47) is simplified as:

⎡ 1 ⎢ ⎡ σ B ⎤ ⎢ EB ⎢ ⎥ ⎢ ν FB σ F ⎢ ⎥ = ⎢− E F ⎢⎣ − ΔPf ⎥⎦ ⎢ ν VB ⎢− ⎣⎢ EV

−

−

ν BF EB 1 EF

ν VF EV

−

ν BV ⎤

E B ⎥⎥ ν − FV ⎥ EF ⎥ 1 ⎥ ⎥ EV ⎦⎥

−1

⎧⎡ 0 ⎤ ⎡γ B ⎤ ⎫ ⎪⎢ ⎥ ⎢ ⎥⎪ ⎨ ⎢ 0 ⎥ + αΔS ⎢γ F ⎥ ⎬ ⎪ ⎢ε ⎥ ⎢⎣γ V ⎥⎦ ⎪⎭ ⎩⎣ V ⎦

(59)

Since ΔPf and ΔS are either measurable or computable, there are three unknowns σB, σF and εV in three equations given by Equation (59), leading to a set of unique solutions. Through symbolic matrix inversion followed by algebraic operations, the combination of Equations (58) and (59) can be converted to the format similar to Equation (55) for the explicit representations of the permeability in two directions. However, because of the length and complexity of the resulting equations, the matrix representation is preferred.

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Computation of Adsorbed Mass for Permeability Application It can be seen from the previous section that the permeability computation requires the determination of the adsorbed mass ΔS. In general, there are two techniques for the computation of the adsorbed mass, namely the rate-based method and the equilibrium approach. The rate-based method is to solve the following equation in the overall mass balance model (Wang et al., 2007a):

⎧ ⎡ L ⎫ ⎤ dS NC ⎪ ⎢ ∫0mi (z , t )dz ⎥ in out ⎪ + Ni + Ni ⎬ = ∑ ⎨M i ⎥ ⎢ L dt i =1 ⎪ ⎪ ⎥⎦ ⎩ ⎢⎣ ⎭

(60)

where Mi is the molecular weight for the i-th component, Niin and Niout are the total molar fluxes of the inlet (z = 0) and outlet (z = L) streams in z-direction for the species i consisting

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of five sub-fluxes, namely diffusive and convective streams in both macro- and meso pores, as well as surface diffusion stream through micro-pores, and mi is mole adsorption or desorption rate for the i-th component with either positive or negative sign given by Equation (15) in the previous section. The equilibrium approach proposed by Shi and Durucan (2004) using Langmuir isotherm is given by:

⎛ Pf Pf 0 ΔV = εL⎜ − ⎜ V ⎝ Pf + Pε Pf 0 + Pε

⎞ ⎟ ⎟ ⎠

(61)

where εL and Pε are referred to as the Langmuir-type matrix shrinkage constants, and subscript 0 refers to initial parameter values. From Equations (54) and (61), the following Shi and Durucan model can be developed:

⎡ Ec f ⎛ Pf Pf 0 ⎡ 3νc f ⎤ ΔPf ⎥ exp ⎢ − − k = k 0 exp ⎢ ε L ⎜⎜ ⎣ (1 − ν ) ⎦ ⎢⎣ (1 − ν ) ⎝ Pf + Pε Pf 0 + Pε

⎞⎤ ⎟⎥ ⎟ ⎠⎥⎦

(62)

where cf is the cleat compressibility with respect to changes in the effective horizontal stress normal to the cleat. It should be noted that Equation (62) developed by Shi and Durucan (2004) is structurally similar to Equation (56) proposed by Gilman and Beckie (2000) with a difference in coefficient. This can be seen by substituting the following two equations into Equation (56):

cf =

1 Ef

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⎛ Pf Pf 0 αΔS = ε L ⎜⎜ − ⎝ Pf + Pε Pf 0 + Pε

⎞ ⎟ ⎟ ⎠

(63)

The difference could be caused by a typographical error in the Gilman and Beckie model. The following two assumptions are necessary for the successful application of Equation (62): (1) The adsorbed phase should be in instantaneous equilibrium with the cleat pressure; and (2) The Langmuir isotherm is valid for the determination of the adsorption equilibrium. Since in transient processes, the cleat pressure could be very different from the pressure in the matrix centre, and the Langmuir isotherm is not accurate for heterogeneous specimens, Equation (62) should be used with caution.

Dynamic Permeability Variations From Equations (45) and (54), one obtains:

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F. Y. Wang, Z. H. Zhu, P. Massarotto and V. Rudolph

dσ 3ν αE dK = −3 F = dPf − dS (1 − ν )E f (1 − ν )E f K Ef

(64)

Divide both sides in Equation (64) by dt, we obtain the dynamic permeability model for isotropic systems under the assumptions described by Equation (44) as:

dPf dK 3kν αkE dS = − (1 − ν )E f dt (1 − ν )E f dt dt

(65)

Since dS/dt in Equation (65) can be evaluated by solving Equation (60) and dPf/dt is measurable, Equation (65) can be solved numerically for a specified initial permeability K0. Although we have only demonstrated the development of the dynamic permeability model for the isotropic systems, the procedure can be extended to the anisotropic systems with the matrix stress-strain models described by Equation (47) or (49). Of course, the explicit representations will be much more complicated. The matrix representations are recommended for anisotropic systems.

Incorporation of Geo-Mechanical Properties into Mass Transfer Model The newly developed permeability models represented by Equations (55), (57), (58) and (65) contain both Young’s moduli for coal matrix (E) and fracture (Ef), for which values must be provided to computer simulations. For the isotropic systems, this can be done through the analysis of the compressibility variations with external stress. The compressibility factor Cp is the reciprocal of the bulk modulus defined as:

Et 3(1 − 2ν ) 1 Cp = Kt Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

Kt =

(66)

where the subscript t stands for “total”. A typical Cp-σ curve is shown in figure 8 for a coal specimen under our study. The measured Cp-σ data can be represented by an empirical polynomial equation as follows:

Cp =

1 a + bσ + cσ 2 + dσ 3 = K t 1 + eσ + fσ 2 + gσ 3 + hσ 4

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303

Figure 8. Variations of Compressibility Factor with Average Stress.

The parameters a-h in the above empirical equation are estimated as: a = 1.4703, b = 0.2981, c = 2.4676, d = 0.1530, e = 299.6990, f = 316.4895, g = 304.2036, h = 393.9829. We use the following approximation to estimate the contributions from the cleats and matrix to the compressibility:

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1 1 − (σ / σ max ) (σ / σ max ) = + Kt Kf Km where the subscripts f and m stand for “fracture” and “matrix”, respectively, and σmax is the stress above which the compressibility variation is negligible, that is, the fracture contribution to the compressibility approaches zero. Hereafter, we drop the subscript “m” for notational simplicity. If the Poisson’s ratio in Equation (66) is a constant, can be replaced by Young’s modulus as:

1 1 − (σ / σ max ) (σ / σ max ) = + Et Ef E

(67)

Since the Young’s modulus for coal matrix can be estimated at the high stress level, and the total Young’s modulus is computable using the measurement data, the Young’s modulus of the fractures can be approximately determined using Equation (67). We now show a few numerical results using Equations (66) and (67) for the coal specimen shown in figure 8. The estimated maximum stress σmax is about σmax = 11.0 MPa with the bulk modulus and Young’s

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F. Y. Wang, Z. H. Zhu, P. Massarotto and V. Rudolph

modulus for the coal matrix as 12419.0 and 8279 MPa, respectively. As the external stresses are specified as 2.0 and 4.0 MPa, the ratios of Ef/E are computed as 0.0591 and 0.1415, respectively. This implies that the Young’s modulus for fractures keeps increasing until it approaches the value of Young’s modulus of coal matrix. Consequently, the use of constant Young’s modulus in the permeability models will certainly lead to significant errors. Our experimental rig is designed in such a way that the pressure in fractures can be controlled perfectly. Consequently, the condition of ΔPf = 0 can be assumed to simplify the model in our illustrative examples. Equation (55) is reduced to the following format for the constant Pf:

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⎤ ⎡ S αE K = K (S 0 ) exp ⎢ ∫ − dS ⎥ S ⎥⎦ ⎣⎢ 0 (1 − ν )E f

(68)

Equation (68) provides more accurate prediction than the original model due to the following reasons: (1) Operations are not normally started from zero adsorption load; and (2) both of the Young’s modulus for fractures Ef and the volumetric swelling coefficient are not constants, but functions of the adsorbed gas. As S is small, Ef could be very small leading to a large value of α. On the other hand, Ef increases and α decreases with the increase of S, leading to small variations of K as the system approaches adsorption saturation. This qualitative analysis can be justified using the measurement data shown in table 1. The computed γ for the replacement of He by CH4 under the mean pressure Pm = 2000 kPa is 2.5×10-4 (m3/kg gas adsorbed). However, that for the replacement of CH4 by CO2 is only 1.3×10-4 (m3/kg gas adsorbed), indicating the dependency of α and Ef on the amount of the adsorbed gases. The α values measured in our laboratory (1.3-2.5×10-4) are within the literature data range summarised by Van Krevelen[23]. In order to compute permeability K using Equation (68), it is necessary to determine the Young’s moduli E and Ef in addition to α. This can be done by measuring the compressibility factor of coal specimens. The range of compressibility factor of coal specimens is from 4.0×10-4~1.0×10-1 MPa-1 with a typical curve shown in figure 8. The median value is consistent with the literature value as 2.1×10-4 MPa-1 reported by Van Krevelen23. The lower and upper value sections are used for the estimation of Ef and E, respectively. From the measurements of compressibility for a number of coal specimens performed in our laboratory, the value range of Ef/E is 0.01~0.07 depending the extent of adsorption. It should be emphasised that for the dynamic processes, the values of γ and Ef/E must be changed dynamically according to the total amount of adsorbed gases. For example, if constant values of α and Ef/E are used to estimate the permeability for CO2 replacing CH4 under 400 kPa, the computed permeability values differ from the real permeability (1.9 vs. 2.1 mD). This discrepancy can be effectively eliminated using Equation (13) with adsorption dependent α(S) changing from 2.0×10-4 to 1.3×10-4 (m3/kg gas adsorbed) and Ef(S)/E(S) changing from 0.05 to 0.06. An alternative technique for the determination of porosity/permeability variations for single component systems is based on the Palmer-Mansoori theory (1998). The limitation of the method is identified as: (1) It was originally developed for single component systems; and (2) constant model parameters suggested in the original development lead to significant errors

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and inconsistencies. Further work is needed to extend the Palmer-Mansoori theory to multicomponent systems. With the assumption of negligible ΔPf, the dynamic model of permeability described by Equation (65) can also be reduced to:

dK αkE dS =− (1 − ν )E f dt dt

(69)

Similar to Equation (68), K, Ef and α in Equation (69) are all functions of S, and should also be estimated dynamically. The term dS/dt in Equation (69) is evaluated using Equation (60). Up to this point, the mass transfer model is completely developed with the following four sub-models: (1) fluid flow in cleats described by Equation (7); (2) overall mass balance in coal matrix given by Equation (35); (3) mass transfer rate dS/dt between coal matrix and cleats represented by Equation (60); and (4) permeability variations quantified by Equation (65) for time varying Pf or Equation (69) for constant Pf. These four key model equations should be solved simultaneously together with the computation of other intermediate model equations and physical properties.

PHYSICAL PROPERTIES AND NUMERICAL SCHEME

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Computation of Physical Properties The physical properties should be computed dynamically as part of the simulation objectives rather than pre-specified parameters or averaged values. The main physical properties of gases are estimated using established methods and working equations, eg as recommended and detailed in Reid, Prausnitz, and Poling (1987), which are briefly explained as follows. Diffusion coefficients for binary gas systems at low to moderate pressures are predicted using the Chapman-Enskog theory (Reid, Prausnitz, and Poling, 1987) [p. 581, Equation (11-3.1)]. The pure gas viscosities with effects of intermolecular forces are computed by incorporating the Lennard-Jones potential function into the Chapman-Enskog equation (Reid, Prausnitz, and Poling, 1987) [pp. 292-293, Equations (9-3.9) and (9-4.3)], while the Wilke method is used for the determination of viscosities of gas mixtures (Reid, Prausnitz, and Poling, 1987) [p. 407, Equations (9-5.13)-(9-5.15)]. Since the processes operate under high pressures, it is necessary to account for the effects of pressure on physical properties. The diffusivity variations with pressures can be quantified by using the Takahashi correlation (Reid, Prausnitz, and Poling, 1987) [p. 592, Equations (115.1)-(11-5.5)], showing non-trivial pressure effect. For the pressure range in our studies (5.070.0 bar), the error could be up to 30% using the popular correlations for low to moderate pressures without corrections. The viscosity of a gas is a strong function of pressure near the critical point and at reduced temperature of about 1 to 2 at high pressures. Since the critical values for CO2 are: Tc = 304.1 K and Pc = 73.8 bar, dramatic viscosity changes could happen to CO2 rich mixtures. However, most of our experiments are carried out below 50 bar, so the pressure effect on gas viscosities can be ignored for the moment.

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There is no rigorous method, even for idealised systems, for the computation of surface diffusivities under high pressures without carrying out molecular simulations (Bhatia and Nicholson, 2003). The correlation described by Equation (32) requires two empirical parameters, among which Dμ∞0 is very difficult to estimate. However, it is noteworthy that the surface diffusivity in a cylindrical pore without accounting for the effects of porosity and tortuosity is in the same order of magnitude as the self diffusivity (10-9-10-8 m2/sec). Consequently, self diffusivities at high temperature can be used as the starting point for the estimation of Dμ∞0. A simple method for the computation of the self diffusivity developed Mathur and Thodos (1965) is given by:

Ds = 3.67 × 10

−5

Tc5 / 6Tr3.5 M 1 / 2 Pc2 / 3 Pr0.10

(70)

In our simulations for CO2 sequestration in coal seams, Equations (32) and (70) are used for the estimation of the surface diffusivity. Very recently, a number of reasonably rigorous models for the determination of surface diffusivity have been developed by the authors for the systems under low to median pressures (Wang et al., 2008). These newly developed models require neither empirical parameters, nor molecular dynamic simulations. Further efforts are needed to extend these techniques to systems under high pressures.

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Numerical Scheme Because of the complexity of the problem, the numerical scheme covers a broad range of computational techniques. These mainly include the reduction of partial differential equations (PDE) with moving front into ordinary differential-algebraic equations (DAE); resolution of large-scale, stiff DAE systems; operations of multi-dimensional arrays with up to 4 arguments (k, ι, i, j) where k, ι, i and j indicate the spatial location, pore size, species i and species j, respectively; symbolic differentiations; computations of physicochemical properties, and parameter identification using dynamic optimisation algorithms. A detailed explanation regarding each of the methods exceeds the scope of the current contribution. Only the most important ones are briefly described in this section. The matrix form partial differential equations (PDE) in the mathematical model are discretised by using the method of orthogonal collocation on finite elements (OCFE) to form a set of ordinary differential equations with nonlinear algebraic constraints. This numerical scheme has been well explained by Finlayson (1972), as well as Villadsen and Michelsen (1978). The method can be briefly described as division of the spatial domain into a number of finite elements followed by a further discretisation of each element by using the orthogonal collocation method. We use a simplified matrix model to demonstrate the method. Assume that the matrix model is described by:

∂x 1 ∂ ⎛ s −1 ∂x ⎞ 1 ∂ s (ς Vx ) + R = ⎜ς B ⎟+ ∂t ς s ∂ς ⎜⎝ ∂ς ⎟⎠ ς s ∂ς

(71)

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307

where x and R are column vectors with NC elements in each vector representing component concentration and mass generation rate respectively, B is a square Stefan-Maxwell matrix with NC rows and columns, and ζ is the dimensionless spatial variable defined in [0, 1]. In each finite element, the spatial interval must be converted into [0, 1] in order to apply the orthogonal collocation method. Using the notation IB to replace B-1, the ordinary differential equation (ODE) for the i-th component in the k-th collocation point is represented by:

⎧⎡ s ⎫ ∂x ⎤ s ⎪ ⎢ς 1 IB1 (i,1 : NC ) ⎥ ⎡ς 1 xi (ς 1 )⎤ ⎪ ∂ς k ⎥ ⎪⎪ ⎢ dxi (t , ς k ) 1 ⎥ ⎪⎪ ⎢ # = s A k ⎨⎢ ⎥ + V ⎢ # ⎥ ⎬ + R (t , ς k ) ςς dt ∂x ⎥ ⎪⎢ s ⎢ς ns xi (ς n )⎥ ⎪ ( ) ς i , 1 : NC IB ⎦⎪ ⎣ n n ⎪⎢ ∂ς k ⎥⎥ ⎪⎩ ⎢⎣ ⎪⎭ ⎦

(72)

where n is the number of collocation points, Ak, k = 1,2,…,n are the k-th row in the collocation matrix A, IBk represent the matrix IB at the k-th collocation point, the notation “1:NC” represents “from the column 1 to the column NC”, and the spatial derivative at the kth collocation point can be computed by using:

⎡ A k x1 ⎤ ∂x =⎢ # ⎥ ⎥ ∂ς k ⎢ ⎢⎣A k x NC ⎥⎦

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⎡ x j (ς 1 )⎤ ⎥ k = 1,2,..., n ⎢ x j = ⎢ # ⎥; j = 1,2,..., NC ⎢⎣ x j (ς n )⎥⎦

(73)

The spatial derivatives in the boundary conditions can also be formulated using the collocation matrix A. Although the actual mass balance equations, consisting of molecular and Knudsen diffusions as well as viscous flow within macro- and meso-pores, and surface diffusion in micro-pores, are much more involved than this simpler example, the mathematical principles used in the numerical computations are the same. In numerical computations, IB is treated as a three-dimensional array with k = 1,2,…n, to indicate spatial location, and i, j = 1,2,…,NC to identify components. Computations of mass transfer in micro-pore involve the handling of 4-dimentional arrays with the argument (k, ι, i, j) defined previously. For example, if we identify CH4, CO2 and N2 as the first, second and third component (i, j = 1, 2, 3), respectively, then Cμ(2, 3, 1, 1:3) provides the following information: location specification, which is at the 2nd collocation point; micro-pore size specification, which is in the 3rd size class; component specification, which is for the species CH4; and interaction specification, which interacts with 3 components (including itself). The same strategy is used for Jμ computations. The resulting differential-algebraic (DAE) system for the actual process has the general form as follows:

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F. Y. Wang, Z. H. Zhu, P. Massarotto and V. Rudolph

M ( x)

dx = F ( x) dt

(74)

where M(x) is a singular, sparse and state-dependent mass matrix; and F the vector function. The large-scale DAE system is solved by using ode15s.m in MATLAB. It should be pointed out that there is a discontinuity at the inlet boundary due to the step change of the CO2 concentration. Numerical difficulties are encountered by using the fixed boundary OCFE methods for large step changes. There are two ways to solve the problem, namely (1) increase the number of finite elements; and (2) using special numerical methods for solving problems with moving fronts, described by Finlayson (1972). Compared with method (1), the moving front approach leads to a smaller model order. The method requires the determination of the location of the mass front in the coal matrix. This can be done by using Darcy’s law in the coal matrix represented by Equation (7) with appropriate values of the permeability computed using Equation (68) or (69). As soon as the moving boundary is located, the coal specimen can be divided into two zones before and after the moving front. For small step changes, such as to step up the CO2 mole fraction from 0.8 to 1.0, two fixedsize finite elements can be used for the solution of the problem without leading to severe Gibbs oscillations.

SIMULATION RESULTS USING INTEGRATED MASS TRANSFER MODEL

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Fluid Flow in Cleats Simulations of fluid flow in cleats are performed using the mass balance model described by Equations (7)-(10) and (15), together with the models accounting for the mass transfer between coal matrix and cleats, and permeability variations during adsorption/desorption processes represented by Equations (60) and (69), respectively. We display selected simulation results using the pure distributed parameter model to show the main characteristics of fluid flow in cleats. The experiment generally follows the schemes illustrated in Figures 2 and 3 and the associated text. The downstream (outlet) mole fractions for both CH4 and CO2 for an experiment where CH4 is displaced by CO2 are shown in figure 9. It can be seen that the simulated results agree with the measured data reasonably well. Two 3-dimensional diagrams for CH4 and CO2 variations in cleats are depicted in figures 10a and 10b, respectively. These figures indicate that the variations of gas concentrations with time are much more significant than with spatial locations. This implies that the fluid flow in cleats can be classified as a process with fast dynamics. On the other hand, the fluid flow and molecular diffusion in macro- and meso-pores can be classified as having medium dynamics, and the surface diffusion within micro-pores is very slow. This will be demonstrated and analysed in following sections.

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Figure 9. Outlet Mole Fractions in a CO2 Flush Experiment.

Figure 10a. Dynamic Behaviour of CH4 in Cleats During a Flush Experiment.

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Figure 10b. Dynamic Behaviour of CO2 in Cleats During a Flush Experiment.

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Mass Transfer in Coal Matrix A wide range of numerical simulations have been carried to investigate mass transfer in coal matrix, based on the model described by Equations (16)-(39). We report here the results for a typical coal specimen under the conditions of 400 kPa mean pressure and temperature of 25oC. The nominal values in the adsorption-diffusion model are listed in table 2. In contrast to fast dynamics depicted in figures 9 and 10 for fluid flow in cleats, mass transfer processes in the coal matrix are slow. In particular, the lower the residual CH4 remaining in the coal specimen, the slower the mass transfer rates. In order to get a detailed view of the concentration variations within the coal matrix at later stages of methane extraction, we carry out simulations with the initial conditions, set as yCH4 = 0.20 and yCO2 = 0.80, and the upper stream boundary condition as yCO2 = 1.0. From a practical point of view, these conditions would not be tolerable for an ECBM production well, but could represent conditions within the reservoir somewhere between the CO2 injection well and the CH4 production well. For the present purposes, we simply wish to demonstrate model methods and their validity. Two diagrams showing the simulated variation of CH4 and CO2 within the coal matrix are presented in figures 11a and 11b, respectively, for one hour of operation. From these two figures, it can be seen that even after one hour, there is still a long way to go to complete the CO2 sequestration process. In addition to slow dynamics, another notable feature of the concentration within the coal matrix is the high degree of non-uniformity. There are large differences between the concentrations in the upstream and downstream parts of the specimen. This implies that the hybrid distributed-lumped parameter model using averaged properties within the coal matrix suggested by Gilman, and Beckie (2000) is likely to lead to significant errors unless model parameters are appropriately adjusted. Although we are unable

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311

to measure the concentration profiles inside coal specimens at this stage, the predicted time scales agree with the overall measurement data very well. Table 2. Nominal values in adsorption-diffusion model

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Gas property Compressibility factor Self diffusivity Ds (m2/sec) Surface diffusivity Dμ (m2/sec) Knudsen diffusivity in Meso-pores DK (m2/sec) Binary diffusivity DAB for CH4-CO2 (m2/sec) Saturated adsorption Concentration Cμs (mol/m3) Langmuir parameter b (mPa-1) Parameter η in LangmuirFreundlich equation Mean-size adsorption Energy E(〈w〉) (J/mol) Pore property Half-width w (m) Porosity ε Pore size distribution f(w) α in ε/τ = εα (Eqn. 5)

CH4 0.95 3.51×10-8 3.31×10-12 6.27×10-6

CO2 0.76 4.65×10-9 1.60×10-13 3.79×10-6

4.81-7.87×10-6 2.44×103

2.12×103

0.27

1.45

0.89-1.0

0.91-1.0

1.30×104

1.80×104

Micro-pores 4.5×10-10 0.13 Lognormal or Γ

Figure 11a. CH4 Dynamics in Coal Matrix (1 hour operation).

Meso-pores 6.5×10-9 0.01 Lognormal 1.4

Macro-pores 4.0×10-8 0.02 1.4

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F. Y. Wang, Z. H. Zhu, P. Massarotto and V. Rudolph

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Figure 11b. CO2 Dynamics in Coal Matrix (1 hour operation).

Simulations for longer operating times were carried out, which provide additional physical insights to the process. Mole fraction surfaces of CH4 and CO2 in the coal matrix for two hour operation are depicted in figures 12a and 12b, respectively. Notably, at the down stream boundary, the mole fraction of CH4 does not decrease monotonically with time. It increases first followed by a gradual decrease. A consistent but opposite trend can be observed for the CO2 trajectory, decreasing initially, followed by a gradual increase. This phenomenon has also been observed in our laboratory. The existence of an interior maximum or minimum point is normally caused by two opposing factors. It can be shown that the gas phase diffusion and viscous flow within macro- and meso-pores due to partial pressure gradients promote transport of CO2 in the z-dirsection. On the other hand, because of different affinities of CH4 and CO2 to coal “counter-adsorption diffusion” can be observed in micropores. The counter-adsorption diffusion prevents the fast transport of CO2 along the zdirection due to the replacement of the adsorbed CH4 by CO2 in micro-pores, leading to hindered CO2 transport dynamics. The overall result of these two opposing factors is the dynamic trajectories providing interior extreme points at the down-stream boundary, as shown in figure 12.

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313

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Figure 12a. CH4 Dynamics in Coal Matrix (2 hour operation).

Figure 12b. CO2 Dynamics in Coal Matrix (2 hour operation).

Two diagrams showing the variations of CH4 and CO2 fluxes in coal matrix for the two hour operation are presented in figures 13a and 13b, respectively. The much higher CO2 flow rate shown in the figures was due to the inlet boundary conditions, in which CO2 and CH4 mole fractions were 1 and 0, respectively. The two figures also show that CH4 flow rate increases along the z-direction, whereas CO2 flow rate changes in the opposite direction. This verifies the counter-sorption theory. Because of the relatively higher adsorption affinity of

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F. Y. Wang, Z. H. Zhu, P. Massarotto and V. Rudolph

CO2, CH4 molecules originally adsorbed on the coal surface were gradually replaced by CO2, leading to the adsorption of CO2 and desorption of CH4. The simulation results for various molar fluxes can be explained physically, providing an improved insight into the countersorption phenomena.

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Figure 13a. CH4 Flux in Coal Matrix (2 hour operation).

Figure 13b. CO2 Flux in Coal Matrix (2 hour operation).

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315

The fractional uptake for species i at time t is defined as:

fri (t ) =

[ε C (t ) + (1 − ε ) C t

i

t

μi

t

μi

[ε C (t ) + (1 − ε ) C t

i

f

]− [ε C (0) + (1 − ε ) C (0) ] (t ) ] − [ε C (0) + (1 − ε ) C (0) ]

(t )

t

e

f

i

t

e

μi

t

i

e

μi

t

(75)

e

for adsorption and

fri (t ) =

[ε C (t )) + (1 − ε ) C (t ) ]− [ε C (t ) + (1 − ε ) C (t ) ] [ε C (t ) + (1 − ε ) C (t ) ]− [ε C (0) + (1 − ε ) C (0) ] t

i

f

t

i

f

t

t

μi

μi

f

f

t

e

e

t

i

i

t

t

μi

e

μi

e

(76)

for desorption. In Equations (75) and (76), tf stands for the final time of operation, C and

Cμ are the spatially averaged concentrations in the gas phase and adsorbed phase,

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respectively. The fractional uptakes for CH4 and CO2 are depicted in figure 14. The curves show nearly linear fractional uptake dynamics in the specified operational range. From Equations (75) and (76), it can be seen that the fractional uptakes should be bounded by 0 ≤ fr ≤ 1. However, figure 14 indicates a value of fr slightly greater than 1 for CH4 early in desorption. This is caused by numerical oscillations near the discontinuous points of the boundary conditions, also known as the Gibbs phenomenon (Kreyszig, 1999).

Figure 14. CH4 and CO2 Uptake Dynamics.

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F. Y. Wang, Z. H. Zhu, P. Massarotto and V. Rudolph

Significance of Pore Size Distributions Compared with models based on mean pore sizes, the incorporation of pore size distributions into dynamic models described in the sub-section titled “Surface diffusion model (Flux 6)” may lead to about 10 times longer computing times. It is therefore important to analyse the necessity to adopt the pore size dependent methods. Selected computational results from a study comparing the simplified approach using the mean micro-pore size and the effective averaging strategy based on distribution density functions are shown in table 3. Both approaches provide similar results for the adsorption energy, E, and the adsorbed phase concentrations, Cμ. However, there are big differences in computed values of Langmuir parameters bi for both gases, and the surface diffusivity Dμ for CO2. This is because both b and Dμ are exponential functions of E as shown in Equations (30) and (32) respectively, and E is very sensitive to micro-pore sizes as depicted in figure 5. It can also be seen from table 3 that the ratios of Maximum {b} over b(〈w〉) for both CH4 and CO2 are very large. Consequently, it can be concluded that if a significant amount of micro-pores lies within the size (radius) range from 3.2 to 4.2 Å correlated to the peaks of the adsorption energy curves, simplified methods based on mean sizes could lead to unacceptable errors. It is interesting to note that although the pair {b(〈w〉), 〈b(w)〉} are very different from each other, Cμ(〈w〉) and 〈Cμ(w)〉 are reasonably close to each other. This is because the processes are operated under a relatively high pressure (400 kPa), where the errors cancel for the extended hybrid Langmuir-Freundlich isotherm, given by Equation (26). However, in low pressure operations, there is an unacceptable inconsistency for Cμ using different computational approaches. Under any pressures, the pair {Dμ(〈w〉), 〈Dμ(w)〉} are very different. This demonstrates the necessity for using pore size dependent computations. Table 3. Values Based on Mean Micro-Pore Size and Size Distributions Property

CH4

CO2

Cμ(〈w〉) (mol/m3)

49.3

7.36×102

〈Cμ(w)〉 (mol/m3)

47.1

6.92×102

(Cμ(〈w〉)-〈Cμ(w)〉)/ 〈Cμ(w)〉×100%

4.8%

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E(〈w〉) (J/mol)

1.80×104

4

1.75×104

1.30×10

〈E(w)〉 (J/mol)

1.32×10

(E(〈w〉)-〈E(w)〉)/ 〈E(w)〉×100%

-1.5%

3 -1

6.4% 4

2.8%

6.36×10

-4

2.73×10-3

〈b(w)〉 (mol/m )

2.13×10

-3

1.83×10-2

(b(〈w〉)-〈b(w)〉)/〈b(w)〉 ×100%

-70.1%

Max{b}/b(〈w〉)

25.2

b(〈w〉) (mol/m )

3 -1

2

-85.1% 37.8%

3.32×10

-12

1.60×10-13

〈Dμ(w)〉 (m /sec)

3.30×10

-12

2.03×10-13

(Dμ(〈w〉)-〈Dμ(w)〉)/ 〈Dμ(w)〉×100%

0.6%

Dμ(〈w〉) (m /sec) 2

-26.9%

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Prospective Industrial Applications of the Integrated Model The integrated modelling framework has far reaching implications in industrial applications. A few application examples are listed as follows. 1. Prediction and physical insights for the dynamic behaviour of CO2 geo-sequestration processes. 2. Improved process design through the determination of optimal operational conditions. 3. Formation of a basis for model order reduction applicable to optimal control of accelerated residual methane recovery (Wang et al., 2007b). 4. Process diagnosis through identifying likely causes for quality problems and process deviations.

COMPUTATIONAL RESULTS OF GEO-MECHANICAL PROPERTIES The most important objective in the study of geo-mechanical properties is the estimation of the values of Young’s modulus and Poisson’s ratio along three directions. For this, the external stress induced dimensional changes of the coal specimen need to be measured. The experiments reported here consisted of four different types of operations, namely (1) uni-axial operations under axial stress variations from 0.3-1.5 MPa; (2) anisotropic operations under axial stress variations from 23-34 MPa; (3) isotropic operations under 3-D stress variations from 0-50 MPa; and (4) uni-axial failure tests. The nine parameters in Equation (47), namely νBF, νBV, νFB, νFV, νVB, νVF, EB, EF, and EV, can be determine through the minimization of the differences between the measured and computed strains with specified stress values. The Bulk modulus defined as K = p/(ΔV/V) with p as the hydrostatic pressure can be computed directly from the measurement data.

