PERMEABILITY AND THE STRUCTURE OF POROSITY IN PARTICULATE MATERIALS

Transcription

1 PERMEABILITY AND THE STRUCTURE OF POROSITY IN PARTICULATE MATERIALS A thesis submitted for fulfilment of the requirements for the award of the degree of Doctor of Philosophy from The University of Newcastle Australia by Timothy James Donohue BE(Mech) (with Hons 1 st class) Department of Mechanical Engineering Centre for Bulk Solids and Particulate Technologies June, 2008 i

2 DECLARATION This work contains no material which has been accepted for the award of any other degree or diploma in any university or other tertiary institution and, to the best of my knowledge and belief, contains no material previously published or written by another person, except where due reference has been made in the text. I give consent to this copy of my thesis, when deposited in the University Library, being made available for loan and photocopying subject to the provisions of the Copyright Act Timothy James Donohue ii

3 ACKNOWLEDGEMENTS The work for this thesis has been carried out with the Centre for Bulk Solids and Particulate Technologies at the University of Newcastle. I would like to thank the directors, Professor Alan Roberts and Professor Mark Jones, who was also my co-supervisor, for providing the opportunity to study within the Centre. Over the course of my studies both Mark and Alan have been very helpful and they have offered kind words of encouragement when needed. I would also like to thank my principal supervisor, Dr Chris Wensrich, for his invaluable support. Chris has provided me with much advice over the years ranging from the finer points of mathematics to more general life experience. Without his help and support this thesis wouldn't be what it is today so I am very grateful. The technical staff at Tunra Bulk Solids must also be acknowledged as they have offered support at different stages of my research. Every member of staff was always more than happy to offer help when I needed it. I would like to thank the academics within the Faculty here at the University for helping me with aspects of this research and also for helping me along the way during my undergraduate years. More generally, I would also like to thank the school teachers I have had along the way as they laid the platform for my future studies and helped to ingrain my current passion for knowledge. Lastly, and most importantly, I would like to acknowledge my family and friends. My friends have been more than happy to discuss my work with me over a cold beer, which was particularly helpful in achieving clarity. My family though, and in particular my wife Carly, have been instrumental through the course of my research in keeping me focused, happy and sane. To them I offer many thanks. iii

4 TABLE OF CONTENTS Chapter 1 - INTRODUCTION... 1 Chapter 2 - LITERATURE REVIEW Introduction Ergun Equation Kozeny-Carman Equation Pore Size Models Continuum Models Computational Methods Conclusions Chapter 3 - PERMEABILITY, VOID RATIO AND VOID SIZE Introduction The Meaning of Particle Diameter Discrete Particle Size Distributions Fibrous Particles A Brief Word on Particle Density Conclusions Chapter 4 - PERMEABILITY OF SPHERICAL PARTICLES Introduction Experimental Method Materials Lead Shot Glass Beads Results Mono-size mixtures Binary mixtures Distributed Mixtures Other results Conclusions Chapter 5 - SIMULATION OF SPHERICAL PARTICLES Introduction iv

5 5.2 Existing Simulation Methods Lennard-Jones Simulation Model Lennard-Jones Energy Potential Lennard Jones Simulation Method Hertz-Gravity Simulation Model Hertz-Gravity Energy Potential Hertz-Gravity Simulation Method Packing efficiency results Algorithm Mono-size mixtures Binary mixtures Distributed mixtures Radial Distribution Function Mean free path length Permeability Conclusions Chapter 6 - PERMEABILITY OF FIBROUS PARTICLES Introduction Materials Results Rigid Fibres Non-rigid Fibres Past Literature Results Permeability Predictions Conclusions Chapter 7 - SIMULATION OF FIBROUS PARTICLES Introduction Rigid Fibres Energy Potential Simulation Algorithm Non-Rigid Fibres v

6 7.3.1 Energy Potential Simulation Method Simulation Method Fibre Representation Fibre Orientation Packing efficiency Rigid Fibres Non-Rigid Fibres Mean Free Path Length Permeability Conclusions Chapter 8 - TORTUOSITY Introduction Numerical Method Axial Direction Radial Direction Tortuosity Spherical Particles Assemblies Fibrous Particle Assemblies Permeability Spherical Particles Fibrous Particles Conclusions Chapter 9 - CONCLUSIONS REFERENCES APPENDIX A: MATERIAL PHOTOS APPENDIX B: ERROR EQUATION vi

7 Nomenclature a Dynamic specific surface area [m 2 ] a ~ Acceleration vector [m/s 2 ] A Cross-sectional area [m 2 ] AR Aspect ratio [-] A E Viscous coefficient in Ergun equation [-] A LJ Coefficient of repulsive force in Lennard-Jones equation [-] b Kozeny constant [-] b 0 Kozeny constant (independent of tortuosity) [-] B E Inertial coefficient in Ergun equation [-] B LJ Coefficient of attractive force in Lennard-Jones equation [-] c Shape factor dependent on porosity [-] C Constant proportional to Reynolds number that [-] represents the transition from viscous to inertial flow d Particle diameter [m] d p Diameter of tortuous path through assembly [m] D Diameter of test chamber [m] dp dx Pressure gradient [Pa/m] dp dx dp dx dp dx flow form drag deflection Pressure loss due to viscous flow Pressure loss due to drag effects of the material Pressure loss due to energy absorption of the fibres [Pa/m] [Pa/m] [Pa/m] e ~ t Unit vector in the tangential direction [-] e ~ n Unit vector in the normal direction [-] E Young's modulus [Pa] E A Axial stiffness [Pa] E B Bending stiffness [Pa] f v Coefficient of drag [-] vii

8 f b Inertial contribution to coefficient of drag from changing [-] passage cross-sectional area and inter-passage branching f c Inertial contribution to coefficient of drag from [-] curvature of the passages f v,l Modified coefficient of drag [-] I Inertial term coefficient [-] I A Second moment of area [m 4 ] k Permeability coefficient [m/pas] k d Dimensionless permeability [-] K d Coefficient of denominator [-] K Factor dependent on array geometry [-] K 0 Modified Bessel function of the second kind [-] K 1 Modified Bessel function of the second kind [-] L e Average length of streamline through packed bed [m] L Total length of packed bed [m] m Repulsive index of Lennard-Jones equation [-] MR Mass ratio [-] M ~ Marker point coordinate [-] n Attractive index of Lennard-Jones equation [-] Q Volumetric flow rate [m 3 /s] r~ Rotational direction of fibre [-] r e Mean entry pore radii [m] r ij Distance between the i th and j th particle [m] Re Reynolds number [-] Re m Modified Reynolds number [-] S Area of particle surface per unit volume of packed bed [m -1 ] S 0 Total surface area per unit volume [m -1 ] U A Energy due to axial strain [J] U B Energy due to bending strain [J] U G Energy due to gravity [J] U H Energy due to Hertz contact strain [J] viii

9 U ij Energy between the i th and j th particle [J] v ~ Velocity vector [m/s] V Velocity [m/s] V i Distribution by volume [-] x~ n Particle coordinate [-] X d Mean free path length [m] X i Distribution by number [-] α Angle between the direction of flow and the normal of [degrees] an element exposed to flow β Parameter of gamma distribution [-] γ Angle used to calculate velocity component [degrees] δ Radius of curvature [m] ε Void ratio [-] ε eff Effective void ratio [-] θ Parameter of gamma distribution [-] µ f Fluid viscosity [Pas] ρ b Bulk density [kg/m 3 ] ρ s Solids density [kg/m 3 ] ρ f Fluid density [kg/m 3 ] τ Tortuosity [-] υ Poisson's ratio [-] ψ Shape factor [-]! Angle used to calculate velocity component [degrees]! Packing efficiency [-] ix

10 Abstract Permeability is an important property that arises in many fields of study. The ability to predict the permeability for a particular material is necessary as it affects the design of many materials handling and storage solutions. There are an abundance of equations that predict permeability for specific applications, but the underlying theory for these equations remains constant. Key factors affecting permeability that appear in many equations are the pore space, individual pore size, and pore connectivity. Many existing equations seek to quantify these factors in some form, with void ratio, particle diameter and tortuosity the most commonly used. Each of these factors is investigated throughout this thesis to further investigate their influence on permeability. These factors are investigated with specific reference to two equations; the Ergun equation and the Kozeny-Carman equation, and with specific reference to two types of materials; spherical particle mixtures and fibrous particle mixtures. Numerical simulation methods are used to build assemblies of spherical and fibrous particles. The assemblies of particles are used to extract fundamental information regarding the pore size and connectivity. The average size of the individual voids can be found as well as the average length the flowing fluid takes through the voids of the material. The use of the simulated assemblies to find material properties such as these allows for new insight into the structure of these types of packed beds. This new insight allows for an improvement in the way permeability is characterised for the materials studied in this thesis. x

11 Chapter 1 - INTRODUCTION Permeability is defined as the ability of a material to transmit fluids, and is a property that has been widely studied amongst a diverse range of fields. Permeability is a fundamental property, specific to each material, and past literature has seen a great deal of work in predicting and quantifying its value for different materials under different conditions. Generally a material becomes less permeable the tighter it is packed, meaning less fluid can pass through the material per unit time. This concept of permeability being governed by the space that exists within the material is an important concept that lies at the core of this thesis. Some examples that illustrate the importance of permeability are: In the field of geology an understanding is necessary in the flow characteristics of hydrocarbons in gas and oil reservoirs, and of groundwater in aquifers The blowing of air is used to improve the discharge behaviour of fine powders out of silo hoppers In the textile industry a high speed flow of air through the pores of the fabric is used to remove water from the fabric Many filters and air cleaning elements are concerned with minimising their resistance to flow The permeability of biomass fuels has a strong influence on the production of ethanol/methanol during industrial fermentation. The permeability of a bulk solid is an important parameter in determining the pneumatic conveying characteristics The interest in gaining an understanding of permeability is certainly not a new one; rather it dates back to 1856 with the formation of Darcy s Law [16]. Since then, it has long been established that permeability is dependent on many factors including particle size and distribution, particle shape, surface roughness, and moisture content as well as 1

12 consolidation condition. Due to the complexity that these factors bring the permeability is often measured experimentally or determined through estimation using one of the many empirically derived formulas. While there has been much work in the area of permeability prediction the formulas have not yet evolved past the empirical stage for any materials other than spherical particles due to the complex and often random packing that exists. This thesis is primarily concerned with permeability as it relates to bulk solids handling. The topic of bulk solids handling is varied and wide reaching, but specifically permeability is a property that features prominently in the design of materials handling systems such as pneumatic conveying and storage bins. As a result of this, the permeability is frequently expressed as a function of the bulk density with no real depth of understanding into the flow characteristics. Scope The scope of this thesis is to obtain a greater understanding of the role of basic material properties on the behaviour of permeability. Basic material properties include particle diameter, density, shape, and void structure. This thesis attempts to do so by re-examining existing empirical equations on a variety of materials in order to improve the equations and in doing so gain a greater understanding of the relationship between void space/structure and permeability. The main content of this thesis involves the use of numerical simulations, in which particles, including those of a spherical and fibrous shape, are simulated to represent a physical packing. The advantage that the simulations offer is that data can be extracted from the models that have previously not been available from physical experiments. The discovery of this new data from the models is then used to expand and modify existing equations that better reflect the effect of the basic material properties on the permeability. 2

13 Thesis outline Chapter 1 introduces the topic of permeability and its relevance to the field of bulk solids handling. Chapter 1 also includes an outline of this thesis. Chapter 2 covers the main ideas relevant to this thesis and includes a literature review of past research on these ideas. The empirical equations that are studied in this thesis will be presented as well as their applications and limitations. Chapter 3 discusses in detail, citing specific examples, the effects of particle diameter, particle shape and material composition on permeability. The materials studied range from mono-size spherical particles to stringy/compressible materials such as cotton wool. The relative short comings of general empirical theoretical equations are presented. Chapter 4 presents results for the packing efficiency and permeability for assemblies of spherical particles. The assemblies range from mono-size particles, binary mixtures and assemblies with a narrow particle size distribution. The purpose of this chapter is to present results for comparison with the simulation algorithms presented in the following chapter. In Chapter 5 two alternate optimisation algorithms for simulating packed spherical particles are presented. Comparison of the simulated assemblies is made with experimental results in the form of packing efficiency. The radial distribution function is also studied and compared with experimental results and one other simulation method. A further algorithm is used to extract mean free path length data from the simulated assemblies. The theoretical equations are then modified to include the mean free path length rather than the particle diameter. Chapter 6 presents experimental results for fibrous materials for the packing efficiency and permeability. Rigid fibres of two different aspect ratios are studied, as well as three aspect ratios of non-rigid fibres. Dimensionless permeability is compared to past literature, and as well aspect ratio versus packing efficiency is compared to empirical equations. The purpose 3

14 of this chapter is to present results for comparison with the simulation algorithm presented in the following chapter. Chapter 7 discusses an extension of the algorithm presented in Chapter 5 to simulate a bed of fibrous particles. The algorithm is used to study the effect of aspect ratio and orientation on the packing efficiency of the assembly. The packing efficiency from the simulation algorithm is also compared to experimental results and past literature. The mean free path length is found for each of the fibrous assemblies and used to calculate the permeability using the equations presented in Chapter 5. Chapter 8 explores the concept of tortuosity and its relevance to permeability. An algorithm is presented that maps a path through the void space of an assembly of particles in both the axial and radial directions. Using the tortuosity of an assembly allows further modification of theoretical equations to predict permeability. Chapter 9 summarises the thesis and draws conclusions based on the research. A brief discussion on each conclusion is also included as well as an outline of further work that can be carried out as an extension of this thesis. 4

15 Chapter 2 - LITERATURE REVIEW 2.1 Introduction Permeability is a broad term that has received a significant amount of attention from a diverse range of fields. As stated in Chapter 1 the earliest published work is Darcy s Law [16], which is the basic law governing permeability. Published in 1856, Darcy s Law states the rate of flow of a fluid through a porous structure is directly proportional to the pressure gradient causing flow. dp V = k (2.01) dx In the Darcy equation, dp/dx is the pressure loss across the sample length, k is the coefficient of permeability, and V is the superficial gas velocity defined as; Q V = (2.02) A where Q is the volumetric flow rate of the fluid and A is the cross-sectional area of the chamber being permeated. The application of the Darcy equation is limited to viscous flow in which the velocity of the flowing fluid is low enough such that there are no inertial effects. Following Darcys Law it was first suggested by Dupuit [25] and then Forchheimer [29] that a non-linear relationship exists between pressure loss and the fluid velocity at high speed flows. This led to the establishment of the Forchheimer equation 1 which introduced the term I as the inertial term coefficient. 1 Forchheimer's law is a vectorial quantity, but is shown here in scalar form as it is more relevant to this work 5