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Uni-Axial Operations For the uni-axial experiments, only the external stress along one direction was varied from 0.3 to 1.5 MPa with constant external stress on the other two directions. The Young’s modulus and Poisson’s ratio can be directly computed. For example, νBF and EB are computed as follows:

ν BF =

∂ε F ∂σ B ; EB = ∂ε B ∂ε B

(77)

The values of the directional Young’s modulus and Poisson’s ratios along three directions are shown in table 4 for four 80 mm cube coal samples. It can be seen from table 4 that most computed data make physical sense although there are a few abnormal points where Poisson’s

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F. Y. Wang, Z. H. Zhu, P. Massarotto and V. Rudolph

ratios are negative or very large. These could be caused by the anisotropic properties and the limitations of the measurement technique using the stain gages. Table 4. Uni-Axial Operations under Axial Stress Variations from 0.3-1.5 MPa Property

Sample 1

Sample 2

Sample 3

Sample 4

Average

νBF

-0.0008, 0.3261 -0.0632, -0.0945 0.3372, 0.0712 0.1587, 0.1066 0.2709, 0.1756 0.1261, 0.0131 1017.4, 4702.8 2684.7, 2440.3 1835.4, 1704.4 321.4, 2052.5 1771.9, 848.4 810.3, 700.2

0.0530, 0.0790 0.1559, 0.0530 0.3203, 0.3808 0.1565, 0.7276 -0.0016,0.0187 0.1309, 0.0071 889.5, 1621.1 1895.3, 3853.5 401.0, 477.8 431.8, 833.5 1201.9, 3062.3 153.5, 159.1

0.2497, 2.5183 0.4693, 0.0730 0.4135, 0.0926 0.7268, 0.3049 0.1038, 0.1001 0.0610, 0.0868 1277.0, 2407.9 1833.4, 4572.6 494.2, 1592.0 1515.1, 5111.7 -4394.6, 1953.1 198.2, 529.9

-0.02529, 0.7952 3.2236, 0.2336 0.2944, 0.02715 -0.2317, 0.6802 0.07415, 0.00187 0.03591, 0.02553 1013.9, 16703.3 2411.4, 2018.4 497.9, 472.5 -153.7, 243.1 853.7, 1935.4 185.7, 162.9

0.2110

νBV νFB νFV νVB νVF EB (MPa) EF (MPa)

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EV (MPa) K (MPa)

0.5063 0.1944 0.3287 0.0632 0.0608 3704.1 2713.7 934.4

The uni-axial stress-strain curves with the stress imposed in the face cleat direction for Sample 2 are shown in figure 15. The stress occurs in the face cleat direction and the strain in this direction is positive (contraction) and has the highest absolute value. In contrast, the strains in the other two directions are negative (extension) with smaller absolute values. These trends are as expected and consistent with geo-mechanical theory. Since the experimental measurements are carried out under low stress levels, the results are used to estimate the contributions from the fractures in order to estimate the ratio of Ef/E for the computation of the permeability using Equations (55) and (65) for the pressure varying systems, or Equations (68) and (69) for the pressure invariant systems.

Anisotropic Operations The experimental procedure for anisotropic operations is similar to uni-axial operations with the difference in the starting stress levels. The starting stress for uni-axial operations is close to zero (0.3 MPa), whereas that for anisotropic operations is about 23 MPa. Consequently, the model parameters can also be estimated by using the definitions, which are

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319

listed in table 5. The computed data are quite consistent with that reported in the literature, but it may be noted that the data quality is higher for anisotropic over uni-axial operations. This observation is important to the design of improved experimental procedures. The anisotropic stress-strain curves with the stress imposed to the vertical direction (orthogonal to the bedding plane) for Sample 1 are depicted in figure 16. Figure 16 shows a typical stress-strain pattern for many solid materials, with notable contraction in the stress direction and much smaller extension in the other two directions

Figure 15. Uni-Axial Stress-Strain Curves for Sample 2.

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Table 5. Anisotropic Operations under Axial Stress Variations from 23-34 MPa Sample 3

Sample 4

Average

0.1795

Sample 2 0.1927

0.2557

0.1797

0.2019

νBV

0.2320

0.1071

0.2858

0.1696

0.1986

νFB

0.3782

0.6235

0.4622

0.4239

0.4719

νFV

0.4913

0.3085

0.3715

0.3935

0.3912

νVB

0.0898

0.1728

0.1709

0.2038

0.1593

νVF

0.3521

0.2233

0.4660

0.3404

0.3455

EB (MPa)

5311.3

4362.6

3568.6

3781.5

4256.0

EF (MPa)

6048.1

5421.3

3889.7

4500.8

4965.0

EV (MPa)

4554.6 3025.0 15782.0 2739.2

2833.0 2075.2 2675.3 1587.6

4299.3 2647.2 7835.7 3991.9

3581.5 1976.6 8368.6 2572.4

3817.1

Property

Sample 1

νBF

K (MPa)

3590.4

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F. Y. Wang, Z. H. Zhu, P. Massarotto and V. Rudolph

Figure 16. Anisotropic Stress-Strain Curves for Sample 1.

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Isotropic Operations Under isotropic operation conditions, the external stresses along three directions change simultaneously in uniformly incremental step from 0 to 50 MPa. Unlike uni-axial operations, the values of the Young’s modulus and Poisson’s ratios cannot be computed directly using the definitions but require the use of curve fitting techniques and optimization algorithms to estimate parameter values. The results for the four coal samples are listed in table 6. Although most estimated values of the parameters lie within the expected ranges, the method suffers from some repeatability problems, as can be seen from the results in table 6. The error sources probably include both experimental procedures and the value extraction methods. The stress-strain curves for isotropic operations with simultaneous stress variations in three directions for Samples 3 and 4 are shown in figures 17 and 18, respectively. The behavior in high stress regions reflects the mechanical failure of the samples, so the data are not used for stresses above 50 MPa. It can also be seen from figures 17 and 18 that, for both samples, the strain variation is greater in the vertical direction than other two directions under the same stress variations. This implies that the coal is more compressible in the vertical direction. The samples show roughly equivalent compressibility in face cleat and butt cleat directions.

Advances in Integrated Modeling of Mass Transport and Geo-mechanics… Table 6. Isotropic Operations under 3-D Stress Variations from 0-50 MPa Property νBF σ = 0-20 σ = 21-40 σ = 41-50 νBV σ = 0-20 σ = 21-40 σ = 41-50 νFB σ = 0-20 σ = 21-40 σ = 41-50 νFV σ = 0-20 σ = 21-40 σ = 41-50 νVB σ = 0-20 σ = 21-40 σ = 41-50

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νVF σ = 0-20 σ = 21-40 σ = 41-50 EB (MPa) σ = 0-20 σ = 21-40 σ = 41-50 EF (MPa) σ = 0-20 σ = 21-40 σ = 41-50 EV (MPa) σ = 0-20 σ = 21-40 σ = 41-50

Sample 1

Sample 2

Sample 3

Sample 4

0.4810, 0.1256

0.1656, 0.2980 0.2725, 0.2848 0.1707

0.2729, 0.1320 0.03959, 0.3956 0.00026

0.01304, ~0 0.06912, 0.2437 0.2859

0.03788, 0.5000

0.5000, 0.4325 0.3895, 0.3617 0.5000

0.5000, 0.5000 0.5000, 0.1643 0.4929

0.4089, 0.4330 0.5000, 0.3305 0.5000

0.5000, 0.3274

0.1558, 0.1680 0.3516, 0.3176 0.5000

0.1989, 0.09708 0.5000, 0.1093 0.5000

~0, 0.5000 0.08743, 0.2142 0.1074

0.07679, 0.3508

0.5000, 0.5000 0.4019, 0.4420 0.2378

0.5000, 0.5000 0.0775, 0.4583 0.1320

0.4792, 0.1343 0.5000, 0.3032 0.5000

0.4905, 0.4997

~0, ~0 0.3595, ~0 0.3200

~0, ~0 0.5000, 0.04247, 0.4994

~0, 0.5000 0.1555, 0.05316 ~0

0.1465, 0.1914

~0, 0.5000 ~0, 0.3211 0.0047

~0, ~0 0.02215, 0.5000 0.4494

0.4235, 0.1987 0.3299, 0.5000 0.2110

4987.4, 5152.0

3053.9, 5061.0 5494.2, 4672.3 6355.5

1502.4, 4085.2 5615.3, 4706.8 3526.4

5438.2, 7052.3 5225.6, 5130.5 2598.5

5769.0, 6119.6

5406.8, 7401.9 6926.1, 7341.1 6687.8

1977.0, 3724.3 4705.1, 5665.0 6323.3

6709.0, 4954.6 5551.9, 6274.0 5167.1

4528.9, 5278.3

2527.1, 2428.3 3972.0, 3854.2 3692.0

2246.7, 6591.1 4149.9, 8939.2 9969.8

2233.6, 2309.7 4361.0, 4156.3 8058.9

321

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F. Y. Wang, Z. H. Zhu, P. Massarotto and V. Rudolph

Figure 17. Isotropic Stress-Strain Curve for Sample 3.

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Figure 18. Isotropic Stress-Strain Curve for Sample 4.

Uni-Axial Failure Test In these tests the peripheral stress is kept at a constant value and the axial stress is increased in steps (from 0 to 13 MPa) until the sample fails. This differs from the isotropic tests above in which the radial and axial stresses are increased together. It also differs from uni-axial tests with small stress variations (0.3-1.5 MPa) shown in table 4 and figure 15, in which the specimens were not damaged. Since the geo-mechanical models represented by Equations (42), (47) and (49) are developed under the assumption of the elastic deformation in the linear stress-strain region, which may not be valid near the failure points, data presented in table 7 should only be used for rough estimation and general information.

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Table 7. Uni-Axial Failure Test σV Range (MPa) 0-1 1-3 3-5 5-7 7-9 9-11 11-13

νVB S3

S4

νVF S3

-0.0213 -0.0821 -0.0393 -0.0145 -0.0086 -0.0301 -0.0909

0.0430 0.0043 0.0138 0.0153 0.0154 0.0152 0.0071

0.1290 0.1140 0.2877 0.1407 0.2941 0.3124 0.1553

S4 0.0049 0.1490 0.2043 0.0.2466 0.2878 0.3072 0.2776

EV(MPa) S3 533.0 990.0 1675.0 877.6 1941.9 2076.1 5592.4

S4 285.7 930.6 1356.9 1618.4 1785.2 1898.1 1721.3

K (MPa) S3 209.4 331.5 742.8 334.8 905.9 964.2 1992.5

S4 100.6 366.4 578.4 730.9 854.0 933.7 802.1

MODEL ORDER REDUCTION In the development of the integrated mass transfer model, a number of model simplification techniques have already been applied. These include the incorporation of the effective medium theory with the smooth field approximation (EMT-SFA) procedure to compute the mass transfer in meso-pores with broad pore-size distribution, and the approximation of a two dimensional (2-D) system by two 1-D systems. Additional model order reduction techniques are described in this section, which may permit simpler and faster outputs suitable for industrial applications.

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Development of Effective Diffusivity for Model Simplification There are three types of diffusivities in the system, namely diffusivities in micro- and meso-pores, and surface diffusivity. A very common practice is the development of a unique effective diffusivity to replace all of the three diffusivities (Wang et al., 2007a; b). It should be pointed out that the effective diffusivity developed through the combination of different diffusivities is different from the effective EMT-SFA diffusivity defined by Equations (18) and (19). Normally the effective diffusivities are estimated using experimental data and in general they are many orders of magnitude less than Knudsen and molecular diffusivities. They are sometimes called “pseudo diffusivities” (Yang, 1997). There are no convincing physical explanations for the huge discrepancies between the effective and mechanistically based diffusivities. We present an analysis of this inconsistency. The derivation of the effective diffusivity for a single component system consisting of two pore types, namely micro- and macro-pores is presented below. This could be extended to multi-components, but we use a one component system here for notational simplicity. The overall mass balance is given by:

εt

∂C μ ε t ∂ 2 C ∂ 2Cμ ∂C + (1 − ε t ) = D 2 + (1 − ε t )Dμ +" ∂t ∂t ∂z τ ∂z 2

(78)

324

F. Y. Wang, Z. H. Zhu, P. Massarotto and V. Rudolph Partial derivatives of Cμ are:

∂C μ

∂C μ ∂C ∂t ∂C ∂t 2 ∂ C μ ∂C μ ∂ 2C ∂C ∂ ⎛ ∂C μ ⎜ = + ∂z 2 ∂C ∂z 2 ∂z ∂z ⎜⎝ ∂C =

(79)

⎞ ⎟⎟ ⎠

Substitute Equation (79) into (78), after algebraic operations, we obtain:

∂C ∂C μ εt (1 − ε t )Dμ ∂ ⎛⎜⎜ μ ⎞⎟⎟ Dμ 2 D + (1 − ε t ) ∂z ⎝ ∂C ⎠ ∂C ∂ C ∂C τ ∂C +" + = 2 ∂C μ ∂C μ ∂z ∂z ∂t ε t + (1 − ε t ) ε t + (1 − ε t )

(80)

∂C

∂C

Consequently, the effective diffusivity should be defined as:

∂C μ εt Dμ D + (1 − ε t ) τ ∂ C De = ∂C ε t + (1 − ε t ) μ

(81)

∂C

Equation (81) can be written in a more general form to account for the separated contributions from macro- and meso-pores as follows:

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De =

∂C μ ε ma ε Dma + me Dme + (1 − ε t ) Dμ τ ma τ me ∂C

ε t + (1 − ε t )

∂C μ

(82)

∂C

Since the total porosity εt is defined by Equation (5) without counting the micro-porosity, it is very small for coal specimens. If we use εt = 0.03 to perform illustrative computations, the effective diffusivity could be in the range of 10-10 to 10-13 m2/s for the proper combinations of the values of εt, τ and ∂Cμ/∂C. The concentration (or partial pressure) dependency of the effective diffusivity is partly accounted for through the derivative term ∂Cμ/∂C. Through the development of the effective diffusivity, the diffusion coefficients in three pore types, Dma, Dme and Dμ are reduced to a single diffusion coefficient De, leading to a significant model simplification. The overall effective diffusivity can be estimated using Equation (82) without experimental measurements. It can be seen from Equation (82) that the effective diffusivity may be very different from the individual diffusion coefficients. This provides a mathematical explanation for the significant difference between the effective diffusivity and the Knudsen diffusivity observed by Gilman and Beckie (2000).

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325

The mass transfer rate between coal matrix and cleats given by Equation (15) is simplified as follows by using the effective diffusivity:

mi = mi =

Di , e

λ

2

Di , e

λ2

(C (C

i

− C f ,i ) +

i

− C f ,i ) +

Kc

μ cm λ2 Kc

μ cm λ2

(P

c

(P

c

− Pf )C i for Pc > Pf − Pf )C f ,i for Pc < Pf

(83)

Although models based on the effective diffusivity are much simpler than that based on the individual diffusion coefficients, they are unable to quantify the relative importance of different diffusion streams in various pores. In order to keep mathematical rigor, numerical accuracy and physical insights, individual diffusion coefficients should be used to compute mass transfer within the coal matrix. The effective diffusivity is only used for the computation of mass transfer between coal matrix and cleats.

Black-Box Modelling The simplest modelling method is the so-called “black-box” strategy, which depends only empirically fitting measured data and requires no physical understanding of the systems under study. The black-box permeability model is used as an illustrative example to demonstrate the main concept and procedure of the model development. The same technique can be applied to any process with arbitrary complexity provided there are sufficient experimental data. The dynamic permeability variations during a process in which CO2 replaces CH4 and operating at a mean pressure of 400 kPa are shown in figure 19. The experimentally measured dynamic behaviour is compared with simulation results based on two different models, namely the mechanistic model and the black-box ARX (Auto-Regressive Exogeneous) model. The mechanistic model used is that described by Equations (69) and (60), for the evaluations of dK/dt and dS/dt, respectively, and the single input, single output black-box ARX model is given by (Ljung, 1987):

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y (t ) + a1 y (t − 1) + " + a na y (t − n a ) = b1u (t − 1) + " + bnb u (t − nb ) + e(t )

(84)

where u and y are input and output variables, respectively, e is the disturbance, a and b are time varying coefficients identified online, na and nb are defined as prediction and control horizons.

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F. Y. Wang, Z. H. Zhu, P. Massarotto and V. Rudolph

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Figure 19. Dynamics of Permeability Variations.

From figure 19, it can be seen that both mechanistic and black-box models predict the trend very well. However, the measured data show significant oscillations before reaching the final steady state, which are not predicted by the mechanistic model. The oscillations could be caused by the following three mechanisms: (1) the relative permeability due to different properties of CH4 and CO2; (2) damped mechanical vibrations around the equilibrium position; and (3) locally higher adsorption densities near the cleats. Further work is needed to understand the oscillation mechanisms and to incorporate these into the mechanistic model. Since the black-box ARX model described by Equation (84) is a linear modelling strategy, and the measured dynamic behaviour shows nonlinearity, two different ARX model structures are used in the time intervals 0-25 and 25-60 minutes, respectively. Although multiple black-box models are able to predict the oscillation behaviour, the parameters in the models do not possess any physical significance. If the main purpose of the model is to gain some physical insights regarding the process, then black box models are unsuitable and mechanistic models are required. For example, the dramatic permeability drop shown in figure 19 is caused by large E/Ef ratio incorporated in Equations (68) and (69). At the early stage of adsorption, the Young’s modulus of fractures, Ef, is about two orders of magnitude smaller than that of the coal matrix. Consequently, very fast dynamics can be observed. Unfortunately, this important insight cannot be explored or even identified using the blackbox model represented by Equation (84). There are two ways to address the mass transfer between coal matrix and the fluid stream in a cleat, namely pure distributed parameter model and hybrid distributed-lumped parameter model. In the hybrid distributed-lumped parameter model, the fluid flow in cleats is represented by the distributed parameter model and the adsorbed gas variations in the matrix are described by the lumped parameter model following Gilman and Beckie (2000). Compared with the pure distributed parameter model, the hybrid modelling technique reduces

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computational load considerably but sacrifices some physical insights. The effective diffusivities need to be adjusted based on the measurement data.

Hybrid Distributed-Lumped Parameter Model In this sub-section, we describe a practical model simplification technique with applications to depressurization processes for the accelerated recovery of residual methane. A simplified mass transfer model under depressurization conditions has been developed by the authors (Wang et. al, 2007b). The model is classified as a hybrid distributed-lumped parameter model with the fluid flow in fractures described by partial differential equations based on Darcy’s law, and the concentration dynamics within the coal matrix represented by ordinary differential equations. This model is significantly simpler than the integrated model described by pure partial differential equations for mass transfer in both cleats and coal matrix developed by the authors (Wang et al., 2007a). Two major differences between the current model and the one developed by Gilman and Beckie (2000) are identified as: (1) total pressures in both of the coal matrix and fracture change dynamically; and (2) two mechanisms for gas release from the coal matrix, namely diffusive and convective mass transfer, are taken into account instead of diffusion only mass transfer. The model is applicable to multi-component systems with any gas mixtures. However, as a prelude to more complicated CO2-CH4 systems, the model validation was carried out for He displacement of CH4 under a series of controlled pressure reductions using the TTSCP. Good agreement between measured and predicted values has been achieved, which show that, at the early stage of depressurization, convective (Darcy) flow induced by pressure gradient between the coal matrix and fractures plays a major role in mass transfer. The dominant mechanism switches from convection to diffusion when the pressures on each side approach each other. This discovery – which is at odds with current convention - is important for the optimal design of CBM and ECBM processes using depressurization technology (i.e. almost all practical systems). For depressurization operations, Pc > Pf, the lumped parameter model can be developed by using averaged values over the specimen length described as:

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εt

∂C μ ,i ∂Ci dCi = − mi + (1 − ε t ) ∂Ci ∂t dt

mi =

Di ,e

λ

2

(C

i

− C f ,i ) +

L

∫ C dz ; C C = 0

L

Kc

μ cm λ2

L

f

∫C = 0

f

(P

c

− Pf ,i )Ci

dz

L

Equation (85) can be represented by a matrix format as:

(85)

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F. Y. Wang, Z. H. Zhu, P. Massarotto and V. Rudolph

D e (C − C f ) +

1

m=

λ

m=

2

1

λ

2

D e (C − C f ) +

⎡ D1,e ⎢ 0 De = ⎢ ⎢ 0 ⎢ ⎣ 0

Kc

μ cm λ2

(P − P )C

Kc

μ cm λ2

c

f

(P − P )C c

f

for Pc > Pf for Pc < Pf

f

0 ⎤ 0 0 ⎥ ⎥ % # ⎥ ⎥ " DNC ,e ⎦ "

0 D2,e 0 0

C = [C1 , " , C NC ] ; C = [C f ,1 , " , C f ,NC ] T

T

(86)

If the Stefan-Maxwell analysis is applied to macro- and meso-pores (Krishna and Wesselingh, 1997), the format of the effective diffusivity becomes a matrix with the dimension of NC × NC represented as follows:

D eff

⎡ ⎡ ∂C μ ,1 ⎢ ⎢ ⎢ ⎢ ∂C1 ⎢ ⎢ = ⎢ε t I + (1 − ε t )⎢ 0 ⎢ ⎢ ⎢ ⎢ # ⎢ ⎢ 0 ⎢ ⎢ ⎣ ⎣

−1

0 ∂C μ , 2 ∂C2 0 "

⎤⎤ " 0 ⎥⎥ ⎥⎥ ⎥⎥ # ⎥⎥ × 0 ⎥⎥ % 0 ⎥⎥ ∂C μ ,NC ⎥⎥ 0 ∂C NC ⎥⎦⎥⎦

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⎡ ∂C μ ,1 ⎡ ⎢ ⎢ Dμ ,1 ∂C1 ⎢ ⎢ ⎢ε ⎢ −1 ε me −1 0 ⎢ ma B ma B me + + (1 − ε t )⎢ τ me ⎢τ ma ⎢ # ⎢ ⎢ ⎢ ⎢ 0 ⎢ ⎢ ⎣ ⎣

0 Dμ , 2

∂C μ , 2 ∂C2 0

"

(87) ⎤ ⎥ ⎥ ⎥ 0 # ⎥ ⎥ % 0 ⎥ ∂C μ , NC ⎥ 0 Dμ , NC ∂C NC ⎥⎦ "

0

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

In the cases where the operational pressures are reduced, a quick estimation of the mass transfer rates between the coal matrix and fractures can be conducted as an alternative technique to Equation (85). In this method, the equilibrium concentration of the adsorbed gas under the final lower pressure is computed using Equation (26), followed by the quick estimation of the transfer rate as follows:

mi ≈

dC μ ,i dt

= τ q (C μ ,i _ L − C μ ,i )

(88)

where Cμ,i_L indicates the adsorbed phase concentration under the low pressure for component i, and τq is the time constant, which must be estimated using experimental data. Compared

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with Equation (88), the computation of mi using Equation (85) is more accurate and also has a more defensible physical basis, but is significantly more complicated. This comparison will be further examined in the next section. It can be seen that lumped parameter model described by Equations (85)-(88) are used to replace the distributed parameter model given by Equations (16)-(36) for the computation of mass transfer in coal matrix with five fluxes (Fluxes 2-6). This leads to a significant reduction of computing time. The hybrid distributed-lumped parameter model provides satisfactory results for the simulation of the depressurization operations and is also able to predict the dynamic behaviour with acceptable accuracy. Nevertheless, two main advantages for the use of the pure distributed parameter model can be identified as: (1) the effective diffusivities can be computed using Equation (82) without the incorporation of adjustable parameters; and (2) the dynamic variations of CH4 and CO2 in the coal matrix can be investigated in detail, which could be important for the development of improved operational strategies.

APPLICATION OF SIMPLIFIED MODEL TO DEPRESSURIZATION OPERATIONS

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Problem Statement The amount of residual methane in coal seams after the production of coal-bed methane (CBM) is not trivial. For example, even under the best circumstance, the CBM recovery is typically only 50-60% before low gas flow rates make further recovery uneconomical (Harpalani, 2005). The low recovery is due to the small driving forces and low mass transfer fluxes between the coal matrices and fractures (cleats). Residual methane is both a valuable energy resource and a hazard if the coal is mined after gas drainage. Even using ECBM technology, a significant amount of residual methane still remains in coals prompting interest in how this may also be extracted. Consequently, the development of effective techniques for the accelerated recovery of residual methane after primary CBM or secondary ECMB operations is of both theoretical and practical significance. An obvious option to achieve nearly complete methane recovery is the depressurization technique. In this section, we apply the hybrid distributed-lumped parameter model to depressurization conditions to examine the improved recovery of residual methane. The fluid flow in fractures is modeled using Darcy’s law described by partial differential equations, whereas the variations of gas and adsorbed phase concentrations within the coal matrix are represented by a lumped parameter model represented by ordinary differential equations as explained in the previous section. The formats of the sub-models for physical properties and adsorption isotherms are the same as that in the pure distributed parameter model. The simplified model is validated using TTSCP, which provides accurate dynamic measurements of systems properties under a series of controlled pressure reductions. The model provides physical insights into the observed behavior. The relative importance of the convective and diffusive flows is quantified through numerical simulations, which are essential to the proper design of depressurization operations. We report a series of depressurization operations on a He-CH4 system. He gas is used to displace CH4 adsorbed in the 80 mm cube specimen under 5100 kPa until 63% CH4 recovery

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F. Y. Wang, Z. H. Zhu, P. Massarotto and V. Rudolph

of the originally adsorbed gas (OAG) is achieved, leaving a CH4 mole fraction as 0.20 in the macro- and meso-pores. The pressure is then reduced to 3100 kPa while measuring gas flow rate, composition, stresses and strains, until a pseudo steady state is reached. Mass balance shows that 82% recovery of OAG is achieved, or about 18% of the originally adsorbed CH4 still remains in the specimen. The second step depressurization, from 3100 to 1100 kPa, reduces the CH4 residual to 6%, thereby increasing overall recovery to 94% of OAG. The final operation is to further reduce the pressure from 1100 to 200 kPa, which brings down the CH4 residual to a negligible level, and cumulative recovery to essentially 100% of OAG. The key measured variables include: inlet and outlet gas pressures, temperatures and compositions, stresses and strains in three orthogonal directions, and overall gas flow rate. Based on these data, the permeability and net-stress in fractures, and mean values can be readily determined. The operational conditions for these three steps are listed in table 8.

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Table 8. Operational Conditions of Depressurization Processes Mean pressure in Fracture (kPa)

Flow Rate

5100

InletOutlet Pressure Drop (kPa)

Permeability

0.14

70

0.18

6.50

3100

0.26

27

1.39

3.95

24.8

1100

0.28

21

5.30

1.35

47.3

200

0.39

20

41.08

0.26

4.2

(NL/min)

Net Stress

(mD)

Operation Time (hrs)

(MPa)

In table 8, the net stress is defined as the external stress minus the pressure in fractures, which is adjusted to mimic the condition of the coal-bed in the depth with the same cleat pressure. The permeability increases significantly in the depressurization processes due to the following two reasons: (1) net stress reduction as shown in table 8; and (2) coal matrix shrinkage induced by desorption. Since the permeability increases more rapidly than the inletoutlet pressure drop is reduced, the flow rate increases as the depressurization sequence occurs.

Simulation results and discussion Experimentally measured and simulated profiles of CH4 mole fraction during the three depressurization processes are partly shown in figures 20-22, respectively. Due to very long operating times, the figures are truncated and the initial time is set to zero for each operation. In the simulations, the maximum adsorbed phase concentration Cμs and the Langmuir parameter b for methane are estimated as 2.44×103 mol/m3 and 6.36×10-4 m3/mol, respectively. From the figures, it can be seen that the second step with the pressure reduction

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from 3100 to 1100 kPa leads to the most profound CH4 composition variation, whereas the third step with the pressure reduction from 1100 to 200 kPa has the fastest dynamics. It should be pointed out that in all experiments, two sampling rates were applied: high sampling frequency in early depressurization operations in order to observe the fast dynamics, and low sampling frequency in later stages due to the slowed down dynamics. In hindsight, the sampling rate switchover from high to low rates was done prematurely and the loss of sufficient discrimination makes it difficult to compute the integral errors over more extended periods of time than ones shown in the figures. This protocol has been changed for future experiments. In order to explain the observed dynamics, selected computational results are shown in table 9 at the early depressurization stages. It can be seen that the convection fluxes are more important than the diffusion fluxes for all of the three cases at the early depressurization stages. This is caused by the fast pressure reduction in fractures and comparatively slow dynamics in coal matrix, leading to a significant pressure gradient between the coal matrix and fractures. Consequently, diffusion only matrix-fracture mass transfer models (e.g. Gilman and Beckie, 2000) are unable to predict accurately the dynamic behaviour of the depressurization operations. The faster dynamics under lower pressures are attributed to the significant increases of overall permeability, matrix permeability, and effective diffusivity shown in tables 8 and 9. The permeability increases are induced by both the net stress reduction and the micro-structure shrinkage during desorption processes, leading to size enlargement for all of the pore types except micro-pores. The increases of the effective diffusivity shown in table 9 are also caused by two major factors: pore size enlargement induced Knudsen diffusivity increase and pressure reduction induced molecular diffusivity increase. The incorporation of these physically based parameter variations leads to good agreement between model predictions and experimental data.

Figure 20. Outlet Methane Composition after Step Pressure Reduction (5100-3100 kPa).

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F. Y. Wang, Z. H. Zhu, P. Massarotto and V. Rudolph

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Figure 21. Outlet Methane Composition after Step Pressure Reduction (3100-1100 kPa).

Figure 22. Outlet Methane Composition after Step Pressure Reduction (1100-200 kPa).

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Table 9. Computed Variables in Various Depressurization Operations Convection Flux (mmole/m3 s)

Matrix Permeabilit y (mD)

(mmole/m3 s) 5.8

9.8

3100

13.2

1100

5.3

Mean Cleat Pressure (kPa) 5100

Diffusion Flux

Recovery

7.6×10-5

CH4 Effective Diffusivity (m2/s) 2.9×10-9

31.3

4.7×10-4

2.1×10-8

82

7.1

1.1×10-3

1.2×10-7

94

(%) 63

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Mass transfer in a coal matrix should be described by a set of multi-dimensional partial differential equations. In this work, it is simplified to a lumped parameter model. Mass transfer between coal matrix and cleats is computed using a two point finite difference method shown in Equation (85). Compared with the more rigorous model reported by the authors (Wang et al., 2007a), the difference is about 30%. Since there are a number of adjustable parameters, such as apertures of macro- and meso-pores, and Archie’s parameter, the error can be reduced to an acceptable level by adjusting selected parameters. Numerical results predicted by the simplified models developed this way are similar to that predicted by more involved models, but the physical significance of the adjusted parameters is sacrificed. It can be shown that all parameters in Equation (85) are computable offline, whereas the parameter τq in Equation (88) must be identified using the measurement data. As a result, Equation (85) predicts the dynamic behavior better than Equation (88) and can be applied where direct experimental information lacking. Furthermore, Equation (85) provides more physical insights than Equation (88). Consequently, Equation (85) is recommended as one of the key model equations, and Equation (88) is treated as a supplementary part of the model. Equation (26) is a general representation for the extended Langmuir and LangmuirFreundlich isotherms for multi-component systems. It is presented in the paper for generality. Because of the big difference in adsorption energies for CH4 (13-22 KJ/mol) and He (only about 2 KJ/mol) as shown by figure 5 and the exponential relationship between Langmuir constant b and adsorption energy, adsorption of He is negligible compared with CH4. The fact that standard Langmuir and Langmuir-Freundlich isotherms for single component systems can be applied to CH4-He systems without introducing unacceptable errors allows further model simplification.

MECHANISMS OF ADSOPTION INDUCED COAL SWELLING It has long been recognized that adsorption in porous materials is accompanied by adsorption induced deformation (Wang et al., 2007a). We show in the previous section that this deformation affects coal permeability significantly, a result suggesting increased research in the relevant areas for CBM and ECBM processes. In spite of the advances achieved in the areas of sorption induced deformation, the mechanisms leading to this practically important phenomenon are only known very approximately. Based on previous research, including

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F. Y. Wang, Z. H. Zhu, P. Massarotto and V. Rudolph

comprehensive mechanistic studies carried out at The University of Queensland, we propose three major mechanisms in this section.

Adsorption Induced Carbon Bond Extension: Basal Plane Enlargement Through molecular orbital theory calculations using the software package Gaussian 98, Zhu, Lu and Wang (2005) have determined carbon-carbon (C-C) bond extensions during hydrogen adsorption processes. In that work, three types of adsorption sites on single layer graphite are investigated: (1) the “on-top site” directly above a carbon atom (called t site); (2) the “bridging site” above a C-C bond (called b site); and (3) the “middle hollow site” above a hexagon (called m site). These different sites are graphically shown in the original paper (Zhu et al., 2005). The following two models were proposed in the previous study: model 1-A has an odd electron number with an open-shell electronic state, and model 2-B has an even electron number with a closed-shell electronic state. A very important and consistent observation is that the adjacent C-C bond of the anchoring carbon of model 1-A is extended from ca. 1.42 Å before H adsorption to ca. 1.51 Å after H adsorption. This implies about 7% C-C bond extension in the H adsorption process. Of course, the single C-C bond extension does not completely contribute to the overall swelling. Because of the complexity of graphite and pore arrangements in coals and to space confinement and energy dissipation, only a small part of the bond extension shows up as overall coal size enlargement. However, there seems little doubt that the C-C bond extension during adsorption is an important mechanism for adsorption induced coal dimensional change. It should be noted that the C-C bond extension is mainly along the basal plane. The edge-site plane deformation will be explained in the next section.

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Adsorption Induced Solvation Pressure Variation: Edge-Site Plane Deformation Ustinov and Do (2006) believe that the solvation pressure due to adsorption of fluids in porous materials is the cause of elastic deformation of an adsorbent. Such a deformation contributes to the Helmholtz free energy of the whole adsorbent-adsorbate system due to accumulation of compression or tension energy by the solid. This means that in the general case, the solid is not just a source of the external potential field for the fluid confined in the pore volume, but should be thermodynamically treated as contributing to a combined solidfluid system. They presented analysis of nitrogen adsorption isotherms and heat of adsorption in slit graphitic pores accounting for the adsorption deformation by means of non-local density function theory (NLDFT). The solvation pressure ps is defined as follows:

p s = ΔΠ − pb =

Π ss + Π sf − Π 0 − pb

(89)

where pb is the bulk pressure, Π0 the initial pressure at zero loading acting between pore walls separated by an initial distance H0, Πss the pressure acting between pore walls at non-zero loadings with pore size variations, and Πsf the pressure acting on pore walls by fluids. Although the analysis was carried out for idealized systems at low temperature (77K), the

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single pore deformation data reported by Ustinov and Do (2006) are very useful for the study on the mechanisms of adsorption deformation. We assume that the Young’s modulus of coal matrix is 3 GPa. There is about 13% pore size enlargement for a pore with initial width of 6.1 Å. The deformation is more pronounced for larger pores with the same Young’s modulus. For example, 21% pore size enlargement for a pore with initial width as 8.4 Å is estimated based on the computations carried out by Ustinov and Do (2006). Through a comparative study between basal plane and edge-site plane deformations, it can be shown that although edge-site plane deformations are more significant than for basal planes, the extension along both needs to be taken into account.