16 dp dx 1 k 1 I 2 = V + V (2.03) It can be seen that the Forchheimer equation is based on the Darcy equation with the addition of a second term to account for local inertia effects such as direction changes and drag forces. The equations by Darcy and Forchheimer are based on a theoretical background and have been applied to a range of materials with success. However, there has been a great deal of work to quantify these formulas to include specific material properties. The influences of these other factors on the permeability are not as rudimentary to establish as accurately as the velocity but are no less important. Formulaic expressions for these other factors will be covered shortly but it is deemed important that a brief explanation of the ideas are covered first. The properties of the flowing fluid that affect the pressure gradient are the fluid viscosity and the fluid density. These properties are trivial to find, with the only consideration worth discussing being the case of compressible gases. However, if isothermal flow is considered and the pressure loss over the material sample is very small compared to the inlet pressure, as it is in most cases, then compressibility can be ignored. The most important factor in the discussion of the pressure loss is obviously the solid that is being permeated. It is however quite a large and complex topic, which will be discussed in two main categories. a) void ratio and void size b) particle size and shape The void ratio (ε) of the material can be defined using the relationship between bulk density (ρ b ) and solids density (ρ s ); 6

17 ! " #! b = 1 (2.04) s The bulk density is the apparent density of the material in its sample container, while the solids density is the density of the particles. The void ratio is an indication of the closeness of the packing, with a smaller void ratio indicating a tighter packing. While the application of the above formula is straightforward, the determination of the particle density that appears in Equation (2.04) is not. For solid particles there are a variety of tests available to measure the particle density including an air comparison pycnometer and a density tube. For the case of porous particles the particle density must reflect the fact that the flowing fluid often does not pass through the pores of the particles. The consideration is discussed in more detail in the following chapter. Granular materials are rarely spherical, mono-size mixtures that represent the ideal case. The particle size and shape are important factors influencing permeability, and can be quite hard to define and can be subjective for more complex shapes. For non spherical particles a term that is frequently used is the shape factor (ψ) which is defined by [8]; surface area of sphere with same volume as particle! = (2.05) surface area of particle The shape factor indicates the 'sphericity' of the particle, and is based on finding the equivalent sphere diameter, with a sphere obviously having a sphericity of 1. In practice, the shape factor is usually calculated via experimental techniques as the surface area of a particle can be quite difficult to accurately measure. When a shape factor is used it is often an indication of the average sphericity of the material holding no specific details of the range of sphericity within the material. When the shape of a particle deviates largely from a sphere, such as for fibrous particles, the shape factor can still be experimentally found but has no physical meaning. 7

18 In any discussion of permeability it is necessary to discuss the units used to describe permeability. There has been a range of units used in the past; however, certain terms have had multiple meanings depending on the field of use. Intrinsic permeability, or often just permeability, measures the contribution of the porous medium to the flow of fluids. It is independent of the fluid being passed through the medium, and the SI units are m 2. Conversely, the term specific permeability is the permeability of a material relative to a specific fluid, and has SI units of m 2 /Pa.s. Experimental results for permeability in this thesis have all been presented relative to air unless stated otherwise. 2.2 Ergun Equation The most well known and robust equation that predicts the pressure gradient is the Ergun equation [26], which is based on the more general Forchheimer s Law (2.03) but with the material properties void ratio (ε), particle diameter (d), viscosity (µ f ) and density (ρ f ) explicitly included. dp dx 2 2 µ f V (1 #! ) " f V (1 #! ) = AE + B 2 3 E (2.06) 3 d! d! The Ergun equation was developed with the understanding that the pressure loss was attributable to four main factors, two concerning the fluid and two concerning the solid; 1) rate of fluid flow 2) viscosity and density of the fluid 3) closeness and orientation of the packing 4) size, shape, and surface of the particles It was also noted that the pressure loss was due to both viscous and kinetic effects. To find the coefficients of these viscous (A E ) and inertial (B E ) terms the method of least squares was used over 640 experiments covering a wide range of sample materials and using a range of fluids. The sample materials consisted of various-sized spheres, sand and pulverized coke while the gases that were used included carbon dioxide, nitrogen, hydrogen 8

19 and methane as well as air. The method used to find the coefficients A E and B E was to rewrite the equation in a linear form; 2 3 dp d! " f Vd = A 2 E + BE dx µ V (1 #! ) µ f (1 #! ) f (2.07) Equation (2.07) can be summarized further to; Re f v = AE + BE (2.08) (1 "! ) Where f v is equal the left hand side of Equation (2.07) and is known as the coefficient of drag and Re is the Reynolds number that is given by! Vd µ f f. A plot was then created for varying Reynolds numbers where Re (1 "! ) (x-axis) was plotted against f v (y-axis). The slope of the line of best fit was then the coefficient B E while the y-intercept of the line was the coefficient A E. The Reynolds numbers that were studied in the formulation of this equation were for creeping flow (low Re. number) to approximately The final values for A E and B E determined by Ergun are 150 and 1.75 respectively. Explicit information regarding the range of void ratios covered in the study is not given. Ergun was able to derive Equation (2.06) by realizing that the pressure loss was the sum of viscous and kinetic effects, and that each of these terms were affected differently by the two parameters particle diameter and void ratio. An important point noted by Ergun is that orientation of the particles in a packed bed is not susceptible to mathematical formulation, and as such the equation makes no effort to distinguish between varying orientations of the particles. This can be considered a weakness of the equation, although it is known that for laminar flow shape plays little part in the resistance to flow [5]. A shape factor has been defined already (2.05) and the Ergun equation can be modified to include this shape factor: 9

20 dp dx µ f V (1 $! ) 1.75# f V (1 $! ) = + (2.09) " d! " d! Where the diameter term, d, is multiplied by the shape factor, ψ. Care must be taken though if trying to calculate a shape factor analytically, as it has been shown that this procedure is inadequate for wide particle size distributions [52]. Since the establishment of the Ergun equation there have been many applications of the equation and also attempts to expand the practicality of the equation. Handley and Heggs [32] These workers studied a variety of particle shapes and attempted to correlate the results to Equation (2.08) above. The first shape that they studied was spherical particles and expanded the correlation for a greater range of Reynolds numbers, plotting results up to Re = The porosities for the spherical particles represented dense random packing (! = ). Creating a plot by utilizing a form of the equation similar to Equation (2.08), and fitting a line of best fit by using the method of least squares, the authors found the coefficients to be 1.24 and 368 which represent a 30% decrease in the inertial term and 245% increase in the viscous term compared to Ergun. The authors do not discuss any reasons to account for the seemingly large differences, except for the fact that their coefficients are formulated specifically for spheres whereas Ergun attempts to account for a wide variety of shapes. The other shapes that were studied were cylinders, rings and plates. For these shapes the most important result to come from the study was that the authors confirmed the pressure drop was a result of the summation of the viscous and inertial terms. This was again done by plotting a line of best fit as described already. A summary of the results for each of the materials is as follows; 10

21 Cylinders and rings The porosity was in the range of 30-35% for the cylinders, and in the range of 55-75% for the rings. The Reynolds number studied was up to approximately The pressure loss for both these shapes due to the viscous term was greater than that for spheres. Plates The plate mixtures varied in size and in the spacing between plates, with porosities varying between 30-65%. The Reynolds numbers studied for the plate mixtures were quite large, ranging up to In contrast to the cylinders and rings, the pressure loss for the plates due to the viscous term was less than that for the spheres. With these results the authors concluded that the nature of the Ergun equation could predict the pressure loss for a variety of shaped particles, although the coefficients depend on the packing size, shape and orientation. Kuo and Nydegger [44] This paper presented results from an investigation into the flow resistance of WC 870 ball propellant grains, and sought to find a correlation between the drag of the permeating gas and the Reynolds numbers, in a similar form to that of Equation (2.08). A large part of the study was expanding the range of Reynolds numbers that the equation was valid for, with the study conducting tests on Reynolds numbers up to The porosity range that was studied was quite narrow, being only between <! < The test apparatus was set up specifically to mimic a 30-caliber system, and as such the chamber diameter of the test apparatus was quite narrow (7.7 mm) compared to the average diameter of the particles (0.826 mm). The authors detailed a very accurate experimental process in the paper, and then presented a correlation that represented the best fit from 220 data points; 0.87 & Re # f v = $! (2.10) % 1' ( " 11

22 A plot is then created to compare Equation (2.10) with an extension of the Ergun equation at higher Reynolds numbers, and as expected with the different coefficients there is a distinct difference between the two equations. The Kuo and Nydegger equation predicts higher drag at low flow rates and lower drag at higher flow rates. However, the authors do offer a number of reasons for the discrepancies in the two equations; i. Particle size the authors mention that their work has been formulated for particles with a size distribution whereas the work of Ergun was for mono-size particles ii. Particle shape The authors compare their work that includes particles of varying shape to Ergun s work, and argue that Ergun used all spherical particles. However, it is well known that Ergun used various shaped particles iii. Wall effect It is known that for spherical particles that are packed into a cylindrical container the porosity of the bed is greater next to the wall, and so it would appear that the permeability should be higher in this region as well. Various workers [9, 14] have formulated an empirical equation to account for the wall effect although there is no single correlation that is generally accepted. Throughout existing literature the common acceptance is that for situations where the ratio of the diameter of the test chamber is less than 10 times the particle diameter there will definitely be some wall effect, while having this ratio above 20 will negate the wall effect. For this study the authors were quick to note that small size of the test chamber relative to the particles would have some effect on the flow resistance through the granular bed. However, the authors reinforced the fact that they were trying to simulate the flow resistance in a 30-caliber system and as such had no control over the size of the test chamber. The authors concluded their paper by validating the Ergun equation in that it was successful in determining a correlation between the resistance to flow and the Reynolds number. They also stated that their form of the coefficient of drag may not be suitable for other types of flow due to the specific nature of the experimental conditions in obtaining the equation. 12

23 Macdonald et al [52] These workers analysed a wide range of data and sought to find alternate values of the coefficients A E and B E in the form of the Ergun equation given in (2.06). The range of materials they studied varied in Reynolds numbers, and included both viscous and inertial flow regimes. The porosity range that they tested the Ergun equation over was wide representing the range 0.36 <! < In their study of the coefficients they included the particle roughness and proposed alterations to the form of the Ergun equation. They concluded that the viscous coefficient is independent of particle roughness, although suggesting it should be 180 rather than 150, but that the kinetic term coefficient should be dependent on particle surface roughness. It was suggested that smooth particles should bear a coefficient of 1.8, while the roughest particles should have a coefficient of 4. However, it is not mentioned in the study how to determine how rough a particle surface is. Lastly, the authors briefly discussed the dependence on porosity in the Ergun equation. They concluded that the porosity function was adequate, but proposed that it could be improved by replacing the ε 3 term with ε 3.6. They validate this conclusion by noting the standard deviation in the coefficients A E and B E is reduced by approximately half if this change is made. Jones and Krier [39] Permeability tests were carried out in this study for high Reynolds numbers in which the ratio of the diameter of the test chamber to the diameter of the spherical particles ( D d ) was varied. These researchers also investigated the coefficient of drag and were able to find values for the coefficients of the viscous and inertial terms. The variation in the D d ratio that was studied was approximately An interesting result of this study can be seen when a plot of Re (1 "! ) against f v is made. The authors have split the results into two separate lines of best fit, one with results from D d less than 10 and the other with all the 13

24 results from D d greater than 20. This indicates that according to their results the diameter of the test chamber must be at least 20 times greater than the diameter of the particles to negate the wall effect. The coefficient of drag that these workers formulate is of the form; 0.87 & Re # f v = $! (2.11) % 1' ( " Equation (2.11) shows that the inertial term coefficient remains the same as in the Ergun equation but the viscous term coefficient is considerably different. If the above equation is rearranged into the same form as the Ergun equation it is of the form; dp dx f V (1 #! ) " f V (1 # 0.13 f 1.13 µ! ) µ = (2.12) d! d! The materials used in this study were spherical glass beads of varying diameters, with the porosity in this study ranging from 36-44%. The application of this study was to expand the use of the Ergun equation, as it was argued by the authors that the Ergun equation is only valid to Re = With a modification of the equation it was shown that the equation has been extended significantly, with this equation being valid up to a Reynolds number of Liu et al [48] These workers used a two dimensional model for flow through porous media and attempted to include a factor to account for the wall effect. Their study looks at the fundamentals of flow through porous media, and they consider the path the fluid takes as it passes through the material. They also consider the effects of the viscous and inertial terms separately. For viscous dominated flow the pressure loss is given by; 2 dp (1 "! ) µ V = 85.2 (2.13) 11/ 3 2 dx! d 14

25 It can be seen that the main difference between the above equation and the viscous term in the Ergun equation is of course the coefficient and the index of the void ratio. The reason for the difference is that the authors have theorized that the total length of the path that the fluid takes through the material varies with the void ratio, and hence the void ratio index is higher which has the effect of reducing the coefficient. The formulation of an equation for the inertial dominated flow was also approached differently to existing literature. The authors argued that the pressure drop is affected by two factors when the flow rate is increased; mixing effect and curvature effect. This meant that the coefficient of drag was of the form; f = f + f (2.14) v b c Where f b and f c denote the inertial contributions to f v from changing passage crosssectional area and inter-passage branching and from the curvature of the passages respectively. The authors then used a correlation to represent the transition of flow between the viscous and inertial flow regimes similar to one used in [47] that it is used for flow in a helical pipe. This correlation was used to complete the final form of the coefficient of drag so that it had the form; f v, L 2 m Re = AE + BE Re m (2.15) 2 C + Re 2 m Where C is a constant that is proportional to the critical Reynolds number that represents the transition from viscous to inertial flow that must be found experimentally, and f, is the modified coefficient of drag due to the different form of the porosity function shown in Equation (2.13). The parameter Re m is a modified Reynolds number given by; v L 15

26 Re m 1/ 2 1/ 2 1+ (1 # " ) d! f V = (2.16) 1/ 6 (1 # " )" µ f The authors also formulated theoretical expressions for the wall effect for both the viscous and inertial terms. However, the equations including the wall effect will not be presented here as it is known that the wall effect can be greatly reduced, if not eliminated, under the correct experimental conditions [39]. Writing the equation in the form of the Ergun equation, and leaving it in terms of Re m for simplicity, results in the following expression; dp dx µ f V (1! " ) = / 3 d " 2 µ f V (1! " ) / 3 d " 2 Re m Re 2 m Re 2 m (2.17) To verify their equation the authors used a combination of data from existing literature and their own experimental data. The model was first compared with spherical particles packed to dense random packing in the viscous flow regime, and compares favourably with the experimental data. The authors then go on to compare the model with experimental data that is affected strongly by wall effects, with D / d ratios in the range of , and higher Reynolds numbers (up to 3000). The comparisons with this data show that the correlation for the wall effect that the authors have made provides reasonable results. Lastly, the authors presented some results for foam cylinders that were machined into set dimensions so that high porosities could be studied. The porosity was about 93% and the Reynolds number was less than 100. Once again the model showed excellent results in predicting the pressure loss of the experimental data. As a final note in their study of high porosity materials they commented that a higher porosity has the effect of increasing the wall effect. Comparisons to the Ergun equation The preceding reviews of past researchers have all presented their work in the form of an empirical equation that can be related back to the Ergun equation. To compare the equations with each other, they have all been plotted on the same graph of the Reynolds number versus the pressure gradient. To compare all equations in the same fashion, a 16

27 velocity range was selected that produced a range of Reynolds numbers. This velocity was used in each of the equations to predict the pressure drop. Conditions were selected that were suitable for all of the equations; spherical particles packed to dense random packing with the flowing fluid being air at room temperature and standard atmospheric conditions. Two graphs were produced, one for low Reynolds numbers in which all of the equations were valid and one for higher Reynolds numbers where only a few of the equations were valid. Pressure Gradient (kpa/m) Ergun Handley and Heggs Kuo and Nydegger Macdonald et al (Smooth) Macdonald et al (Rough) Jones and Krier Liu et al Reynolds Number Figure 2-1: Comparison of similar equations to that of the original Ergun equation for low Reynolds numbers The maximum Reynolds number in Figure 2-1 is 1400, the limit of the Ergun equation. The upper limit of the Macdonald et al equation has been assumed to be the same as the Ergun equation as the paper does not state the range of Reynold numbers that the equation is valid for. As Figure 2-1 shows, all of the correlations, with the exception of the Macdonald et al equation for rough particles, are somewhat similar. In fact if the Macdonald et al correlation is excluded, the highest value at Re = 1400 over predicts the smallest value by only 25%. 17

28 The small range of the Liu et al equation is also noticeable, but this is because their study was focused on viscous flow Pressure Gradient (kpa/m) Ergun Handley and Heggs Kuo and Nydegger Jones and Krier Reynolds Number Figure 2-2: Comparison of similar equations to that of the original Ergun equation for higher Reynolds numbers Figure 2-2 shows the correlations for the higher Reynolds numbers. It must be noted that the Jones and Krier equation is valid to Re = but it has been truncated here so that some resolution is maintained at the low end of the scale. The Ergun equation has been extrapolated to the higher Reynolds numbers for comparison although it is not actually valid at these Reynolds numbers. It can be seen that as expected the Ergun equation grossly over predicts the Jones and Krier equation, which closely predicts experimental data at these high Reynolds numbers. At the range of Reynolds numbers that the three equations are valid over (excluding Ergun equation), it can be seen that there is reasonable agreement, with the highest value over predicting the lowest value by 28%. 18

29 Kyan et al [42] and Belkacemi and Broadbent [1] The paper by Kyan et al took a somewhat different approach that was based on the Ergun equation but modified for the case of fibres. The pressure drop was a result of three factors: dp dx dp dp dp = + + (2.18) dx dx dx flow form drag deflection The flow pressure loss was the same as the viscous pressure loss in the Ergun equation, but it was argued that the inertial term from the Ergun equation can be ignored for a fibrous bed, and two more terms specific to fibres were included. The form drag term is to account for drag effects of the material while the deflection term represents the possibility that a fibre will absorb some energy of flow. A number of new parameters are needed for this approach though including the ratio of the viscous drag of the fluid to the elastic force of the fibre and the knowledge of the effective porosity of the bed rather than the actual porosity. The effective porosity (ε eff ) was defined as: volume of flow region! eff = (2.19) total volume The important idea to come from this paper is that in high porosity beds of fibres a significant portion of the pore space occupied by the fluid is not flowing. It was later shown by Dullien [24] that the claim made by Kyan et al that non-linear behaviour is due to deflection is unfounded. However, what Kyan et al did provide is a useful empirical correlation for fibrous beds. Belkacemi and Broadbent took into account the criticisms of Dullien and modified the equation of Kyan et al so that the pressure drop is due to a contribution of viscous forces, inertial forces and forces related to fibre deformation. The main difference of this model is that drag coefficient is estimated by assuming the air flows through a bed of cylindrical particles whereas Kyan et al estimated the drag coefficient by using a simple relationship 19

30 that considered neither the irregularities of these particles, their interaction, or the influence of the fluid. The authors needed to find the parameters of the model by using the least squares method, and in doing so it allowed the calculation of the bending stiffness for each of the different fibre types. Having obtained the parameters for the model, the authors compared the results to their own experimental data and to the data of Kyan et al. Both sets of results provided a good correlation between theory and experiment. The most noteworthy result of the model is that the authors state that the bending stiffness predicted from the equation compares well with literature values. 2.3 Kozeny-Carman Equation The other prominent equation that is synonymous with permeability prediction is known as the Kozeny-Carman equation [8]. The derivation was initiated by Blake [4], who considered viscous and streamline flow. In a later paper by Kozeny [41], and apparently without knowledge of Blake s paper, Kozeny used the same principles as Blake to deduce a similar equation. However, Kozeny s equation includes a better explanation of the nature of the flow and the terms that make up the equation. For this reason the work of Kozeny has been used more in the literature and is the more commonly referenced work. The derivation of the Kozeny-Carman equation is based on Darcy s Law (2.01) and stemmed from the early work of Kozeny [41]. Kozeny considered only viscous flow and assumed that the pore space was equivalent to a bundle of parallel capillaries with a common hydraulic radius. The equation in its most compact form is; k 3! = (2.20) 2 bs Where ε is the void ratio, b is known as the Kozeny constant and S is the particle surface for unit volume of the bed. To form the equation into a more manageable expression the term S 0 is used which represents the surface area per unit volume; 20

31 S = S (1 " ) (2.21) 0! Also assumed is that the path of a stream line through the pore space would be tortuous, with an average length L e, greater then the length of the test piece, L, defining the tortuosity to be L e /L. With the introduction of tortuosity it was Carman who made the realisation that if the stream line follows a tortuous path then the true pore velocity must be higher than the pore velocity parallel to the direction of flow. This led to the important result; 2 & L # b = $ e! b0 (2.22) % L " Using Equations (2.20), (2.21) and (2.22), and the fact that S0 = 6 d for spherical particles, the final form of the Kozeny-Carman equation for spherical particles can be written as; k 2 d ( = 36b 0 3 ( 1' ( ) 2 & $ % L L e #! " 2 (2.23) The Kozeny constant b is found empirically and so this makes Equation (2.23) somewhat subjective, although studies suggest it is approximately constant over a range of tests. Carman [9] conducted an experiment to find the value of the tortuosity by using colour bands in water flowing vertically downwards through a bed of spheres. The author observed that the path made an almost constant angle of 45 o with the axis of the test chamber, with very little deviations. On these results it was assumed that the track an element of fluid follows through a packed bed corresponds to an average inclination of 45 o, and therefore the value for L e L is 2. In his later book [8], Carman also tabulated values for b 0 for various sectional shapes, and found that they lie in the range of Combining these two results and using Equation (2.22) it can be seen that a value of approximately 5 for the Kozeny constant (b) is relatively independent of shape. 21

32 It must be noted that in the original formulation of the Kozeny equation it was intended that the Kozeny constant was in fact a constant. However, it has since been shown that it is not, as it varies with porosity and also particle shape. The above explanation also implies that the tortuosity is independent of void ratio, whereas intuitively it would seem it depends on void ratio. For this reason many workers have included the tortuosity into the Kozeny constant term, that is, they have conducted experiments and found the value of b rather than b 0 that appears in Equation (2.23). There have been a number of investigators who have studied the Kozeny-Carman equation more thoroughly and in particular to find the value of the Kozeny constant for different shapes. A few of the more comprehensive and relevant studies will be reviewed here. Coulson [14] This study presented results on the flow of fluids through packings of spheres, cylinders, cubes, plates and prisms, for which surface and shape can be accurately specified. The study consisted of measuring the pressure loss from passing oil through a bed of particles. With the experimental results they then used the Kozeny-Carman equation to calculate values for the Kozeny constant. The wall effect was again discussed in this paper, with the author pointing out that the arrangement of the spheres adjacent to the wall will be different from that of the main body of the packing. A correction term was included in the equation for pressure loss, although the author also pointed out that this term is empirical. For the range of experiments conducted on spheres, Coulson concluded that the value of the Kozeny constant should be 4.8 ± 0.2. For other shapes the results are best viewed on a graph. 22

33 6.5 6 Kozeny constant Cylinders 1/8" x 1/4" Cylinders 1/4" x 1/4" 4 Cubes Prisms 3.5 Plates 1/4" x 1/4" x 1/16" Plates 1/4" x 1/4" x 1/32" Void Ratio Figure 2-3: Reproduction of the graph that appears in [14] Figure 2-3 shows that the Kozeny constant lies in the limited range of between 3.2 and 5.8 for most of the results. The graph also shows that the Kozeny constant not only depends on the shape of the particle but also the void ratio. The graph also shows that the closer the particle is to having an aspect ratio of 1 (e.g. cubes) then the less the Kozeny constant varies with void ratio. This result is logical as the more a particle varies from an aspect ratio of unity then the more important the orientation of the particles becomes The paper also included an interesting discussion on the application of the equation to shapes such as plates. It was noted that the equation involving S assumes that all surfaces of the particles are wetted by the fluid. While this is true for spheres, which have just one point of contact with another particle, it is not for particles such as plates which have an area contact. Coulson makes the point that in calculations for the Kozeny constant using the specific surface for such particles the value for the constant will be too small. 23

34 Sullivan and Hertel [74] This paper also determined the value of the constant for spherical particles but more importantly explored the effect of orientation in the case of fibres. For the Kozeny constant they investigate an approximation put forward by Fowler and Hertel [30]; 3 b = (2.24) (sin 2!) AV Where! is the angle between the direction of flow and the normal to an element of surface exposed to the flow. While this formula does not come from an explicit theoretical derivation it does provide some excellent predictions. Using this equation for spheres, fibres parallel to flow and fibres perpendicular to flow the respective values for the constant are 4.5, 3 and 6. Experiments were conducted by passing a light mineral oil through beds of glass spheres and the resulting average value for b was 4.5 ± This is in good correlation with their predicted value of 4.5 and also with previous workers [9, 14]. The range of void ratios that these measurements were made over was not stated. Measurements were also made for beds of glass wool. The cases of fibres aligned parallel to the test chamber axis and the fibres aligned perpendicular to the test chamber axis were considered. The fibre diameters were measured using a microscope so that the surface areas could be found. For the case of fibres packed perpendicular to flow results were presented for a void ratio of approximately 0.80, with the average value of b being For the fibres packed parallel to flow the void ratio varied from and the average value of b was Once again these are in excellent agreement with Equation (2.22). No results were given in this paper for random orientations of fibres. Wiggins et al [79] The research involved in this paper is concerned only with fibrous materials. A number of fibrous materials were studied in which the permeability of the fibres to water or benzene 24

35 was measured. The materials studied were 0.4 mm diameter glass fibres, 0.1 mm copper wire of length 5 mm, glass wool, fibre glass and Celanese yarn. The void ratios encountered for these materials were in the range The researchers measured the specific surface of the fibres and compared that with the specific surface calculated from the Kozeny-Carman equation. The difference from the two methods was typically in the range of 5-10%. From their results, it is possible to calculate the value of the Kozeny constant for each of the fibres. Calculating the average for all of their results yields b = 5.1± For all of the packings considered in this study the fibres were orientated randomly. Davies [17], Happel [34] and Lord [50] In contrast to the results in [79], these researchers found that at high porosities the Kozeny constant increases greatly for beds of fibres. Davies studied beds of fibres used for air filters and obtained an empirical equation. It is not stated whether or not the flow is perpendicular or parallel to the direction of the fibres. Happel used a free surface model to predict resistance to flow relative to cylinders. The theory is developed on the basis that two concentric cylinders can serve as the model for fluid flowing through an assembly of cylinders. The inner cylinder consists of one of the rods in the assembly and the outer cylinder a fluid envelope with a free surface. Employment of appropriate boundary conditions enables solutions to be obtained for the Navier-Stokes equations by the use of this model. The model can be used to find the Kozeny constant for the cases of flow parallel and perpendicular to the cylinders. The results for Davies and Happel are shown in the table below. 25

36 Table 2-1: Results for the Kozeny constant using different methods Void Ratio Happel (Parallel) Happel (Perpendicular) Davies Lord also conducted experimental results for different types of fibres, and they were packed in a random arrangement, although the author noted that they tended to orientate perpendicular to flow. For these experiments Lord was also able to calculate values of the Kozeny constant for varying void ratios by experimentally determining the specific surfaces of the fibres. The values that Lord found were largely in agreement with the results in Table 1, although the values at the higher end of porosity were not as large. For the five different fibres, the value of the Kozeny constant at porosity of 0.80 was approximately 5, while at a porosity of 0.99 the value of the Kozeny constant showed a larger variation ranging from 17.9 to 31.3 for the different fibres. Comiti and Renaud [15] These workers investigated both the Ergun equation and the Kozeny-Carman equation. The material that was studied was parallelepiped particles. They considered the form of the Ergun equation presented in Equation (2.06) and found the coefficients for parallelepiped particles. They validated their experimental method by studying spherical particles and obtaining values of 141 for A E and 1.63 for B E, which is in agreement with Ergun s 150 and For the parallelepiped particles the average value of the coefficient of the viscous term was found to be 165, which is half way between Ergun s value and that of the Kozeny-Carman equation. However, the coefficient of the inertial term was found to be in the range of 2.73 B < 12. 2, values which are obviously greater than for spheres and < E 26