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Adsorption Induced Carbon-Metal Bond Extension There are a significant number of metal atoms adsorbed on the inner surfaces of coals. It has been demonstrated experimentally (Zhang et al., 2004) that the Pt-C bond is reduced from 2.62 Å to 2.02 Å after the removal of the atomically adsorbed hydrogen in the interface between the metal particles and the carbon support. This represents a change of 30%. It may be surmised that this also happens to other metal-carbon bonds resulting in similarly significant dimensional changes. Further studies are necessary in this area and we are pursuing detailed studies on metal-carbon bonds using an ab initio method to provide accurate predictions of metal-carbon bond lengths (following e.g. Zhu and Lu, 2004). The three mechanisms coexist, playing different roles. We speculate that the adsorption induced carbon bond extension quantified by the authors (Zhu et al., 2005) could be the dominant mechanism for coal permeability variations. Edge-site plane deformation due to the solvation pressure change (Ustinov and Do, 2006) and the carbon-metal bond extension (Zhang et al., 2004; Zhu and Lu, 2004) in adsorption processes may mainly affect mass transfer in micro-pores with relatively small effects on cleat permeability. We reason as follows: (1) The length of the basal plane is larger than that of the edge-site plane, leading to more extensive stress transmission and strain propagation; (2) Due to the softness of the edgesite plane and carbon-metal bonds, deformation may cause porosity relocations within the coal matrix before reaching the cleat zones. In contrast to the edge-site plane, the stiffness and high density of the basal plane provide relatively smaller opportunity for the deformation to be accommodated within the coal matrix. It should be pointed out that microscopic level studies on adsorption deformations need to be carried out to explore the deformation mechanisms, for example using SANS/SAXS. The aforementioned quantitative values cannot be directly applied to macroscopic level coal swelling/shrinkage in sorption processes due to the complexity of coal structures. However, the mechanistic understanding on adsorption induced coal size changes will certainly help to develop physically based and validated models in the near future.

CONCLUSION Through comprehensive simulation and experimental studies, the following conclusions can be drawn.

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336

F. Y. Wang, Z. H. Zhu, P. Massarotto and V. Rudolph 1. An integrated modelling approach to the dynamics of coupled mass transfer and variations of geo-mechanical properties in porous media with particular applications to CO2 sequestration in coal seams is developed. The process under study can be depicted by a spectrum of representations ranging between a very simple black-box model based on input-output data only, and a highly complex mechanistic model consisting of several sub-models described by a large scale partial differential equation (PDE) system, along with adequate parameter identification techniques. The development of an intermediate “upper-middle class” model is described in detail in this article. The model is represented by a system of matrix partial differential equations incorporating the Stefan-Maxwell analysis, geo-mechanical analysis, online computations of physical properties, determination of pore size dependent parameters, and the EST-SFA procedure. 2. The overall model integrates key sub-processes represented by a number of submodels. These include fluid flow in cleats, dynamic coal permeability-dimension variations induced by gas adsorption and external stress variations, fluid flow and diffusion in macro- and meso-pores, adsorption and surface diffusion within micropores. Furthermore, parameter estimation is done through thermodynamic computations, molecular simulations, and online identification. Most important parameters are computed using well established equations with a minimum of empirical curve-fitting. Consequently, the overall model, which has been constructed from literature based sub-models, provides a more complete and flexible representation. 3. A number of model order reduction strategies are developed and demonstrated using illustrative application examples. In particular, the hybrid distributed-lumped parameter model with significantly reduced model complexity may be successfully applied to staged depressurization processes for the accelerated methane residual recovery from deep coals. Physical insights related to accelerating the recovery include the establishment of new adsorption equilibrium, intensification of both diffusive and conductive mass transfer, and stress/adsorption induced coal structure variations. The simplicity of this reduced model, makes it particularly relevant for scoping studies related to system optimization and for process control. 4. It has been identified that the convective mass transfer between the coal matrix and fractures is more significant than the diffusion flux at the early depressurization stage. This observation is very important for the design of optimal depressurization conditions. However, the convective mass transfer in the coal matrix is ignored in many reported models, such as the bi-disperse models (e.g. Shi and Durucan, 2003; Cui et al., 2004). 5. Three main mechanisms of coal swelling and pore size changes during adsorption and desorption processes are analysed through molecular simulations and thermodynamic computations. We speculate that the adsorption induced C-C bond extension is responsible for the dimensional changes of the basal planes, which could be the dominant mechanism for permeability variations, and the interactions between adsorbate and adsorbent molecules are the driving forces leading to the dimensional variations of the edge-site planes. Notable C-metal distance changes during adsorption/desorptin processes may lead to dimensional variations of both basal and edge-site planes.

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6. The macroscopic level model is validated using a true tri-axial stress coal permeameter (TTSCP), which provides accurate dynamic measurements of gas flowrates, compositions, temperatures and pressures in three orthogonal directions. 7. Misleading results could be obtained using the mean size based approximations for micro-pores due to highly nonlinear characteristics of the functions connecting pore size with adsorption energy, adsorption energy with Langmuir parameters, and adsorption energy with surface diffusivity. Consequently, the simplified methods based on mean micro-pore sizes should be used with caution.

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REFERENCES Adler, P.M. Porous media: geometry and transport. Butterworth-Heinemann: Boston, 1992. Bae, J.S., Bhatia, S.K. High-pressure adsorption of methane and carbon dioxide on coal. Energy & Fuels. 2006; 20: 2599-2607. Bhatia, S.K.; Nicholson, D. Hydrodynamic origin of diffusion in nanopores. Physical Review Letters. 2003; 90: 016150-1 – 016150-4. Box, G.E.P. & Draper, N.R. Empirical Model-Building and Response Surfaces. Wiley: New York, 1987. Clarkson, C.R.; Bustin, R.M. The effect of pore structure and gas pressure upon the transport properties of coal: a laboratory and modelling study. 2. Adsorption rate modelling. Fuel. 1999; 78: 1345-1362. Cui, X.J.; Bustin, R.M.; Dipple, G. Selective transport of CO2, CH4 and N2 in coals: insights from modelling of experimental gas adsorption data. Fuel. 2004; 83: 293-303. Do, D.D.; Wang, K. A new model for the description of adsorption kinetics in heterogeneous activated carbon. Carbon. 1998; 36: 1539-1554. Do, D.D.; Do, H.D. Non-isothermal effects on adsorption kinetics of hydrocarbon mixture in activated carbon. Separation and Purification Technology. 2000; 20: 49-65. Finlayson, B.A. The Method of Weighted Residuals and Variational Principles: with Applications in Fluid Mechanics, Heat and Mass Transfer. Academic Press: New York, 1972. Finlayson, B.A. Numerical Methods for Problems with Moving Fronts. Ravenna Park Pub: Seattle, 1992. Gamson, P.D.; Beamish, B.B.; Johnson, D.P. Coal microstructure and micropermeability and their effects on natural gas recovery. Fuel. 1993; 72. Gilman, A.; Beckie, R. Flow of coal-bed methane to a gallery. Transport in Porous Media. 2000; 41: 1-16. Goodman, R.E. Introduction to Rock Mechanics (2nd edition). Wiley: New York, 1989. Hangos, K.M.; Cameron, I.T. Process Modelling and Model Analysis. Academic Press: San Diego, 2001. Harpalani, S. Gas flow characterization of Illinois coal. Final Technical Report, ICCI Project Number: 03-1/7.1B-2, 2005. Jackson, R. Transport in Porous Catalysts. Elsevier: Amsterdam, 1977. Jakubov, T.S.; Mainwaring, D.E. Adsorption-induced dimensional changes of solids. Phys. Chem. Chem. Phys. 2002; 4: 5678-5682.

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Karger, J.; Ruthven, D.M. Diffusion in Zeolites and Other Microporous Solids. Wiley: New York, 1992. Kreyszig, E. Advanced Engineering Mathematics (4th edition). John Wiley & Sons: New York, 1999. Krishna, R.; van Baten J.M. Diffusion of alkane mixtures in zeolites: Validating the MaxwellStefan formulation using MD simulations. J. Phys. Chem. B. 2005; 109: 6386-6396. Krishna, R.; van Baten, J.M. Describing binary mixture diffusion in carbon nanotubes with the Maxwell-Stefan equations. An investigation using molecular dynamics simulations. Ind. Eng. Chem. Res. 2006; 45: 2084-2093. Krishna, R.; Wesselingh, J.A. The Maxwell-Stefan approach to mass transfer. Chemical Engineering Science. 1997; 52: 861-911. Levine, J.R. Coalification: The evolution of coal as source rock for oil and gas. in AAPG Studies in Geology No. 38 “Hydrocarbon from Coal” edited by Law, B.E.; Rice, D.D., pp. 39-77. American Association of American Geologists: Tulsa, OK, 1993 (ISBN 089181-046-3). Ljung, L. System Identification: Theory for the User. Prentice-Hall: Upper Saddle River, NJ, 1987. Massarotto, P. 4-D coal permeability under true triaxial stresses and constant volume conditions. PhD thesis, The University of Queensland, Australia, 2002. Mathur, G.P.; Thodos, G. The self-diffusivity of substances in the gaseous and liquid states. AIChE Journal. 1965; 11: 613-616. Morsi, B.I.; Schroeder, K.T. Sequestration of carbon dioxide in coal with enhanced coal-bed methane recovery – A review. Energy & Fuels. 2005; 19: 659-724. Palmer, I.D.; Mansoori, J. How permeability depends on stress and pore pressure in coalbeds: A new model. SPE Reservoir Evaluation & Engineering. 1998; September issue: 539-544. Reid, R.C.; Prausnitz, J.M.; Poling, B.E. The properties of gases & liquids (4th edition). McGraw-Hill: New York, 1987. Sahimi, M. Flow and Transport in Porous Media and Fractured Rock: From Classical Methods to Modern Approaches. VCH: Weinheim, 1995. Shi, J.Q.; Durucan, S. A bidisperse pore diffusion model for methane displacement desorption in coal by CO2 injection. Fuel, 2003; 82: 1219-1229. Sotirchos, S.V. Multicomponent diffusion and convection in capillary structures. AIChE Journal. 1989; 35: 1953-1961. Sotirchos, S.V.; Burganos, V.N. Analysis of multicomponent diffusion in pore networks. AIChE Journal. 1988; 34: 1106-1118. Steele, W.A. The Interaction of Gases with Solid Surface. Oxford: Pergamon, 1974. Tsotsis, T.T.; Patel, H.; Najafi, B.F.; Racherla, D.; Knackstedt, M.A.; Sahimi, M. Overview of laboratory and modelling studies of carbon dioxide sequestration in coal beds. Ind. Eng. Chem. Res. 2004; 43: 2887-2901. Ustinov, E.A.; Do, D.D. Effect of adsorption deformation on thermodynamic characteristics of a fluid in slit pores at sub-critical conditions. Carbon. 2006; 44:2652-2663. Van Krevelen DW. Coal: typology – physics – chemistry – constitution (3rd edition). Amsterdam: Elsevier, 1993. Villadsen, J.; Michelsen, M.L. Solution of differential equation models by polynomial approximation. Prentice-Hall, Englewood Cliffs, NJ, 1978.

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Wang, F.Y.; Bhatia, S.K. A generalised dynamic model for char particle gasification with structure evolution and peripheral fragmentation. Chemical Engineering Science. 2001; 56: 51-65. Wang, F.Y.; Zhu, Z.H.; Massarotto, P.; Rudolph, V. Mass transfer in coal seams for CO2 sequestration. AIChE Journal. 2007a; 53: 1028-1049. Wang, F.Y.; Zhu, Z.H.; Massarotto, P.; Rudolph, V. A simplified dynamic model for accelerated methane residual recovery from coals. Chemical Engineering Science. 2007b; 62:3268-3275. Wang, F.Y.; Zhu, Z.H.; Rudolph, V. Molecular transport in nanopores with broad pore-size distribution. AIChE Journal. 2008; DOI 10.1002/aic.11520. Wang, K.; Do, D.D. Multicomponent adsorption, desorption and displacement kinetics of hydrocarbon on activated carbon – dual diffusion and finite kinetic model. Separation and Purification Technology. 1999; 17: 131-146. White, C.M.; Smith, D.H.; Jones, K.L.; Goodman, A.L.; Jikich, S.A.; LaCount, R.B.; DuBose, S.B.; Ozdemir, E.; Morsi, B.I.; Schroeder, K.T. Sequestration of carbon dioxide in coal with enhanced coal-bed methane recovery – A review. Energy & Fuels. 2005; 19: 659-724. Yang RT. Gas Separation by adsorption processes. London: Imperial College Press, 1997. Zhang, Y.; Toebes, M.L.; van der Eerden, A.; O’Grady, W.E.; de Jong, K.P.; Koningsberger, D.C. Metal particle size and structure of the metal-support interface of carbon-supported platinum catalysts as determined with EXAFS spectroscopy. J. Phys. Chem. B. 2004; 108:18509-18519. Zhu, Y., Lu, X., Zhou, J., Wang, Y., Shi J. Prediction of diffusion coefficients for gas, liquid and supercritical fluid: application to pure real fluids and infinite dilute binary solutions based on the simulation of Lennard-Jones fluid. Fluid Phase Equilibria. 2002; 194197:1141-1159. Zhu, Z.H.; Lu, G.Q. Comparative study of Li, Na, and K adsorptions on graphite by using ab initio method. Langmuir. 2004; 20: 10751-10755. Zhu, Z.H.; Lu, G.Q.; Wang, F.Y. Why H atom prefer the on-top site and alkali metals favour the middle hollow site on the basal plane of graphite. J. Phys. Chem. B. 2005; 109: 79237927.

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In: Progress in Porous Media Research Editors: Kong Shuo Tian and He-Jing Shu

ISBN: 978-1-60692-435-8 © 2009 Nova Science Publishers, Inc.

Chapter 7

INFLUENCE OF THE INTERFACIAL DRAG ON PRESSURE LOSS FOR TWO PHASE FLOW AND COOLABILITY IN POROUS MEDIA Werner Schmidt AREVA NP, P.O.Box 3220, D 91050 Erlangen, Germany

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ABSTRACT This article investigates the influence of the interfacial drag on the pressure loss of combined liquid/vapour flow through porous media. This is motivated by the coolability of fragmented corium with internal heat sources, which are expected during a severe accident in a nuclear power plant. Due to the decay heat in the particles cooling water is evaporated. To reach steady states the out-flowing steam must be replaced by in-flowing water. The pressure field inside the porous structure determines the water ingression, and in effect the overall coolability. Typically, correlations for the dryout heat flux of porous media are adjusted to measurements where water ingression is from a pool positioned above. In these models the nature of the two-phase flow is included in corrections of the permeability and passability, achieved by simple functions of the void fraction. However, already configurations with possible water ingression from below demonstrate that this treatment is insufficient. The drag between the liquid and the vapour phase supports the co-current water inflow from below, and hinders the counter-current flow from above. Thus, the interfacial drag must be explicitly included in the modeling. This necessity is already seen in the measured pressure loss of simple isothermal air/water flow in porous media as well as for boiling particle beds with water ingression from below. Based on such experiments, two models with explicit consideration of the interfacial drag from the literature are discussed. Different flow patterns for the drag coefficients are included with the most advanced of these models. This article proposes some modifications on this model with respect to the small particle sizes that are expected in the reactor application. Furthermore, modifications to the formulation in the annular flow regime are necessary, as here the original model yields unreliable results. Based on the friction laws referred, the models are applied to typical reactor invessel and ex-vessel configurations in two-dimensional geometry. An enhanced overall coolability of the particle bed is already reached by the 2-D nature of the arrangement.

342

Werner Schmidt Furthermore, the supporting influence of the realistic modeling by explicit consideration of the interfacial drag in the enhanced models is shown.

Keywords: porous media, two-phase flow, interfacial friction, coolability, severe accident.

NOMENCLATURE Latin symbols

Db

m

diameter off the gas bubbles

dp

m

particle diameter

Fp

N/m

Fi

N/m

F∗ g

-

j

3

volumetric particle-fluid drag force

3

volumetric gas-liquid drag force dimensionless force

m/s m/s

gravitational constant

2

superficial velocity of the fluids

jr

m/s

K Kr

m -

P∗ p

-

dimensionless pressure gradient

Pa

pressure

relative velocity ⎛

j j ⎞ ⎜⎜ jr = g − l ⎟⎟ α 1 α⎠ − ⎝

permeability

2

relative permeability of the fluids

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Greek Symbols

α β ε η ηr μ σ ρ

-

void, gas volume fraction in the pores volume fraction

m

porosity passability

m

relative passability of the fluids

kg/m s

dynamic viscosity

N/m

surface tension density of the phase

kg/m

3

Indices g l

gas,vapour liquid

Influence of the Interfacial Drag on Pressure Loss for Two Phase Flow…

343

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1. INTRODUCTION The flow of fluids through porous structures is of central interest in various scientific and technical fields. Among these are environmental phenomena, as groundwater flow and the propagation of pollutants or oil in the ground, as well as technical applications, for example in the chemical industry, in catalytic reactors or in heat exchangers. In many of these applications more than one kind of fluid has to be considered. If these are immiscible, each fluid must be treated as a separate phase. Additionally, chemical reactions (either exo- or endo-thermal) as well as evaporation or condensation may yield phase change processes between the fluids. In the most common configurations two fluids, one liquid and one vapour component, have to be considered. In view of reactor safety of a nuclear power plant (NPP) such processes with two-phase flow in porous media are at least in discussion since the severe accident in the nuclear power plant TMI-II in Harrisburg (USA) on March 28, 1979. Even after the chain reaction is stopped by regular or emergency shutdown heat is still produced due to ongoing nuclear α , β and γ -decay processes in the fission products. This decay heat yields a specific residual power of 100 to 300 W/kg in the fuel. This power must be removed, even in a very unlikely severe accident with failure of all normal and emergency cooling systems. If this is not possible the reactor core will dry-out, heat-up and melt down. In contact with cooling liquid (water) at least some parts of the melt (corium) will be fragmented and quenched, yielding a porous structure. This may occur either due to reflood in the core region, in the lower head or, after failure of the reactor pressure vessel (RPV), in the reactor cavity. Long term coolability of the corium must be reached to avoid environmental pollution with radioactive material, remelting and further destruction, especially the failure of safety barriers. Therefore, the cooling potential of such porous structures with internal power sources must be observed. The knowledge about the coolability of strongly destroyed core structures and relocated core material in the late phase of severe accidents in nuclear power plants is the central point in the evaluation of safety margins. This directly raises in the questions of accident management (AM). Effective cooling and heat removal of the high specific power is reached by evaporation of water. Due to the expected non-availability of forced coolant flow a steady state, and thus coolability, is reached if the produced steam can be discharged and the evaporated water is replaced by inflow driven by natural forces. In a pure 1-D configuration with a water reservoir on top of the porous bed, the up-flowing steam and the down-flowing water are in counter-current flow configuration. Overall coolability is reached, if evaporated water can be replaced every position. Thus, a limit of coolability, corresponding to the dryout heat flux (DHF), is determined by the counter-current flooding limit for this flow pattern. A significantly higher DHF is reached, if at least some fraction of the water can flow into the porous bed via the bottom, as sketched in figure 1 (b). This requires less water influx from the top, enabling higher steam fluxes and thus an increased DHF.

0110 111 00 1011 00 01 1011 0011 11 00111 000 00 11 00 11 10000 00 11 0 1 00 11 00 11 0 1 00 11 1 0 111 000 00 11 00 11 00 11 0 1 1011 1 0 00 11 00 000 111 1 0 11 00 00 11 0 1 1100 00 0 1 0 1 00 11 0111 01101011 1110111 00 11 000 00 00 11 1 0 10 00 01 01111 000 00 01 010110100111

(a) top-fed

Water

Werner Schmidt

Debris

344

(b) top- and bottom-fed

Figure 1. One-D geometries.

The overall coolability depends on several parameters: •

porosity

The pore volume fraction (porosity) defines the contents as well as the flow cross sections for the fluids. Thus, the greater the porosity the better coolability will be. For granular particle beds porosities in the range of 0.2 to 0.5 are reached, depending on the size distribution and shape. •

particle size

The pore size is defined by the effective particle diameter. Greater diameters yield greater flow paths and thus increased coolability. For debris of fragmented corium particle diameters of 1 to 5 mm are expected.

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•

system pressure

The system pressure has two concurring aspects of influence on the steam properties. Fist of all, the steam density increases with increasing pressure. Thus, more steam mass can be transported in the limited volume flow, yielding a strong increase in coolability until a pressure of approximately 5 bar is reached. For higher pressures the coolability slowly decreases due to the counter effect of the decreasing latent heat of the evaporation process. •

bed height

A further aspect of the influence of the configuration is the overall bed height. For very shallow configurations the coolability is significantly increased due to capillary forces at the interface from the particle bed to the pure water environment, even for 1-D top-fed conditions. This influence can be neglected in the deep bed limit that is reached for beds higher than about 50 times the bed particle diameter (see [14, 17, 8]). Thus, with the expected grain sizes the deep bed limit is already reached for bed heights above 25 cm. Relating this

Influence of the Interfacial Drag on Pressure Loss for Two Phase Flow…

345

value to the expected melt height in reactor safety applications shows that the deep bed limit must always be applied.

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•

coolant inflow

Significantly more power can be removed if at least some fraction of the water can flow into the porous bed via the bottom, as e.g. sketched in figure 1 (b). This requires less water influx from the top, enabling higher steam fluxes and thus an increased DHF. This previous discussion already shows that the definition of a DHF depending on particle bed parameters is not possible. The maximum power that can be removed by evaporation of water significantly depends on the flow boundary conditions, especially the water inflow paths. In particular, water inflow from below supports the overall coolability because then at least some fraction of the liquid has not to pass the bottle-neck of the countercurrent-flooding limit that is reached near the top. As consequence, investigations of the debris coolability during severe accidents in a NPP require detailed analyses. Several experimental and theoretical programs on debris coolability have been performed, especially in the early 1980's. The objective of most of these programmed was to determine the maximum power that can be removed from a heated particle filled column with a water reservoir on the top. The internal heating is produced either by resistance heaters (e.g. Hu and Theofanous [9]), by inductive heating (e.g. Hofmann [8]), or by direct neutron irradiation in a reactor (DCC-experiments [13]). A local dryout is detected by the temperature increase at thermocouples located inside the particle bed. Based on these data, correlations for the friction laws of liquid and vapour were deduced. These correlations have then been used by e.g. Lipinski [10] in a model to calculate the critical heat flux, that corresponds to the maximum bed power before reaching a local dryout anywhere in the bed. It has to be noted here, that this critical heat flux is only relevant under pure 1-D top-fed conditions, as already discussed. Besides the boiling bed experiments, other authors (e.g. Chu et al. [3] or Tutu et al. [19]) used pressure loss measurements in isothermal air/water flow experiments to investigate the drag forces in the particle bed. New experimental programs to investigate the coolability of particulate corium have been begun in recent years. The central aim of the SILFIDE experiments at EdF [5] is the measuring of effects resulting from the two-dimensional geometry. Unfortunately, this experimental program was stopped. A different attempt was made in the DEBRIS experiments performed at IKE [15]. These experiments are oriented towards specific investigations of the exchange terms. The friction and heat transfer laws are observed for boiling beds, as well as during the quenching of hot dry particles.

2. ONE-PHASE FLOW THROUGH POROUS MEDIA Solving the complete set of Navier-Stokes equations for the flow through disordered porous media is a difficult and complex task. Therefore a macroscopic treatment via pressure loss correlations is required. The first attempt was by Darcy [4] who showed that for laminar

346

Werner Schmidt

flow the pressure loss is proportional to the fluid volume flux. Dividing this flux by the cross section yields the superficial velocity j . Ergun [6] extended Darcy's law to higher velocities by adding the quadratic Forchheimer term to include a ''turbulent'' friction, also including inertia effects:

Δp μ ρ = j+ | j |j−ρg l K η

(1)

where L is the length over which the pressure difference Δ p is measured, μ is kinetic viscosity and ρ density of the fluid. The parameters K and η are called permeability and passability, respectively, and depend on the specific parameters of the porous media. According Ergun they are calculated by:

ε 3 d p2 K= A(1 − ε ) 2

and η =

ε 3dp B (1 − ε )

(2)

where the Ergun-constants A and B are taken from pressure loss measurements in granular debris. Typically, A=150 and B=1.75 are used. Andrade et. al. [1] demonstrated that the pressure loss equation given above is a sufficient phenomenological model for the friction of flow in porous media over a wide range of Reynolds numbers. Thus, it is adequate to use equation (1) for steady state or near steady state flow through porous media.

3. TWO-PHASE FLOW THOUGH POROUS MEDIA 3.1. Simple Models

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To consider the impact of the smaller effective cross section due to the existence of an additional phase for two-phase flow in porous media the permeability K and the passability η have to be extended by a function depending on the volume fractions. In these relative permeabilities and passabilities the influence of the other fluid phase on the pressure loss is included.

Vapour : K → KK rg (α )

Liquid :

K → KK rl (1 − α )

η → ηη rg (α )

(3)

η → ηη rl (1 − α )

(4)

Inserting this definition into the momentum conservation equations of the fluids leads to:

G μl G ρ G G − ∇pl = ρ l g + jl + l | jl | jl K K rl ηη rl for the liquid, and

(5)

Influence of the Interfacial Drag on Pressure Loss for Two Phase Flow…

μg G ρ G G G − ∇p g = ρ g g + jg + g | jg | jg K K rg ηη rg

347

(6)

for the vapour phase. In these classical models the interfacial friction between the liquid and gas is not included explicitly. Several approaches for the relative permeability and passability, based on dryout heat flux experiments, are found in the literature. The most commonly used approach has the form:

K rg (α ) = α n

η rg (α ) = α m

(7)

K rl (α ) = (1 − α ) n

η rl (α ) = (1 − α ) m

(8)

The relative permeability K r was determined experimentally for laminar flow 3 conditions by Brooks and Corey [2]. Simplifying their results, Lipinski used K r =β ( β = phase fraction in the pores) and assumed the same exponent for the relative passability η r =β 3 too [10]. Based on his own dryout experiments yielding a smaller dryout heat flux, Reed [12] proposed an exponent m=5 for the relative passabilities. This exponent was also used by Lipinski in later publications [11]. The dryout heat flux calculated with this approach fits the experimental values for top-fed particulate bed configurations. This heat flux was measured for beds heated either by resistance or inductive heating, as well as in-pile by neutron irradiation in a reactor. In the experiments usually the measuring procedure was iterative. By increasing the bed power, and observing the largest value before the temperature in any part of the bed starts to rise, the critical heat flux was determined. Theofanous criticised this measurement method [9], and explained that the resulting dryout heat flux values would be too high, because the power was increased further before the dryout was detected. Additionally, he pointed out that in most of the experiments the ratio of the test section diameter to the particle diameter was too small, yielding an increased water inflow along the walls. So he proposed an exponent m=6 to increase the friction between the fluids and the particles, yielding a smaller dryout heat flux. A summary of the classical models is given in table 1.

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Table 1. Relative permeability K r and passability η r in the classical formulations

K rg

η rg

K rl

η rl

Lipinski

α3

α3

(1 − α ) 3

(1 − α ) 3

Reed

α3

α5

(1 − α ) 3

(1 − α ) 5

Theofanous

α3

α6

(1 − α ) 3

(1 − α ) 6

348

Werner Schmidt

3.2. Models Including Explicit Interfacial Friction A more detailed look at the equations offers a significant disadvantage of the simple modeling. As Tutu [19] already pointed out in his isothermal air/water experiments, an explicit consideration of the interfacial friction is necessary. In his experiments he used a onedimensional test column filled with stainless steel spheres. A defined air mass-flow was injected from below into the water filled test column. In steady state the gas is sparkling through the test section. The pressure loss inside the bed was measured. Additionally, the level swell of the water may be used to determine the void fraction inside the bed. As there is G no net water flow ( jl = 0 ), equation (5) yields − ∇p l = ρ l g . Thus, the resulting pressure field remains constant, independently of the existence or the magnitude of the gas flux. This is not reliable and disproved by the experiments (see e.g. figure 3). This directly indicates the influence of the interfacial friction. The experiment and the results are discussed in detail in section 4.1. So, equations (5) and (6) have to be extended by the interfacial friction, yielding the liquid equation:

G G Fi μl G ρl G G − ∇pl = ρ l g + | jl | jl − jl + ηη rl 1−α K K rl

(9)

and the vapour equation:

G μg G ρ g G G Fi G − ∇p g = ρ g g + | jg | jg + jg + K K rg ηη rg α

.

(10)

3.2.1. Schulenberg Model

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Based on similar isothermal air/water experiments Schulenberg and Müller [16] correlated their data, and deduced an equation for the interfacial friction from the measured pressure loss and the liquid momentum equation (9):

⎛ jg ρK jl ⎞ ⎟⎟ Fi = 350(1 − α ) α l ( ρ l − ρ g ) g ⎜⎜ − ησ ⎝ α (1 − α ) ⎠

2

7

(11)

Inserting this formulation for the interfacial drag into the vapour equation and assuming a 3 relative permeability of K rg =α , Schulenberg and Müller determined the relative passability η rg for the vapour phase. For the liquid phase they assumed the same exponents as Reed. Their result is:

K rg (α ) = α

3

⎧0.1α 4 α ≤ 0.3 η rg (α ) = ⎨ 6 ⎩ α α > 0.3

(12)

Influence of the Interfacial Drag on Pressure Loss for Two Phase Flow…

K rl (α ) = (1 − α ) 3

η rl (α ) = (1 − α ) 5

349 (13)

3.2.2. Tung/Dhir Model A completely different approach has been proposed by Tung and Dhir [18]. Based on visual observation in air/water flow experiments of Chu et al. [3] they defined flow pattern ranges for bubbly, slug and annular flow. Besides a sub-pattern in the bubbly flow regime, only relevant for particle sizes greater than 12 mm, the flow pattern limits are given in table 2. Tung and Dhir defined a weighting function for the transition region between these flow patterns, such that the result is continuous to the first derivative at the connecting points:

W (ξ ) = ξ 2 (3 − 2ξ ) , where ξ =

α − αi α i +1 − α i

(14)

Based on the definition of the permeability K and passability η in equations (2), Tung and Dhir used geometrical arguments, and inserted an effective porosity and particle diameter that is exposed to the gas flow. This guided them to the following formulation for the relative permeability and passability of the gas for the different flow patterns: Particle gas drag F pg

0≤α ≤α 3 (bubbly and slug flow) 4/3

⎛ 1− ε ⎞ ⎟⎟ α 4 K rg =⎜⎜ − ε α 1 ⎝ ⎠

2/3

⎛ 1− ε ⎞ ⎟⎟ α 4 and η rg =⎜⎜ − ε α 1 ⎝ ⎠

(15)

α 3 ≤α ≤α 4 (transition) α 3 ≤α ≤1 (pure annular flow)

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4/3

⎛ 1− ε ⎞ ⎟⎟ α 3 K rg =⎜⎜ ⎝1− εα ⎠

2/3

⎛ 1− ε ⎞ ⎟⎟ α 3 and η rg =⎜⎜ ⎝1− εα ⎠

(16)

Similar ideas for the liquid part are based on the visual observation that the liquid is always in contact with the particles like a film. So they remain the particle diameter unchanged and only included an effective volume fraction. By this they deduced: Particle liquid drag F pl

K rl =η rl = (1 − α ) 4

(17)

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Werner Schmidt

It should be noted that the above given exponents are increased by 1 from 3 to 4 compared to the original paper, because the momentum equations (9) and (10) are divided by the volume fraction in this work, in contrast to the formulation in [18]. Table 2. Flow regime bounds of the Tung / Dhir model

α1

min (0.3,0.6(1 − γ ) 2 )

transition

π

α2

bubbly flow

≈ 0.52

6

slug flow

α3

0.6

α4

π 2 6

transition

≈0.74

annular flow

Interfacial drag Fi

Additionally to the drag at the solid matrix, Tung and Dhir also deduced a correlation for the liquid/vapour interfacial friction Fi . For bubbly and slug flow this drag force is based on an expression for the drag on a single bubble or slug, multiplied by the number of bubbles or slugs per unit volume. A detailed description can be found in the original paper [18]. Here only the results are provided. The interfacial drag is defined by:

Fi = C1

μl

(1 − α ) jr + C2 2

Db

((1 − α ) ρ l + α ρ g ) Db ε

(1 − α ) 2 | jr | jr

(18)

Here the relative velocity j r is defined by:

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jr =

jg

α

−

jl (1 − α )

(19)

Based on visual observation, Tung and Dhir defined the diameter of the bubbles or slugs by:

Db = 1.35

σ g ( ρl − ρ g )

The friction coefficients are given separately for bubbly and slug flow:

0≤α ≤α 1 (bubbly flow)

(20)

Influence of the Interfacial Drag on Pressure Loss for Two Phase Flow…

C1 = 18α

and

C2 = 0.34(1 − α ) 3 α

351 (21)

α 1 ≤α ≤α 2 (transition) α 2 ≤α ≤α 3 (slug flow) C1 = 5.21α

and

C2 = 0.92(1 − α ) 3 α

(22)

α 3 ≤α ≤α 4 (transition)

α 4 ≤α ≤1 (pure annular flow) In the annular flow regime the interfacial drag is modeled in a manner similar to the particle-gas drag by using the relative velocity between the gas and the liquid. This yields:

Fi =

μg K K rg

(1 − α ) jr + (1 − α )α

ρg | j r | jr . ηη rg

(23)

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The relative permeability and passability to be inserted here are the same as for the gasparticle drag and are given in equation (15). As before, the transitions between the flow regimes are to be interpolated using the interpolation function defined in equation (14).