37 even greater than what appeared in [52]. In an attempt to explain the mechanisms of flow they considered a capillary type approach similar to Kozeny. In doing so, the tortuosity was able to be calculated through the use of Poiseuille s equation [61] and the Nikuradse equation [58] to calculate the friction factor. The equation for pressure loss that they produced is: dp dx (1 $! ) 3 (1 $! ) 2 = 2# µ f a V # " a V 3 f (2.25) 3!! Where τ is the tortuosity and a is the dynamic specific surface area. By fitting this formula to experimental results they found that the tortuosity was between 1.40 <! < for spherical particles, which is in excellent agreement with the value of 2 found by Carman [9]. Using the same method the range of tortuosities found for the parallelepiped particles was 2.01 <! < The larger range for the tortuosity in the results for the parallelepiped particles is most likely due to the larger range of void ratios ( ). 2.4 Pore Size Models The existence of these models is a result of the belief that the parameters void ratio and particle diameter cannot accurately predict permeability for all types of materials. These discussions are concerned with the argument that the knowledge of the nature of the pore space is essential in determining the permeability. Childs and Collis-George [12] These workers attempted to define a model in which the permeability is predicted from the pore size distribution. The authors argue that the pore space of a material such as sand is continuous, but presents a set of caverns of various sizes each of which is connected to several others by narrower channels. The caverns or pores have a certain size distribution which may be of Poisson type. The distribution function of these pores is derived from the moisture characteristic. Using this method the authors obtained reasonable results for beds 27

38 of unconsolidated sand and dust. However, the constant in their equation is obtained from empirical correlation. Bo et al [5] This paper comments on the anomalies in existing correlations between particle size and permeability and shows that they are in part due to a lack of definition in particle size. The study uses different sizes of glass ballotini beads to blend varying size distributions and then measures the porosity and permeability of the mixtures. Comparisons to the Kozeny- Carman equation were also made, which predicted moderate results, but more importantly the authors noted how much variation there was in the permeability for varying distributions with the same size range. The authors then go on to note that this effect could be predicted better by an equation that relates pore size distribution to permeability, although noting that there are limited methods available for measuring the pore size distribution of a material. Lukasiewicz and Reed [51] This paper is concerned with the permeability of differing size distributions of alumina powders and a model to describe their behaviour. The authors found the pore size distribution by using mercury penetration porosimetry, and also found the porosity and liquid permeability for each sample. The permeabilities of the samples were plotted along with correlations using the Kozeny-Carman equation with the results not being very good. The authors put this down to the fact that the Kozeny-Carman equation was derived from the hydraulic radius concept, and concluded that this concept is not able to represent the relatively wide pore size distribution that was present in the materials tested. The authors also noted that using mercury porosimetry the pore volume is assigned to entry radii rather than actual pore radii; although they later show that excellent results are achieved using the entry radii. The equation that they use is; 28

39 k! r c 2 e = (2.26) Where r e is the mean entry pore radii and c is a shape factor depending on pore geometry Hydraulic radius Linear mean radius k/! (10-10 cm 2 ) Pore Radius Squared (10-10 cm 2 ) Figure 2-4: Reproduction of the figure that appears in reference Lukasiewicz and Reed The experimental permeability was then divided by the void ratio and plotted against the square of the mean entry pore radii and the shape factor was found from the inverse of the slope. The figure has been reproduced from the reference and can be seen in Figure 2-4. As the figure shows, the relationship between the two parameters in the graph is linear for the equation put forward by these workers, which means that the shape factor c is constant over the number of tests carried out. The same curve for the Kozeny-Carman equation, which is based on hydraulic radius, shows that the Kozeny constant is not constant for the same tests. 2.5 Continuum Models The approaches of Ergun and Kozeny-Carman are based on conduit flow where certain geometric assumptions are made. An alternative method is the continuum approach where the assumption is made that the flow is a continuum at the scale of interest. A particular area of interest that is relevant to this work is the use of these models with fibrous materials, in which the general approach has been to idealise the porous medium as a 29

40 matrix of rods and then to solve Stokes equation for a particular configuration. The discussion of these models will be dealt with in three categories; flow parallel to an array of rods, flow perpendicular to an array of rods and flow through three dimensional random arrays. Much of the theoretical work in this area utilized a unit cell method. This consisted of having rods arranged in a periodic pattern like a square (Figure 2-5a) or triangular array (Figure 2-5b), and defining a unit cell a polygon with a rod at the centre so that the flow in the cell is the same as that in the array. This approach can be used for flow parallel or perpendicular to the rods. Figure 2-5: Example of a) Square array and b) Triangular array Parallel Flow In the section on the Kozeny-Carman equation the free surface model of Happel [34] was discussed. This method was quite simple in that the liquid associated with each cylinder is lumped into a concentric cylinder with zero drag on the surface. This makes the mathematics simpler but the accuracy of the solution can be questioned as the model does not represent an actual array. As shown in Table 2-1 Happel used this method to obtain results for the Kozeny constant for flow both parallel and perpendicular to the rods. A more comprehensive approach was that of Drummond and Tahir [23]. They used a method of distributed singularities to find the flow field in square, triangular, hexagonal and rectangular arrays. Their paper presents the results in terms of cylinder drag, but elsewhere [37] it has been converted to permeability by applying a force balance on the fibre. The form of their equation is; 30

41 k 2 2 = ( d 2) & (1 ' ( ) # $ ' ln(1 ' ( ) + 2(1 ' ( ) ' +! 4(1 ' ( ) K % 2 " (2.27) Where K depends on the array geometry. It has been shown [37] that generally the permeability is over predicted by this equation but it can be argued that this should be the case as the fibres are rarely all aligned parallel to flow. Perpendicular Flow The unit cell method described previously has also been used for flow perpendicular to fibres. Hasimoto [33] used a Fourier series method to solve Stokes equations for a square array for circular cross-sectional fibres and also elliptical fibres, finding a similar result between the two shapes. Sangani and Acrivos [67] used the method of least squares to solve the flow through square and hexagonal arrays, while Drummond and Tahir [23] used the method discussed in the previous section for flow through square arrays. All of these results agree closely, and it was also pointed out [37] that the permeability from these results is almost exactly half compared to when the fibres are parallel to flow. This finding agrees with the results of Sullivan and Hertel [74] discussed previously who found the Kozeny constant for flow parallel to fibres is half for that for flow perpendicular to fibres (Kozeny constant appears in denominator of Kozeny-Carman equation). As an alternative to the regular arrays discussed so far, Yu and Soong [67] devised a method to deal with inhomogeneity. Their method consisted of dividing the medium, which consisted of parallel fibres normal to the flow, into equal sized compartments. The compartments were to contain at least one fibre and were to be homogeneous. The permeability was found for each compartment and the total permeability was found by summing the individual ones according to probability theory. They then used this model to calculate the maximum difference in permeability that could be achieved by varying the uniformity of the medium, with the result being that inhomogeneity can increase the permeability by as much as 50%. 31

42 Flow through three dimensional random arrays A completely different method to the unit cell approach is a method commonly referred to as swarm theory [37]. The earliest application of this approach was by Spielman and Goren [72] who used it to model the permeability of randomly orientated fibres. The idea behind the analytical approach was that flow could be modelled through a fibrous porous medium by the flow around a single cylinder surrounded by an infinite homogeneous porous medium. To solve the single cylinder problem, they used the superposition of the Stokes equations and Darcy s Law. To expand this method so that the permeability could be estimated for a random orientation of fibres in three dimensions they allowed the permeability to have different values in the directions parallel and perpendicular to the flow, and then averaged over all directions. The paper by Spielman and Goren provides a rigorous mathematical derivation in which they found the flow field and drag on an infinitely long cylinder. The equivalent permeability has been determined elsewhere [37] and will be presented here; 1 4(1 ' ( ) = k ( d 2) & ( d 2) # K1$! % k " & ( d 2) # K 0 $! % k " (2.28) where the permeability (k) features in implicit form and K 0 and K 1 are modified Bessel functions of the second kind. The fibres are assumed to be perfectly circular in crosssection, and unlike previous equations no information is given for the use of the equation when this is not the case. The authors state the above correlation to be valid for a void ratio greater than Jackson and James [36] also presented a method to calculate permeability for three dimensional arrays of fibres. The argument that they considered is that the permeability of a random medium such as steel wool is equivalent to the permeability of a cubical lattice formed of the same material. The flow resistance of the lattice was estimated by adding the 32

43 resistance of rods across the flow to rods aligned with the flow. The individual resistances were calculated firstly from Happel s [34] method and then updated [37] with the method of Drummond and Tahir [23]. The equation of permeability is; 2 3( d 2) k = (! ln(1! " )! 0.931) (2.29) 20(1! " ) The above equation agrees quite well with the equation of Spielman and Goren. 2.6 Computational Methods With the rapid improvement in computing power in recent years there has been much work in modeling different aspects of the mechanics of granular media. The particular area relevant to this thesis is the modeling of flow through porous media. The Lattice Boltzmann Method (LBM) has been used widely for modeling flow through porous media and as such a brief introduction and some of its uses will be covered here. Lattice Boltzmann Method The LBM is a numerical scheme for simulating fluid flows and modeling physics in fluids. LBM originated from the Lattice Gas Automata (LGA) method in which space, time, and particle velocities are all discrete. The method produces correct Navier-Stokes hydrodynamic behaviour and has been used by many workers for varying purposes. Some of these purposes include simulating viscous flows [54], multiphase and multi-component fluids [70], and chemical-reactive flows [10] to name a few. When using the LBM the porous medium is represented as a collection of solid obstacles. At solid boundaries, a bounce-back condition is imposed on any incident fluid particle, reflecting it back into its incoming direction. The resolution of this method needs to be adequate to accurately represent the porous medium which places an upper limit on the size of the porous medium that can be studied. There are a number of boundary conditions that can be applied depending on the use of the LBM. 33

44 i) Periodic boundary conditions these are typically intended to isolate bulk phenomena from the boundaries of the real system. They allow departing populations on outward pointing links to re-enter the lattice via corresponding inward pointing links on the opposite boundary. ii) No-slip boundary conditions these are appropriate when there is no net fluid motion at the wall. The implementation simply consists of reversing any populations that sit on a boundary node iii) Pressure Conditions these can be used when it is desirable to impose a given velocity profile at the inlet. At the outlet either a given pressure or a no flux condition normal to the wall is imposed. One of the earliest uses of the LBM was used to verify Darcy s law for flow through three dimensional porous media [73]. The authors first studied the flow of a fluid in a square cross-section pipe, comparing the numerical flow with the theoretical one. In doing so, they found the resolution of the lattice sites to accurately simulate the flow. They then carried out simulations in a 32 3 cubic lattice through porous media of three different void ratios. The result that they found was that a linear relationship existed between the flow rate and the pressure gradient for all three porosities, hence confirming Darcy s law. The LBM method was later used to study the how the permeability of a porous medium varies with void ratio [7]. The authors studied what they called a penetrable-sphere model in which the spheres were allowed to overlap. They argued that while this model is clearly artificial, it captures much of the geometric complexity observed in natural porous media. The simulations in this study were carried out on a 64 3 cubic lattice, with the authors noting that any void region must be at least 4 lattice units to ensure that the continuum equations are modeled correctly. Simulations were performed for void ratios in the range 0.02 <! < 0.98 and calculations for the permeability were made. The authors then used the form of the Kozeny-Carman equation shown in Equation (2.18) with a Kozeny constant of 6. The prediction of the Kozeny-Carman equation shows an excellent agreement with the results from the LBM. 34

45 The work of Heijs and Lowe [35] also used the Kozeny constant to verify flow through a porous media using the LBM. The authors studied a random packing of spheres that were packed to a volume fraction of 0.60, and found a value for the Kozeny constant. They varied the size of the spheres (in lattice units) and found that the Kozeny constant varied also. Not surprisingly, as the resolution of the lattice increased they found that the value for the Kozeny constant got closer to 5.0, the generally accepted value for spheres [8]. These authors also applied the LBM to a clay soil, obtaining the structure of the clay soil by computed tomography. The pore space for the clay soil was considerably more complex than for the spheres, with the authors finding that there was only one continuous pore network connecting the bottom to the top. This lead to values for the Kozeny constant being in the range of 1.0, although the authors noted the there are errors associated with both the estimation of k and the representation of the pore structure. In their final discussion of the study, the authors state that the result for the permeability for the soil calculated at the highest resolution was consistent with an experimental value for this type of soil, however this is not quantified with any numbers. Boundary Element Method To a lesser extent the boundary element method (BEM) has also been used in the calculation of permeability. The main advantage of the BEM is that after transforming the governing equations into boundary integral equations using the fundamental solutions, only the boundaries need to be discretised to obtain solutions at the boundaries. The solution in the interior space can be retrieved in the post processing phase. The BEM has been used recently to investigate the permeability of fibres and how it varies with porosity [11]. To model the fibres, a geometric model was used in which fibres were placed and then perturbed according to some set conditions. The main parameters that determined the fibre geometry were the minimum inter fibre distance and the desired porosity. The authors used the BEM to calculate the permeability of the fibres and then used these results to calculate values of the Kozeny constant for the various porosities. A number of assemblies were modelled for each void ratio, and the authors noted a wide 35

46 scatter in the computed permeability. Following this observation, the authors argued that while permeability is a strong function of porosity, the use of porosity alone cannot determine the permeability of fibrous materials. The authors did however show a good correlation between the Kozeny constant and the minimum inter fibre distance within each assembly. The values for the Kozeny constant that they found were between 8 < B < 16 for the porosity range 0.45 <! < The authors also showed that increased non-uniformity in fibre packing leads to a decrease in permeability. 2.7 Conclusions It can be seen that there has been much information and research published in the area of fluid flow through beds of granular media. Much of the early work was focused on finding empirical equations for different flow conditions, while the trend in recent years has been more towards the use of computational techniques, which can allow data to be found that would not have been otherwise possible using physical experiments. There is a lot of literature based on the models that are of a geometric nature (Ergun and Kozeny-Carman), which are based on Darcy s law and the capillary flow assumption of Kozeny. Throughout this literature there has been considerable discussion on various parameters such as dimensionless coefficients, the ratio of the viscous term to the inertial term and the dependence of the pressure loss on the void ratio. There is also past literature that is concerned with predicting the permeability of a material using continuum models, where the assumption is made that the material is continuous at the scale of interest. For the case of fibres, these types of models are accurate in predicting the permeability, although the main disadvantage of these models is that the fibres need to be packed in regular ordered arrays. An area that has received significantly less attention is the use of pore size models in the prediction of pressure loss. It seems logical that the pore space should be used as a measure of the permeability, as the inclusion of parameters particle diameter, particle shape and void ratio in equations act only as a measure of the pore size. It has been established that there is 36

47 not a complete understanding of how the pore size affects permeability and this is an area that is to be investigated more thoroughly in this thesis. 37