3.2.3. Modifications of the Tung/Dhir Model Tung and Dhir compared their model to the measured pressure gradients, as well as the void fractions of the isothermal air/water experiments of Chu et al. [3]. They found a good agreement with the experimental data. But, Chu et al. used relatively large particles of d p =5.8 mm, d p =9.9 mm and d p =19 mm in their experiments. As already could indicated in the paper [18] for d p =5.8 mm, the Tung/Dhir model is not well applicable for smaller particles. Examing at the diameter of the gas bubbles used by Tung and Dhir, this discrepancy can easily be understood. Inserting the density difference between air and water as well as the surface tension into equation (20) yields a bubble diameter of Db =3.75 mm. Thus, the applied bubble diameter become larger than the pores for small particles, in contradiction to the geometric ideas of the model. Additionally, the assumption of gas bubbles in the pores becomes questionable for small particles. This requires some modifications of the original Tung/Dhir model to extend it to smaller particle diameters. The first point to be modified, is the diameter of the gas bubbles or the slugs. This diameter strongly influences the interfacial drag in bubbly and slug flow, as could be seen in equation (18). Based on the assumption of a cubic arrangement of spherical particles a maximum diameter of Db =d p ( 2−1) may be deduced for a bubble. To obtain a max connection to the original Tung/Dhir model, a modified bubble diameter is simply defined by:

352

Werner Schmidt

⎛ σ Dbm = min ⎜ 1.35 , 0.41d p ⎜ − g ( ) ρ ρ l g ⎝

⎞ ⎟ . ⎟ ⎠

1

original modified

0.9

annular flow

0.8 0.7

void α

0.6

slug flow

0.5 0.4 0.3 0.2

bubbly flow

0.1 0

0

2

4

6

8

10

particle diameter dp [mm]

12

14

(24)

⎧π/6 : d p < 8mm ( d − 8mm) + α 1 ⎪ α 1m = ⎨ 5 p : d p > 8mm ⎪⎩α 1 ⎧π/6 ( d − 8mm) + α 2 ⎪ α 2m = ⎨ 5 p ⎪⎩α 2

: d p < 8mm : d p > 8mm

⎧π/6 ( d − 8mm) + α 3 : d p < 8mm ⎪ α 3m = ⎨ 5 p : d p > 8mm ⎪⎩α 3 ⎧π/6 ( d − 6 mm) + α 4 ⎪ α 4m = ⎨ 5 p ⎪⎩α 4

: d p < 6mm : d p > 6mm

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

Figure 2. Flow pattern map for the modified Tung/Dhir model.

The second point to be modified are the flow patterns ranges utilized in the model. The original bounds are presented as dashed lines in figure 2. As can be seen, for particle diameters less then 3 mm no pure bubbly flow is expected. This fits the usual understanding. But, for this report a steeper reduction of the bubbly flow regime with decreasing particle diameter is proposed. On the other hand, annular flow in the picture of Tung and Dhir is via gas tubes wiggling along the pores. With decreasing particle size, these pores will get smaller, leading to smaller tube diameter. For small particles this ''tube picture'' becomes questionable. In the annular flow regime fixed gas channels in the order of particle size may be established. Inside a representative control volume, including several pores - in the mean - annular flow may be established even for smaller void fractions. This is also conform to the flow pattern observations characterised by Haga et al. [7] using tracer particles. They reported a channel like configuration for 2 mm spheres down to a void fraction of α =0.3 . An enlargement of the annular flow range for particles less than 6 mm is proposed in this report. Consequently, the bounds for the slug flow regime have also to be modified, as this pattern becomes also questionable for small particles. Slugs will prefer flow paths in the wake of predecessors. Thus, the interfacial friction is then less than for independent slugs. The modifications to the flow pattern bounds are provided in figure 2 by the solid lines. Additionally, further modifications to the interfacial drag in the annular flow regime are necessary. As already discussed above, for smaller particles, channel flow is established. Compared to the classical picture of Tung and Dhir, the interfacial area between gas and liquid is reduced. This motivates a decrease of the drag for decreasing particle diameter. By adaptation to the experimental results of Tutu et al. [19], discussed in detail in the next −3 2 is proposed for particles smaller than 6 mm chapter, a multiplicative factor of d p /6 x10 in this work. Additionally, as can be seen in the formulation of the friction term for annular flow in equation (23), this drag decreases linearly to zero when the void fraction reaches the limit α →1 . This decent seems to be too weak compared to usual correlations. So, in this

(

)

Influence of the Interfacial Drag on Pressure Loss for Two Phase Flow…

353

work an additional multiplier of (1−α ) is proposed to get a more realistic decrease of the interfacial drag in the annular flow regime for increasing void fraction. This leads to the following modified formulation of the interfacial friction in the annular flow regime: 2

⎞ ⎛ μg ρg Fi m = ⎜ (1 − α ) j r + (1 − α )α | jr | jr ⎟ * ⎟ ⎜ KK ηη rg rg ⎠ ⎝ ⎧⎛ d p ⎞ 2 ⎟ ⎪⎜ (1 − α ) 2 ⎨⎜ 6 x10 −3 ⎟ ⎠ ⎝ ⎪ 1 ⎩

: d p < 6mm

(25)

: d p > 6mm

In general, all modifications proposed above influence mainly the formulation of the interfacial drag. As will be seen in the next chapter, in general it is not easy to separate the different friction contributions in the experiments. Interfacial and particle drag are superimposed, but can be separated for specific conditions with no net water flow. On the other hand, for boiling beds - as in the reactor application - this splitting is not possible. One has to rely on the assumption, that the friction laws are valid. Comparisons to experiments with boiling beds are presented in the next chapter. Especially the DEBRIS experiments described in chapter 4.2.1 support the above proposed modifications of the interfacial drag.

4. APPLICATION OF THE MODELS

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4.1. Comparison to Isothermal Air/Water Flow Experiments Isothermal air/water experiments are a sufficient method to investigate the friction laws of two-phase flow in porous media with a simple experimental setup. By fixing the water and air flow rates through a vertical test column filled with particles, defined steady state conditions, either for co- or for counter-current flow, may be established. As the gas flow rate from the bottom to the top is fixed, a constant void fraction establishes over almost the whole bed height. Therefore, the capillary pressure is constant and its gradient is zero. So, the same pressure gradient acts on both fluids. This pressure gradient in the test column can easily be measured. Additionally, the void fraction in the bed can be determined, either by visual observation through transparent vessel walls, or by a change of the water level on the top. The experimental data can be compared to theoretical results of the models to verify the friction laws. For this, the momentum conservation equations (9) and (10) are used in a dimensionless form. Dividing by gε ( ρ l −ρ g ) yields:

(1 − α ) P* = (1 − α )

ρl g + F pl* − Fi * g (ρl − ρ g )

(26)

354

Werner Schmidt

α P* = α

ρg g + F pg* + Fi * g (ρl − ρ g )

(27)

with

P* =

F − ∇p , F* = . g (ρl − ρ g ) gε (ρ l − ρ g )

(28)

The two friction terms of the particle-fluid drag are combined in the force F p . Eliminating * the normed pressure gradient P yields:

α (1 − α )+α Fpl* −(1 − α ) Fpg* − Fi * = 0

(29)

Inserting the friction laws presented in the previous chapter provides an equation with the three unknowns j g , jl and α . Fixing one velocity, usually liquid, and varying the void fraction, the other velocity can be calculated easily by selecting the physically relevant root. Additionally, adding the two equations (26) and (27) yields an equation for the pressure gradient:

P* =

(1 − α ) ρ l + αρ g

ρl − ρ g

(30)

So, sets of j g , jl , P and α may be calculated that can directly be compared with the experimental values. A comparison of calculated results with experimental air/water data measured by Tutu et al. [19] is shown in figure 3 and 4. Air was injected into the bottom of a water filled test column of stainless steel spheres ( d p =6.35 mm). The superficial velocity of the air was varied, corresponding to the mass flux through the test column. For the established steady states the pressure gradient was measured by the difference of two pressure taps. Additionally, the void fraction in the bed was determined by the increase of the liquid level above the bed. In figure 3 the dimensionless pressure gradient is plotted as function of the superficial velocity of the gas. The figure shows that the pressure gradient becomes less than one, indicating a pressure loss due to the gas flow. The classical models, which do not explicitly consider the interfacial friction, cannot reproduce this behaviour. This can already be seen in equation (26). As there is no net water flow the superficial velocity jl =0 leads to Fpl* =0 . With Fi * =0 this yields P * =ρ l /( ρ l −ρ g )≈1 . In these models the pressure field must always be equal to the hydrostatic pressure, uninfluenced by the gas flux. This behaviour is not verified by the experimental data. The measured pressure first decreases with increasing gas flow rate, although the water is still the continuous phase. This can only be explained by including the drag of the up-flowing gas on the liquid. The models of Schulenberg and Tung/Dhir, which include an explicit formulation for the interfacial friction, show this characteristic qualitatively. *

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+ F pl* + F pg*

Influence of the Interfacial Drag on Pressure Loss for Two Phase Flow…

355

As already mentioned in the last chapter, the interfacial friction law in the annular flow regime of the Tung/Dhir model must be modified to obtain a reliable trend for large voids up to one. This can also be seen in figure 3. With increasing gas flow rate, the pressure gradient first strongly decreases below the hydrostatic one due to the drag of the up-flowing gas bubbles and slugs on the liquid. Later, with further increased gas flow, the friction between the phases decreases again for void fractions greater than 0.5, as the interfacial area between the gas and the water decreases. This explains the increase in the normed pressure gradient for higher gas fluxes. A limit is reached when the gas replaces all the water. This is the case when the pressure loss due to the gas flow is equal to the hydrostatic head of the water, independent of the model. This limit is not reached for the original Tung/Dhir model. To compensate this, 2 it is proposed in the previous chapter to multiply the interfacial friction by (1−α ) in the annular flow regime, to obtain a cubic decrease for large void fractions. The comparison with the experimental data, especially the development for high gas fluxes in figure 3, strongly supports this modification of the Tung/Dhir model.

1

0.9

*

p [-]

0.8

Lipinski (p*≈1) * Reed (p ≈1) Hu/Theofanous (p*≈1) Tung/Dhir mod.Tung/Dhir Schulenberg p* Exp. Tutu (1984)

0.7

0.6

dp = 6.35 mm ε = 0.38 j0L = 0.0 mm/s

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

0.5

0.4

0

0.2

0.4

Figure 3. Dimensionless pressure gradient flow.

0.6

P*

0.8

jG [m/s]

1

1.2

1.4

for an isothermal air/water experiment with no net water

356

Werner Schmidt

1 0.9 0.8

void α [-]

0.7 0.6 0.5 0.4 Lipinski Reed Hu/Theofanous Tung/Dhir mod.Tung/Dhir Schulenberg α Exp. Tutu (1984)

0.3 0.2

dp = 6.35 mm ε = 0.38 j0L = 0.0 mm/s

0.1 0

0

0.2

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

Figure 4. Void fraction

α

0.4

0.6

0.8

jG [m/s]

1

1.2

1.4

in the case of figure 3.

Similar conclusions are drawn from the corresponding void data in figure 4. Although the experimental data points are not up to the annular flow regime, the modified Tung/Dhir model fits the experimental points better. This is due to the decreased bubble diameter in the slug and bubbly flow regime ( Db =2.6 mm modified instead of 3.75 mm in the original form), yielding an increased interfacial friction. Additionally, as already described for the pressure gradient, the tendency in the annular flow regime for the original Tung/Dhir formulation is again not reliable. The one-phase flow limit with α =1 should be reached at the same gas velocity for all models, independent of the interfacial drag. The modified formulation, with decreased interfacial friction in the annular flow regime, fulfills this condition. For the special case without net water flow in the porous medium ( jl =0 ), an enhanced analysis is possible. The interfacial friction can be deduced directly from the measured pressure gradient and void fraction via the liquid momentum equation given in (26). The results for the described data, together with the results of the various models, are given in figure 5 as function of the void fraction. While the interfacial friction is always zero for the classical models, a principle agreement of the enhanced models with the experimental data is seen. But, especially the local maximum at higher void (higher j g ) in the original Tung/Dhir formulation seems to be not verified by the experiment. This hump originates in the annular flow regime. Furthermore, as already mentioned, the linear decrease of Fi * for α →1 is unreliable. These faults are eliminated by the modifications of the Tung/Dhir model proposed in chapter 3.2.3, which then yields the best estimate for the experimental data.

Influence of the Interfacial Drag on Pressure Loss for Two Phase Flow…

357

Measurements for larger particles with a diameter of d p =9.9 mm (Chu et al. [3]) and d p =12.7 mm (Tutu et al. [19]) yield similar results. A comparison of the experimental data with the different models is given in Schmidt [14]. For smaller particle diameters, which are to be expected during a severe accident in a nuclear power plant, the models show greater deviations compared to the experimental data. Results of Tutu et al. for d p =3.18 mm are presented in figures 6 and 7. The described modifications of the original Tung/Dhir model - especially the modifications of the flow pattern limits and the modified slug size - are of decisive influence for smaller particle sizes. E.g. the annular flow regime starts at a void fraction of 0.45, while the pure bubbly flow disappears, obviously because the pores are too small for the bubbles. 0.18

dp = 6.35 mm ε = 0.38 j0L = 0.0 mm/s

0.16

*

Lipinski (Fi =0) * Reed (Fi =0) * Hu/Theofanous (Fi =0) Tung/Dhir mod.Tung/Dhir Schulenberg F*i Exp. Tutu (1984)

0.14

0.1

*

Fi [-]

0.12

0.08 0.06 0.04 0.02 0 0

0.2

0.4

void α [-]

0.6

0.8

1

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

Figure 5. Experimentally and theoretically deduced dimensionless interfacial friction in the case of figure 3.

The influence of the interfacial friction is plotted in figure 6. The decrease in the pressure gradient with increasing gas flow rate is only reproduced by the enhanced models including the interfacial friction explicitly. But the further measured increase for higher gas fluxes is steeper than expected from these models. Additionally, at the void development, the modified Tung/Dhir model only yields satisfying results. This is seen even better for the interfacial friction, given in figure 8. Especially for voids greater then α =0.5 the measured interfacial drag is zero. This can be explained in the flow pattern picture with the transition to a channellike configuration with minor contact of gas and water, as described by Haga et al. [7]. The interfacial friction is significantly reduced because of the smaller interfacial area in this flow pattern. In the modifications to the Tung/Dhir model proposed in the previous chapter, this reduction is included in the redefined lower limit for the annular flow regime and in the

358

Werner Schmidt

reduction of the interfacial friction with decreasing particle diameter. As such small particulate debris has to be expected during a severe accidents in a nuclear power plant, the modifications gain importance for these applications. Chu et al. [3] also performed corresponding experiments with a net water flow in cocurrent as well as in counter-current configuration. The co-current results, with water and air injection into the bottom, confirm the above results in principle. As an example, results for a fixed water superficial velocity are given in figure 9. As, per definition, the net water flow is fixed, the void limit of α =1 can not be reached in this case. Increasing the gas mass flux yields an increasing pressure gradient, that presses the fluids through the porous structures, as can be seen in figure 9. More interesting with respect to debris coolability is the counter-current flow configuration, where water is added to the top and exhausted from the bottom of the test section, yielding a top-to-bottom flow. Then, the bottom injected gas and exhausted water are in counter-current mode, similar to the case of a boiling bed with a coolant pool on the top. Again, for fixed water mass flux, the air inlet rate is varied. Results of the measured pressure gradient and void fraction, as well as the corresponding theoretical results are given in figure 10. In contrast to the co-current case, an upper limit for the gas superficial velocity exists, corresponding to a maximum gas mass flux. This limit is the counter-current flooding limit. No larger gas fluxes than this limit are possible in such a configuration. From figure 10 it can be seen that this value corresponds to a maximum void fraction. This fraction cannot be one, because a certain amount of the cross-section or volume is necessary for the water flow. A further increase of the gas flow rate is not possible because this would hinder the water flow.

1

dp = 3.18 mm ε = 0.39 j0L = 0.0 mm/s

0.8

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

*

p [-]

0.9

0.7

Lipinski (p*≈1) Reed (p*≈1) Hu/Theofanous (p*≈1) Tung/Dhir mod.Tung/Dhir Schulenberg * p Exp. Tutu (1984)

0.6

0.5

0

0.2

Figure 6. Dimensionless pressure gradient particles ( d p

= 3.18 mm).

0.4

P*

0.6

jG [m/s]

0.8

1

for an isothermal experiment filled with smaller

Influence of the Interfacial Drag on Pressure Loss for Two Phase Flow…

359

1

0.8

void α [-]

0.6

0.4

0.2

0

dp = 3.18 mm ε = 0.39 j0L = 0.0 mm/s 0

Figure 7. Void fraction

0.14

0.2

α

0.4

0.6

jG [m/s]

1

dp = 3.18 mm ε = 0.39 j0L = 0.0 mm/s

*

0.1

0.8

in the case of figure 6.

Lipinski (Fi =0) * Reed (F i =0) * Hu/Theofanous (F i =0) Tung/Dhir mod.Tung/Dhir Schulenberg F *i Exp. Tutu (1984)

0.12

Lipinski Reed Hu/Theofanous Tung/Dhir mod.Tung/Dhir Schulenberg α Exp. Tutu (1984)

*

Fi [-]

0.08

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

0.06

0.04

0.02

0 0

0.2

0.4

void α [-]

0.6

0.8

1

Figure 8. Experimentally and theoretically deduced dimensionless interfacial friction in the case of figure 6.

360

Werner Schmidt

1.2

p* , α [-]

1

0.8

0.6 Lipinski Reed Hu/Theofanous Tung/Dhir mod.Tung/Dhir Schulenberg p* Exp. Chu (1983) α Exp. Chu (1983)

0.4

dp = 9.9 mm ε = 0.40 j0L = 9.15 mm/s

0.2

0

Figure 9.

0

P*

0.1

and

α

0.2

0.3

0.4

0.5

0.6

jG [m/s]

0.7

0.8

0.9

1

1.1

1.2

for co-current air/water flow with constant water flow rate.

1 0.9 0.8

0.6 0.5

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

*

p , α [-]

0.7

0.4 Lipinski Reed Hu/Theofanous Tung/Dhir mod.Tung/Dhir Schulenberg p* Exp. Chu (1983) α Exp. Chu (1983)

0.3 0.2

dp = 9.9 mm ε = 0.40 j0L = -3.89 mm/s

0.1 0

Figure 10.

0

P*

0.1

and

α

0.2

0.3

0.4

0.5

0.6

jG [m/s]

0.7

0.8

0.9

for counter-current air/water flow with constant water flow rate.

1

1.1

Influence of the Interfacial Drag on Pressure Loss for Two Phase Flow…

361

4.2. Comparison to Experiments with Boiling Debris Beds

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In contrast to the isothermal air/water experiments, the local flow rates vary inside a boiling bed, even in a one dimensional configuration. The local gas flux, driven by buoyancy forces, is determined by the integrated steam flux, and thus by the bed power integrated from the bottom. For a homogeneous power distribution the steam mass flux is given by ρ g j g ( z )=Q z/LH , where Q is the volumetric power density and LH = hg − hl is the specific latent heat of the evaporation. In a steady state, the corresponding local water flux is directly calculated by the mass conservation equation. Assuming water inflow just from a pool above the bed, the liquid mass flux follows from ρ l jl ( z )=− ρ g j g ( z ) . The dryout heat flux of 1-D boiling debris corresponds to the counter-current flooding limit. This limit is reached near the top, where the steam flux as well as the down-flowing water flux are highest. The friction at this bottleneck - the location of highest steam fraction - is decisive for the coolability of the whole bed.

4.2.1. The DEBRIS Experiment For specific investigations of the exchange terms, the friction laws, as well as the heat transfers in boiling particulate beds, experiments have been conducted in the DEBRIS facility at IKE, University of Stuttgart [15]. The experimental setup is sketched in figure 11. A ceramic cylinder of 12.5 cm diameter and a height of 60 cm is filled with oxidised steel spheres. These particles are heated inductively to represent the decay heat. The power distribution is nearly homogeneous. 64 thermoelements are distributed in the test column to detect local dryout. Along the bed height, 8 pressure tubes are connected to differential pressure transducers to allow pressure gradient measurements at 7 different levels. The coolant flowing into the porous region comes from a water pool above the bed. Optionally, an adjusted water inflow rate from below can be injected. With this facility dryout experiment, as well as experiments on quenching of dry, hot particles are possible. Besides the direct measurements of the dryout heat flux by increasing the bed power until the first local temperature raise is detected, experiments with steady states in boiling beds can be used to deduce the friction laws. The produced up-flowing steam accumulates from bottom to top while the evaporated water must be replaced by water inflow. So, for a steady state the water and steam fluxes at each level are defined by the total power below. The measured pressure gradients can again be used to compare the various friction models with the experimental data. All the measured values for different heating power can be collected in one plot. Figure 12 shows the experimental data for 6 mm particles at a pressure of 1 bar. The pressure gradient, adjusted by the hydrostatic head of water, is shown versus the superficial velocity of the steam j g . Because of the statistical character of the measurements, an error in the order of 500 to 1000 Pa/m must be assumed for the experimental data. Additional to the superficial velocity of the steam j g the corresponding superficial liquid velocity jl is provided as second x-axis in the plots. In the top part of figure 12 the results for a pure top-fed configuration are shown. The steam and the water fluxes are always in countercurrent mode, as can be seen by comparing the different x-axises. With fixed water injection from the bottom, co-current mode occurs in lower particle bed regions, as can be seen on the jl -axis in the lower part of the figure.

362

Werner Schmidt

R e f lu x C o n d e n se r

O v e r f lo w s

P L 7 d p 7

5 2 4 9

P L 6 d p 6

3 7 3 3

P L 4 d p 4

2 9 2 5

P L 3 d p 3

2 1 1 7

P L 2

13

d p 2

n tr lu m ssu 1 0 0

o l e re , m m

9

P L 1 d p 1 1

5

P L 0

z

5 1

6 4 0

4 8

4 7

4 6

4 4

4 1

d p 5

C o V o P re H =

5 0

4 5

P L 5

5 6

4 2 4 0 3 8 3 6 3 4 3 2 3 0 2 8 2 6 2 4 2 2 2 0 18 16 14 12 10 8 6 4 2

4 3

3 5

4 9 0 4 4 0

3 1 2 7

3 9 0 3 4 0

2 3 1 9

2 9 0 2 4 0

15

1 9 0

11

1 4 0 7

3

5 9 0 5 4 0

3 9

9 0 4 0 0

C o n V o lu T e m H = 5

tro l m e p ., 0 m m

@ F &

a m b ie n t r e s p p l7

W a t e r In je c t io n

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

Figure 11. DEBRIS experimental setup.

In addition to the experimental data, results of calculations based on the different models for the given conditions are also plotted in figure 12. In principle, the development of the pressure gradient shows the same behaviour as in the counter-current air/water flow of figure 10. Again, it can be seen that the classical models, without interfacial friction, cannot reproduce the experimental data, and must be rejected. Only the enhanced models, including an explicit formulation of the interfacial drag, fit qualitatively to the experimental results. Numerous data points have been measured for small vapour velocities. In this range, the enhanced models show no significant difference. For higher gas fluxes, where stronger difference between the models are observed, it was difficult to establish steady states. Only a few data points with a larger spread could have been measured in this range. Based on this data alone, no final conclusion can yet be drawn to finally prove the model formulations. The development of the classical models to obtain the measured dryout heat flux, corresponding to the maximum j g , can also be seen in figure 12. The first approach by Lipinski yields a dryout heat flux higher than measured for top cooled particulate beds. By increasing the friction in the relative passability, as done by Reed, the measured critical heat flux is better reproduced. In figure 12 this is seen by the smaller maximum reachable gas velocity. As mentioned, Hu and Theofanous [9] criticised the published dryout heat flux data due to the measuring procedure, and consequently introduced even stronger particle fluid friction. This adaptation process can also be seen in the isothermal air/water experiments shown in figure 10. By inspecting the pressure gradient development, it can be seen that this

Influence of the Interfacial Drag on Pressure Loss for Two Phase Flow…

363

adjustment does not represent the measured pressure losses. These can only be explained by including the interfacial friction, similar to the case without net water flow.

-∇p-ρLg [Pa/m]

0

+ +++ -1000 + + + + -2000 + + +++ + ++ -3000 + -4000

+ + + ++ + ++ ++ + + + + + +

p = 1 bar ε = 0.4 dp = 6 mm j0L = 0.0 mm/s

-5000

-6000

+

+

0

+

0.2

0

0.4

0.6

-0.2

-0.4

Lipinski Reed Hu/Theofanous Tung/Dhir mod.Tung/Dhir Schulenberg DEBRIS exp.

+

0.8

1

jG [m/s]

1.2

-0.6

jL [mm/s]

1.4

1.6

-0.8

(a) without water injection from below 0

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

-∇p-ρLg [Pa/m]

-1000

-2000

+ ++ ++

-3000

-4000

p = 1 bar ε = 0.4 dp = 6 mm j0L = 0.18 mm/s

-5000

-6000

+ + ++ + + ++ + + +++++ + + + + ++ ++

0

0.2

+ 0.4

0

Lipinski Reed Hu/Theofanous Tung/Dhir mod.Tung/Dhir Schulenberg DEBRIS exp.

+

0.6 -0.2

0.8

1

jG [m/s] -0.4

jL [mm/s]

1.2

1.4 -0.6

(b) with water injection from below Figure 12. Pressure gradient in steady states in the DEBRIS experiments for

d p =6 mm.

1.6 -0.8

364

Werner Schmidt

-∇p-ρLg [Pa/m]

0

+ + + + +++ + + + + ++ -1000 ++ + + ++ ++ +++ + ++ + +++ + + -2000 + + + +

++ + +

+ + + + + ++ + +

-3000

p = 1 bar ε = 0.4 dp = 3 mm j0L = 0.0 mm/s

-4000

-5000

0

Lipinski Reed Hu/Theofanous Tung/Dhir mod.Tung/Dhir Schulenberg DEBRIS exp.

+ 0.2

0.4

0

-0.2

0.6

jG [m/s]

0.8 -0.4

jL [mm/s]

(a) without water injection from below

+ + + + ++ + + ++

0

-1000

+ +

+ +

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

-∇p-ρLg [Pa/m]

+

+ + + + + + +

-2000

-3000

p = 1 bar ε = 0.4 dp = 3 mm j0L = 0.2 mm/s

-4000

-5000

0

0.2

+

0.2

0.4 0

jG [m/s] jL [mm/s]

0.6

Lipinski Reed Hu/Theofanous Tung/Dhir mod.Tung/Dhir Schulenberg DEBRIS exp.

0.8 -0.2

-0.4

(b) with water inflow from below Figure 13. Pressure gradient in steady states in the DEBRIS experiments for

1

d p =3 mm.

Influence of the Interfacial Drag on Pressure Loss for Two Phase Flow…

365

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In principle, the results with additional water feeding from below show the same behaviour, as can be seen in the lower plot. The pressure field obtained by the enhanced models again shows the basic behaviour. These models fit well in both configurations, the counter-current ( j l kNa+ > kK+ and kMg2+ > kCa2+ > kBa2+ can be estimated. In addition divalent cations lead to higher rates of formaldehyde disappearance as compared to the monovalent cation. Figure 3 shows the change in rate constant of formaldehyde disappearance as a function of the ionic radius of the hydrated cation. It is seen that there is a linear relationship for both monovalent and divalent cations. The slope of the line increases with cation valence, thereby confirming the influence of this parameter on reaction kinetics. These results clearly confirm that the nature of the cation modifies the kinetics of formaldehyde disappearance where two parameters (ionic radius and valence of the hydrated cation) play an important role. [21

Figure 3. Influence of the size of hydrated cations on the rate of disappearance of formaldehyde: ♦ KOH; ■ NaOH; ● LiOH; ◊ Ba(OH)2; □ Ca(OH)2; ○ Mg(OH)2.

426

Fernando Pérez-Caballero, Anna-Liisa Peikolainen and Mihkel Koel

The influence of the basic catalyst type has also been studied by Horikawa et al. [22] via small-angle X-ray scattering. In their study minimal difference in RF carbon aerogels micropore volumes were observed as a product of different catalyst species and ratios. However, a greater influence on the mesopore volumes of these aerogels was observed, demonstrating a strong dependency on the catalyst species and ratio employed.

Double-Step Base-Acid Catalysis Pekala’s method may use both basic and acidic catalysts to obtain gels with dimensional stability capable of surviving the ScCO2 drying step. It has been shown by Barral et al. [23] that basic media activates ortho- and para-sites of the phenolic compound and leads to fast formation of hydroxymethyl substitutions. After this process the formation of clusters starts slowly, with hydroxymethyl groups forming ether and methylene bridges between aromatic molecules. Methylene bridges are formed as hydroxymethyl groups reacts directly with the aromatic ring. Ether bridges are the product of the condensation of two hydroxymethyl groups. The second process occurs faster if reaction is performed under acidic conditions. These base and double catalysed steps are summarised in Figure 4.

ACIDIC CONDITIONS Basic catalysts are typical in aerogel preparation but acid catalysed polymerisations are also known commonly utilising aqueous HCl or acetic acid. The reaction of R with F and the subsequent condensation of the resulting hydroxymethyl resorcinol with R, are electrophilic aromatic substitutions, and therefore proceed efficiently by acid catalysis, via a convenient one pot synthetic approach [24]. a) OH

(CH2OH)n

OH

(CH2OH)m

OH

OH CH2

HCHO

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OH

OH CH2 O CH2

and/or

base

fast

b)

(CH2OH)m

slow

n=1, 2, 3. m=0, 1, 2.

OH

(CH2OH)n

OH

OH

(CH2OH)m

HCHO

OH

OH CH2

(CH2OH)m

OH

OH CH2 O CH2

and/or

base

acid

fast

fast

Figure 4. Schemes on steps of formation of resole polymers by a) base catalysed process and b) double step process.

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There is developed and reported an HCl-catalyzed gelation process in CH3CN, which is completed in about 2 h at room temperature or in 10 min at 80 °C. The final aerogels are chemically indistinguishable (by IR and 13C CPMAS NMR) from typical base-catalyzed samples. In analogy to phenol-formaldehyde resin formation, the mechanism may involve oquinone methide intermediates (hence the red color prevailing throughout the process). The effect of aging is attributed to further reaction and incorporation of more formaldehyde into wet gels, followed by syneresis (reaction with one another of dangling oligomeric appendices on the skeletal framework). The advantage of using organic acids (acetic acid) as catalysts, is that metallic impurities in the carbon aerogel may be avoided. [24,25] Figure 5 summarizes the proposed mechanism for the HCl-catalyzed gelation of resorcinol with formaldehyde. Protonation of formaldehyde (eq 1) is followed by nucleophilic attack by the π-system of resorcinol (eq 2) leading to hydroxymethylation. Subsequently, protonation of a hydroxymethyl group (eq 3) forms –OH2+, a good leaving group, that may cleave either unimolecularly (eq 4), leading to o-quinone methide-type intermedates (also referred to as o-quinomethanes), or bimolecularly (eq 5) after direct attack at the –CH2OH2+ carbon by the π-system of another resorcinol molecule. The latter process results directly into -CH2- bridge formation between phenyl rings at three possible positions. [24] O H

OH

H2O

H OH

H

H

+

C H

OH

OH OH

+ OH

OH

OH

+

H -H2O

C H

-H

+

(1)

OH

(2)

OH

+

H

OH

H

OH

H2O+ +

H OH

H2O+

O

OH -H2O

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(3)

OH

OH

-H

H2C

+

OH

(4)

o-quinone methide OH

H2O+

OH

OH OH

-H2O , -H OH

+

HO

OH

OH

(5)

428

Fernando Pérez-Caballero, Anna-Liisa Peikolainen and Mihkel Koel OH

O

OH

H2C

OH

OH

HO

OH

(6)

OH

OH OH OH

OH

OH

HO

O H

HO

OH

HO

OH

+

H, H , -H2O

OH HO

OH

HO OH HO OH

(7)

Figure 5. The steps of the proposed mechanism for the HCl-catalyzed gelation of resorcinol with formaldehyde. [24]

TAILORING GEL STRUCTURE The physical properties of the aerogel are dependent on the structure of the original gel. The size of the clusters and pores and the surface area of the gel can be tailored by varying the concentrations of the reagents (here expressed as molar ratios). Tailoring of the gel structure can be achieved via altering the molar ratio of aromatic monomer to cross-linking monomer (in the example of the resorcinol-formaldehyde system R/F), and aromatic monomer to catalyst (R/C). Further control may be exerted via control of the amount of solvent used, nominally expressed as the percent of the solid fraction, or occasionally as the molar ratio of solvent to aromatic monomer (W/R), catalyst to solvent (C/W) or solvent to catalyst (W/C).

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R/F Ratio R/F molar ratio for optimal cross-linking has been found to be in the range of 0.4 - 0.7 [26], with a 0.5 ratio the preferred embodiment. It was found that specific surface area was independent of R/F ratio, whilst a strong dependence for mesopore volume and pore size distribution was observed and divided into two groups. Mesopore volume and average pore size was found to be be small for R/F ratios ≤ 0.34, whilst increasing the ratio to values ≥ 0.40 resulted in an increase in the mesoporous size. It was assumed that the high amount of formaldehyde remaining in the sol-gel induces a collapse of mesopore structure and decreases the mesopore volume in the resulting aerogel. In case of systems composed of R/F ratios = 1.0, the resulting sol did not gel enough to allow for production of porous aerogel material. [27]

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R/C Ratio

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The greatest impact on gel structure is typically the amount of catalyst used in the polymerisation step, typically expressed by the ratio of aromatic compound to the catalyst employed. Figure 8 summarises cluster formation in a system composed of a fixed R/W ratio, and varying catalyst amount. Catalyst concentration determines particle size and the morphology of the cross-linked gel structure. During the first stage of the gelation process, each particle grows individually at the catalytic site; subsequent aggregation generates the interconnected structure initially, resulting finally in the cross-linked porous RF hydrogel structure. If catalyst concentration is high, particle growth points are densely packed. Thus a small amount of the reactant is consumed per particle and consequently the size of the particles remains small. Contrastingly, if catalyst concentration is low, particle growth initiation points exist sparsely. Therefore a large amount of the reactant is consumed per particle and consequently the particles size increases as a function of decreasing catalyst concentration. Likewise, the pore size becomes relatively small if the catalyst concentration is high, and vice versa. [28]

Figure 6. Depiction of the effect of R/C ratio on the growth of aerogel particles; (a) at high and (b) low catalyst concentration.

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Fernando Pérez-Caballero, Anna-Liisa Peikolainen and Mihkel Koel

For resorcinol-formaldehyde systems, the initial pH of the mixture is maintained in the range 6.5 - 7.4 to form a transparent gel. As the reaction progresses, and formaldehyde is consumed, the pH decreases as increasing numbers of hydroxymethyl (-CH2OH) groups are formed. [29] If the cluster size (determined by catalyst concentration) is much smaller than the wavelength of visible light aquagels will be transparent in appearence. For catalyst ratios 25 ‹ R/C ‹ 400, in resorcinol-formaldehyde systems, the resulting gels obtained will be transparent. For R/C ≤ 25, gels become increasingly opaque, frangible and hard. For R/C ≥ 400, gels are opaque and soft. Monodisperse distribution of aerogels has been obtained under the conditions of R/C ≤ 75. At high catalyst concentrations (i.e. R/C ≈ 50), the resulting aerogels have high BET specific surface areas (SBET ca. 850 m2 g-1) and large mesopore volumes (Vmeso ca. 3 - 4 cm3 g-1). At low catalyst concentrations, SBET and Vmeso decrease with increasing R/C. The specific surface area and the volume of the mesopores have maximum values at R/C ≤ 100. [27]

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R/W Ratio As a result of too high a system water content in the sol (e.g low R/W ratio) gel formation may be hindered. Zhang et al. [30] have studied the influence of phloroglucinol content in the sol and its impact on volume shrinkage during drying, specific surface area and pore volume in the resulting porous aerogel products. This research demonstrated that gels formed at low reactant concentrations are discrete with weak junctions existing between particles. As a result, such hydrogels collapse as gel-bound water is replaced with acetone, inducing significant volume shrinking. In contrast, networks of hydrogels formed at higher phloroglucinol concentrations, are bi-continuous and hydrogel structure collapse does not occur during the solvent displacement step. During hydrogel formation initial nucleation-andgrowth followed by spinodal composition occurs leading to small and large sized mesopores. At low aromatic compound concentration, large sized mesopores are not formed. Increasing the amount of aromatic compound induces the formation of large sized mesopores. The cause of it, as assumed by the authors, is that the large sized mesopores in hydrogels formed by spinodal decomposition collapse to form closed pores by severe volume shrinking at low aromatic compound concentration due to weak junctions of the hydrogels. No influence of R/W on the BET specific surface area of organic aerogels has been observed. [27,31] Figure 7 depicts the influence of the solid fraction amount on the gelling time: at higher values the gelling occurs faster. The gelling time also depends on the temperature at which the sol is cured. [31, 32] The mesoporous properties of RF hydrogels however, are not greatly influenced by the gelation temperature. [33] The procedure of controlling the pore size distributions of RF aerogels is proposed as follows. First one should choose low R/C to prepare the aerogels with sharp mesopore size distributions. For example, one obtains RF aerogels of pore radii 2.5 - 4.0 nm by adjusting R/W at R/C=12.5. To prepare RF aerogels of larger pore radius, R/W is changed at larger R/C. One can prepare the aerogels with pore radii ranging from 2.9 to 6.0 nm at R/C=25, from 3.9 to 6.9 nm at R/C=75 and from 4.5 to 9.2 nm at R/C=100. It is not possible, however, to obtain aerogels of mean pore radii less than 2.5 nm or over 9.2 nm. In the first case the amount of distilled water used in the sol-gel polycondensation should be decreased but resorcinol cannot be dissolved in the solution.

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Figure 7. Effect of solid fraction of resorcinol on gelation time depending on gelation temperature. [31].

When trying to prepare RF aerogels of mean pore radii over 9.2 nm, R/C must be over 100, in which case monodisperse pore size distribution is not achieved. [26]

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THE CHOICE OF THE SOLVENT— WATER, ALCOHOL, ACETONE, ACETONITRILE There are two aspects related to solvent used in gelation process: the first is related directly to solubility of reactants, and the second is related to obtaining the aerogel with saving the unique structure of gel what means proper drying process. If the liquid which permeates the pores of the resulting gel is water, the gel is termed an “aquagel”. If it is an alcohol, the gel is termed an “alcogel”. [34] Water is very often used as the solvent; however, it has disadvantages that must be taken into account. In case of water the direct supercritical drying is impossible to use because high critical parameters of it (the critical temperature and pressure of water are 374.2 °C and 220.5 bar [35], respectively). At such high temperature and pressure water is extremely reactive and the gel decomposes. The approach is that after the gel being formed, exchange water in the pores with another solvent that has lower critical parameters, or use this other solvent in the original solution. In order to dry a hydrogel supercritically, the water inside the pores must be exchanged for another solvent which can be removed later through the supercritical state, or it must be miscible with the fluid that is used for supercritical drying of the gel. In case of resorcinol-formaldehyde hydrogel, the water is exchanged for acetone or methanol [36], because they are miscible with CO2 that is used for venting the solvent from the pores of the gel via supercritical state.

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DRYING OF THE GEL The pore texture of dried aerogels depends on the drying process. Several more or less expensive methods (supercritical drying, freeze-drying, evaporative drying) are proposed [36].

Supercritical Extraction

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To turn the gel into aerogel, the water from the pores of the gel is removed by extracting the gel with supercritical carbon dioxide (ScCO2). The idea of drying the gel over the supercritical state is to eliminate the solvent from the sol-gel without generating a two-phase system and the related capillary forces, which would destruct the structure of the gel when the solvent evaporates at ambient temperature and pressure. Prior to this step water in the pores of the gel is replaced with acetone. Acetone, apart from water, is miscible with ScCO2 and the removal of the liquid from the gel is thus successful. The supercritical extraction is the most used method for removing the solvent from the pores of the gel because it better preserves the original pore texture of the wet gel. The process can be performed by venting the solvent above its critical point (generally high temperature) or by prior solvent exchange with another solvent (CO2) followed by supercritical venting (lower temperatures). This is possible through compressing and heating the sol-gel above the critical pressure and temperature of the solvent (for CO2: Tc = 31.1°C, Pc = 73.8 bar) and then by decompressing it down to atmospheric pressure following lowering the temperature to the room temperature, maintaining the solvent in gas phase without any condensation.

Figure 8. Phase diagram depicting the physical state as a function of temperature and pressure. [34]

Organic Aerogels as Precursors for the Preparation of Porous Carbons

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On Figure 8 there is a pressure/temperature curve representing the regime how the sol-gel mixture (point A at room pressure and temperature) can be pressurised and heated to reach the supercritical state (point B) and then depressurised and cooled to reach again room conditions (point C); during this operation, the solvent vaporisation curve (V) is never crossed: so, at no time any two-phase solvent system appears, and finally, only a low pressure solvent vapour is present in the porous aerogel that is further filled with air by diffusion as the aerogel is highly porous with open pores. [34] Nevertheless, supercritical drying does not always guarantee the absence of shrinkage: when the wet gel is constituted of small nodules (< 20 nm), a volume loss is observed during drying. This volume loss is due to residual surface tension problems whose effect is aggravated when the pores are small or when the material density (and mechanical resistance to tensions) is low. When the wet gel is constituted of larger nodules, both pore volume and pore size can be fixed independently, the residual capillary forces being without effect. Very large pore volumes (total pore volume VTotal > 3 cm3 g-1) can be obtained if the dilution ratio is large enough. However, these extremely porous aerogels are also extremely brittle and difficult to handle without damage.[37] Rapid supercritical extraction– A partial differential equation is derived that describes the pressure developed in the pores of a gel during the rapid supercritical extraction process. A comparative analysis of the strains caused by syneresis and expansion of the fluid, respectively, suggests that the latter is the dominant effect for this process. Experimental results indicate that the rate of leakage from the mould is equal to the rate of volumetric expansion of the fluid, so this was used as the boundary condition for the calculation. An analytical solution is obtained for the strain produced in a purely elastic gel. The strain is found to develop most rapidly at high temperatures, where the thermal expansion of the fluid increases sharply. The model predicts a temperature dependent heating rate that can be used to avoid irreversible strains by compensating for the increase in thermal expansion coefficient. [38] Although it leads to the largest pore texture range, the supercritical drying step imparts to organic aerogels an uncompetitive price-to-performance ratio as compared with other materials for any given application and thus limits their commercial viability. The costs of drying increase if the pretreatment of the gels (solvent exchange) is necessary. Therefore, alternative methods for the fabrication of aerogels have been tested: especially welcome are the techniques of conventional ambient pressure drying.

Freeze Drying Cryogels are obtained when the gels are dried through sublimation i.e freeze-drying. The expansion of water during freezing must be eliminated to avoid destroying the gel structure. For this, however, acetone cannot be used because its freezing point is lower than the temperature of the freeze-drying process. In view of these constraints, water was exchanged with t-butanol for the freeze-drying method. After the seven-day incubation the hydrogel gel rods were removed from the glass vials and immersed in three times their own volume of the replacement solvent for 3 days, renewed with fresh solvent every day. Prior to freeze drying a pre-freezing period is necessary. The gel rods together with the excess t-butanol were precooled for 2 h at 20 ºC. The frozen gel rods were then transferred into the freeze-drying

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apparatus, where the temperature was reduced to -45 ºC. At this temperature the pressure was slowly reduced to 1 mbar and the frozen solvent was sublimated. The diameter of the RF cryogel rods hardly changed in diameter while drying. [39] Monoliths are very difficult to obtain by freeze-drying, and the appearance of huge channels due to ice crystal growth at high dilution ratio hinders the fabrication of low density materials. Moreover, gels with small pores do not remain frozen throughout drying, which leads to surface tensions and shrinkage. Although generally replaced by more complicated techniques, evaporative drying is suitable when dense carbons are needed or when the only selection criterion is the pore size: all pore sizes are reachable. [36] Freeze-drying of organic aqueous gels generally leads to broken samples. Keeping monoliths is particularly difficult with water, and expensive solvent exchanges do not always resolve the problem. Nevertheless, direct freeze-drying (i.e. without solvent exchanges) can be used when monoliths are not required. The pore texture of the gel is not preserved when the pores are small (< 40 nm): confinement inside the pores leads to solvent fusion and surface tension overpressures, the solvent being partly removed by evaporation. When the pore size is large enough, no fusion occurs and the material texture after drying and after pyrolysis is comparable to that of an aerogel of the same starting composition. This seems nevertheless limited to dense materials (ρbulk > 0.3 cm3 g-1 for the dried organic gels and the carbon materials). Increasing the dilution ratio does not lead to very light carbon materials with a regular texture: at high solvent amount, the growth of ice crystals reorganizes the pore texture into very large channels, which makes the dried gel and further carbon material extremely fragile and inhomogeneous. [36]

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Drying at Ambient Temperature and Pressure If a gel is dried by normal means, such as exposing to ambient conditions or placing it in an oven, the product is termed as “xerogel”. In drying in inert atmosphere, the gel rods were filtered before drying, then placed in an oven in a flow of high purity nitrogen to ensure an inert atmosphere during the process. The temperature is raised slowly to 65 ºC and maintained for 5 h. The temperature was then increased to 110 ºC and maintained for another 5 h. The diameter of the RF xerogel rods was only about 50 % of that of the original hydrogel. Evaporative drying without pre-treatment of an aqueous gel leads to a monolithic shrunk material. Nevertheless, the pore texture is generally not completely destroyed despite the capillary forces exerted upon the pores during drying. Large pore volumes (up to 2.4 cm3 g-1 and pore sizes are still obtained by this drying technique when the size of the polymer nodules is large enough to stop the shrinkage due to capillary forces. The problem of producing lowdensity, high-surface area materials at ambient pressure primarily arises from the large capillary pressures that occur during drying (a result of the small pore sizes) and low stiffness (a result of the low-density). More recently it has been found that when resorcinol and furfural are polymerized in isopropanol in the presence of hexamethylenetetramine (HMTA), the latter acts not only as a catalyst but also as a cross-linking reagent. This affects the size and stacking of the organic aerogel particles as they grow and thus enhance a network strength of alco-gels. Carbon

Organic Aerogels as Precursors for the Preparation of Porous Carbons

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aerogels prepared by this method have nano-particle structure typical to the aerogels prepared with CO2 or isopropanol supercritical drying technique. [40]

Use of Surfactants to Minimise Shrinking on Drying Aerogels may also be successfully fabricated by a microemulsion-templated sol–gel polymerisation method. Due to the formation of a microemulsion in the sol–gel process, the network of the gels and aerogels can be controlled by the surfactant concentration. It has been shown that the network particle size decreases with an increase in the surfactant to resorcinol (S/R) molar ratio [41]. Using a suitable S/R molar ratio, drying of the organic gels with appropriate network strength resulted in minimal shrinkage on heating at ambient pressure. This small shrinkage was attributed to the following three properties in the gels obtained: an appropriate specific macropore volume; appropriate network strength; and the decrease in surface tension. Generally the organic and carbon aerogels prepared via microemulsiontemplated sol-gel polymerisation have a microstructure similar to that of typical aerogels prepared by supercritical drying technique. Due to the transformation of macroporosity into mesoporosity during carbonisation, carbon aerogels produced in this manner have more inherent mesoporosity than their organic aerogel precursors. Recently, Lee and Oh have prepared porous carbons by a surfactant-templated sol–gel polymerisation (STSGP) [42]. In this method, surfactant molecules are incorporated on to the surface of Na2CO3-catalyzed resorcinol–formaldehyde (RF) sols, resulting in a decrease of the surface tension at the pore water–resin interface, minimising pore collapse during the drying step. Analogously, via a microemulsion-templated sol–gel polymerisation method (MTSGP), gel cluster size and thus RF gel pore size may be simply controlled by the molar ratio of surfactant to resorcinol (S/R) using a fixed monomer concentration [41]. This enables the use of ambient pressure drying of the aerogel, providing good mesoporosity and little shrinkage in the final product. This is the first advantage of using the MTSGP method. The second advantage of the MTSGP method is that an organic gel with a three-dimensional nanonetwork can be formed simply via the addition of surfactant. There is no need to add Na2CO3 as the surfactant itself acts as catalyst. The pore size and the final density of the dried material are directly related to the polymer nodule size which itself depends on the synthesis conditions (e.g. R/C ratio or system pH). Contrary to supercritical drying or freeze-drying, the dilution ratio has no effect on the pore texture, the nodule size being quasi-independent from the solvent amount in the considered diameter range. As a consequence, samples with large pore volume (> 2.5 cm3 g-1) and small pores (< 50 nm) cannot be produced by evaporative drying. In this case, supercritical drying (when monoliths are needed) or freeze-drying must be used. [36] Using surfactants above their critical micelle concentration (CMC) forms micelles allowing the preparation of ordered mesoporous carbons (OMCs) negating the use of wasteful inorganic templates [43,44]. The most common synthetic route to OMCs involves preparation of ordered mesoporous silica (OMS) templates, followed by impregnation of the OMS pores with a carbon precursor, carbonisation, and subsequent removal of the hard template. The removal of the OMS templates is typically performed through treatment of the inorganic carbon composite with with HF or caustic solution, removing the inorganic component, producing an inverse carbon replica of the original siliceous OMS template. The novel OMC synthesis route reported by Tanaka et al. [45], avoids the use of hard templates, and thus the

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Fernando Pérez-Caballero, Anna-Liisa Peikolainen and Mihkel Koel

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number of preparation steps and the cost involved in producing these materials was reduced. The strategy utilised an organic–organic interaction between a thermosetting polymer and a thermally decomposable surfactant to form periodically ordered nanocomposites. The thermosetting polymer after carbonisation remains as the carbonaceous pore wall. Resorcinolformaldehyde (RF) and triethyl orthoacetate (EOA) were used as the carbon co-precursors and triblock copolymer Pluronic F127 was used as a surfactant. [45] Three gel drying approaches for the preparation of RF materials are compared by scanning electron microscopy (Figure 9). Material obtained by ScCO2 extraction exhibits a compact structure (Figure 9(a)); individual microspheres are smaller than in the RF xerogel (Figure 9(c)). The freeze-dried RF gel presents a more loose structure (Figure 9(b)), and is the finest of three samples presented. It consists of very small spheres of diameter approximately 30 nm, arranged in a filament-like structure to form a three-dimensional matrix. Macroscopically, these rodlike materials are brittle. Material dried at high temperature in nitrogen (Figure 9(c)) also presents a somewhat more compact structure, consisting of more or less uniform microspheres of diameter 0.05 - 0.10 µm. [39]

Figure 9. SEM images of the RF gels. (a) RF aerogel, (b) RF cryogel, (c) RF xerogel, scalebar = 1 μm.

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From porosity studies of different gels (e.g RF gels: aerogel, cryogel, xerogel, and carbon aerogel) RF cryogels presented the highest surface areas and total pore volumes. Interestingly pore size distributions of all the RF gels studied in this investigation were very similar [39].

Evaporative Drying Vacuum Drying Organic xerogels can be obtained by vacuum drying without any sample pre-treatment, i.e. without solvent exchange prior to drying. Unsealed flasks may be simply kept at 60 ºC whilst pressure is progressively reduced from 105 Pa to 103 Pa over a period of five days. Finally the samples are then heated to 150 ºC at 103 Pa for three days. [36] These long drying steps may certainly be shortened or even replaced by classical convective drying techiques, although drying rate has been shown to influence the final pore texture of the material. Microwave Drying Microwave drying offers certain advantages over conventional approahces (i.e. rapid drying and reduction in energy consumption). In addition, microwave drying is gaining importance in several fields on an industrial scale [46], at the expense of supercritical and freeze-drying methods which require expensive and sophisticated machine hardware. When different microwave devices are compared, multimode microwave ovens (i.e., MOM) were found to dry samples in the shortest times. Thus, in the case of organic gels synthesised at pH 7 and 9, only 6 min is required to dry the resulting gel. When a unimode microwave oven is used (i.e., MOU), the drying time depends on the power employed. Higher powers are required for shorter drying times.[47] In the case of conventional and vacuum drying, the time necessary to ensure complete drying is more or less the same. However the mass loss is lower in the initial steps when vacuum drying is employed. The mesoporous properties of gels prepared by microwave drying can be maintained at lower catalyst concentrations. As was presented by Tonanon et al. [48], unlike gels prepared at molar ratio R/C = 100 and 200, the mesoporous structure could not be obtained at R/C = 50.

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DIFFERENT SHAPES FOR AEROGELS It has been recognized that the formation of carbon aerogel (CAs) in small spheres (micro-beads) may greatly expand the use of these materials, as well as reduce the process time and equipment costs in applications where they would be appropriate. For example CA microbead supported catalysts could be recovered from an industrial process in a more simple manner than the corresponding powder supported catalyst. First reported by Mayer et al. the fabrication of CA microspheres involves stirring of the aqueous organic phase in mineral oil at elevated temperature until the dispersed organic phase polymerises and forms a non-sticky organic gel spheres [49]], which retain their morphology and structure upon pyrolysis. A similar process was investigated by Dresselhaus et al. [50]. The inverse phase suspension polymerisation of resorcinol and formaldehyde monomers with Na2CO3 as a

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Fernando Pérez-Caballero, Anna-Liisa Peikolainen and Mihkel Koel

catalyst in a peanut oil medium followed by supercritical drying was explored. The RF aerogels were synthesized by suspension polymerisation. The stirring speed greatly affected the size of the resulting aerogel spheres. The productive micro-beads were easily poured out from the reaction container. Experimental results indicated that it was easy to avoid the accumulation of polymerisation heat during gelation, and easy to remove products from the reaction container, via this synthetic approach. Sol–gel microspheres with diameters ranging from about 30 – 1000 µm could be obtained. After drying of the sol–gel spheres under alcohol supercritical drying conditions, aerogel spheres with a bulk density of 0.8 - 1.0 g cm-3 were prepared, and by subsequently pyrolyzing them, CA spheres with surface areas of 250 – 650 m2 g-1 were obtained. Yamamoto et al. have reported the synthesis of spherical carbon cryogels through the inverse emulsion polymerisation of aqueous solution of RF [51]. However, this method does not provide monodisperse droplets of a RF solution, hence polydisperse carbon cryogel beads are formed [52]. The key to the sequential formation of regular sized droplets of the RF solution in the oil may be achieved via the use of an injection apparatus. This method prevented the aggregation of RF solution droplets and consequently, monodisperse RF hydrogel microspheres (m-RHMs) were obtained. m-RHMs were subsequently washed with cyclohexane and ethanol respectively at least three times each, to remove any residual silicone oil, and then immersed in t-butanol to exchange the solvent in the pores before freeze-drying. The beads were dried at -20 °C under a reduced pressure and then carbonized under an inert atmosphere to obtain monodisperse carbon cryogel microspheres. The carbonisation temperature was kept at 1000 °C. RF foam preparation differs considerably from that of methacrylate-based foams, and these differences profoundly affect the encapsulation process. One important difference is that the typical gelation time of conventional RF aerogel formulations is many hours. Because successful encapsulation is possible only if gelation of the shell phase occurs within 15 to 20 min, a means of accelerating gelation is needed that can maintain the desired foam properties (e.g. transparency and low density). Another important difference is that RF polymerisation occurs in an aqueous medium rather than an organic (‘‘oil’’) medium. As a result, an oil-in-water-in-oil system is needed to create RF aerogel shells rather than the water-in-oil-in-water system used for methacrylate foam shells. To accomplish this, the gelation time may be reduced from several hours to several minutes by the addition of acid following the base-catalyzed RF particle growth[53]. However, additional ‘‘annealing’’ of the gel for at least 20 h was needed to maximize cross-linking and minimize swelling in exchange of solvents. Increasing the molar ratio of formaldehyde to resorcinol from 2 to 3 also helped to increase cross-linking. Densification of the foam shells due to dehydration during curing was greatly reduced by selection of appropriate immiscible oil phases and by saturating the exterior oil phase during the annealing stage. Using this approach shells have been produced possessing diameters of ~2 mm, with wall thicknesses ranging from 100 to 200 mm and foam densities approaching 50 mg cm-3. Thermosetting polymer nanospheres and nanowires have been prepared by a NaOH catalysed polymerisation of resorcinol and formaldehyde in the presence of cetyltrimethylammonium bromide as a core template together with/without trimethylbenzene (TMB) as an additive and further converted into their carbonized forms.

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Figure 10. Morphology of the CA spheres (left) and photograph of cross-sectional view of an Activated Carbon Aerogel bead (right) (SEM) [50,55].

This approach is based on controlling the interaction between RF polymers and cationic surfactant molecules through the ionization of phenolic hydroxyl groups of resorcnol species. The polymer composites were structurally modified by the additional use of TMB as a micelle modifier with high effectiveness [54].

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ORGANIC AEROGELS WITH ADDED FILLERS OR MODIFIERS A common method for the preparation of the three-dimensionally ordered macroporous solids utilizes close-packed arrays of monodisperse spheres (typically silica or polystyrene) as templates. In this approach, the interstitial regions of the colloidal crystals are infused with a reaction solution of the material that will eventually comprise the replicate structure. Once the composite forms, the template is removed, by either chemical or thermal means, yielding the macroporous replicate solid with periodic voids in the 0.1 – 1 µm size range. A variety of macroporous materials have been prepared by this method, including silica [56,57], metal oxides [58,59], metals [60,61] carbon [62,63], polymers [64,65] as well as other materials [66,67]. A promising extension of the template approach towards ordered porous materials of higher complexity is the development of methods that allows one to tailor composition and porosity of the wall structure defining the macropores. Since organic sol–gel chemistry provides a straightforward method to control the textural porosity and the composition of the aerogel matrix, utilization of this technique in the preparation of ordered macroporous solids should provide control over the properties of wall structure in these materials [44].

POROUS MATERIALS FROM BIOMASS DERIVED POLYSACCHARIDES Nature provides a wide range of biosynthetic sugar polymers termed polysaccharides. These renewable resources are readily available, inexpensive and functionally rich (e.g. -OH, -C(O)OH), -NH2). These naturally occurring products of metabolism perform a wide range of functions in the native environment, including membrane and cell wall functions, sources of energy and as sequestering agents for water, nutrients and metals in the cell environment. [68]

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Polysaccharides are known to self associate or order into particular structures, physical forms or shapes upon inducement of an aqueous gel state. Exploitation of this ability for the preparation of novel porous aerogel and xerogel materials is now beginning to be exploited. Native polysaccharides present neglible surface and pore structure, and therefore in applications where diffusion and surface interactions (i.e. chromatography) are critical to function, polysaccharide use is therefore limited. Recently attempts to generate porous materials from naturally occuring polysaccharides have began to appear in the research literature. In the late 1990’s Glenn et al. and Te Wierik et al. independently demonstrated that the low surface area of native starch could be expanded to generate xerogel materials with surface areas (SBET) between 25 and 145 m2 g -1 depending on preparative route. [69] Glenn et al. demonstrated that a highly expanded starch material could be obtained from retrograded starch gels by successively exchanging water for a series of lower surface tension solvents, consequently followed by drying via supercritical carbon dioxide (ScCO2) to yield a low density aerogel product. More recently research at York has demonstrated the potential of corn starch (~ 73 % amylopectin) in the production of low density, high surface area starch xerogels (SBET ~ 120 m2 g-1) for applications in normal phase chromatography separations. [70] Furthermore, our most recent research at York has demonstrated the tuneable microwave assisted preparation of high surface area (SBET > 180 m2 g-1), highly mesoporous starch derived materials (Vmeso > 0.6 cm3 g-1; > 95 % mesoporosity) [71]. This research demonstrated that the key to the formation of the porous polysaccharide form in starch was the induction of metastable polysaccharide gel states. Investigation via the aqueous adsorption of methylene blue probe dye demonstrated the porous state exists and is a product of the polysaccharide gel formation. Significantly we also demonstrated that the key mesopore forming polysaccharide in starches appears to be the linear α(1→4) polyglucopyranose, amylose, as opposed to the other branched starch polysaccharide, amylopectin (Figure 11).

Figure 11. Relationship between mesopore volume, polysaccharide ordering (determined via FT-IR measurements) and % amylopectin content (by mass) in synthetic mixtures of α-D-polysaccharides.

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Figure 12. Mesoporous polysaccharide supported CdS quantum dot supported material (MS(130oC)), demonstrating mesoporous pore structure.

Aqueous, (typically thermally assisted) gelation of a given polysaccharide yields a porous gel state, which upon extraction of gel bound water via a solvent exchange process (normally for a lower surface tension alcohol (e.g. ethanol)) produces a low density porous polysaccharide xerogel. ScCO2 of the alcohol saturated gel produces the corresponding aerogel gel materials. Direct TEM analysis of the nanopore strucutre of mesoporous starch materials still represents a challenge due to media transparency in the electron beam. By applying strategies employed in the preparation of CdS quantum dot materials, pore structure, and morphology at the mesopore scale may be observed (Figure 12). Using CdS quantum dot preparation as a novel contrast agent, a number of slit shaped pores are observed and the mesoporous polysaccharide material appears to be constructed from nano polysaccharide crystallite associations potentially formed upon cooling, as a product of entropic gain, stabilising in the presence of water to generate the mesoporous aquagel network. Our generic approach optimised at York for Starch based systems is summarised in Figure 13. The reassociation and reorganisation of the polysaccharide chains during recrystallisation or retrogradation following gelation generates a porous state which can be maintained by solvent exchange and ScCO2 drying strategies employed in classical RF aerogel preparation, presenting material morphology similar at the micro and nano meter scale to conventional aerogel materials. There are a number of other literatures reports referring to the preparation of high surface area materials from polysaccharides such as cellulose and chitosan. Jin et al. have reported the preparation of cellulose xerogels via dissolution and crystallisation of cellulose from calcium thiocyanate aqueous solutions. [72] The resulting products produced via a solvent exchange drying step gave highly porous xerogels and aerogels (as visualised by SEM imaging (Figure 14), composed of approximately 50 nm wide cellulose micro fibrils

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presenting surface areas of 160 - 190 m2 g-1. However limited detailed discussion was made regarding mesopore or micropore characteristics of these materials.

Figure 13. Schematic Diagram depicting the major process steps in the preparation of the porous starch form (MS1). [Aquagel photograph courtesy of Dr. Kris Milkowski, Green Chemistry Centre, University of York, York, UK].

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Figure 14 SEM images of cellulose aerogel materials preparaed by Jin et al. dried via a rapid freeze drying approach using liquid nitrogen.

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Figure 15: High surface area porous cellulose aerogel materials as presented by Cai et al. prepared from an aqueous solution of 6 wt% cellulose/4.6 wt% LiOH/15 wt% urea, regenerated with either A) H2O, B) 10 % EtOH, C) 30 % EtOH, D) 50 % EtOH, E) 70% EtOH, or F) 90 % EtOH, and dried from ScCO2.

In a recent communication, Cai et al. reported the preparation of highly porous and strong cellulose aerogels via an aqueous alkali hydroxide/urea gelation route, followed by drying with ScCO2; materials presented surface areas approaching 400 m2 g-1, and pore sizes in the range 7 to 70 nm. [73] These materials demonstrated strong preparation dependent properties, allowing textural property direction via tuning of the gelation conditions used. Again interconnected fibrous pore networks similar to those presented by Jin et al. were observed via SEM imaging (Figure 15). Cai et al. also investigated the dimensional stability of such cellulose aerogel films, demonstrating high mechanical strength, presenting a Young’s modulus of between 200 – 300 MPa depending on preparation conditions used. Porous cellulose materials reported by Gavillon and Budtova presented surface areas above 200 m2 g-1, with pore size ranging from 6 to over 10,000 nanometres, with the majority of porosity lying in the macropore range. [74]Similar preparative routes (employing aqueous

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acetic acid) have been employed for the preparation of porous chitosan microspheres with surface area of 110 m2 g-1. [75] The utilisation of the acidic polysaccharides pectin and alginic acid to prepare porous materials, has as yet received minimal attention. Recent investigations by Quignard et al. have demonstrated the utilisation of the alginates, as base materials for the preparation of nanoporous solids. Quignard et al., have been particularly successful at producing aerogel products from the gelation of sodium alginates with divalent cations such as Ca2+. [76] Subsequent solvent exchange with ethanol and ScCO2 drying produced a porous material and millimetre sized beads were possible. Such materials presented high surface area materials (> 300 m2 g-1), possessing very broad pore size distributions. [77] However little discussion was made regarding pore size or volume of these metal alginate aerogels. Quignard et al., have also demonstrated the use of metal alginate aerogels for the preparation of photo luminescent materials, prepared via the gelation of sodium alginate in the presence of trivalent lanthanide ions such as Eu3+ and Tb3+. [78] Materials produced in this way, exhibited high surfaces areas (> 380 m2 g-1). Acidification / neutralisation of sodium alginate gels was found to yield aqueous alginic acid gels (and upon ScCO2 drying the aerogel product) with high surfaces areas (SBET ~ 391 m2 g-1), and broad pore size distributions ranging from 3 to over 60 nm. [77] Current research at York is looking at means of utilising a range of different polysaccharide for the preparation of novel xero- and aerogel materials. ScCO2 drying of alcohol saturated polysaccharide gels is proving to be a very tool to efficiently maintain the porous state generated in the aqueous gelation of the native polysaccharide. We are also looking at means to exploit the surface chemistry and textural properties of these porous materials. Excitingly, we have recently published the use of porous starches in the preparation of supported palladium nanoparticle heterogeneous catalysts, which were found to be useful reusable catalysts in C-C bond forming approaches such as the Heck, Sonogashira and Suzuki reactions. [79]The unique textural properties (i.e. a hierarchical pore structure) of these starch xerogels, allowed the preparation of sub 5 nm Pd nanoparticles, presenting a biomodal size distribution believed to be reflective of the micropore and mesoporous structure. The porous polysaccharide support acted as both a stabilising support for small metallic nanoparticle growth but also as the site for the deposition and metal cation reduction, negating the use of conventional reducing agents. The preparation of porous materials (e.g. aerogels) from polysaccharide is a relatively new area and is quickly receiving increased attention partly due to enhanced interest in biodegredable and environmentally friendly materials. Utilisation of polysaccharides in the preparation of aerogel materials represents an interesting alternative to the materials prepared via conventional polymerisation and co-condensation techniques. Furthermore, the use of polysaccharides derived from biomass potentially adds value to inexpensive and typically waste products from industry such as the food sector.