48 Chapter 3 - PERMEABILITY, VOID RATIO AND VOID SIZE 3.1 Introduction The previous chapter has outlined various models that can be used in the prediction of the permeability of granular materials. While the accuracy of each model varies depending on the circumstances, they each share common practical issues regarding the definition of relevant parameters in real bulk solid materials. Each model is based upon material properties such as particle size, particle shape and void ratio, and often for real materials these properties are ill defined. This chapter illustrates these issues using a selection of real bulk solid materials. These issues are demonstrated using the two most common equations used for predicting permeability; Ergun, and Kozeny-Carman. These two models have been discussed in detail in the previous chapter. They are similar in that they both rely on the knowledge of both the particle diameter and the porosity of the material. These two properties are vital; the particle diameter is used to represent the average size of the voids while the porosity represents the amount of voids within the material. There is a class of materials where this characterisation is successful, for example near-spherical non-porous particles that are uniform in size. However, for a large number of real bulk materials these properties are not as easily defined. Shape factors and adjustments can be made, however this is often a subjective process for materials that are far from traditional. In this chapter, specific examples have been chosen to illustrate the motivation behind this thesis. These materials have been chosen explicitly to demonstrate the issues involved in estimating permeability in irregular materials. We will examine two classes of materials; (near) spherical materials with discrete particle size distributions (lead shot) and cellulose based fibrous materials (cotton wool, sugar cane mulch). 38

49 The near spherical materials demonstrate how easy it is to become confused by the notion of particle size, while the fibrous materials demonstrate a myriad of issues ranging from the difficulty in measuring and defining the void ratio, to the actual meaning of particle size. 3.2 The Meaning of Particle Diameter As discussed in the previous chapter, the two most common models used for estimating the permeability of real materials are Ergun s equation, and the Kozeny-Carman equation. These equations are stated as follows; dp dx 2 2 µ f V (1 #! ) " f V (1 #! ) = (3.01) d! d! 2 3 d! k = (3.02) 180! ( 1" ) 2 In these equations, the particle diameter refers to a representative size of an individual (or average ) particle. If a material consists of near-spherical particles, the particle diameter is simply the average width of the average particle. While this may be a precise definition, the reality is that the particle size in these equations is nothing more than a scale to characterise the dimensions of the pore space within the material. From this characterisation, the models have been calibrated (in the case of the Ergun equation) to suit a large class of materials. A difficulty arises whenever we attempt to deal with a material that cannot be characterised by these simple terms. As discussed earlier, we can attempt to apply shape factors when the particles are not round, although even this carries the assumption that the shape factor captures the equivalent dimensions of the void space within the material. In other words, if a material consists of particles with an average diameter of d, and a shape factor ψ, then we effectively assume that the assembly has a void space of equivalent dimensions to an assembly of spherical particles of diameter ψd. For approximately spherical particles this may or may not be true, however there are many situations where this concept is far from reality and there is no conceivable way a shape factor can capture 39

50 the true nature of an assembly. We will illustrate this issue through the use of a number of case studies Discrete Particle Size Distributions The simplest situation where the notion of particle size can become problematic is the case of discrete particle size distributions. Consider a binary mixture of two sizes of lead shot with diameters of 2.1 mm and 4.3 mm, shown in Figure 3-1a and Figure 3-1b. It is clear that the Ergun and Kozeny-Carman equation should have no problems in dealing with either of these materials individually, as the particles are uniform and near spherical. Figure 3-1: (a) Lead shot 2.1 mm (b) Lead shot 4.3 mm Defining a mass ratio to describe the composition; mass of l arg e particles MR = (3.03) total mass of particles The composition of the binary mixture can be varied from 100% small particles to 100% large particles. Doing so results in the following permeability results (see section 4.4.2). 40

51 Permeability (m 2 /kpas) Mass Ratio Figure 3-2: Permeability of binary mixtures The difficulty of these types of typical binary mixtures, with reference to the Ergun and Kozeny-Carman equation, is that to predict the permeability a particle diameter is needed. From an earlier discussion, the role of the particle diameter in the equations is to specify the dimensions of the void space. Since this example involves near spherical particles it should still be possible to describe the void space by the diameter term, but there is no clear means of determining what this term should be. A shape factor would be of no use in this situation as the particles are near spherical, which leaves no avenue for determining a particle diameter for use in the permeability equations. This simple case highlights an obvious deficiency in the current methods of permeability prediction Fibrous Particles For the case of fibrous particles the actual meaning of the particle diameter takes on a new role. These materials are generally classified by the diameter of the smallest dimension, which is often inadequate for determining the dimensions of the void space. For example, cotton wool is one material that illustrates this particular difficulty. The fibre diameter of common domestic cotton wool is relatively uniform, and one method of finding the diameter is to use scanning electron microscopy; 41

52 Figure 3-3: Scanning electron microscopy images of cotton fibres The two images above show that the diameter of the cotton fibres is relatively uniform and consistent between fibres. Averaging the diameter over 10 samples yielded a diameter of 17.2 ± 1.50 microns. The void space of granular media with the above particle diameter would be of an entirely different structure compared to fibrous materials with this diameter. The use of the fibre diameter in the permeability predictions would result in values for permeability that differ greatly from experimental values for the permeability of cotton wool. Typically a shape factor is fitted empirically in these circumstances; Permeability (m 2 /kpas) 0.02 Data Ergun Kozeny-Carman Void Ratio Figure 3-4: Permeability predictions for cotton wool (ψ=0.68) 42

53 A shape factor of 0.68 results in the Ergun and Kozeny-Carman equations accurately predicting the permeability for the cotton fibres over a range of void ratios. However, this shape factor has been fitted empirically and so most likely holds no physical meaning. Presenting the definition for the shape factor again; surface area of sphere with same volume as particle! = (3.04) surface area of particle If it is assumed that the cross-section of the fibres is circular (a close approximation) and since the diameter of the fibres is known, it is possible to calculate an average length of the fibres. The aspect ratio that corresponds to a shape factor of 0.68 is 11, so according to the shape factor the average length of the fibres is about 0.2 mm. This indeed confirms that the empirically fitted shape factor holds no physical meaning as it is quite clear that the average length of cotton fibres in domestic cotton wool is much longer than 0.2 mm. This inadequacy of the shape factor exacerbates the absence of a suitable method of finding a particle diameter that accurately describes the void space within a material. To further illustrate this point, another more complex material where the particle diameter is not immediately obvious is sugar cane mulch, commonly referred to as Bagasse. This material consists of fibres that vary in both length and width (see Figure 3-5). 43

54 Figure 3-5: Photo showing the bagasse fibres The image clearly shows that there is not a distinct particle (fibre) diameter that can be selected, and any type of image analysis would only result in a range of diameters without a method for selecting one. One possible method to deal with such non-uniform materials is by finding the hydrodynamic diameter of the material by classifying it according to the terminal velocity. The advantage of this method is that the hydrodynamic diameter is calculated regardless of the complexity and distribution of shape. The method relies on experimentally determining the distribution of terminal velocities, and relating this to effective diameter through the following equations [31] (note the absence of fibre length); Where; * f ' $ V ( = ( ) % + " t + +, gµ f. s!. f () & d* d* #! 1 (3.05) 1 3 & g( f (( s ' ( f )# d * = d hed $ 2! (3.06) % µ " The resulting hydrodynamic diameter that is found from these equations is the hydrodynamically equivalent (fictitious) diameter of a sphere. Figure 3-6 shows a typical 44

55 apparatus that is used to determine the distribution of terminal velocities. The apparatus consists of a series of chambers through which air is drawn using a vacuum system. The lower most chamber forms a venturi with the highest air velocity within the apparatus. Following this venturi is a long (4 diameters) section of constant diameter with the lowest air velocity. The final chamber converges to a smaller diameter which leads to a vacuum system. The method involves starting the vacuum system at a prescribed air flow rate (found from the removable Pitot tube) which draws a sample of the material into the test chamber through the venturi. The lower two chambers aim to separate the material into individual fibres and classify the fibres in terms of terminal velocity. The high air velocities in the lower section of the apparatus serve two purposes; the separation of material into individual fibres, and the retention of fibres that are not drawn through the system. The constant diameter section forms a barrier to fibres that have a higher terminal velocity than the fixed air velocity due to the vacuum system. Fibres with lower terminal velocities rise up through the chambers and exit the apparatus. The sample is repeatedly drawn into the constant velocity section and the smaller fibres (lower terminal velocity) are removed. After a period of time steady state is achieved whereby all of the fibres that have a lower terminal velocity than the air velocity in the constant velocity chamber have been drawn out of the system and the vacuum system is switched off. The remaining material is weighed to determine the percentage of the mass lost to the vacuum system. By conducting this process over a range of air flow rates a distribution of terminal velocities (and hence hydrodynamic diameters) can be found. 45

56 Figure 3-6: The experimental apparatus used to determine the hydrodynamic particle size distribution for sugar cane mulch fibres 46

57 A number of tests over a range of air velocities reveal terminal velocities of 0.9 m/s (5% mass lost) and 4.7 m/s (95% mass lost) for the Bagasse. For each terminal velocity the equations 2.02 and 2.03 can be solved for a corresponding hydrodynamic diameter. The results of this can be seen in Figure 3-7. Mass Lost vs Diameter 100% 80% Mass Lost 60% 40% 20% 0% Diameter (mm) Figure 3-7: Cumulative fibre hydrodynamic diameter distribution by mass for sugar cane mulch Figure 3-7 shows the bagasse fibres have a range of hydrodynamic diameters ranging from 0.3 mm up to 1.8 mm. This range of diameters is due to two effects; actual variation in particle size, and dramatic variation in cross-sectional shape. The average fibre diameter from these results on the basis of 50% by mass is approximately 0.6 mm. This diameter can be used in combination with a shape factor to fit the Ergun and Kozeny-Carman equation to experimental results over a range of void ratios. 47

58 Permeabiliity (m 2 /kpas) Data Ergun Kozeny-Carman Void Ratio Figure 3-8: Permeability predictions for bagasse (ψ=0.45) Figure 3-8 shows that with the use of a shape factor the two equations can predict the permeability of the Bagasse over a wide range of void ratios. This shape factor that has been found empirically to achieve these results is It is impossible to even attempt to substantiate this shape factor in practical terms, as the material does not lend itself to the direct calculation of a shape factor. This further illustrates the previous point that the shape factor is an entity that exists only in an empirical sense. Further to this point, without a definitive method for choosing a particle diameter, shape factor or not, the reliance on the particle diameter term to describe the void space is unsatisfactory. The two examples of cotton wool and Bagasse illustrate the importance of the particle diameter for any theoretical equations to be applied to fibrous materials. The downfall in the equations is the lack of an explicit parameter representing the average void space; rather this is implied through the use of the particle diameter. As the particle diameter for fibrous particles is subjective and application dependent then correctly choosing a particle diameter that implicitly defines the average void space is impossible as there is no sound method available. Without this correct particle diameter the equations can only be used empirically. 48

59 3.3 A Brief Word on Particle Density For the simplest case of non-porous materials the particle density is straightforward to determine, with methods such as Archimedes principle or a density tube commonly used. However, porous fibres must be treated with some caution in the determination of particle density as the fluid flowing during a permeability test often does not pass through the centre of the fibre. This means that the volume that exists within the centre of the fibre must be included in the volume of the fibre. An example of this difficulty can be illustrated by the cotton wool and Bagasse discussed previously. Cellulose is the constituent material of both cotton wool and Bagasse. Performing a typical test to determine the particle density of these materials will result in finding the density of cellulose. One possible method for estimating the particle density of such materials is to obtain an image of the cross-section of a typical fibre. Figure 3-9: Cross-sectional images of (a) Cotton fibres; and (b) Bagasse The images show that the Bagasse fibres are a lot more porous than the cotton wool, which just have an area in the centre of the fibre that is porous. The Bagasse however, has a complex internal structure. On one level, large capillaries are observed that travel the length of the fibre. On a smaller scale a matrix type construction is apparent with thin walls layered throughout the fibre. By estimating the ratio of porous area to solid area it is possible to obtain an estimate for the particle density of these materials. 49

60 3.4 Conclusions This chapter has outlined the difficulties associated with theoretical equations that are based on material properties to predict permeability such as the Ergun equation and the Kozeny- Carman equation. These types of equations are based on the idea that the particle diameter represents the average void size while the porosity of the material represents the total amount of void space. While successful for materials with near spherical particles of uniform size these equations must be applied with some caution for materials that do not fit into this category. The most important deficiency in the current equations is the reliance on the particle diameter to accurately describe the average size of the voids. The simple case of a binary mixture of two near spherical particles sizes illustrates the case where an alternate method needs to be used to describe the size of the voids. However, there is no clear method that can be used for materials that fall into this class. Fibrous materials are a different class of materials that are also problematic. Two different materials were considered that illustrated two different methods of finding the fibre diameter; a simple imaging technique for fibres with a uniform diameter, and one utilising hydrodynamics for a material showing a large range in the diameter of the individual fibres. The application of the Ergun and Kozeny-Carman equations to these materials clearly illustrated that the shape factor is nothing more than empirical. Without a reliable particle diameter term ( d or! d ) the dependence of the shape factor to describe the average void space is unsatisfactory. It has been discussed [14] that the failure of the shape factor can be put down to the fact that it is assumed the entire surface area of the particles is wetted by the fluid as occurs with spheres that have only one point of contact. This could certainly be the case as with fibrous particles there could be entire walls of contact. As a final note to the chapter, the pitfalls of dealing with porous cellulose fibres are discussed. The determination of the particle density for these materials must be treated 50

61 cautiously as fluid flowing during a permeability test often does not pass through the centre of the fibre. A possible method for dealing with such situations is presented. It is clear that regardless of the method chosen to estimate permeability, there are crucial parameters that must be determined for any given material. They are; - void shape/size (defined by the particle size) - void ratio There is a whole class of materials, a selection of which have been discussed in this chapter, where the Ergun equation and Kozeny-Carman equation exhibit numerous problems in their application. The majority of these problems, if not all of them, lie with the fact that the particle diameter is the sole descriptor for the average void size. As the particle diameter is such a difficult, and in some cases subjective, parameter to define then the failure of these types of equations is always going to occur. A more suitable method would be not to imply the average void size, but to actually use it as a parameter in the equations. While measuring this type of parameter in the laboratory would present an arduous task it would be relatively simple to find through a numerical simulation of particles. If the average void size could be found for any given material then the problems outlined in this chapter would be overcome. 51

62 Chapter 4 - PERMEABILITY OF SPHERICAL PARTICLES 4.1 Introduction The previous chapter has discussed in detail how common theory uses the particle diameter to define the average size of individual voids and the problems associated with this method. In an effort to eliminate all of these difficulties associated with such factors as particle shape and orientation this chapter deals strictly with assemblies of spherical particles. A useful method for studying these systems is to use a numerical method to simulate assemblies of particles. As with any numerical simulation, it is essential to have a set of experimental data to validate the model. The case of spherical particles is an obvious starting choice for such a model for two reasons; 1) The simplicity in the application of permeability equations to assemblies of spherical particles and; 2) The fact that spheres are the simplest particle shape to model With this in mind, the underlying objective of this chapter is to generate a specific set of experimental results that can be used to validate computer simulations in further chapters. The experimental results throughout this chapter are used to gain a greater understanding into the deficiencies of commonly used theories in permeability prediction. The set of materials for which these theories are valid are those materials where the particle diameter and particle density can be found definitively. The materials selected for study in this chapter that meet these criteria are non-porous and near spherical, with the materials being lead shot and glass beads. The data presented in this chapter is for spherical particle mixtures that range from monosize to mixtures of varying distributions. The majority of the data presented in this chapter 52

63 has been found experimentally by the author throughout the course of this research, but to ensure a wide range of mixtures are covered a brief summary of results for other distributions is also presented. 4.2 Experimental Method The permeability of the materials throughout this chapter was measured using a test rig which made use of two pressure measurements from within the bed of material. The materials that were tested within this chapter were granular, which required the use of porous pads to prevent any particles from lodging in the holes of the permeable plates, see Figure 4-1. The pressure loss through these porous pads was not negligible which necessitated the need for the pressure loss to be found within the bed of material. The permeability test rig is compact, with a maximum bed height of approximately 300 mm, and accurate in the pressure loss measurement due to the use of two pressure measurements. The bed height between pressure measurements is fixed at 200 mm. A schematic of the test rig can be seen in Figure 4-2 and a photo showing the test rig can be seen in Figure 4-3. The air supply was provided by an Ingersoll Rand MM 37 SE rotary compressor (37kW, 5.9 m 3 /min) with a subsequent air dryer. It was controlled by a choked flow array that was controlled by digital means. The range of mass flow rates that were available using the choked flow array was kg/s in increments of approximately kg/s. The pressure measurements were made using two separate devices, a digital pressure indicator and a pressure transducer, and were calibrated as per their specifications. For each permeability test, the sample of material was weighed and the poured into the test chamber, ensuring that the height of the material was above the top pressure measurement port. The test chamber was then loosely vibrated to ensure random close packing was achieved. The bed height was then measured to calculate the void ratio. The top permeable plate was then lowered into position and secured so that there was no bed expansion. The 53

64 air flow was then increased in regular increments while taking the two pressure measurements at each air flow increment. Figure 4-1: Permeable plate with porous pad Figure 4-2: Experimental test rig using two pressure measurements 54

65 Figure 4-3: Experimental set up for permeability measurement utilising two pressure measurements 55

66 4.3 Materials A range of materials was used to obtain results for spherical particles. Each of the materials that were selected was chosen for their spherical shape and solid nature. The two different types of materials that were used were lead shot and glass beads. For each of these materials there was a range of sizes and distributions that were studied within the material type Lead Shot Lead shot was used as it was spherical and uniform. Five sizes were selected in the range of mm. The density of lead is known to be approximately kg/m 3, however, the lead shot purchased is alloyed with a small amount of Antimony during the manufacture which lowers the density. Each of the different sizes had slightly different densities due to this process. The properties are summarised in Table 4-1. Table 4-1: Lead shot properties Lead Shot Diameter (mm) Density (kg/m 3 ) A photo showing the smallest size of the lead shot, 2.1 mm, can be seen in Figure 4-4. The underlying grid is 2 mm in size, confirming the diameter of the particles. The sphericity of the lead shot is also evident, with a large majority of the particles appearing circular indicating that the lead shot is a good approximation of spherical particles. It should be noted that the other sizes of the lead shot are just as uniform in size and also show good sphericity, with the complete set of photos for all the sizes appearing in Appendix A. 56

67 Figure 4-4: Photo showing size and sphericity of 2.1 mm diameter lead shot The lead shot was used to study mono-size mixtures, binary mixtures and distributed mixtures. Mono-size All of the five sizes of the lead shot were used individually to create monosize mixtures. For the largest size particles the ratio of the test chamber to the particle diameter was 23 (>20), so the wall effect as discussed in Chapter 2 could be neglected. The wall effect for all smaller particle sizes could then also be neglected. Binary The smallest and largest sizes of the lead shot were used for the binary mixtures. A mass ratio (MR) was used to define the composition of the binary mixture which was given by; Mass of big particles MR = (4.01) Combined mass of particles The MR was varied from 0 to 1 in increments of The density of the binary mixture was estimated by interpolating between the upper and lower limit for the density according to the MR. 57

68 Distributed The distributed mixtures were composed by mixing certain amounts of each of the sizes of the lead shot. Due to the limited sizes of the lead shot each of the distributions were not continuous. Once each test was completed, the mixtures were then separated by the use of different size sieves. In total five mixtures were studied, the distributions of each can be seen in Figure Cumulative Distribution Lead 1 Lead 2 Lead 3 Lead 4 Lead Glass Beads Particle Diameter (mm) Figure 4-5: Distributions of the lead shot mixtures Spherical glass beads were also used to obtain experimental results. The glass beads were purchased with a mean particle diameter and standard deviation. Three sets of these glass beads were used with the properties summarised in Table 4-2. Table 4-2: Properties of glass bead mixtures Mean Particle Diameter (mm) Standard Deviation (mm) Glass Beads Set Glass Beads Set Glass Beads Set

69 As Table 4-2 shows, the distributions for the glass beads are quite narrow. The cumulative distributions are shown in Figure 4-6 which are found using the parameters in Table 4-2 with the knowledge that the glass beads are distributed normally. 1 Cumulative Distribution Glass 1 Glass 2 Glass Particle Diameter (mm) Figure 4-6: Distributions of glass bead mixtures A photo showing the range of sizes and sphericity of the glass beads can be seen in Figure 4-7. The particles are once again quite spherical and it is evident that there is a range of different sized spherical particles that make up the mixture. Photos of the other two sizes of glass beads also appear in Appendix A. Figure 4-7: Photo showing the glass beads of mean diameter 2.05 mm on a 2 mm grid 59

70 4.4 Results Permeability results were obtained for each of the lead shot and the glass bead mixtures. The results will be presented here in three categories; mono-size mixtures, binary mixtures and distributed mixtures Mono-size mixtures For each of the five sizes of the lead shot a permeability test was carried out in which the void size and pressure drop was measured, with the results in Table 4-3. Table 4-3: Results for mono-size mixtures of lead shot Diameter (mm) Packing Efficiency (%) Permeability (m 2 /kpa-s) It is a well known fact that the packing efficiency of mono-size spherical particles will pack to approximately 64% when placed randomly into a container [28]. This is known as dense random packing for spherical particles. The data in Table 4-3 confirms this for the monosize mixtures of lead shot, further confirmation of the sphericity of the lead shot. Contrary to most particle assemblies, the test chamber did not need to be shaken to achieve dense random packing due to the heavy weight of the lead shot. As expected, the permeability results show that an increase in the particle diameter results in an increase in permeability. This is because while the void ratio remains constant the average size of the voids within the mixture increase Binary mixtures During each of the permeability tests for the binary mixtures of lead shot the void ratio was measured for each mass ratio, which can be seen in Figure

71 Void Ratio Mass Ratio Figure 4-8: Void ratio versus mass ratio for binary mixtures The void ratios for the two pure mixtures, at 0 and 1 mass ratio, were taken from the monosize results in section 4.4.1, and are packed to dense random packing. However, for all of the binary mixtures the particles are packed better than dense random packing. As Figure 4-8 shows, the void ratio starts to decrease as increasing amounts of larger particles are added. By introducing larger particles, effectively a group of smaller particles are replaced by one big particle, and although this makes the pores directly surrounding this one larger particle slightly bigger, there will be less of them as we have one large solid mass occupying a space where there were previously a number of voids. As more large particles are introduced, the smaller particles are able to reside in the interstitial void spaces of the larger particles. At the point where the void ratio starts to increase again the volume of the interstitial pore space of the large particles is greater than the volume of the smaller particles needed to fill them and hence the void ratio starts to increase. The maximum packing efficiency of the binary mixture occurs in the range of 65-75% mass ratio of large particles, with a more exact number not being available due to the increments that were tested. This is in good agreement with the literature [82, 3], which states that the maximum packing efficiency occurs at a value of 73% large particles by volume. In this 61

72 case, the volume ratio and mass ratio are almost identical due to the very small change in densities between the sizes of the lead shot. Permeability (m 2 /kpas) Mass Ratio Figure 4-9: Permeability results for each binary mixture Figure 4-9 shows the measured permeability at each mass ratio. As predicted, the permeability started at some relatively low value when the sample was 100% small particles, and increased to a relatively higher value when the sample was 100% large particles. However, it was surprising to find that there was a slight decrease in permeability when a certain amount of the larger particles were introduced into the mixture. The reason for this can be explained by considering pore size and frequency. As the larger particles are added to the mixture, the average pore size increases minimally but since the larger particles occupy a larger volume then the pore frequency decreases. While the packing efficiency increases to some maximum then decreases back to dense random packing as more of the large particles are added to the composition, the permeability increases slowly until this maximum value of packing efficiency and then increases rapidly. This can be explained by the average void size, which obviously increases at a similar rate to the permeability as the mass ratio increases Distributed Mixtures The results for the eight distributed mixtures that were tested can be seen in Table

73 Table 4-4: Void ratio and permeability results for the distributed mixtures Void Ratio Permeability (m2/kpa-s) Glass Set Glass Set Glass Set Lead Mix Lead Mix Lead Mix Lead Mix Lead Mix The void ratios for all of the mixtures were less than 0.36, indicating that the particles were packed better than dense random packing. This is obviously a result of the distributed nature of the mixtures as the smaller particles can fit in the voids of the larger particles. For all of the mixtures in Table 4-4 the void ratios are only slightly lower than dense random packing for mono-size particles, and this is a result of the narrow size distributions. It is also interesting to note that while the void ratio between all of the mixtures does not change much, the permeability varies by a factor of approximately 3. This is once again a further indication that the average size of the voids is the determining factor in the permeability rather than the void ratio. 4.5 Other results The study of how the packing efficiency varies according to the distribution of the spherical particles has received experimental attention in the past as it does not lend itself to theoretical analysis. The paper by Yerazunis et al [82] provides a summary of work done in the area of binary mixtures. This paper noted that experimental results indicated that the packing fraction of binary mixtures of spheres is greater than that of either of the single particles and exhibits a maximum value, which is a function of the diameter ratio, in the range of 0.73 volume fraction large spheres. More recent work is in good agreement with this value achieving a value of [3]. 63

74 Experimental data in the area of continuous distributions was also sought as the results for the glass beads discussed in section do not exhibit packing efficiencies substantially greater than dense random packing due to the narrow distribution. Many granular materials have a range of particle sizes that can be described by a normal distribution. Figure 4-10 shows the results for tests conducted Sohn and Moreland [71] Sohn and Moreland d=500 um Packing Efficiency Geometric Standard Deviation Figure 4-10: Packing efficiency results for sand mixtures of varying distributions The data shown in Figure 4-10 is for sand particles having a mean sphericity of 0.86 throughout the mixtures. For the above set of results the experimental procedure included tapping the test chamber until no further reduction in volume was possible. These results for the normally distributed mixtures provide a further basis of comparison for the simulation method presented in the following chapter. 4.6 Conclusions The behaviour of spherical particles is predictable in terms of both packing efficiency and permeability, making assemblies of spherical particles ideal to study due mostly to the lack of orientation effects owing to the particle sphericity. The packing efficiency of the monosize mixtures confirmed the well known 64% for dense random packing while the distributed mixtures also behaved in a predictable manner. 64

75 Results for the permeability of mono-size assemblies of varying particle diameters, binary mixtures of varying compositions, distributed mixtures that are both discrete and continuous, and also packing efficiency results for normally distributed mixtures were generated. This data provides a range of conditions under which spherical particles have been packed. In the following chapter numerical simulation methods are introduced that reproduce physical assemblies of particles and allow more information to be found than experimental packings allow. Spherical particles were selected as a starting point for the numerical simulation model as they are geometrically simple to model and behave in a predictable manner experimentally. 65

76 Chapter 5 - SIMULATION OF SPHERICAL PARTICLES 5.1 Introduction Dense random packing of particles is an area that has received a significant amount of attention from a number of fields. The case of spherical particles is arguably the simplest packing to analyse. For the case of dense regular packings the packing efficiency can be calculated analytically. It is quite simple to calculate a value of 74% for face and body centred cubic structures and a value of 68% for the hexagonal close packed structure. However, the determination of the packing efficiency of dense random packing for spheres is not susceptible to mathematical formulation. A number of workers have conducted experiments to find the value of the packing efficiency for dense random packing. Scott [68] conducted experiments using 1/8 inch steel balls in a variety of differently shaped containers and measured the packing efficiency by weighing the container with the particles in air and then filled with water. Upper and lower limits were established for the packing efficiency of spherical particles being and respectively. Similar results were obtained by Finney [28] of ± and Scott and Kilgour [69] of ± using a computer analysis of the packing. From these results it has since been established that dense random packing is known to be approximately 64%. Conducting these types of physical studies is obviously very laborious and time consuming and so the trend towards computer simulations is a natural progression. This chapter is concerned with the simulation of spherical particles that aims to reproduce these physical packings in order to extract data on the structure of such packings. There has been extensive work on the simulation of particles from a number of workers, with each method used for different purposes. This chapter starts with a brief overview of the methods that have been used in the past and their relative advantages and disadvantages, with a specific focus on models using the optimisation of an objective function. The particular method that this research utilises is based on the optimisation of a specific 66