CARBON AEROGELS Carbon aerogels have a well-developed and controlled micro- and mesoporosity as well as presenting large surface areas. Carbon aerogel structure consists of a network of

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interconnected nanosized primary particles. Micropores are related to the intraparticle structure, whereas mesopores and macropores are produced by interparticle structure and arrangement. In cases where different processes are responsible for forming the intra- and interparticle structure, it is, possible to control the concentration of micropores and mesopores independently, presenting a useful mechanism by which to control the textural properties and is considered one of the main advantages of carbon aerogels [80]. The fine-tuning of the synthesis conditions allows direction of pore structure and textural morphology of the organic aerogel materials introduced earlier in this chapter.

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POSSIBLE RAW MATERIALS FOR CARBON AEROGELS There is a wide selection of raw materials for making porous carbons such as sucrose, polydivinylbenzene, furfuryl alcohol, and among these phenolic resins are advantageous as the structure of the organic precursor may be successfully maintained during processing. There are also a number of routes employing inorganic precursors or templates for the preparation of nanoporous forms of carbon. The unique nanoporous structure of carbidederived carbon (CDC) with its narrow pore size distribution and possibility to fine-tune the pore size, has been used in the preparation of textured nanoporous carbon material where material structure and crystallinity depending noticeably on the chlorination temperature and the metal carbide employed. At temperatures up to 900 °C, amorphous carbon with a high specific surface area ~ 1300 m2 g−1 may be produced via this route. At higher chlorination temperatures, increased graphitisation and reduction in the specific surface area was observed [81]. Hard template methods for the synthesis of ordered mesoporous carbons most famously have used ordered inorganic mesoporous materials such as MCM-48 [82] or SBA-15[83]. The introduction of the carbon precursor to the pores is typically achieved either by chemical vapour deposition or by liquid infiltration of an organic precursor (e.g. furfuryl alcohol). The material is then heat treated to induced carbonisation / graphitisation, with the inorganic template removal typically achieved with aqueous HF or caustic treatment to yield the inverse carbon replica. The emulsion of a molten mesophase pitch after foaming because of blowing technique, a thermal stabilization up to 1000 °C is accomplished to get large macropores [84]; even carbon nanofibers have been introduced during the process to reinforce the resulting foam [85]. Other low-cost carbon precursors, such as coal, coal tar pitch and petroleum pitch, have also been used to prepare various foams [86]. Nanotube aerogels were dimensionally strong and electrically conductive. Furthermore, ensembles of such nanotubes are useful aerogel precursors: they form electrically percolating networks at very low volume fractions and elastic gels in concentrated suspensions through van der Waals interactions mediated inter-tube cross-linking. With adding proper surfactant the foam is formed, cooled down and freeze-dried. Further these dried solid materials are pyrolysed under a nitrogen atmosphere and thermally treated at 1200 ºC [87]. In another example after the gelation the surfactant was replaced (washed out) with water solution of polyvinyl alcohol (PVA). [88]. As well as assisting surfactant removal, the PVA was found to reinforce the gel structure. The electrical conductivity of the resulting aerogels was found to depend on several factors, including nanotube and PVA content, and the drying process

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employed (i.e. freeze-drying or supercritical extraction). Freeze-dried samples were consistently less conductive than the supercritically dried samples believed to be the result of reduced mechanical distortion in the latter examples. The preparation of nanoporous forms of naturally occurring polysaccharides, as mentioned earlier in this chapter, negates the use of conventional templates reducing overall resource consumption and process steps. In this regard, the conversion to more thermally and chemically stable carbonised porous polysaccharide derived materials may also add to the potential environmental benefits, as this process may represent a form of carbon sequestration particularly if the polysaccharide is derived from fast growing plants. [89] Herein lies the interface between the idea of Green Chemistry and modern materials science; adding value to biomass via the generation of novel porous materials, demonstrating application advantages, whilst at the same time generating as minimal as possible carbon footprint. Such materials, initially developed from carbonised forms of porous starch xerogels at the Green Chemistry Centre, York, have been termed Starbons®, as they present chemical properties intermediary between Starch (Star..) and conventional activated carbons (…bons) (Figure 16). [90]

Figure 16. (A) Porous starch monolith, and (B) Porous Starbon® monolith prepared via a thermally assisted acid catalysed decomposition of the porous starch precursor.

Due to melting of starch before decomposition, direct thermal degradation of expanded starch yields non-porous carbonaceous materials. An acid catalysed thermal decomposition of

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mesoporous starch enables the maintenance and preparation of predominantly mesoporous materials at relatively low temperatures (Figure 17). The lack of a (classical) template avoids wasteful processing steps and harmful chemicals, and allows materials to be prepared at a temperature of choice (e.g. 300 – 1000 oC); this enables tuneability of surface chemistry amenable to potential facile post modification strategies. The surface chemistry of these materials is summarised in Figure 18.

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Figure 17. Textural properties of Starbons® as a function of the pyrolisis preparation temperature; [PV = Pore volume, SA = BET Surface area, PD = BJH Pore Diameter].

Figure 18. Surface chemistry properties of Starbons® materials as a function of pyrolisis preparation temperature.

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Furthermore, as a consequence of preparation dependent surface chemistry, the hydrophilicity – hydrophobicity properties may be moderated, generating the possibility of designer material synthesis. [91] Figure 19 demonstrates the transition from a predominantly “Starchy” polysaccharide type materials presenting aliphatic and carbonyl type functionality through to material presenting extended conjugated and aromatic systems as a function of the pyrolisis preparation temperatures for these Starbons® materials via solid state 13C CP MAS NMR analysis. This analysis was complemented by TEM analysis, which indicated that the acid catalysed decomposition of the porous starch form occurred at the surface and within the mesopores (Figure 20).

Figure 19. Solid state 13C CP MAS NMR investigation of the variation in chemical nature of Starbons® as a function pyrolysis preparation temperatures from the parent starch material through pyrolysis temperatures in the range of 100 – 450 °C.

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Figure 20. TEM images of Starbons® prepared at a range of temperatures – (A) mesoporous starch precursor; (B) 100 °C; (C) 150 °C; (D) 220 °C; (E) 300 °C; and (F) 450 °C.

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Variability in surface chemistry as a function of preparation temperature not only provides a mechanism for control over the hydrophobic – hydrophilic properties of these materials, but also allows for relatively straight forward surface chemical transformations to be conducted (i.e. bromination, fluorination or amination), which would otherwise be challenging for conventional mesoporous carbon materials, prepared typically at much higher temperatures. One possibility currently under investigation at York, is the utilisation of different polysaccharides in the preparation of these Starbon® materials. If new novel forms of different polysaccharides can be developed and transformed into similar carbonised equivalents, such materials may present different porous characteristics as compared to the mesoporous starch form used in first generation Starbons preparation. This will potentially add another dimension to this mesoporous carbon synthesis approach. We are now looking at the exploitation of ScCO2 dried starch (and other polysaccharides) aerogels in the preparation of 2nd generation Starbons® material technology.

PYROLYSIS The main procedure to obtain porous carbons and porous carbon aerogels is pyrolysis usually performed under a non-oxidative atmosphere / inert atmosphere (He, Ar or N2). Commonly a tubular furnace is used, allowing material preparation temperatures as high as 1200 ºC. Stepwise heating programs using heating rates of between 5 – 10 °C min-1, using final isothermal hold at the desired preparation temperature of typically between 1 - 4 hours. After pyrolytic material preparation step is complete, the furnace is allowed to cool to room temperature upon which the sample may be collected for use or analysis.

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Figure 21. SEM image of a typical carbon aerogel.

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For organic aerogel precursors the textural properties and material morphology is basically retained upon completion of this pyrolysis step. Aerogel precursors upon pyrolysis undergo mass loss, as the thermal evolution of typically chemisorbed H2O and CO2 proceeds with increasing furnace temperature. This results in the reduction of oxygen content, generation and extension of conjugated and aromatic systems and ultimately at temperatures in excess of 1200 °C the formation of developed graphitic systems. Such chemical and physical transition in the material result in a volumetric shrinkage and densification relative to their pre-carbonized organic aerogel precursor. [26, 44, 92] The resulting porous material is mainly composed of carbon. Pore size may be tailorable, as is material density and specific surface area via selection of the organic aerogel employed as the parent porous material. Figure 21 depicts the surface morphology, as visualised by SEM imaging, of a typical carbon aerogel material as prepared by the authors and co-workers in Tallinn, Estonia.

Impact of Temperature Program on the Structure of Carbon Aerogels Research in Estonia by the authors and co-workers demonstrated the preparation and investigation of studies carbon aerogels prepared at three different carbonisation temperatures: 700, 780 and 1000 ºC. Comparison of materials prepared at different final preparation temperatures, namely 700 ºC and 1000 ºC, demonstrated that lower carbonisation temperatures provided materials with more developed micropore volume and surface area than for materials prepared at higher pyrolysis temperatures. At increasingly higher pyrolysis / preparation temperatures, the material particle size becomes increasingly smaller resulting in a reduction in the micropore content (i.e. micropore diameter) that is accessible to the nitrogen probe molecule employed in gas adsorption / desorption investigations [93,94] The similarity between such porous carbon and their organic aerogel precursors in terms of

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bulk density confirms that organic aerogels may be transformed via this route to the corresponding carbon aerogel material without a structural collapse and loss of surface and porosity. Optimisation of the pyrolysis program allowed the production of carbon aerogels characterised by increased specific surface area and microporosity. The study of the influence of the pyrolysis temperature and program on the final structure of carbon aerogels revealed the importance of this parameter in the monitoring of the final properties of carbon aerogels. In our previously reported work [95] the pyrolysis temperature program employed utilised a stepwise increase in temperature featuring distinct isothermal steps (i.e. 30 min) using a heating rate of 10 ºC min-1. Gels are then held at the final pyrolysis temperature (from 700 to 1000 °C) for some hours, and then the furnace is cooled to room temperature under its own thermal mass. Calculation of the mass loss at different temperatures obtained from thermogravimetric analysis (see TGA curve on Figure 23) it was possible to optimise the pyrolysis temperature program based on minimal evolution of degradation products, resulted in a slowing down of the program rate for certain temperature intervals and taking into consideration the exothermic nature of reactions at temperatures around 570 °C. It appeared that there is no need for a final temperature as high as 1000 °C, and good results would be achieved by using the proposed temperature program because the change in mass was not noticeable at that high temperature. For the resorcinol-formaldehyde aerogels thermal analysis demonstrated that at temperature = 300 °C an exothermic reaction occurs. Therefore the heating rate in this region was reduced – samples were kept at 300 °C for some time and then slowly heated to 550 °C at a heating rate of 2 °C min-1.

Figure 22. Stepwise temperature program for the pyrolysis of MR-F aerogels proposed.

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Figure 23. A Thermogravimetric diagram of a MR-F sample.

Intensive reactions are taking place at that temperature, and this is a next step in program to hold the temperature for some time. After that the temperature raised at a rate of 10 °C min-1 up to 780 °C. The material preparation temperature was taken as the temperature at which no further intensive reactions were observed to occur. The choice of the final temperature depends on the requirements for pore size and specific surface area [96]. Above 780 °C, as the gel mass loss was very small, more time was needed to increase the activation of the gel, though no considerable changes were expected above the final temperature used in this study. There was a mass loss of greater than 80 % at temperatures < 750 °C with most exo- and endothermic reactions observed taking place at temperatures between 400 and 650 °C. [97] Surface areas and pore volumes for gels pyrolysed using common linear temperature programs and the stepwise program demonstrated that optimisation of the temperature program led to materials with much higher Langmuir specific surface area and overall porosity than those obtained via the non-optimised program.

ACTIVATION RF based carbon aerogels have a network structure of primary carbon particles, providing a predominantly mesoporous structure, with these carbon particles presenting few micropores relative to their parent organic aerogels which are considered mainly mesoporous materials and have limited developed microporosity accessible to the nitrogen [98]. The activation

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procedure is well know for common carbon materials and can be easily applied to the preparation of carbon aerogels generating increased microporosity and specific surface area of aerogel. Activated carbon aerogels (ACs) produced in this manner presented a bimodal pore size distribution of uniform micropores and mesopores [99]. Conventionally prepared carbon aerogels provide predominantly mesopores, resulting in a relatively small surface area in comparison with microporous carbons. Without change in the skeletal carbon gel structure, production of micropores by activation of carbon aerogel, giving a unique carbon material with a bimodal pore structure could increase the useable surface area for EDL (electric double-layer) formation and a high internal resistance [100]. Carbon materials are generally activated under a CO2 flow at high temperatures. Recently, Alar Jänes et al. have reported very interesting results using water vapour as activating agent at temperatures between 950 °C and 1150 °C. The BET specific surface area of activated samples increased from 1358 m2 g-1 up to over 2200 m2 g-1. The electrochemical characteristics obtained showed that some of these materials could be used for compilation of high energy density and power density non-aqueous electrolyte supercapacitors with higher power densities than aqueous capacitors [101]. Sánchez-Montero et al. analyzed the properties of carbon fibers activated with supercritical CO2, a strong development in the porosity as well as in specific surface area was found though with different pore size distributions for supercritical CO2 and low temperature CO2 activation. Activation with supercritical CO2 affords a widening of the narrow porosity, giving rise to activated carbon fibers with wider micropores [102].

Figure 24.The BET surface area of carbon aerogels activated with water or carbon dioxide [95].

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Effect of Modifiers on Activation The precursors of porous carbon material were synthesized through polycondensation of resorcinol (R) with formaldehyde (F) catalyzed by KOH. Then the RF gel without removing of the KOH was dried at 110 °C at ambient pressure directly; the dried RF gel was carbonised in a tubular furnace under flowing nitrogen from room temperature up to 700 °C. During the carbonising process, KOH in RF gel serves as activating agent with the aim of producing pores to increase surface area. Then the furnace was allowed to cool down to room temperature under nitrogen flow and the obtained sample was subsequently washed with deionized water, transferred into 0.1 M HCl and stirred for 1–2 h, and washed with deionized water again until pH of the washing solution was about 6 - 7. Finally, the sample was dried at 110 °C. [103]

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APPLICATIONS AND FURTHER PROSPECTS Aerogel materials have demonstrated an incredible versatility of applications and not only at the lab scale. Until recently, aerogels were considered prohibitively expensive (around €5000 m-3 for powder of silica aerogels for example), but the constant progress achieved in the chemical processes involved and the development of aerogel applications are likely to bring about significant reductions in cost, especially with optimised processes operated at very large scale[104]. This success of silica aerogels gives hope that organic aerogels and related carbon aerogels will find wider applications. The highly porous structure of aerogels where material structure is composed of very small particles linked in a three-dimensional network (with many "dead-ends") make them very useful insulators. The transport of thermal energy through the aerogel occurs through a very tortuous path and is not particularly effective. Sealing of the material under vacuum makes it even less effective. The preparation of highly porous aerogels derived from biomass and polysaccharides may potentially provide useful, inexpensive and ultimately biodegradable insulation and packaging materials. All of the aerogel varieties have unusual acoustic and mechanical behaviour, which could be exploited for various applications. In acoustics, aerogels have been implied in sound absorption (e.g. anechoic chambers) and in the construction of efficient ultrasonic devices [105]. Aerogels are also known to absorb kinetic energy by plastic and elastic deformations and as such have found use in the collection of particulate matter like cosmic or cometary dust during experiments in space (NASA’s Stardust Project) [106] Organic aerogels possess excellent dielectric properties and can be used for microwave electronics and as high voltage insulators. On the other hand, carbon aerogels are electrically conductive, and as such have found application as electrode materials for batteries and capacitors. Aerogels have been recognized for many years as excellent catalysts and catalyst supports. The great advantage of synthetic carbon aerogels as supports for fuel cell catalysts in comparison with common activated carbons is that the pore size can be moderated and optimised for optimum performance. The porosity of the aerogel can be altered via chemical

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synthesis modification with further distribution of Pt particles on its surface allowing good contact with the ionically conducting polymer electrolyte. Further improvement in the cell performance can be achieved by optimisation of the aerogel structure and catalyst layer composition. The unique structure of the aerogel supported Pt catalyst having large pores, even distribution of Pt particles inside the porous carbon structure, and less tendency for agglomeration and sintering during the cell operation, make this catalyst the most promising novel catalyst for fuel cell operation [107]. In this regard the preparation of carbon aerogels via synthetic polymerisation routes or porous polysaccharide aerogels presents a promising alternative to the use of conventional carbons, which often suffer from impurities such as sulphur, which when applied in a fuel cell, act to reduce the activity and ultimately the lifetime of the solid catalyst as the surface metal species (e.g. nickel) becomes poisoned. The high porosity and very large surface areas available in typical aerogel materials, is leading to applications as filters, absorption media for desiccation and waste containment, and as encapsulation media. Potentially the most intriguing use of carbon aerogels is in hydrogen storage. Molecular hydrogen can be stored as the physisorbed species on low weight materials with a large specific surface area: carbon nanotubes, activated carbons (ACs), carbide-derived carbons (CDCs) and other carbon nanostructures and alkali-doped carbon nanotubes and nanostructures. One operational requisite is the reversible adsorption at room temperature and at low and medium pressures. Thermodynamic estimations indicate that for this to occur, the binding energy of molecular hydrogen to those surfaces should be about 300 - 400 meV/molecule. [108] The binding energy of a hydrogen molecule to a typical graphitic surface is only about 100 meV/molecule. Hence, it is necessary to increase the binding energy by a factor of three or four. A possible way to achieve this is to generate interactions of the molecule with two or more surfaces at the same time. This can be achieved inside nanopores (e.g. micropores). A graphene slitpore gives a simple model for the pores existing in nanoporous carbon materials such as ACs and CDCs. That slitpore consists of two parallel graphene layers separated a certain interlayer distance. The model predicts that in order to reach the US Department of Energy hydrogen storage targets for 2010, the nanopore widths should be equal to or larger than 0.56 nm for applications at low temperatures, -193 °C, and any pressure, and about 0.6 nm for applications at 27 °C and at least 10 MPa [109]. The use of naturally occurring polysaccharide derived aerogel materials, presents an interesting alternative to conventionally prepared materials. The self-assembly of the starch polysaccharides into mesoporous xero and aerogel forms and there subsequent conversion into their carbon aerogel equivalents (e.g. Starbons®) presents an efficient and cost effective route to novel impurity free carbon aerogel materials, potentially ideal for future fuel cell applications. The temperature dependent surface chemistry of these materials also presents readily available functionality upon which post modification strategies can be applied. Research at York has demonstrated the utility of acidified versions of these Starbons®, carrying -SO3H groups in a variety of organic transformations. Interestingly the preparation temperature dependent activity of such solid acid catalysts has been demonstrated in the aqueous phase esterification of potential future biorefinery products such as succinic acid [110]. These approaches to mesoporous aerogel preparation may go in some way to achieving the aims of sustainability and green chemistry within modern materials science.

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a)Pekala, R. W. J Mater Sci. 1989, 24, 3221-3227. b)Pérez-Caballero, F.; Peikolainen, A.-L.; Uibu, M.; Kuusik, R.; Volobujeva, O.; Koel, M. Micropor Mesopor Mat. 2008, 108, 230-236.c)Budarin, V.; Clark, J. H.; Hardy, J. J. E.; Luque, R.; Milkowski, K.; Tavener, S. J.; Wilson, A. J. Angew Chem Int Edit. 2006, 45, 3728-3786. Gommes, C. J.; Roberts, A. P. Phys Rev. 2008, 77, 041409. Pekala, R. W. US Patent 4,873,218. 1989. Peikolainen, A.- L.; Pérez-Caballero, F.; Koel, M. Oil Shale. 2008, 25, 2008, 348-358. Nguyen, M. H.; Dao, L. H. J Non-Cryst Solids, 1998, 225, 51-57. Wu, D.; Fu, R. J Porous Mat. 2005, 12, 311-316. Albert, D. F.; Andrewa, G. R.; Mendenhall, R. S.; Bruno, J. W. J Non-Cryst Solids. 2001, 296, 1-9. Wu, D.; Fu, R.; Zhang, S.; Dresselhaus, M. S.; Dresselhaus, G. J Non-Cryst Solids. 2004, 336, 26-31. Pekala, R. W.; Schaefer, D. W. Macromolecules. 1993, 26, 5487-5493. Tamon, H.; Ishizaka, H. J Colloid Interf Sci. 1998, 206, 577-582. Kim, H.- J.; Kim, J.- H.; Kim, W.- I.; Suh, J. S. Korean J Chem Eng. 2005, 22, 740744. Ruben, G. C.; Pekala, R. W. J Non-Cryst Solids. 1995, 186, 219-231. Pekala, R. W.; Alviso, C. T.; Lu, X.; Gross, J.; Fricke, J. J Non-Cryst Solids. 1995, 188, 34-40. Zhang, R.; Lu, Y.; Zhan, L.; Liang, X.; Wu, G.; Ling, L. Carbon. 2002, 41, 1660-1663. Biesmansa, G.; Mertensa, A.; Duffoursb, L.; Woignierb, T.; Phalippoub, J. J Non-Cryst Solids. 1998, 225, 64-68. Yamashita, J.; Ojima, T.; Shioya, M.; Hatori, H.; Yamada, Y. Carbon. 2003, 41, 285– 294. Mukai, S. R.; Tamitsuji, C.; Nishihara, H.; Tamon, H. Carbon. 2005, 43, 2618-2641. Li, W.; Lu, A.; Guo, S. Carbon. 2001;39, 1989–1994. Li, W.; Reichenauer, G.; Fricke, J. Carbon. 2002, 40, 2955–2959. Lee, J.; Kim, H.; La, K. W.; Park, D. R.; Jung, J. C.; Lee, S. H.; Song, I. K. Catal Lett. 2008, 123, 90-95. Grenier-Loustalot, M.- F.; Larroque, S.; Grande, D.; Grenier, P.; Bedel, D. Polymer. 1996, 37, 1363-1369. Horikawa, T.; Hayashi, J.; Muroyama, K. Carbon. 2004, 42, 1625-1633. Barral, K. J Non-Cryst Solids. 1998, 225, 45-50. Mulik, S.; Sotiriou-Leventis, C.; Leventis, N. Chem Mater. 2007, 19, 6138-6144. Brandt, R.; Petricevic, R.; Pröbstle, H.; Fricke, J. J Porous Mat. 2003, 10, 171-178. Tamon, H.; Ishizaka, H.; Araki, T.; Okazaki, M. Carbon. 1998, 36, 1257-1262. Tamon, H.; Ishizaka, H.; Mikami, M.; Okazaki, M. Carbon. 1997, 35, 791-796. Yamamoto, T.; Nishimura, T.; Suzuki, T.; Tamon, H. J Non-Cryst Solids. 2001, 288, 46-55. Pekala, R. W. US Patent 4,997,804. 1991. Zhang, R.; Li, W.; Li, K.; Lu, C.; Zhan, L.; Ling, L. Micropor Mesopor Mat. 2004, 72, 167-173.

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[31] Kim, S. Y.; Yeo, D. H.; Lim, J. W.; Yoo, K.- P.; Lee, K. H.; Kim, H. J Chem Eng Jpn. 2001, 34, 216-220. [32] Job, N.; Panariello, F.; Crine, M.; Pirard, J.- P.; Léonard, A. Colloid Surface A. 2007, 293, 223-228. [33] Tamon, H.; Ishizaka, H. J Colloid Interf Sci. 2000, 223, 305-307. [34] Perrut, M.; Français, E. In State of the Art Book on Supercritical Fluids, Edited by AINIA; ISBN: 84-87345-68-9; AINIA: Valencia, 2004; pp 129-134. [35] Mandel, F. S.; Wang, J. D. Inorg Chim Acta. 1999, 294, 214-223. [36] Job, N.; Théri, A.; Pirard, R.; Marien, J.; Kocon, L.; Rouzaud, J.- N.; Béguin, F.; Pirard, J.- P. Carbon. 2005, 43, 2481-2494. [37] Shen, J.; Wang, J.; Zhai, J.; Guo, Y.; Wu, G.; Zhou, B.; Ni, X. J Sol-Gel Sci Techn. 2004, 31, 209-213. [38] Scherer, G. W.; Gross, J.; Hrubesh, L. W.; Coronado, P. R. J Non-Cryst Solids. 2002, 311, 259-272. [39] Czakkel, O.; Marthi, K.; Geissler, E.; László, K. Micropor Mesopor Mat. 2005, 86, 124-133. [40] Wu, D.; Fu, R.; Zhang, S.; Dresselhaus, M. S.; Dresselhaus, G. Carbon. 2004, 42, 2033-2039. [41] Wu, D.; Fu, R.; Dresselhaus, M. S.; Dresselhaus, G. Carbon. 2006, 44, 675-681. [42] Lee, K. T.; Oh, S. M. Chem Commun. 2002, 2722–2723. [43] Jun, S.; Joo, S. H.; Ryoo, R.; Kruk, M.; Jaroniec, M.; Liu, Z.; Ohsuna, T.; Terasaki, O. J Am Chem Soc. 2000, 122, 10712-10713. [44] Baumann, T. F.; Satcher, J. H. Jr. J Non-Cryst Solids. 2004, 350, 120-125. [45] Tanaka, S.; Nishiyama, N.; Egashira, Y.; Ueyama, K. Chem Commun. 2005, 21252127. [46] Price, D. W.; Schmidt, P. S. J Microwave Power EE. 1997, 32, 145-154. [47] Zubizarreta, L.; Arenillas, A.; Domínguez, A.; Menéndez, J. A.; Pis, J. J. J Non-Cryst Solids. 2008, 354, 817-825. [48] Tonanon, N.; Wareenin, Y.; Siyasukh, A.; Tanthapanichakoon, W.; Nishihara, H.; Mukai, S. R.; Tamon, H. J Non-Cryst Solids. 2006, 352, 5683-5686. [49] Mayer, S. T.; Kong, F.- M.; Pekala, R. W.; Kaschmitter, J. L. US patent 5,908,896. 1999. [50] Liu, N.; Zhang, S.; Fu, R.; Dresselhaus, M. S.; Dresselhaus, G. Carbon. 2006, 44, 2430–2436. [51] Yamamoto, T.; Endo, A.; Eiad-ua, A.; Ohmori, T.; Nakaiwa, M.; Soottitantawat, A. AIChE Journal. 2007, 53, 746-749. [52] Yamamoto, T.; Sugimoto, T.; Suzuki, T.; Mukai S. R.; Tamon, H. Carbon. 2002, 40, 1345–1351. [53] Lambert, S. M.; Overturf, G. E.; Wilemski, G.; Letts, S. A.; Schroen-Carey, D.; Cook. R. C. J Appl Polym Sci. 1997, 65, 2111–2122. [54] Fujikawa, D.; Uota, M.; Yoshimura, T.; Sakai, G.; Kijima, T. Chem Lett. 2006, 35, 432433. [55] Kim, T. Y.; Jin, H. J.; Park, S. S.; Kim S. J.; Cho, S. Y. J Ind Eng Chem. DOI: 10.1016/j.jiec.2008.07.004. 2008. [56] Velev, O. D.; Jede, T. A.; Lobo, R. F.; Lenhoff, A. M. Nature. 1997, 389, 447-448. [57] Holland, B.T.; Abrams, L.; Stein, A. J Am Chem Soc. 1999, 121, 4308-4309.

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[87] Leroya, C. M.; Carna, F.; Backova, R.; Trinquecostea, M.; Delhaes, P. Carbon. 2007, 45, 2317-2320. [88] Bryning , M. B.; Milkie , D. E.; Islam, M. F.; Hough, L. A.; Kikkawa , J. M.; Yodh, A. G. Adv Mater. 2007, 19, 661-664. [89] Titirici, M. M.; Thomas, A.; Antonietti, M. New J Chem. 2007, 31, 787-789. [90] Budarin, V. L.; Clark, J. H.; Luque, R.; Macquarrie, D. J.; Milkowski, K.; White, R. J. PCT/EP2007/052482, UK. 2007. [91] Budarin, V. L.; Clark, J. H.; Luque, R.; Milkowski, K.; Tavener, S. J.; Wilson, A. J. Angew Chem Int Edit. 2006, 45, 3782–3786. [92] Budarin, V. L.; Luque, R.; Macquarrie, D. M.; Clark, J. H. Chem–Eur J. 2007, 13, 6914-6919. [93] Pérez-Caballero, F.; Peikolainen, A.- L.; Uibu, M.; Kuusik, R.; Volobujeva, O.; Koel, M. Micropor Mesopor Mat. 2008, 108, 230-236. [94] Hanzawa, Y.; Hatori, H.; Yoshizawa, N.; Yamada, Y. Carbon. 2002, 40, 575-581. [95] Marsh, H.; Wynne-Jones, W. F. K. Carbon. 1969, 7, 559-566. [96] Pérez-Caballero, F.; Peikolainen, A-L.; Koel, M.; Herbert, M.; Galindo, A.; Montilla, F. The Open Petroleum Engineering Journal. 2008, 1, 42-46. [97] Moreno-Castilla, C.; Moldano-Hódar, F. J. Carbon. 2005, 43, 455–465. [98] Pérez-Caballero F.; Peikolainen, A.- L.; Koel, M. Proceedings of the Estonian Academy of Sciences. 2008, 57, 48–53. [99] Zhou, J.; He, J.; Ji, Y.; Zhao, G.; Zhang, C.; Chen, X.; Wang, T. Acta Phys-Chim Sin. 2008, 24, 839-843. [100] Yang, W.; Wu, D.; Fu, R. J Appl Polym Sci. 2007, 106, 2775-2779. [101] Fang, B.; Binder, L. J Power Sources. 2006, 163, 616-622. [102] Jänes, A.; Kurig, H.; Lust, E. Carbon. 2007, 45, 1226-1233. [103] Sánchez-Montero, M. J.; Salvador, F.; Izquierdo, C. The Journal of Physical Chemistry C. 2008, 112, 4991-4999. [104] Zhu, Y.; Hu, H.; Li, W.; Zhang, X. Carbon. 2007, 45, 160-165. [105] Perrut, M.; Français, E. US Patent 5,962,539. 1999. [106] Lorraine, P. W.; Smith, L. S. US Patent 5,655,538. 1997. http://stardust.jpl.nasa.gov/ [107] Smirnova, A.; Dong, X.; Hara, H.; Vasiliev, A.; Sammes, N. Int J Hydrogen Energ. 2005, 30, 149-158. [108] Li, J.; Furuta, T.; Goto, H.; Ohashi, T.; Fujuwara, Y.; Yip, S. J Chem Phys. 2003, 119, 2376-2385. [109] Cabria, I.; López, M. J.; Alonso, J. A. Carbon. 2007, 45, 2649-2658. [110] Clark, J. H.; Budarin, V., Dugmore, T.; Luque, R.; Macquarrie, D. J.; Strelko, V. Catalysis Communications. 2008, 9, 1709-1714.

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In: Progress in Porous Media Research Editors: Kong Shuo Tian and He-Jing Shu

ISBN: 978-1-60692-435-8 © 2009 Nova Science Publishers, Inc.

Chapter 10

REDUCING POLYCYCLIC AROMATIC HYDROCARBONS BY POROUS MATERIALS Jian Hua Zhu*,1, Shi Lu Zhou1, Ying Wang2, Ling Gao1,2, Zhi Yu Yun1 and Jia Hui Xu1 1

Key Laboratory of Mesoscopic Chemistry, School of Chemistry and Chemical Engineering, Nanjing University, Nanjing 210093, China 2 Ecomaterials and Renewable Energy Research Center (ERERC), Nanjing University, Nanjing 210093, China

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ABSTRACT Adsorption and catalytic cracking of polycyclic aromatic hydrocarbons (PAHs) such as fluoranthene (Flu) and anthracene (Ant) in microporous zeolite NaY, NaZSM-5 and NaA, along with amorphous materials such as alumina, silica and activated carbon, were investigated for the first time to study the selective reduction of PAHs by zeolite in tobacco smoke. Reduction of PAHs in the smoke by zeolite was observed depending on the type of zeolite and PAH; And the elimination mechanism of PAHs was discussed in this article, which resulted in not only the direct adsorbing or cracking the PAHs species, but also the suppressing the formation of PAHs by zeolites in the burning cigarette. Zeolites trapped the intermediates/precursors that were necessary for the formation of PAHs so that the thermal formation of the carcinogenic compounds was thus extinguished, leading to the reduction of PAHs in smoke. Besides, the decrease of PAHs in smoke by zeolites seemed unique because neither tobacco combustion nor the levels of all smoke products were interfered. Moreover, the presence of zeolites additive lowered the mean toxicity of the cigarette smoke as proven by Ames assay and CHO cell assay, which was beneficial for the protection of environment and public health.

Keywords: zeolite; polycyclic aromatic hydrocarbon; tobacco smoke; adsorption; reduction; protection of environment.

*

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1. INTRODUCTION Owing to their specific ability to recognize, discriminate and organize molecules with precisions that can less 0.1 nm, zeolites have well-known application in petrochemical and chemicals industries as well as laundry detergents as the adsorbent, ion-exchanger and catalysts or catalyst carrier [1]. They have the potential application of the microporous storage media of gases such as hydrogen and are of increasing interest as the hosts for nanotechnology applications [2]. Recently zeolites play an increasingly important role in environment protection and the life science. Apart from slow-releasing drug, NO-releasing [3], nontoxic medical diagnosis tools and clotting enhancers [2], a noteworthy example is the reduction of carcinogenic compounds in tobacco smoke [4-7]. Smoking is a well known global problem of environmental pollution to cause serious health hazard, nonetheless it is difficult to remove the unwanted compounds in smoke with selectivity because tobacco smoke contains thousands components in vapor phase and particulate phase, hindering the distinguishing and trapping of harmful constituents target. Among these toxic and carcinogenic agents, polycyclic aromatic hydrocarbons (PAHs) were considered to be the most relevant smoke constituents in the 1960s and a lot of work was thus undertaken to reduce their levels in smoke. In general interfering with the burning rate of the cigarette, or using a conventional cigarette filter, can reduce the level of all smoke products, including PAHs, for instance as the ‘tar’ yields of cigarettes have fallen from an average of about 35 mg to about 10 mg over the last forty years or so, through the use of conventional cigarette filters, the yields of PAHs in smoke have also fallen by a similar proportion [8]. However, it was realized by the 1980’s that key cigarette design changes to reduce PAHs in smoke would also increase nitrosamines, which some now regard as more important in disease generation [9]. Later, Meier and Siegmann tried a new method to reduce PAHs in cigarette smoke, and they directly attached NaY zeolite as the catalyst onto the tobacco fibers of blend cigarette [4]. When the hot zone in the burning cigarette approached the zeolite, the zeolite would be thermally activated and catalyze the elimination of PAHs along with other harmful compounds. Owing to the shape selectivity of zeolite Y, as they reported [4], the PAHs molecules larger than the free pore diameter of 0.74 nm did not react noticeably in the mainstream of smoke that inhaled by smoker, while the mean reduction of PAHs, including fluoranthene (Flu), anthracene (Ant) and benzo[a]pyrene (B[a]P) could reach around 40% in side-stream (the tobacco smoke emitted in between puffs) [4]. However, Yong et al tried to use mesoporous materials MCM-48 as the additive to mix with tobacco or add into filter, but only got a similar performance, if not inferior, in reducing the PAHs in main stream smoke [10]. These preliminary study results indicate the complexity of eliminating PAHs by the ordered porous materials whose efficiency seems not proportional to their pore size. At the same time, the adsorption and catalytic degradation of volatile nitrosamines and tobacco specific nitrosamines by zeolites were detailedly investigated [11-15]. The zeolite with small pore could trap the bulky nitrosamines due to the electrostatic interaction between the cation of zeolite and the N-NO functional group of nitrosamines [16-18], which spurs us to assess the elimination of PAHs by zeolite and other porous materials such as amorphous silica and activated carbon, emphatically on the selective adsorption of PAHs in gas stream by these porous materials.

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One objective of the study is to assess whether zeolites can be used in cigarettes to selectively reduce the levels of PAHs in cigarette smoke. Although using zeolite catalyst in cigarette is expected to reduce the risks of smoking [4, 5, 7], there are still many uncertainties concerning how the zeolite reduces the PAHs in tobacco smoke, which will hinder the actual application of zeolites. Firstly, the role of zeolite adsorption in the reduction of PAHs is unclear though the adsorption precedes the catalytic decomposition. In our previous research [16-20], it was found that zeolites could adsorb bulky nitrosamines through a special way inducing nitrosamines to insert their N-NO group into the channel, consequently even the KA zeolite with a pore aperture of 0.3 nm still traps those nitrosamines with a much larger molecular diameter [6,19]. Is there similar effect in adsorption of PAHs? Can zeolite adsorb those large PAHs molecules? Secondly, usually sodium form of Y and ZSM-5 zeolites are expected to be inactive for catalytic cracking, how is the contribution of zeolite catalysis in the reduction of PAHs? Although fresh tobacco smoke containing light carboxylic and other acid might be acidic enough to protonate NaY prior to reaction [4], there was no clear evidence to support such protonation up to date. In contrast NaY and NaZSM-5 exhibited a considerable activity in degradation of nitrosamines indeed even in the inert atmosphere like helium or nitrogen [15, 21], could these zeolites with sodium cation catalyze the cracking reaction of PAHs? Thirdly, the PAHs in the cigarette smoke form from smaller precursor molecules during combustion [22], for instance most of the PAHs in smoke are formed by pyrolysis and pyrosynthetic reactions of terpenes, paraffins, sugars, amino acids and many other tobacco components and reactions involving primary hydrocarbon radicals [23]. Hence it seems possible to reduce the amount of PAHs in smoke through the suppressing the formation or reducing the concentration of such precursors that are necessary for the formation of PAHs. However, experimental evidences are desirable to support this opinion. Well then, what causes the reduction of PAHs level in smoke by use of zeolite additive is unknown yet, which prompts us to examine the adsorption and catalytic cracking of Ant and Flu, the typical compounds of PAHs, on the zeolites with different pore structure and acidbasicity, in order to understand how the zeolites trap and decompose the carcinogenic constituent. Fourthly, a more important uncertainty is the effect of the presence of zeolites in cigarette on the mean toxicity of the smoke. To avoid the risk that reducing one group of substances in smoke may produce other undesirable effects, it is necessary to evaluate the holistic impact of biological active chemicals in smoke. Consequently, two in-vitro genotoxicity tests using the Ames/Salmonella assay and the Chinese hamster ovary (CHO) cell assay were employed to assess the overall effects of the zeolites additive on the cigarette smoke.

2. EXPERIMENTAL 2.1. Materials Zeolite NaY, NaZSM-5 and NaA were commercially powders, HZSM-5 or HY sample was obtained from parent zeolite by ion exchange method [6]. HZSM-5 zeolite was prepared by conventional ion exchange method with aqueous solution of NH4NO3, at the solid/solution ratio of 1:15, performing at 353 K for 2 h and repeating for 6 times. The obtained sample was

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washed, filtrated and dried at 393 K followed by calcination at 823 K. HY sample was acquired in the similar process except the calcination of NH4Y at 773 K. Table 1 lists the properties of these samples. Copper was incorporated in the porous support using ‘drying impregnation’ [24]: 0.456 g Cu(NO3)2⋅3H2O was dissolved in 40 ml H2O and 5 g NaY added, the mixture was stirred strongly and heated up to half dryness, then dried at 373 K overnight, finally the product was ground to 100 mesh and calcined at 773 K for 6 h in order to convert Cu(NO3)2 to copper oxide. The resulting sample contained 3% (w/w) CuO and was denoted as 3%CuO/NaY. Dimethyl sulfoxide, methanol, cyclohexane and anhydrous sodium sulfate were the agents with AR purity. B[a]P, Ant and Flu were purchased from Sigma and dissolved in cyclohexane. The silica gel was provided by Qingdao Haiyang (China), and its pore size range was 8.0-10.0 nm. γ type alumina was the product of Nanjing Inorganic Factory (China) with a surface area of about 120 m2 g-1. The activated carbon with a surface area of about 904 m2 g-1 and the average pore range of 3.0-7.0 nm was also used as the adsorbent in comparison with zeolite. Virginia type tobacco fibers were the gift of Ningbo Cigarette Factory (China). The test cigarette containing 3% (w/w) of zeolite was hand-rolled in the way reported previously [4, 5], in which the zeolite additive was pefectly mixed with the tobacco finber. Two mechine-made test cigarettes, with different trade marks but containing 3% (w/w) of zeolite NaA [7], were machine manufactured in the normal process of cigarette production in which the additives consisting of zeolite NaA were added into alginate-based slurry and sprayed onto the tobacco before cigarette manufacturing [25], while the control cigarettes were prepared in same procedure without addition of zeolites. Table 1. Relevant parameters of zeolites and their efficiency in reduction of PAHs

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Zeolite

Si/Al

Pore size (nm)

Surface area (m2g-1)

Pore volume (cm3g-1)

Particles size (μm)

Removal of PAHs in the smolder smoke of handmade cigarette (%) B[a]P Ant Flu

NaA

1

0.40

800

0.28

3-4

4.0

6.4

13.2

NaZSM-5

12.5

0.54~0.56

354

0.11

7~8

21.7

14.0

26.6

HZSM-5

12.5

0.54~0.56

346

0.10

7~8

11.1

12.1

25.8

NaY

2.9

0.74

766

0.31

1-2

21.4

9.5

17.7

HY

2.9

0.74

550

0.30

1-2

5.8

17.0

25.3

2.2. Calculations on the Structure of B[A] P, Flu and Ant Optimized calculations of three PAHs molecules were completed using Gaussian 98 Revision A.7 programs [26] to perform ab initio quantum chemical calculations at B3LYP/ 631G level. The input files were compiled and the output files viewed both in Gauss View 2.1. Gaussian 98W for UNIX was run on the high performance computer ShuGuang3000 (School of Chemistry & Chemical Engineering, Nanjing University). Theory method/basis group used was B3LYP/6-31G(d).

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2.3. Adsorption of Ant and Flu on Zeolite Owing to the high boiling points and toxicity of PAHs, adsorption of Ant or Flu was performed with the instantaneous adsorption method in a flow reactor connected with a Varian 3380 gas chromatography (GC) [11, 24], instead of conventional gravimetric adsorption system. 5-mg zeolite (20~40 mesh) was added in a piece of stainless steel tube (φ3×150 mm) and connected between the vaporization port and the separation column of the gas chromatograph. To study the actual adsorption behavior of zeolite in smoke that is hydrogen-rich [23], hydrogen (99.99%) was chosen as the carrier gas and the adsorbent was directly heated to 503 K without thermal pretreatment. The solution containing Ant or Flu was pulse injected (5 μl for Ant and 10 μl for Flu each time) on the sample, gaseous effluent was separated by a SE-30 packed column (φ3×1500 mm) and analyzed by the thermal conductivity detector in the on-line Varian 3380 GC. The Chromatography Workstation v.6 provided by Varian was used to collect data and to calculate the ratio of PAHs to solvent through integrating the area of GC peaks of cyclohexane and the PAHs. The decrement in the ratio of solute to solvent represents the amount of PAHs adsorbed by zeolite.

2.4. In-Situ FTIR Investigations

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The infrared measurements of PAHs adsorbed in zeolites were performed in a home-built IR cell with CaF2 windows [20]. A set of Bruker 22 FT-IR spectrometer was used for in situ FTIR measurement, typically the spectrometer operated at 4 cm-1 resolution collecting 40 scans in a single-beam mode. For the IR measurements of the adsorbed species, the zeolite was firstly pressed into plates under the pressure of 300 kgf cm-2, resulting in a disc with density of 15 mg cm-2. The plate was carefully put into the sample holder of the IR cell and slowly heated up to 773 K in N2 flow for 2 h. Background spectra were collected on the activated adsorbate-free sample at 503 K to exclude the overlap of infrared adsorption features that originate from the zeolite structural vibrations and the adsorbed surface species. Ant or Flu solution was introduced into the cell to contact the sample at 503 K, and the sample should be purged with nitrogen flow for 10 min to remove the physical adsorbed PAHs prior to recording the FTIR spectrum at given temperature.

2.5. Reaction of Ant and Flu on the Zeolites 50-mg zeolite (20~40 mesh) was activated at 773 K for 2 h and then the cyclohexane solution of Ant (0.017 M) or Flu (0.02 M), 0.5 ml each time, was pulse injected at 773 K. Nitrogen (99.99%) was chosen as the carrier gas to ensure the reaction performed in inert atmosphere. The gaseous effluent was collected by ice-trap, separated by a SE-30 packed column and analyzed by the thermal conductivity detector in Varian 3380 gas chromatography. The conversion of reactant was determined by external standard method. Thermogravimetric analysis (TGA) [27] was used to determine the amount of coke formed in reaction, and the used catalyst was heated in air from 293 K to 1073 K with a heating rate of 5

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Jian Hua Zhu, Shi Lu Zhou, Ying Wang et al.

K min-1 using a NETZSCH STA 449C apparatus. Repeated experiments showed that the variation of dada was below 5%.

2.6. Temperature Program Desorption (TPD) of Acetylene 100 mg of NaA zeolite (20-40 mesh) was activated at 873 K for 2 h then cooled to 293 K. Blank TPD was carried out to 873 K to confirm there was no any desorption occurring. 0.83mmole acetylene was introduced to contact with the activated sample at 293 K. After the sample was purged by a helium flow at 293 K for 0.5 h or more, TPD test was performed at the rate of 8 K min-1 up to 873 K and the acetylene liberated was detected by an “on line” Varian 3380 gas chromatography combined with external standard method. Temperature program desorption (TPD) of 1, 3-butadiene was carried out in the similar manner.

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2.7. Analysis of PAHs in Smoke To collect the PAHs in smoldering smoke, 4 cigarettes were smoldered in the glass-made chamber with fishtail shape, similar to that designed by Caldwell and Conner [28], and the flow rate of air was controlled as 1500 ml min-1 with a 60% relative humidity [29]. Fouradsorption tubes contained 40 ml dimethyl sulfoxide was used to absorb the smoldering smoke, as shown in scheme 1. The PAHs trapped in the Cambridge filter or dimethyl sulfoxide solution [29, 30] was extracted by cyclohexane because of the better solubility of PAHs in it, and then the extractive solution was concentrated at 313 K to about 2 ml that would be purified by the column chromatography of silica gel. A glass column (300× 15mm i.d.) was packed with 12 g deactivated silica gel and 5 g anhydrous sodium sulfate covered on the top, on which the PAHs extract was eluted with 150 ml of cyclohexane. The first 20 ml elution was discarded and the 20-150 ml fractions concentrated to about 2 ml. Finally the sample solution was constant-volume to 10 ml. To determine the PAHs content in the main stream smoke, 20 machine-made cigarettes were conditioned at 295 K and 60% relative humidity for 48 h. They were then smoked using a Borgwaldt RM20/CS smoking machine under the standard ISO smoking regime (35 ml puff volume and 2 s puff duration, once every 60 s, [31, 32]). The Cambridge filter pad with trapped particulate phase smoke was extracted in the procedure described above. HPLC analysis was performed on a set of Shimadzu LC-10 AD with fluorescence detector, the excitation and emission wavelength was 296 nm and 405 nm for B[a]P, 330 nm and 287 nm for Ant, and 401 nm and 462 nm for Flu, respectively. A C18 column (5 μm, 200×4.6 mm) and a mobile phase of methanol-water (9:1, v/v) at a flow rate of 1 ml min-1, was used to separate the PAHs at 308 K. Each test was repeated for 3 times and external standard plot method used for quantification [33].

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Scheme 1. The apparatus for the smoldering experiment of cigarette to obtain smolder stream smoke.

2.8. Mutagenicity Assay of the Smoke by the Standard Plate Incorporation of the Ames/Salmonella Assay Five or more doses of smoke samples, up to 0.12 cigarette plate-1, were tested for mutagenicity with the pour-plate method [34], with Salmonella typhimurium tester strains TA98, in the presence of an added exogenous microsomal activation system (±S9 mix, prepared with 10%S9 liver homogenate from Aroclor 1254-inched male Sprague-Dawley rates). The mainstream smoke of sample cigarette was absorbed in dimethyl sulphoxide (DMSO) to reach the concentration of 1 cigarette ml-1 and then to be diluted. All tests were done using triplicate plates and replicated at least twice. In cell survival test, the suspension of Chinese hamster ovary (CHO) cells with the concentration of 106~107 cells ml-1 was used and the concentration of smoke-absorption solution was adjusted to 0.2 cigarette ml-1. Plates were incubated for 1 h at 310 K and counted for surviving colonies.

3. RESULTS AND DISCUSSION

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3.1. Adsorption of PAHs in Zeolite Figure 1 shows the structure simulation of Ant, Flu and B[a]P, in which all atoms are supposed in the same plane without bond twisted. In the molecule of B[a]P, the distance between 7H (the H atom on the 7th carbon atom) and 11H (the H atom on the 11th carbon atom), denoted as D7H-11H, is 0.50 nm, and that of D5H-12H or D6H-12H or D5H-1H is 0.72 nm, while the D3H-9H is 1.05 nm. Figure 1 also delineates the pore structure of zeolite NaA, NaZSM-5 and NaY, and among them NaY possesses the largest pore size, 0.74 nm, while NaA has the smallest one, 0.4 nm (table 1). Consulting the calculation result, B[a]P molecule can enter the pore opening of zeolite NaY but cannot fully enter into the channel of NaZSM-5. The structural parameters of the molecule Flu are below: D7H-10H is 0.50 nm, D2H-6H is 0.72nm and the largest distance D4H-9H is 0.91 nm. Accordingly the Flu molecule can get into the pores of zeolite NaY, but is difficult to enter the channel of NaZSM-5. Of course it is hopeless for Flu to penetrate the pore of zeolite NaA. In the molecular structure of Ant there are only two parameters: D1H-4H is 0.49 nm and D2H-7H is

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Jian Hua Zhu, Shi Lu Zhou, Ying Wang et al.

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0.91 nm. Consulting that the pore diameter of NaA is only 0.4 nm, it is clear that Ant molecule cannot enter the channel of NaA at all. Through comparing the pore aperture of zeolite NaY, NaZSM-5 and NaA, with the minimum molecular diameter of three PAHs molecules, it is expected that all of them can be adsorbed and/or decomposed in the channel of zeolite NaY, while B[a]P and Flu are difficult to enter in the pores of NaZSM-5 but none of them can be adsorbed in the channel of NaA.

Figure 1. Simulation on the structure of (a) B[a]P, (b) Flu and (c) Ant, and the pore structure of zeolite (d) NaA, (e) NaZSM-5 and (f) NaY.

Figures 2 and 3 illustrate the adsorption isotherms of Ant and Flu in zeolite at 503 K where the abscissa is the total amount of adsorbate in the fluid phase (Qt) passed through while the ordinate represents the amount adsorbed by the zeolite (Qa). B[a]P has a minimum molecular diameter similar to Flu, its adsorption is thus omitted. The pore aperture of zeolite played a crucial role in the adsorption of PAHs. As seen in Figure 2, Ant could not be adsorbed in zeolite NaA, coincided with the result of calculation mentioned above, because the pore opening of zeolite was too small for the diameter of Ant. Thus, the α-cage in the structure of NaA zeolite was not available for the adsorption. For the same reason, no

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adsorption of Flu was observed in NaA, either (Figure 3). Both the pore diameter（0.74 nm）and the specific area (766 m2g-1) of zeolite NaY are larger than that of NaZSM-5 (0.540.56 nm and 354 m2g-1), so that Ant could be adsorbed easily by NaY and the amount of adsorption was larger than NaZSM-5. One may argue if the adsorbate condensates over the surface of zeolite, or the Ant only just adsorbs on the external surface of the adsorbent, however, this argument was not justified by the experiment. Zeolite NaA has a similar surface area (about 800 m2 g-1) to that of NaY (766 m2 g-1). In case that the Ant just only condensates on the zeolite, especially over the external surface in the state of pseudo-liquid phase, this phenomenon should appear on zeolite NaA and thus the adsorption amount should be similar to NaY. However, when the Qt of Ant increased to 157 μmol g-1, HY or NaY adsorbed more than 90% of them but none by NaA zeolite. This difference excludes the possibility of condensation of Ant on the external surface of zeolite.

Total quality of Anthracene (µmol g -1)

Figure 2. Adsorption of Ant on various zeolites and porous materials at 503 K.

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Jian Hua Zhu, Shi Lu Zhou, Ying Wang et al.

Total quality of fluoranthene (µmol g -1)

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Figure 3. The adsorption of Flu on NaA, NaZSM-5, and NaY and HY zeolite at 503 K.

Surface acidity of zeolite affected the adsorption of Ant, and acidic zeolites adsorbed more Ant than their basic analogue. There were 0.48 mmol g-1 of acid sites on HZSM-5 and about 20% of them were the strong ones that adsorbed ammonia at 673 K [19], hence HZSM5 showed a stronger adsorption ability of Ant than NaZSM-5 through the whole experiment. Zeolite HY had more acid sites (0.62 mmol/g) [19], and it adsorbed more Ant than NaY once the Qt of Ant overstepped 84 μmol g-1 (figure 2A). Two factors should be taken into account for these differences. Firstly, the proton of HZSM-5 can polarize the adsorbed Ant to Ant+ and enable it to survive in the channel of zeolite for several months [35]. Secondly, the pore aperture of the decationated zeolite is a little bit larger than that of the corresponding Naform. Activation has different impact on the adsorption of Ant in NaY and NaZSM-5 zeolite. For the former, the thermal activation at 873 K to remove most of contaminants from the zeolite adsorbent only made Qa enhanced 5% (figure 2A), whereas for the latter the variation achieved 40%. The reason, as aforementioned, is the different pore diameter of zeolite NaY and NaZSM-5. The slight wider of pore size resulting from the removal of contaminate was negligible for zeolite NaY but crucial for NaZSM-5 where the adsorption of Ant was quite difficult. In the case of adsorbing carcinogenic components in mainstream, zeolite additives usually are added in the filter of cigarette and they should trap PAHs at ambient condition without prior thermal activation, thus zeolite NaY or HY can be chosen for this purpose. Figure 2B displays the adsorption of Ant in some amorphous materials at 503 K. Activated carbon exhibited the highest adsorption capacity of Ant among the samples tested, superior to zeolite NaY when the total quality of Ant exceeded 100 μmol g-1. However, amorphous silica and alumina were inactive for the selective adsorption. To check the impact of mesoporous structure on the adsorption of Ant, two mesoporous silica, MCM-48 and SBA15 were employed in this measurement but they were also inactive. In the instantaneous adsorption experiment, the contact time of Ant with the adsorbent was shorter than 0.1 second. Apart from the geometric factor that determines the adsorption of Ant in porous materials, the adsorbate-adsorbent interaction is important to promote the adsorption of

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PAHs. Amorphous silica and mesoporous SBA-15 and MCM-48 have different pore structures but they are the siliceous material without cation, which may result in their failure in adsorbing Ant, similar to that reported in adsorption of nitrosamines [24, 36, 37]. On the other hand, the sample of 3%CuO/NaY showed a higher adsorption capacity of Ant than NaY zeolite under the same conditions (Figure 2B). It is known that the modifier would occupied the space inside of the channel of zeolite to reduce the pore size and/or pore volume of the zeolite more or less, which should affect the adsorption of Ant. However, the coppermodified NaY sample exhibited a higher adsorption capability than the parent NaY, indicating the possible compensation originating from the strengthened interaction with Ant molecule through the electrostatic affinity of copper cation or delicately modified pore size of zeolite [38]. Figure 3 plots the adsorption of Flu on zeolite. Owing to the larger molecular diameter than the pore size of zeolite NaA or NaZSM-5 (Figure 1), Flu could not be adsorbed by these two zeolites at 503 K. Besides, NaY adsorbed more Flu than HY under the same experimental conditions, just opposite to the adsorption of Ant. When the Qt of Flu reached 74 μmol g-1, NaY adsorbed 98% of it while HY trapped 90%; As the Qt increased to 123 μmol g-1, NaY zeolite captured 99% of it but HY trapped 90%. These differences related to the different structures of Flu and Ant because Flu consisted of five-carbons-member-ring but Ant contained six-carbon-member-ring. Such different structure seemed to have influence on adsorption mechanism that involves the high average oxygen charge of NaY (-0.352 [39]), similar to that reported in the adsorption of benzene in which the amount of benzene adsorbed increased in parallel with the oxygen basicity [40]. It is difficult for Flu to form π bond with zeolite HY due to the low affinity of the hydroxyl groups for the π-electron system [41]. Also, copper modification of NaY failed to improve the adsorption of Flu, and 3%CuO/NaY composite was inferior to NaY in trapping the PAHs and its capability was weaker than HY zeolite as seen in Figure 3. The third strange phenomenon came from the sample of activated carbon; this sample adsorbed more Ant than NaY (Figure 2B) but less Flu than the zeolite. Different adsorption behaviors of activated carbon toward Flu and Ant, in our opinion, cannot be simply attributed to the different molecular size of Flu (0.72 nm) and Ant (0.49 nm), because activated carbon has a continue pore size distribution that should be rise to the occasion. Figure 4 depicts the FTIR difference spectra of zeolite adsorbed Ant at 503 K, in which no Ant adsorbed on NaA and only a faint band emerged on NaZSM-5. The characteristic bands of Ant appeared at 3043 (C-H stretching νC-H), 1619 and 1447 cm-1 (C-C stretching vibration,νC=C), and 1320 cm-1 (C-C-C stretching vibration, νC-C-C) on the spectrum of HZSM-5. For the HY and NaY sample adsorbed Ant, the intensity of 1574（νC=C）and 1342 cm-1（νC-H）bands was higher on the spectrum of the former zeolite due to the larger adsorption capacity of HY. Figure 5 shows the FTIR difference spectra of zeolite adsorbed Flu at 503 K. Absence of Flu trapped in NaA, NaZSM-5 and HZSM-5 was consistent with the adsorption results while the appearance of some tiny hugger-mugger weak bands originated from solvent and contaminants. On the spectrum of NaY adsorbed Flu at 503 K, the 3059 cm-1 band of C-H stretching, 1956 and 1832, 1600 cm-1 bands together with the one centered 1438 cm-1 of C-C stretching vibration (νC=C) were observed, similar to that reported on the matrix isolated Flu [42]. The band intensity of Flu adsorbed on zeolite NaY was obviously larger than that on HY indeed, confirming the larger adsorption capacity of NaY

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shown in Figure 3. Another proof for the confirmation came from the bands located in the range of 1800-2050 cm-1 that indicated a strong electrostatic interaction of adsorbate with the Na+ ions [39], the bands on NaY zeolite was considerable stronger than that of HY.

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Figure 4. FTIR difference spectra of zeolite adsorbed Ant at 503 K.

Figure 5. The FTIR difference spectra of zeolite adsorbed Flu at 503 K.

Unlike nitrosamines, PAHs consist of only C and H without N and O as well as N-N=O group, hence the electrostatic field in zeolite cannot induce PAHs into the channel as that happen in the case of nitrosamines, subsequently the adsorption of PAHs in zeolite is mainly determined in terms of pore aperture of the adsorbent and of the dimensions of the target molecules. Rather, the aforementioned discussion on the molecular dimensions of the PAHs (0.5-0.9 nm, from Figure 1), the pore apertures in the zeolites (0.4-0.74 nm, from Table 1), and whether or not the two sizes are compatible, is only relevant for PAHs in the gas phase. However, the distribution of PAHs in the tobacco smoke was much different from that in laboratorial experiments. In general PAHs were wholly in the smoke aerosol particles with

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the diameters of 150 – 200 nm; even immediately downstream of the burning zone inside the cigarette the aerosol particles reached 5 nm in diameter [23]. Thus, adsorption of such particles by zeolite whose pore size was smaller than 1 nm must be impossible, which may be the real reason why NaY zeolite could not reduce the mean PAHs level of main stream smoke [4]. In fact, even the mesoporous silica MCM-48 additive in the filter of cigarette could not significantly reduce the amount of Ant or Flu in the main stream as Yong et al reported [10]. Considering the different situations of PAHs in instantaneous adsorption and the actual cigarette smoke, it should be vary careful to apply the laboratorial adsorption results to explain the reduction of PAHs by zeolite in smoke.

3.2. Catalytic Cracking of PAHs in Zeolite

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Figure 6 depicts the conversion of Ant and Flu in the cracking reaction catalyzed by various zeolite at 773 K. Concerning the elimination of PAHs in tobacco smoke by zeolite, it should point out that this cigarette catalyst (we denote it as CigCat [24]) is extraordinary because of its one-off catalysis. In the procedure of smoking, the zeolite in hot zone underwent heating, combustion and cooling within a minute and after then the used zeolite would remain in the ash of cigarette [24]. So, only the initial activity of the zeolite in the cracking of Ant or Flu, in our opinion, can be related to the actually catalytic behavior of the CigCat therefore we just discuss the first pulsed reaction result. Nitrogen was chosen as the carrier gas to ensure the reaction performed in inert atmosphere.

Figure 6. Conversion of Ant and Flu on zeolite at 773 K.

No Ant or Flu was converted on NaA at all, due to the small pore aperture of the zeolite. Ant could be cracked into toluene, xylene, para- or meta-ethyltoluene, naphthalene, 1-methyl naphthalene, 2-methyl naphthalene and other fragments on both NaZSM-5 and HZSM-5 zeolites though the conversion was different. For the former the converted reactant was about 5% while this value was doubled more on the latter. Judged on these results it is safe to

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conclude that pore aperture of catalyst and the size of the reactant determine the catalytic activity of zeolite in the reaction of PAHs cracking. For the zeolite NaY or HY with larger pore, the conversion of Ant was relative high, 24% on zeolite NaY and 48% on HY. Most zeolites exhibited higher activity in reaction of Ant than that of Flu whereas NaY zeolite was an exception on which the Flu conversion even exceeded that on HY (Figure 6). On the other hand, coking was serious in this reaction, which depended on the pore/cage structure of zeolite and affected the carbon balance during reaction. After 0.674-mmol g-1 Ant passed through zeolite NaY or HY at 773 K, 7.2% (w/w) or 9.1% of coke formed resulting from the twelve-member ring and super-cage of Y zeolite [43]. Under same conditions, however, the amount of coke detected in HZSM-5 was 1.3% because of the excellent anticoking features of the zeolite [44]. The influence of coking on the cracking of PAHs in zeolite is complex. Coke deposition should further restrict the diffusion of PAHs in the partially occluded channels to deactivate the catalyst, but this impact would be minor on the special CigCat that was only used one-off. More likely, coking necessarily led the consumptions of the reactant PAHs so that the heavy coking property of NaY or HY would be available for enhancing the performance of CigCat. The considerable activity of NaY and NaZSM-5 zeolite in catalytic cracking of Ant (Figure 6), together with the temperature programmed surface reaction results of nitrosamines catalyzed by these two basic zeolites [15], confirmed the catalytic ability of basic zeolite so that the protonation of basic zeolite, as proposed by Meier and Siegmann [4], was unnecessary. To exclude the suspicion if any strong Bronsted acid site survived in the sample, the NaY zeolite was carefully exchanged again with 1M NaCl solutions for six times prior to repeat reactions. The exchanged sample, with a sodium content of 12.55 wt%, also possessed the similar catalytic activity in the reaction within the experimental error. Confirmation on the catalytic function of NaY zeolite is important to extend the application of the zeolite with sodium ion in environmental catalysis, saving a lot of time and energy for ion exchange.

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3.3. Elimination of PAHs in Cigarette Smoke by Zeolite Table 1 lists the reduction of B[a]P, Flu, and Ant in the smoldering smoke of hand-made Virginia cigarette by the zeolite additive of 3 wt%. Literature shows the side-stream plume temperature as being a maximum of about 623 K immediately above the cigarette-burning zone and then rapidly falling [45] while the temperature inside the burning zone of a smoldering cigarette is a maximum of about 1050 K [23]. Surprisingly all zeolite samples enabled the content of B[a]P, Flu and Ant in the smoke to be decreased, even in the case of using NaA as additive that has been repeated for several times to exclude experiment error. B[a]P was reduced about 21% by zeolite NaZSM-5 or NaY, more significant than that by using acidic zeolite HZSM-5 or HY. On the zeolite NaZSM-5 or HZSM-5 more than 12% of Ant or 25% of Flu were eliminated, while on HY zeolite the reduction of both Ant (17%) and Flu (25%) was higher than NaY. Moreover, a considerable amount of Flu (13%) was reduced by NaA zeolite (Table 1), which conflicted with the aforementioned adsorption and catalytic reaction results. It appears that the reduction of PAHs in smoke by using zeolite additive cannot be simply attributed to adsorption and catalytic cracking of PAHs, those zeolites with small pore can exert their function through another way.

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To pursue this point further, the machine-made Virginia cigarette containing NaA zeolite (3 wt.-%) was used to measure the variation of PAHs contents in both mainstream and smoldering smoke and Table 2 lists the results. At first sight the results showing decrease of PAHs in the smoke may look somewhat surprising, because the amount of B[a]P and Ant as well as Flu decreased 5-20% in the mainstream of sample cigarette while in smoldering smoke they were lowered 15-30%. Figure 7 illustrates the tests of sample cigarette B that contains 3 wt.-% of NaA zeolite. The reduction of B[a]P in smolder smoke achieved 12% and the Ant in mainstream was decreased about 10%, and this result was much better than that of MCM-48 added in filter [10]. On the other hand, adding zeolite NaA in cigarette did not change the physical parameter or interfere tobacco combustion as demonstrated in Table 2, the ‘tar’ or carbon monoxide yields and the puff numbers of the test and control cigarettes were almost same within the experimental error. That is to say, the reduction of PAHs in smoke in the presence of zeolites was unique, because three important parameters of cigarette, ‘tar’, carbon monoxide and cigarette burn rates were not affected. It is well known that the PAHs in smoke are wholly resident in the aerosol particles of the smoke; however judged from the unchanged three parameters of sample cigarette, it is clear that addition of the zeolites to tobacco does not reduce the generation of the particulate phase of smoke. Rather, based on these results it can be conclusive that, at least tentatively, reduction of PAHs has been done with selectivity. As a comparison, amorphous silica was also added in the cigarette under the same manufacture procedure and tested in the same conditions. However, only the reduction of B[a]P (18%) was observed in the smolder smoke of the sample cigarette, while the Ant in the main stream was slightly decreased about 2%, similar to the report of MCM-48 [10]. Table 2. Physical parameters, standard yields and reduction of PAHs in mainstream (MS) and smolder smoke (SS) of machine-made Virginia type cigarette Sample cigarette

Without zeolite

Containing zeolite

Reduction of PAHs (%)

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NaA (3 wt.-%) Total cigarette weight (mg)

943

939

-

Tobacco weight (mg)

733

724

-

Tobacco moisture (%)

14.7

14.1

-

Total pressure drop (mm H2O)

98

94

-

Puff number

9.2

8.7

-

TPM (mg/cig)

18.1

17.3

-

NFDPM (“tar”, mg cig-1)

15.1

14.6

-

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Jian Hua Zhu, Shi Lu Zhou, Ying Wang et al. Table 2. Continued

Sample cigarette

Without zeolite

Containing zeolite

Reduction of PAHs (%)

NaA (3 wt.-%) Water (mg cig-1)

1.6

1.4

-

Nicotine (mg cig-1)

1.38

1.34

-

CO (mg cig-1)

14.2

14.1

-

B[a]P (ng cig-1), in SS

146.8

105.3

28.3

11.7

10.1

12.9

18.7

15.6

16.6

7.93

5.89

25.7

4.35

3.66

15.9

0.80

0.64

19.6

in MS Ant (μg cig-1), in SS in MS Flu (μg cig-1), in SS

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in MS

Figure 7. Reduction of PAHs in the smoke of machine-made Virginia sample cigarette B containing zeolite NaA (3 wt%).