77 objective function, with the local optimisation of this objective function leading to dense random packing for mono-size particles. The goal in using simulation methods to reproduce physical packings in this work is to gain a greater understanding of the structure of the materials; specifically the connection between void space, particle diameter and permeability. Traditionally the particle diameter has been used to describe the average void space within the material, although this method fails for materials that are not spherical or near spherical. A more logical choice would be for the particle diameter to be replaced by a direct measure of the average void space. Successfully establishing such a parameter for spherical particles is essential in laying a platform for other materials such as fibrous particles. 5.2 Existing Simulation Methods The packing of particles has been a focus of study for many years. It has much relevance to academia and industry and there are a large amount of different methods that have been used over the years to simulate particle packing. Models based solely on geometric and/or statistical conditions were one of the earliest solutions to the particle packing problems. These methods often made sweeping assumptions such as each sphere touches its neighbours [22], or commonly dealt with only steric constraint without considering external forces such as friction and contact forces [60, 65]. These models are limited to simple shapes such as spheres. The Discrete Element Method (DEM) is an area that has received significant attention from many researchers recently. In this method each particle is described by a set of governing equations that account for all body and contact forces. A number of particles are introduced into a region and their governing equations are simultaneously solved numerically until all motion stops. The set of final coordinates for the particles represent a randomly packed assembly. This approach has been successfully used to model the packing of various distributions of hard spheres [see [49] for a detailed description of DEM]. DEM methods 67

78 have now advanced to the point where complex particles can be modelled from an assortment of idealised shapes such as sphero-cylinders [45], agglomerated spheres [21], ellipsoids [77], convex polygons [43, 53] and super-quadrics [13, 18, 80, 38]. The only major drawback of the method is the huge computational resources that are required to solve the governing equations. Another group of models are those concerned with optimisation. These models commonly tend to optimise, or minimise, some form of objective function. A common use is the Monte Carlo simulation whereby spheres position themselves under the influence of gravitational force by minimizing gravitational energy [75, 76]. In these types of models the particle being introduced continues to move (roll) over the other particles until it is stable. Minimisation of an energy potential can also be used in models where particles are packed by some other means than gravity. The optimization of a Lennard-Jones potential is one such example, which can be used to simulate hard spheres, although many applications of this method have been concerned with finding a global minimum [6, 78]. A global minimum in the packing of mono-size spheres would result in the ordered structure of face centred cubic. The method of the optimisation of an objective function will be used in this thesis. This chapter will cover the implementation of two such simulation methods that use two different objective functions; 1. Lennard-Jones energy potential 2. Gravitational potential that considers the Hertz contact strain by any two particles in contact There are a number of methods available to optimise any given function, and so before discussing the above two methods in anymore detail a brief review of the main optimisation methods is included. 68

79 Simulated Annealing This method exploits an analogy between the way in which a metal cools and freezes into a minimum energy structure and the search for a minimum in a general system. This algorithm is particularly useful in trying to find a global minimum, as it not only accepts changes that decrease the objective function but also changes that increase it. These changes that increase the objective function are chosen according to a certain probability function. Genetic Algorithms These algorithms are optimisation techniques that have been derived from the principles of evolutionary theory. The process starts from a population of randomly generated individuals with each individual having a known 'fitness' that is evaluated according to some objective function. For each new generation multiple individuals are selected from the current population, based on their fitness, and modified to form a new population. The modification step involves recombining existing individuals and also random mutations. This population is evolved through successive generations until a stopping criterion is satisfied. These algorithms are particularly useful in multi-objective optimisation problems. This technique has been applied to determine the lowest energy configurations for Lennard- Jones clusters of particles [20]. Geometry Methods These are known as a direct search method in which function values are used to create and maintain a geometric figure that represents the information known about the objective function at any given iteration. The most well known and used method in this category is the Nelder-Mead simplex method [57]. A set of n + 1 mutually equidistant points in n- dimensional space is known as a regular simplex. In two dimensions the simplex is an equilateral triangle and in three dimensions it is a regular tetrahedron. The best vertex corresponds to the lowest function value. There are then five possible operations that can be 69

80 performed on the simplex; reflection, expansion, outside contraction, inside contraction and shrinking [81]. In the method presented by Nelder and Mead the simplexes can become non-regular which results in a robust and powerful direct search method. Derivative Based Methods These methods are traditionally used when the objective function is smooth and the derivative is easy to calculate. In choosing a search direction, derivative based methods use a first order and/or second order partial derivative of the function. Two of the most popular derivative based methods are the method of steepest descent and Newton's method. In derivative based methods the search direction is found from the derivatives of the objective function. The rate of convergence of the steepest descent method is linear whereas Newton's method is much faster as it is quadratic. A method that is widely used when only the gradient of the function is known is the Quasi-Newton method. In this method the Hessian (matrix of second order partial derivatives) is approximated by a symmetric positive definite matrix which is updated from iteration to iteration. This Quasi-Newton method is much quicker than the method of steepest descent and as a result is commonly used as it does not require the second order partial derivatives to be calculated analytically. 5.3 Lennard-Jones Simulation Model Lennard-Jones Energy Potential Neutral atoms and molecules are subject to two distinct forces in the range of short and large distances. In the long range an attractive force exists while in the short range a repulsive force dominates. The Lennard-Jones potential [46] is a simple mathematical model that can represent this behaviour. For two particles whose centres are located a distance r apart, the potential takes the form; ALJ BLJ U =! (5.01) m n r r 70

81 Energy Potential Radial Separation Figure 5-1: Lennard-Jones energy potential for two spheres having diameter 0.25 units In Equation (5.01), the indices m and n can be chosen to approximate spheres, and A LJ and B LJ are chosen so that the minimum of the energy potential corresponds to two mono-size particles being exactly 1 diameter apart, as shown in Figure 5-1. The coefficients A LJ and B LJ are found by differentiating (5.01) and substituting du dr = 0, r = diameter, and for simplicity B = 1. Doing so results in the following value for A; LJ A LJ n m m! n = d (5.02) The equation for the Lennard-Jones energy potential that ensures the minimum is when two particles are one diameter apart is now of the form; U = n m d r m! n m 1! n r (5.03) The form of the Lennard-Jones equation energy potential in Equation (5.03) can be applied to model particles as the optimisation of this potential will result in the spherical particles 71

82 being closely packed. The values of m and n are chosen to approximate hard spheres which is detailed in the discussion below Energy Potential Larger m Radial Separation Smaller n Figure 5-2: Influence of variation of m and n in the Lennard-Jones energy potential Figure 5-2 shows the Lennard-Jones energy potential for two particles having a diameter of 1 plotted for three separate sets of values for m and n. The energy potential when the radial separation is less than a particle diameter (1) is the repulsive force, so to simulate hard spheres this should be as steep as possible. By increasing the value of m then the repulsive force increases. The energy potential when the radial separation is greater than a particle diameter represents the attractive force. This part of the energy potential should be as shallow and continuous as possible to encourage particles to attract and so the gradient is not as steep as the energy potential nears the minimum. This is done by decreasing the value of n. In summary, for m and n to be chosen to approximate hard spheres m should be chosen as large as possible and n should be chosen as small as possible. This will ensure a rapid increase in the energy potential for particles that are in contact and a smooth gradient in the energy potential for particles being attracted to one another. 72

83 5.3.2 Lennard Jones Simulation Method The simulation method that was used to implement the Lennard-Jones optimisation algorithm was sequential addition. In this method, the first particle was placed arbitrarily in space and its location was fixed. Particles were then introduced one at a time in space and were optimised according to the objective function. The energy of the j th particle at any time was calculated according to the function; j! " 1 ( m" n % & n d 1 U = " # ij m n (5.04) i= 1 ' m rij rij $ Once a particle had been optimised successfully its location was also fixed. This meant that for an assembly of n particles the optimisation process was carried out (n-1) times. For each optimisation process the only variables that could change were the coordinates of the current particle being optimised. The optimisation algorithm used to minimise this potential was a derivative based Quasi- Newton method. The algorithm was implemented in Fortran language, with the optimisation routine being drawn from the IMSL Fortran library. An outline of the algorithm is as follows; 73

84 Initialise parameters FOR i=1 to number of particles Assign random coordinates (x, y, z) to i th particle IF particle number = 1 CYCLE END IF IF i th particle overlaps with any existing particles CYCLE END IF Optimise random coordinates according to objective function using UMING Add coordinates of i th particle to final coordinate array END FOR Output results The advantage of using sequential addition was that it made the simulation much faster than using a global rearrangement process. This is because the objective function only considered the Lennard-Jones energy potential between the current particle and each of the existing particles. The energy potential between two particles that already existed was irrelevant as their locations were fixed. The IMSL subroutine that was used was called UMING, and required the function and its derivative. The derivative of the function shown in Equation (5.05) is; du dx = n! " 1 i= 1 du dr in in drin # dx (5.05) The parameters of interest that affected the simulation results were the particle diameter, and the values of m and n. If the attraction force was too strong then the particles will tend to overlap with each other and conversely if the repulsion force was too strong then the 74

85 particles will not pack tightly enough. Due to the nature of the objective function these values for m and n are also a function of the particle diameter being simulated. 5.4 Hertz-Gravity Simulation Model To simulate particles in a more realistic manner an alternate model was used in which particles were dropped inside some arbitrary container and optimised according to gravity. If a particle came into contact with another particle then the Hertz contact strain energy would be considered between the two particles Hertz-Gravity Energy Potential The energy potential in this simulation method is made up from two sources, the gravitational energy and the Hertz contact energy when two particles are in contact. As with the previous method, local optimisation of this potential should result in dense random packing of the particles. The particles are packed into a hypothetical container using the force of gravity as one would imagine doing an experiment. For an assembly of n-1 particles, the n th particle has a radius of r n, and coordinates; & xn 1 # x $! n = $ xn2! (5.06) $ % x! n3 " This particle is then subject to an energy potential that is composed of two components; U = U G + U H (5.07) Where, U =, and, U G is a gravitational component of the form G mgx n 2 U is a Hertzian contact strain energy of the form U = H x x ) given by; H H ( i n 75

86 H ( x, x i j $! 4 E! 15 1& -! ) = #! 0!! " 2, * d xi & x & * j ) ' ' ( 2.5, * + d 4 ) ' ( 0.5 for for x x i i & x & x j j < r % r i i + r + r j j (5.08) In the above equation, E is Young s modulus, υ is Poisson s ratio, d is the diameter of the spheres and x i! x j is the distance between the centres of the spheres. H U represents the strain energy stored in the deformations at the points of contact between particles. The Hertz contact strain between particles i and j is given by H x i, x ) and is calculated from ( j the work done in overcoming the Hertz contact force. Obviously, H x i, x ) is only nonzero when particles i and j are in contact. ( j To prevent the particles from travelling outside of the theoretical container a constraint needed to be placed on the particles. For this study a cylindrical region with a base at x n2 = 0, and diameter D >> d was chosen to contain the centre of the particles. This region provides the following two constraints; x > 0 n2 (5.09) 2 2 n1 x n 3 x + < (5.10) D 2 By repeatedly adding particles to the system in a sequential manner and optimising their coordinates an assembly of any number of particles can be built. The parameters of interest in this simulation algorithm are the stiffness of the particles, to determine contact energy, and also the density of the particles to determine gravitational energy. These two parameters needed to be selected to facilitate the optimisation while minimising particle overlap to achieve dense random packing. 76

87 5.4.2 Hertz-Gravity Simulation Method The simulation method that was used was once again sequential addition. The starting position for each particle was such that its position was above any existing particles. The energy potential of the particle was then calculated from a gravitational component, a Hertz contact strain component for any particles in contact and subjected to the location constraints if necessary. Once a particle had been optimised successfully its location was fixed, which meant the coordinates of the current particle being optimised were the only variables that could change. This simulation method is directly analogous to pouring particles into a container one at a time meaning that the simulated assemblies were built from the bottom up. The nature of the energy potential for this simulation method does not lend itself towards an analytical treatment of the gradient as there are two separate energy potentials that are evaluated in different circumstances. As a result of this an optimisation routine using an analytical gradient was not used, with a finite difference gradient being used instead. The method of optimisation was the same as in the previous Lennard-Jones model, using the Quasi-Newton method, with the only difference being that a finite difference gradient was used instead of an analytic gradient. The algorithm was implemented in Fortran language, with the optimisation routine (UMINF) being drawn from the IMSL Fortran library. An outline of the algorithm is as follows; Initialise parameters FOR i=1 to number of particles Assign random coordinates (x, y, z) to i th particle Ensure y coordinate is greater than any existing y coordinate Optimise random coordinates according to objective function using UMINF Add coordinates of i th particle to final coordinate array END FOR Output results 77

88 As each new particle was introduced it was necessary to calculate the distance between the new particle and all existing particles in the assembly. For a large assembly of particles, this checking between the new particle and all other particles was the bottleneck of the code. To negotiate this, a neighbourhood regime was introduced in which each particle was located in a certain neighbourhood, and only the particles in this neighbourhood or surrounding ones needed to be checked. This technique had the desired effect of speeding up the simulation dramatically. Figure 5-3 is used in the explanation of the neighbourhood regime. Figure 5-3: Neighbourhood regime illustration The theoretical cylindrical container was divided up into layers, as shown in Figure 5-3, with the thickness of each layer being equal to the particle diameter being simulated. When a particle was introduced, it was given a temporary neighbourhood according to the layer it was currently in, and only the particles in that layer or in the layers immediately above and below that layer were searched for particle contact. Once the optimisation for the current particle was completed, then the current neighbourhood was made permanent for that particle. The implementation of this neighbourhood regime required the use of three arrays; Neighbourhood (NP) The purpose of the array was to store the neighbourhood number of the i th particle. The bottom neighbourhood was 1, with each neighbourhood increasing sequentially as the height increased. The length of the array was the same as the number of particles. 78

89 Neighbourhood List (j, k) This array contained a list of the particle numbers in a particular neighbourhood. The first index of the array (j) referenced the neighbourhood number, and the second index (k) held the current number of particles in the j th neighbourhood. The length of the first index was equal to the number of neighbourhoods, while the length of the second index was equal to the maximum theoretical amount of particles that could fit in one neighbourhood. For example, during a simulation, there might be 5 particles in the 3 rd neighbourhood. This array will hold the particle numbers of each of these 5 particles. The particle number is essentially the order in which the particle has been introduced (particle number 1 was the first particle introduced, particle number 2 was the second particle introduced etc). Neighbourhood List (3,1) = 32 Neighbourhood List (3,2) = 36 Neighbourhood List (3,3) = 43 Neighbourhood List (3,4) = 48 Neighbourhood List (3,5) = 60 The advantage of using this method is that it made it very simple to search through all of the particles in any given neighbourhood. Neighbourhood Count (j) This array contained the number of particles in every neighbourhood. The length of the array was equal to the number of neighbourhoods. When a particle had been optimised, the neighbourhood arrays needed to be updated. The code that was used to do this is as follows; 79