Figure 8 illustrates the lower toxicity of sample cigarette containing zeolite exhibited in Ames and CHO cell assay that is critical for the complete toxicological assessment of smoke. Less mutagenic activity was observed on the sample cigarette containing zeolite and the number of mutants was considerably smaller (Figure 8A). Likewise, the livability of cell in the case of test sample was distinctly higher than that of control cigarette in the CHO test

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(Figure 8B). On the basis of these results, it appears that zeolitic additives attached in cigarette lower the mean toxicity of the smoke of the test cigarette therefore more cells can resist in the smoke absorbed solution. Nonetheless, the sample collection method used in our laboratorial experiments was different from the actual cigarette smoking by people, in which the smoking was continuous so the entire smoke produced from a cigarette within about 10 min was trapped. If the results were used to assess adverse effects caused by smoking, the cigarette-smoking machine should be employed to simulate the actual manner of smoking by people. As mentioned above, NaA zeolite could lower the content of Ant or Flu in tobacco smoke when it was used as the additive in cigarette, but it could not adsorb or crack them in laboratorial experiment. This inconsistence implies that the reduction of PAHs in cigarette smoke by zeolite arise from the multiple functions of the additive. Apart form the direct adsorption or catalytic cracking of PAHs themselves, zeolites can also interdict the synthetic route of PAHs through the adsorption or catalyze decomposition of the intermediate or precursors that are needed to synthesize PAHs. Formation of PAHs involves several steps and needs various intermediate or precursors [22], and Scheme 2 delineates the possible thermal synthetic procedure of B[a]p, the representative carcinogen in PAHs, in which acetylene and 1, 3-butadiene are the necessary precursors. And it is possible, in our opinion, for zeolite NaA to trap the precursors or intermediate compounds like 1, 3-butadiene and/or acetylene to reduce the content of B[a]P in smoke, because the minimum molecular diameter of acetylene (0.334 nm) or 1, 3-butadiene (0.309 nm) is smaller than the diameter of the zeolite. To support our inference, Figure 9 plots the TPD spectrum of 1, 3-butadiene or acetylene on NaA zeolite, in which three desorption peaks indicate the occurrence of adsorption and desorption of these hydrocarbons. Among the 1, 3-butadiene species (0.57 mmol g-1) desorbed from zeolite NaA, 70% of them emerged above 500 K and 46% above 650 K. More acetylene (1.19 mmol g-1) was desorbed from NaA, and 72% of them appeared above 600 K. No doubt both 1, 3-butadiene and acetylene could be strongly adsorbed by NaA zeolite, coincided with the report of Onyestyák [46] and Sheikh [47]. Thus, it is very likely that the zeolite additives adsorbed acetylene and/or 1, 3-butadiene when the precursors were formed in tobacco combustion, together with the catalytic cracking reaction, if any, to lower the final concentration of B[a]P in the smoke. Likewise, another zeolite could remove the PAHs in cigarette smoke through the similar way. For instance the molecular diameter of B[a]P (0.72 nm) was larger than the pore mouth of NaZSM-5 (0.54~0.56 nm), but the amount of B[a]P in smoke was still decreased by the zeolite (Table 1), although further study was desirable to explore the mechanism in detail. On the basis of these results, it is reasonable to infer that reduction of PAHs in tobacco smoke by zeolites is not limited to the adsorption and catalytic decomposition of PAHs themselves, suppression of PAHs formation by zeolite may have an important contribution. Zeolite additive could trap or eliminate the intermediates/precursors to stop the formation of PAHs, which also lead to less carcinogenic compounds in the smoke. Through this way even the zeolites with small pore, in our opinion, still can exert their function to reduce the amount of PAHs in smoke; rather, the discovery of reducing PAHs level of smoke by the small pore zeolite provides a strategy for the environment protection: it is better to use the porous functional materials preventing the formation of bulky pollutants, instead of passively treating the bulky pollutants themselves after their formation.

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Figure 8. (A) Ames and (B) CHO cell tests with the main stream smoke of cigarette sample A.

Figure 9. The TPD profile of acetylene or 1,3-butadiene on NaA zeolite.

Reducing Polycyclic Aromatic Hydrocarbons by Porous Materials

479

Scheme 2. The possible synthetic procedure of B[a]P proposed by Badger et al [22].

4. CONCLUSION

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Some conclusive remarks can be tentatively drawn from these primary investigations: (1) Adsorption of PAHs such as anthracene (Ant) by zeolite in gas stream depended on the geometric matching degree between the adsorbate and adsorbent. NaA failed to adsorb Ant and Flu owing to its small microporous structure, while NaZSM-5 and HZSM-5 exhibited a weak activity for the adsorption and reaction of Ant. With the larger pore size and pore volume, zeolite NaY and HY could efficiently adsorb or catalyze the cracking of both Ant and Flu. Besides, cation affected the properties of zeolite. Acidic HY zeolite trapped more Ant than NaY, whereas existence of sodium ion in zeolite NaY was beneficial for adsorption of Flu. Modification of NaY with copper could slightly improve the adsorption of Ant. Among the amorphous porous materials tested in this paper, activated carbon was the best one with the highest capacity in the adsorption of Ant. However, alumina, silica and mesoporous silica SBA-15 and MCM-48 were inactive to selectively adsorb Ant in gas stream at 503 K. (2) The content of Ant, Flu and B[a]P in cigarette smoke could be reduced more or less by dispersing zeolite on tobacco fibers, which was inconsistent with the results of laboratorial adsorption and catalytic reaction. Also, zeolite NaA still exhibited a considerable ability to lower the PAHs level of tobacco smoke without change in the combustion of tobacco. Less mutagenic activity was observed for the test cigarettes containing zeolite additive in comparison with the control cigarette. Higher cell livability for the CHO test was also found on the test cigarette containing zeolite additive. (3) Besides trapping and catalytic cracking PAHs themselves, zeolite could interdict the synthesis of PAHs by adsorbing and eliminating their precursors to lower the content of PAHs in cigarette smoke. Through this way, even the zeolite with small pore size could make the contribution to reduce PAHs in smoke.

480

Jian Hua Zhu, Shi Lu Zhou, Ying Wang et al.

ACKNOWLEDGMENTS Financial support from NSF of China (20773061, 20873059 and 20871067), NHTRDP973 (2007CB613301), Grant 715-04-0120 and 2008AA06Z327 from the 863 Program of the Ministry of Science and Technology of China, Ningbo Cigarette Factory and Analysis Center of Nanjing University is gratefully acknowledged. We wish to thank Professor Y.M. Zhou and Ms L. Si (Nanjing Normal University), Professor H. Li (Chin. Acad. Med. Sci., Inst. Lab. Animal Sci.), and Nanjing Cigarette Factory for their technical assistances.

REFERENCES

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Maesen, T. The zeolite scene - An overview, Stud. Surf. Sci. Catal. 2007, 168, 1-12. Davis, M. E. Ordered porous materials for emerging applications, Nature 2002, 417, 813-821. [3] Wheatley, P. S.; Butler, A. R.; Crane, M. S.; Fox, S.; Xiao, B.; Rossi, A. G.; Megson, I. L.; Morris, R. E. NO-releasing zeolites and their antithrombotic properties, J. Am. Chem. Soc. 2006, 128, 502-509. [4] Meier, M. W. and Siegmann, K. Significant reduction of carcinogenic compounds in tobacco smoke by the use of zeolite catalysts, Microporous Mesoporous Mater. 1999, 33, 307-310. [5] Xu, Y.; Wang, Y.; Zhu, J. H.; Ma, L. L. and Liu, L. Progress in the application of zeolite in the life science: novel additive for cigarette to remove N-nitrosamines in smoke, Stud. Surf. Sci. Catal. 2002, 142, 1489-1496. [6] Xu, Y.; Zhu, J. H.; Ma, L. L.; Ji, A.; Wei, Y. L. and Shang, X. Y. Removing nitrosamines from main-stream smoke of cigarette by zeolite, Microporous Mesoporous Mater. 2003, 60, 125-138. [7] Gao, L.; Wang, Y.; Xu, Y.; Zhou, S. L.; Zhuang, T. T.; Wu, Z. Y.; Zhu, J. H. Reducing the nitrosamines level of mainstream smoke by zeolite NaA, Clean: - Soil, Air, Water. 2008, 36, 3, impress. [8] Hoffmann, D.; Djordjevic, M. V.; Hoffmann, I. The changing cigarette, Preventive Medicine, 1997, 26, 427-434. [9] Baker, R. R. 2001-A smoke odyssey. Presentation given at the 2001 joint meeting of the CPRESTA smoke & technology study group in Xi’an. [10] Yong, G. P.; Jin, Z. X.; Tong, H. W.; Yan, X. Y.; Li, G. S.; Liu, S. M. Selective reduction of bulky polycyclic aromatic hydrocarbons from mainstream smoke of cigarette by mesoporous materials, Microporous Mesoporous Mater. 2006, 91, 238243. [11] Zhou, C. F.; Cao, Y.; Zhuang, T. T.; Huang, W.; Zhu, J. H. Capturing volatile nitrosamines in gas stream by zeolites: Why and how? J. Phys. Chem. C. 2007, 111, 4347-4357. [12] Cao, Y.; Zhuang, T. T.; Yang, J.; Liu, H. D.; Huang, W.; Zhu, J. H. Promoting zeolite NaY as efficient nitrosamines trap by cobalt oxide modification J. Phys. Chem. C. 2007, 111, 538-548.

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[13] Cao, Y.; Yun, Z. Y.; Yang, J.; Dong, X.; Zhou, C. F.; Zhuang, T. T.; Yu, Q.; Liu, H. D.; Zhu, J. H. Removal of carcinogens in environment: adsorption and degradation of N’nitrosonornicotine (NNN) in zeolites, Microporous Mesoporous Mater. 2007, 103, 352362. [14] Cao, Y.; Shi, L. Y.; Yun, Z. Y.; Zhou, C. F.; Wang, Y.; Zhu, J. H. Novel amorphous functional materials for trapping nitrosamines, Environ. Sci. Tech. 2005, 39(18), 72547259. [15] Wu, Z. Y.; Wang, H. J.; Ma, L. L.; Xue, J.; Zhu, J. H. Eliminating carcinogens in environment: Degradation of volatile nitrosamines by zeolites Y and ZSM-5, Microporous Mesoporous Mater. 2008, 109, 436-444. [16] Xu, Y.; Yun, Z. Y.; Zhou, C. F.; Zhou, S. L.; Xu, J. H.; Zhu, J. H. Adsorption of bulky nitrosamines on zeolite with small pore aperture, Stud. Surf. Sci. Catal. 2004, 154, 1858-1865. [17] Zhou, C. F.; Cao, Y.; Zhuang, T. T.; Zhou, S. L.; Wang, Y.; Ma, L. L.; Shen, B.; Zhu, J. H. Investigation on the adsorption of nitrosamines in zeolites, Stud. Surf. Sci. Catal. 2005, 158, 1003-1011. [18] Yang, J.; Ma, L. L.; Shen, B.; Zhu, J. H. Capturing nitrosamines in environment by zeolite, Mater. Manu. Process. 2007, 22, 750-757. [19] Zhou, C. F.; Yun, Z. Y.; Xu, Y.; Wang, Y. M.; Chen, J.; Zhu, J. H. Adsorption and room temperature degradation of N-nitrosodiphenylamine on zeolites, New J. Chem. 2004, 28, 807-814. [20] Yun, Z. Y.; Zhu, J. H.; Xu, Y.; Xu, J. H.; Wu, Z. Y.; Wei, Y. L.; Zhou, Z. P. An in situ FTIR investigation on the adsorption of nitrosamines in zeolites. Microporous Mesoporous Mater. 2004, 72, 127-135. [21] Shen, B.; Ma, L. L.; Zhu, J. H.; Xu, Q. H. Decomposition of N-nitrosamines over zeolites, Chem. Lett., 2000, (4): 380-381. [22] Badger, G. M.; Buttery, R. G.; Kimber, R.W. L.; Lewis, G.. E.; Moritz, A. G.; Napier, I. M. The Formation of Aromatic Hydrocarbons at High Temperatures. Part I. Introduction. J. Chem. Soc. 1958, 2449-2452. [23] Baker, R. R. Tobacco Production, Chemistry and Technology (Edited by D. L. Davis and M. T. Nielsen), Blackwell London 1999. pp 419-425.7. W. S. Caldwell, J. M. Conner, Artifac formation during smoke trapping: An improved method for determineation of N-nitrosamines in cigarette smoke. J. Assoc. Off. Anal. Chem. 1990, 73, 783-789. [24] Xu, Y.; Liu, H. D.; Zhu, J. H.; Yun, Z. Y.; Xu, J. H.; Wei, Y. L. Removal of volatile nitrosamines with copper modified zeolites, New J. Chem. 2004, 28, 244-252. [25] Wang, Y.; Zhu, J. H.; Shen, B.; Chun, Y. and Xu, Q. H. The additive used for lowering the content of nitrosamines in tobacco smoke and the cigarette containing such additives. CN 99114106, assigned to Nanjing University, China, 1999. [26] Gaussian 98, Revision A.7, Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Zakrzewski, V. G.; Montgomery, J. A.; Jr., Stratmann, R. E.; Burant, J. C.; Dapprich, S.; Millam, J. M.; Daniels, A. D.; Kudin, K. N.; Strain, M. C.; Farkas, O.; Tomasi, J.; Barone, V.; Cossi, M.; Cammi, R.; Mennucci, B.; Pomelli, C.; Adamo, C.; Clifford, S.; Ochterski, J.; Petersson, G. A.; Ayala, P. Y.; Cui, Q.; Morokuma, K.; Malick, D. K.; Rabuck, A. D.; Raghavachari, K.; Foresman, J. B.; Cioslowski, J.; Ortiz, J. V.; Baboul, A. G.; Stefanov, B. B.; Liu, G.; Liashenko, A.;

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Jian Hua Zhu, Shi Lu Zhou, Ying Wang et al. Piskorz, P.; Komaromi, I.; Gomperts, R.; Martin, R. L.; Fox, D. J.; Keith, T.; AlLaham, M. A.; Peng, C. Y.; Nanayakkara, A.; Gonzalez, C.; Challacombe, M.; Gill, P. M. W.; Johnson, B.; Chen, W.; Wong, M. W.; Andres, J. L.; Gonzalez, C.; HeadGordon, M.; Replogle, E. S. and Pople, J. A. Gaussian, Inc., Pittsburgh PA, 1998. Wei, Y. L.; Wang, Y. M.; Zhu, J. H.; Wu, Z. Y. In situ coating MgO on SBA-15: direct synthesis of mesoporous solid base from strong acidic system. Adv. Mater. 2003, 15, 1943-1945. Caldwell, W. S.; Conner, J. M. Artifac formation during smoke trapping: An improved method for determineation of N-nitrosamines in cigarette smoke. J. Assoc. Off. Anal. Chem. 1990, 73, 783-789. Lee, M. L.; Novotny, M. V.; Bartle, K. D. Analytical chemistry of polycyclic aromatic compounds, Academic Press, Inc. 1981. p366 Hoffmann, D.; Wynder, E. L. Short-Term Determination of Carcinogenic Aromatic Hydrocarbons. Anal. Chem. 1960, 32, 295-296. International Organisation for Standardisation: Routine analytical cigarette-smoking machine. Definition and standard condition; ISO 3308, 2000. International Organisation for Standardisation: Tobacco and tobacco products. Atmosphere for conditioning and testing; ISO 3402, 1999. Camargo, M. C. R.; Toledo, M. C. F. Polycyclic Aromatic Hydrocarbons in Brazilian Vegetables and Fruits. Food Control. 2003, 14, 49-53. Maron, D. M. and Ames, B. N. Revised methods for the salmonella mutagenicity test, Mutation Res. 1983, 113, 173-215. Moissette, A.; Marquis, S.; Gener, I.; Bremard, C. Sorption of Anthracene, Phenanthrene and 9,10-dimethylanthracene on Activated Acid HZSM-5 Zeolite. Effect of Sorbate Size on Spontaneous Ionization Yield. Phys. Chem. Chem. Phys. 2002, 4, 5690-5696. Xu, Y.; Jiang, Q.; Cao, Y.; Wei, Y. L.; Yun, Z. Y.; Xu, J. H.; Wang, Y.; Zhou, C. F.; Shi, L. Y.; Zhu, J. H. The synthesis of novel porous functional materials for use as nitrosamine traps, Adv. Funct. Mater. 2004, 14 (11): 1113-1123. Xu, J. H.; Zhuang, T. T.; Cao, Y.; Yang, J.; Wen, J. J.; Wu, Z. Y.; Zhou, C. F.; Huang, L.; Wang, Y.; Yue, M. B.; Zhu, J. H. Improving MCM-41 as nitrosamines trap through one-pot synthesis, Chem. Asian J 2007, 2, 996-1006. Liu, H. D.; Zhou, S. L.; Wang, Y.; Xu, Y.; Cao, Y.; Zhu, J. H. CuO modified NaY zeolite: efficient catalyst for degrading nitrosamines. Stud. Surf. Sci. Catal. 2004, 154, 2527-2535. Su, B. L.; Barthomeuf, D. Adsorption Sites for Benzene in KL Zeolite: An Infrared Study of Molecular Recognition. Zeolites 1995, 15, 470-474. Barthomeuf, D. Acidity and Basicity in Zeolites. Stud. Surf. Sci. Catal. 1991, 65, 157169. Coughlan, B.; Carroll, W. M.; O’Malley, P.; Nunan, J. Benzene in Hydrogen Zeolites. Infrared Spectroscopic and Catalytic Investigation of Variously Exchanged Hydrogen Y Systems. J. Chem. Soc., Faraday Trans. 1, 1981, 77, 3037-3047. Hudgins, D. M.; Sandford, S. A. Infrared Spectroscopy of Matrix Isolated Polycyclic Aromatic Hydrocarbons. 3. Fluoranthene and the Benzofluoranthenes. J. Phys. Chem. A, 1998, 102, 353-360.

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[43] Dejafve, P.; Auroux, A.; Gravelle, P. C.; Vedrine, J. C.; Gabelica, Z.; Derouane, E. G. Methanol Conversion on Acidic ZSM-5, Offretite, and Mordenite Zeolites: A Comparative Study of the Formation and Stability of Coke Deposits. J. Catal. 1981, 70, 123-136. [44] Rollmann, L. D.; Walsh, D. E. Shape Selectively and Carbon Formation in Zeolites. J. Catal. 1979, 56, 139-140. [45] Robinson, D. P. Aerodynamic characteristics of the plume generated by a burning cigarette. In: Proceeding of the International Conference on the Physical and Chemical Processes occurring in a Burning Cigarette. Pp115-150. R.J. Reynolds Tobacco Co, Winston-Salem, North Carolina, 1987. [46] Onyestyák, G.; Rees, L. V. C. The Diffusion and Sorption of Acetylene in Various Zeolites. Appl. Catal. A: General 2002, 223, 57-64. [47] Sheikh, J.; Kershenbaum, L. S.; Alpay, E. 1-butene Dehydro- genation in Rapid Pressure Swing Reaction Processes. Chem. Eng. Sci. 2001, 56, 1511-1516.

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Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

SHORT COMMUNICATIONS

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In: Progress in Porous Media Research Editors: Kong Shuo Tian and He-Jing Shu

ISBN: 978-1-60692-435-8 © 2009 Nova Science Publishers, Inc.

Short Communication A

THERMAL DISPERSION AND RADIATION EFFECTS ON MHD FREE CONVECTION OF A NON-NEWTONIAN FLUID OVER A VERTICAL CONE IN A POROUS MEDIUM A. M. Rashad1, S. M. M. EL-Kabeir2 and Rama Subba Reddy Gorla3 1

Department of Mathematics, South Valley University, Faculty of Science, Aswan, Egypt 2 Department of Mathematics, King Saud University, College of science, AL-Kharj, Saudi Arabia 3 Department of Mechanical Engineering, Cleveland State University, Cleveland, Ohio, USA

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ABSTRACT The present investigation deals with the effects of thermal dispersion, radiation and MHD on free convection boundary layer over a vertical cone in a Non-Newtonian fluid saturated porous medium with variable wall temperature and an exponentially decaying internal heat generation. The coefficient of thermal diffusivity has been assumed to be the sum of molecular and dispersion thermal diffusivity due to mechanical dispersion. Rosseland approximation is used to describe the radiative heat flux in the energy equation. The resulting similarity equations are solved numerically for two cases, one with internal heat generation (IHG) and the other without internal heat generation (WIHG). Numerical results for the dimensionless heat transfer rate in terms of the local Nusselt number are presented in tabular form and in graphical form to illustrate the influence of the various parameters on the temperature profiles with and without heat generation.

Keywords: Heat Transfer, Non-Newtonian Fluid, Porous Medium, MHD, Thermal Radiation, Thermal Dispersion, Heat Generation, Free Convection, Variable wall temperature.

488

A. M. Rashad, S. M. M. EL-Kabeir and Rama Subba Reddy Gorla

NOMENCLATURE A B0 CT Cp d Ds f g K(n) kd ke Mn n Nux q qr

q m′′′

constant the strength of magnetic field temperature difference the specific heat pore diameter the dispersion parameter dimensionless stream function acceleration due to gravity modified permeability of the porous medium the dispersion thermal conductivity effective thermal conductivity magnetic parameter the power law index the local Nusselt number local heat flux radiative heat flux the internal heat generation

qw r R

rate of heat transfer the cone radius radiation parameter

Ra x

the generalized local Rayleigh number

T u, v x, y

temperature velocity component in the x and y directions Cartesian coordinates

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Greek Symbols α β

γ

molecular effective thermal diffusivity coefficient of thermal expansion dispersion coefficient

λ ρ ψ η ν θ σ σ1 χ

the range of exponent fluid density dimensional stream function similarity parameter the modified viscosity of the fluid dimensionless temperature function is the electrical conductivity of the fluid Stefan-Boltzman constant the mean absorption coefficient

Thermal Dispersion and Radiation Effects on MHD Free Convection…

489

Subscripts w

∞

evaluated on the wall evaluated at the outer edge of the boundary layer

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INTRODUCTION The study of flow and heat transfer in an electrically conducting fluid permeated by a transverse magnetic field is of special interest and has many practical applications in manufacturing processes in industry. An understanding of the effect of an applied magnetic field on the flow and heat transfer is useful for the cooling process in the presence of an electrolytic bath. The effects of magnetic field of an electrically conducting Non-Newtonian fluid on non-Darcy axisymmetric free convection over a horizontal surface embedded in a porous medium incorporating the variation of permeability and thermal conductivity were studied by Mansour and Al-Shear [1]. The study of heat transfer in fluid–saturated porous media has gained considerable attention because of its numerous applications in geophysics and energy related problems, such as thermal insulation of buildings, enhanced recovery of petroleum resources, geophysical flows, packed bed reactors, water waste disposal and others. Recent monographs by Ingham and Pop [2], Vafai [3], Postelnicu and Pop [4], and Bejan and Nield [5] give an excellent summary of the work on the subject. A large number of physical phenomena involve free convection driven by internal heat generation. The most important applications are in the field of nuclear energy and also fire and combustion modeling. The boundary layer free convection with internal heat generation in porous media was studied by Postelnicu and Pop [6]. Bagai [7] studied the free convection boundary layers over a body of arbitrary shape in a porous medium with internal heat generation. The effect of thermal dispersion and radiation on free convection flows has many engineering applications such as space technology and processes involving high temperature such as geothermal engineering. A linear dispersion model taking the porosity of the porous medium into account for free convection in a horizontal layer heated from below was introduced by Georgiadis and Catton [8]. Cheng [9] and Plumb [10] considered the flow and heat transfer in porous media by taking thermal dispersion into consideration. An analysis of thermal dispersion effect on natural convection flow over a flat plate in a porous medium was presented by Hong and Tien [11]. The radiation effect on free convection boundary layer flow over horizontal surfaces was studied by Ali et al. [12] using the Rosseland diffusion approximation. This approximation leads to simplification in the expression used for radiant heat flux. Hossain and Alim, [13] investigated the natural convection-radiation interaction on boundary layers along thin vertical cylinders. The problem of free convection radiation interaction over cylinders of elliptical cross section was studied by Hossain et al., [14]. Yih, [15, 16] studied the effect of radiation on natural convection about a truncated cone and a vertical cylinder embedded in a porous medium. In the present paper, we present the effects of thermal dispersion, radiation and MHD on steady free convection over a heated vertical cone embedded in porous medium saturated with a non-Newtonian power law fluid driven by internal heat generation.

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A. M. Rashad, S. M. M. EL-Kabeir and Rama Subba Reddy Gorla

GOVERNING EQUATIONS Let us consider a steady, two-dimensional, free convection boundary layer flow over a heated vertical cone embedded in a non-Newtonian power law fluid-saturated porous medium. The vertical cone is assumed to be heated in such a way that its surface temperature varies in a power law form Tw(x)=T∞+Axλ, where A and λ are constants. Under the Boussinesq and boundary layer approximations, the basic equations of motion, continuity, momentum and energy can be written, respectively as

∂ (ru ) ∂ (rv ) = 0, + ∂y ∂x un =

u

K ( n)

υ

(1)

[gβ cos Ω(T − T

∞

]

) − σB02 u ,

(2)

⎤ 1 ⎡ ∂q r ∂ ⎛ ∂T ⎞ ∂T ∂T ⎟⎟ − − q m′′′ ⎥ , = ⎜⎜α ff +v ⎢ ∂y ⎠ ( ρC p ) ⎣ ∂y ∂y ∂y ⎝ ∂x ⎦

(3)

where r = x sin Ω is the cone radius, x and y are the Cartesian co-ordinates along and normal to the generator, respectively, as shown in figure 1. Here, where u, v are the velocity components in the x, y directions, respectively; T the temperature; σ the electrical conductivity of the fluid; B0 the strength of magnetic field; μ the fluid kinematic viscosity; ρ the density of the fluid; α is the molecular thermal diffusivity; g the acceleration due gravity; β the coefficient of thermal expansion ; ν the modified kinematic viscosity; q ′m′′ the internal heat generation source; Cp the specific heat at constant pressure; n the power law index and K(n) the modified permeability, which is given by

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6 ⎛ nϕ ⎞ K ( n) = ⎟ ⎜ 25 ⎝ 3n + 1 ⎠

n

2 ⎛ dϕ 2 ⎞ ⎟ K (n) = ⎜⎜ ϕ ⎝ 8(1 − ϕ ) ⎟⎠

⎛ ϕd ⎞ ⎟⎟ ⎜⎜ ⎝ 3(1 − ϕ ) ⎠ n +1

n +1

, (Christopher and Middleman, 1965)

6n + 1 ⎛ 16 ⎞ ⎜ ⎟ 10n − 3 ⎝ 75 ⎠

3(10 n − 3) 10 n +11

(4)

, (Darmadhikari and Kale,1985) (5)

where d is the particle diameter and ϕ is the porosity. The boundary conditions are given by y=0: v=0, Tw=T∞+Axλ,

y → ∞ : u → 0 , T → T∞ .

(6)

Thermal Dispersion and Radiation Effects on MHD Free Convection…

491

It is noted that λ=0 corresponds to the case of an isothermal cone. The quantity qr on the right-hand side of energy equation (3), represents the radiative heat flux in the y direction. The radiative heat flux term is simplified by the Rosseland approximation (cf. Sparrow and Cess, [17] and is as follows:

qr = −

4σ 1 ∂T 4 , 3χ ∂y

(7)

where σ1 and χ are the Stefan-Boltzman constant and the mean absorption coefficient, respectively. In the above equations, it can be noticed that the thermal diffusivity is taken to be variable in the energy equation (3), where,

α ff = (α + α d ) ,

(8)

where α is the molecular diffusivity and αd is the diffusivity due to thermal dispersion. Plumb, [7] has expressed for the dispersion thermal diffusivity as: α d = γdu , where γ is the dispersion coefficient, which is a function of the structure of the porous medium. Proceeding with the analysis, we introduce the following similarity variables

ψ = αr (Ra x )

1/ 2

f (η ) , θ (η ) =

T − T∞ 1/ 2 , η = ( y / x )(Ra x ) Tw − T∞

Here, ψ is the stream function given by ru =

(9)

∂ψ ∂ψ , rv = − , and Ra *x is the generalized ∂y ∂x

local Rayleigh number for a porous medium, which is defined as

⎛ gβK (n) cos Ω(Tw − T∞ ) x n Ra = ⎜⎜ υα n ⎝ Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

* x

⎞ ⎟⎟ ⎠

1/ n

.

(10)

In order that similarity solutions of equations (1)-(3) exist, we assume the following expressions for the internal heat generation:

q m′′′ =

k e (Tw − T∞ ) Ra x e −η , 2 x

where ke is the effective thermal conductivity of porous medium. The transformed momentum and energy equations become:

(11)

492

A. M. Rashad, S. M. M. EL-Kabeir and Rama Subba Reddy Gorla

( f ′ )n

= θ − Mnf ′

1⎛ 2⎝

(12)

θ ′′ + Ds ( f ′θ ′)′ + ⎜ 3 +

λ⎞

[

]

′ 4 3 −η ⎟ fθ ′ − λf ′θ + R (CT + θ ) θ ′ + e = 0 n⎠ 3

(13)

⎛ ⎞ σB02α Ra x ⎟⎟ is the magnetic parameter, Ds = γdRa x is the where Mn = ⎜⎜ ⎝ ρgβ (Tw − T∞ ) x ⎠ dispersion parameter. R = 4σ (Tw − T∞ ) 3 / χk e is the radiation parameter and CT = T∞ /(Tw − T∞ ) is the dimensionless temperature ratio. The transformed boundary conditions (6) become: f(0)=0, θ(0)=1, f ′(∞ ) = θ (∞ ) = 0 .

(14)

The surface heat flux, qw and the local Nusselt number Nux, may be written as:

4σ ⎛ ∂T 4 ⎛ ∂T ⎞ ⎟⎟ − 1 ⎜⎜ q w = −k e ⎜⎜ ⎝ ∂y ⎠ y =0 3χ ⎝ ∂y

Nu x =

⎞ ⎛ ∂T ⎞ 4σ ⎛ ∂T 4 ⎞ ⎟⎟ , (15) ⎟⎟ = −(k + k d )⎜⎜ ⎟⎟ − 1 ⎜⎜ ⎝ ∂y ⎠ y =0 3χ ⎝ ∂y ⎠ y =0 ⎠ y =0

qw x hx = , k e (Tw − T∞ )k e

(16)

where ke and h are the effective thermal conductivity and local heat transfer coefficient, respectively. The main physical quantity of practical interest is the local Nusselt number, given by

Nu x (Ra x )

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−1 / 2

4 ⎛ ⎞ = −⎜1 + Dsf ′(0) + R (θ (0) + CT ) 3 ⎟θ ′(0) 3 ⎝ ⎠

(17)

RESULTS AND DISCUSSION The system of equations (12)-(13), along with boundary conditions of Eq.(14) has been solved by using the fourth-order Runge-Kutta method with shooting technique. Numerical computations are carried out for several parameters such as the power law index, magnetic parameter, radiation parameter, wall temperature exponent, thermal dispersion and temperature ratio, respectively given by: n = 0.8,1.0,1.5 Mn = 0.0,1.0,2.0 ,

R = 0.0,0.5,1.0 , λ = 0.0,1 / 3,1 / 2 , CT = 0.0,0.2 , and Ds = 0.0,0.2 . The variation of rate of heat transfer in terms of the local Nusselt number with the power law index n for varying values of R, Mn, λ, CT and Ds is presented in tables 1-6. Also in the tables we

Thermal Dispersion and Radiation Effects on MHD Free Convection…

493

compared our results for Newtonian and non-Newtonian fluids with internal heat generation (IHG) and without internal heat generation (WIHG). The accuracy of numerical results has been verified by comparing the present data for Newtonian fluid (n=1), λ=0, 1/3, 1/2,with the absence of Mn, R, CT, and Ds reported by Cheng [18] and for non-Newtonian fluids with n=0.8, 1.0, 1.5, reported by Yih, [21] and Pop et al, [4]. Tables 1-3 show the variation of the local Nusselt number for dilatant fluids (n=1.5>1), Newtonian fluids (n=1) and pseudoplastic fluids (n=0.8