90 Current particle = i FOR j=1 to maximum number of neighbourhoods IF y coordinate < j*diameter Neighbourhood (i)=j Neighbourhood Count (j)=neighbourhood Count (j)+1 Neighbourhood List (j, Neighbourhood Count (j))= i EXIT FOR END IF END FOR The advantage of this neighbourhood regime was that the relationship between computational time and the number of particles simulated remained approximately linear for any quantity of particles simulated. As Figure 5-4 illustrates the computational time without the neighbourhood regime increases by approximately a power of two. For the relatively small amount of 5000 particles the implementation of the neighbourhood regime results in the simulation being 10 times quicker, and when simulations run into the tens of thousands of particles this method provides a large time saving. 400 Computation Time (s) Non-Neighbourhood Regime Neighbourhood Regime Particles Simulated Figure 5-4: Computation times for neighbourhood and non-neighbourhood regimes 80

91 As a final note on the neighbourhood scheme, the method detailed here is different to the one commonly used by DEM algorithms. For the DEM simulations the neighbourhood scheme typically involves keeping a list of particles that are in the proximity of each particle. This method is used as the neighbourhood for each particle does not need to be updated every iteration as the simulation is modelling a real process, i.e. a particle cannot instantaneously move large distances. This is in contrast to the simulation approach used here as the optimisation routine is attempting to place a particle with no regard to the distance it moves, hence the need for the method discussed in this section. As discussed previously, the objective function was made up of the gravitational energy, the Hertz contact energy for any two particles in contact and also a penalty function to restrict the particles from travelling outside the theoretical container. Pseudo code showing the calculation of the objective function is shown over page. 81

92 i = Number of current particle being optimised U1 = 0; U2 = 0; U3 = 0 This part is the Hertz contact energy a = temporary neighbourhood of the current particle being optimised b = Neighbourhood Count (a) FOR j = 1 to b cpn = Neighbourhood List (a, j) r = distance between particle (cpn) and particle (i) IF r < radius (cpn) + radius (i) U1 = U1 + energy due to Hertz contact energy END IF END FOR The above FOR loop is repeated for the (a-1) and the (a+1) neighbourhoods This part is gravitational energy IF y coordinate > 0 U2 = energy due to gravitational energy ELSE U2 = much larger energy to prevent particles travelling below zero END IF This part is the penalty function IF particle is outside container U3 = energy due to penalty function END IF U = U1 + U2 + U3 82

93 83

94 5.5 Packing efficiency results Algorithm For any given assembly created by the simulation algorithms, the exact coordinates of every particle are known as well as the radius of each particle. To calculate the packing efficiency a probabilistic approach was used. This method involved choosing a random particle within the assembly, then choosing a random direction, and calculating the coordinates of the point with the given direction and length. The length of the new point was varied between zero and the container diameter in regular increments, with the length only increasing to the next increment once a certain amount of random points were chosen. For each sample point it was tested whether or not the coordinates of this new point lay inside a particle within the assembly. This process was carried out many times and the chances of finding a particle for any given length were averaged out over the number of times this process was done. This method was used as it gave the packing efficiency of the assembly as seen from any particle in the assembly. The probability fluctuated at small radial distances but as the radial distance increased the packing efficiency converged. For the random packings of particles the packing efficiency started to show some consistency at a radial distance greater than approximately 5 diameters. The code for this method can be seen on the following page. A much simpler method was also employed to find the packing efficiency. This method involved choosing a large number of points randomly inside the container, with the packing efficiency then being given by the ratio of points that lie inside a particle to the total number of points. The packing efficiency returned by this method was the packing efficiency that the previous method tended towards at the larger radial distances (see Figure 5-5). 84

95 Increment = 0.05*diameter steps = container diameter/increment FOR i = 1 to steps Length = i*increment FOR k = 1 to number of attempts Choose a random particle within assembly Choose a random (x, y, z) direction Calculate coordinates of new point given direction and length IF coordinates of new point lie outside container CYCLE END IF FOR j = 1 to number of particles r = distance between new point and j th particle IF r < radius (j) Hit (i)=hit (i) + 1 EXIT FOR END IF END FOR Total (i) = Total (i) + 1 END FOR END FOR FOR i = 1 to steps END FOR Volume Solid Fraction (i) = Hit (i) / Total (i) 85

96 To test this method of finding the void ratio, assemblies of body-centred and face-centred cubic structures were made using approximately particles for each. The locations of the coordinates for these structures are well known and so to generate the coordinates for these two structures is a simple task. The reason for studying these two structures is that the packing efficiency is well known as it can be found theoretically. The results from the algorithm for finding the void ratio described above compared with the theoretical values can be seen in Table 5-1. Table 5-1: Packing efficiency comparisons between model and theoretical results Face-centred cubic Body-centred cubic Theoretical packing efficiency! 2 6! 3 8 Theoretical packing efficiency (4 d.p) Algorithm packing efficiency (4 d.p) As Table 5-1 shows the void ratio algorithm provides an excellent agreement with the theoretical results. The above results were calculated with an increment of 5% of the particle diameter and was structured in such a way that attempts were made to find a particle for each step size increasing from the particle radius to 30 times the particle diameter. When the data from the algorithm is plotted against radial distance, it shows the chance of finding a particle at any given radial distance from another particle. The packing efficiency is the final value that the plot tends towards. The value shown in Table 5-1 is the average value in the region from 15 to 30 particle diameters shown in Figure

97 Packing Efficiency Face-centred Cubic Body-centred Cubic Radial Distance Figure 5-5: Packing efficiency of ordered structures Mono-size mixtures Using the algorithms discussed in sections 5.3 and 5.4 assemblies of particles were simulated. For each of the algorithms certain parameters needed to be set to produce results that are comparable to any results achieved in physical experiments. Since this chapter is concerned with finding information about average pore space and size only the final assemblies are of any interest. This meant that certain parameters in each of the simulation methods (for example, particle stiffness) could be calibrated indirectly through the packing efficiency, specifically dense random packing. Dense random packing stipulates that for any random packing of mono-size particles the packing efficiency should be approximately 64% [68]. Using this result the parameters in each of the simulation methods were calibrated to produce dense random packing. The results for the mono-size mixtures will be presented in two sections according to each simulation algorithm. 87

98 Lennard-Jones packing efficiencies The Lennard-Jones algorithm was implemented to produce assemblies of particles of varying number and particle diameter. The implementation of the code in Fortran included outputting the results to a graphics file using the Virtual Reality Modelling Language (VRML) 2.0 specification. This allowed an image to be viewed for each of the simulations. Due to the nature of the energy potential being optimised, the particles always formed a large sphere as all of the individual particles were attracted to the existing mass of particles (specified by the coordinates of the 1 st particle), as shown in the following figure. Figure 5-6: Lennard-Jones simulation of particles The parameters of interest in the Lennard-Jones simulations were the indices m and n in Equation (5.03). If these parameters were not chosen correctly then the local optimisation did not lead to dense random packing; two common errors were the particles not packing 88

99 tightly enough or the particles packing too tightly due to particle overlap. It was found that the bigger the particle the stronger the repulsion force had to be while the magnitude of the n attractive force decreased. While the attractive force (! 1 r ) relies solely on the value of n meaning a decrease in n is a direct decrease in attractive force with no influence from the value of m, the same is not true for the repulsion force as its dependence on m and n is more complicated. U repulsion m! n n d = (5.11) m m r As seen in Equation (5.11), the coefficient of the term ( n m ) will decrease as m increases, while the repulsive term ( m n d! r m ) will increase as m increases. This means that for a constant n there will be a range of m where the repulsive force decreases with m and another range where the repulsive force increases with m. With such a dependence on the parameters m and n the optimisation routine does not lend itself to calculations involving varied particle sizes. However, for assemblies of mono-size particles the parameters could be chosen to simulate dense random packing for varying size diameters. The following table shows the sizes simulated (in program units) and the resulting packing efficiencies. Table 5-2: Mono-size results for Lennard-Jones simulations Diameter Packing Efficiency Value of m Value of n As Table 5-2 shows, the correct selection of the parameters m and n results in the dense random packing of the particles over a range of particle sizes. The parameters were chosen using the method discussed in section to approximate hard spheres. The limit to the 89

100 variation that could be achieved between m and n was computing power, as too high a value for m or too low a value for n resulted in calculations involving extremely large numbers. As discussed previously in this chapter the simulations presented in this chapter are used for analysis of the final assembly only, which meant that as long as the mono-size spherical particles packed to dense random packing the method of selection of the parameters was not important. For this method, the parameters were chosen by selecting a value of n = 3 (to ensure an adequate attractive force) and increasing the repulsive force to eliminate particle overlap until dense random packing was achieved. As the table shows, the value of m had to increase with particle size, effectively meaning the particles became harder as they got bigger. With any simulation method it is inevitable that there will be some degree of particle overlap due to the simulation method and/or numerical accuracy. In the Lennard- Jones simulations it was found that the particle overlap was small enough to become negligible, as it was in the range of 0.01% volume overlap Hertz-Gravity packing efficiencies The Hertz-gravity algorithm was implemented in Fortran to also produce assemblies of particles of various sizes. The obvious difference in this method was that the assembly of particles was modelled to reflect an actual physical assembly. An example of an assembly can be seen in Figure 5-7, with the particles having been packed into a theoretical cylindrical container. The diameter of the cylindrical container shown in Figure 5-7 is 25 particle diameters wide to avoid any wall effects discussed in Chapter 2. 90

101 Figure 5-7: Particles packed into a theoretical cylindrical container The selection of parameters in the Hertz-Gravity simulation algorithm followed much the same path as in the Lennard-Jones method. The two parameters of interest in this method are Young's modulus and the gravitational constant (a single factor representing gravity, particle density and particle volume). If the purpose of the simulation was to study how certain particles behave under different circumstances then the exact material properties would be needed, however as stated previously only the final assembly is of any importance here. This leaves the luxury of being able to choose the parameters with the sole requirement of mono-size spherical particles producing dense random packing. The results of using this method to simulate assemblies of particles can be seen in Table

102 Table 5-3: Results from Hertz-Gravity simulation model Diameter Young s Modulus Gravitational constant Packing Efficiency (%) e e e e As can be seen in Table 5-3, the range of particle diameters that were simulated vary over a magnitude of It was found that the for each particle diameter the easiest way to simulate dense random packing was to vary the particle stiffness. While this is obviously not possible in the real world it should be remembered that the means to simulate dense random packing is not the point of interest in this study, only the final assemblies of particles. In contrast to the Lennard-Jones model, as the particles increased in size the required particle stiffness decreased. The reason for this in the Hertz-Gravity simulation model is because of the constant gravitational energy for each particle size. Theoretically the gravitational constant should increase with particle size and the particle stiffness should remain constant, but this produced irregular results. It is interesting to note that as the particle diameter increased by a factor of 10, the particle stiffness decreased by a factor of 10, though this has little physical meaning. The range of particle sizes was greater than what could be simulated using the Lennard-Jones model, which is most likely a result of the simpler nature of the objective function. Because of this wider range of particle sizes that can be simulated with this algorithm, the Hertz-Gravity model will be the used to simulate wide distributions discussed later in this chapter. Whereas the particle overlap using the Lennard-Jones model was less than 0.01% by volume, the particle overlap with the Hertz-Gravity was in the range of 0.30% by volume. This amount of overlap was conceded to attain dense random packing. The reasons for this amount of overlap are thought to be primarily as a result of the constraints of sequential addition, but also because of the smaller surface area of the assembly available to each new 92

103 particle being introduced compared with the Lennard-Jones model. With the Lennard-Jones model each new particle is introduced in space and can be anywhere within 360 degrees of the existing assembly, whereas in the Hertz-Gravity model each new particle has to be introduced within a cylinder above the existing particles. As a result the particles have a greater number of possible positions in the Lennard-Jones model to be placed to produce a lower energy according to the objective function. The last point to consider with the Hertz-Gravity model is the homogeneity in the vertical direction. It was found that if the gravity function was not adequate then the assembly would not be homogeneous in the vertical direction. The gravity function was chosen so that the 1 st order differential was continuous at all values of the function. This resulted in a homogeneous assembly as seen in Figure 5-8. Figure 5-8: Graph showing assembly homogeneity in the vertical direction It can be seen that the number of particles less than 1 diameter in the vertical direction is much greater than for other layers, and the reason for this is because these particles are 93

104 dropped onto an even surface. For all other layers the particles are dropped onto a corrugated surface, meaning that not as many particle centres can fit into each layer. Figure 5-8 shows that there is an approximately constant number of particles that lie in each layer of the container in the vertical direction, indicating homogeneity is achieved with the Hertz-Gravity simulation algorithm Binary mixtures Binary mixtures were simulated using both the Lennard-Jones and Hertz-Gravity simulation methods. The simulation results were compared with the experimental results presented in Chapter 4. To ensure a valid comparison, the same ratio between the two particle sizes was used as well as the same mass ratios for each mixture. A comparison between the two simulation models and the experimental results can be seen in Figure 5-10, with only the results shown for each mass ratio in increments of 10% to make the figure clearer. Figure 5-9: HG binary simulation (left, MR=0.47) and LJ binary simulation (MR=0.71) Figure 5-10 shows the simulation results and it can be seen that the simulation mass ratios have not reproduced the exact mass ratios as used in the laboratory experiments, and this is due to the limitations of the simulation models. Each mass ratio from the binary mixtures 94

105 was converted into a number ratio, and the particle that was introduced sequentially into the assembly was given a radius according to the probability that the number ratio implied. For a large enough assembly of spheres, this method obviously reproduces the correct mass ratio. However, the final mass ratio was calculated by including only those spheres that were in the volume of the assembly that was used to calculate the packing efficiency. 0.7 Packing Efficiency (%) Experimental Data Lennard-Jones Model Hertz-Gravity Model Mass Ratio Figure 5-10: Comparison of binary mixture packing efficiencies with simulations Regardless of these slight limitations Figure 5-10 shows an excellent comparison between the experimental results and both sets of simulation results. The correct trend is produced with both models, and the mass ratio corresponding to the maximum packing efficiency from both models is approximately 70% mass ratio of large particles to total particles. This result shows good agreement with [82] who found that the maximum value for packing efficiency is relatively independent of the diameter ratio and approximately 73% volume fraction large spheres. It should be noted that in this study the volume ratio and mass ratio vary by less than 2% for the experimental results due to the almost identical densities between the lead shot sizes, and in the simulation models the volume ratio is the same as the mass ratio as the algorithms assume a constant density between particle diameters. 95

106 5.5.4 Distributed mixtures Typically when distributions are given, whether they are normally distributed or log-normal distributions, they are distributions by volume. To be able to represent these distributions accurately in the simulation algorithm it is necessary to convert each volume distribution into a distribution by number. This is necessary as particles are introduced sequentially into the assembly and the radius needs to be chosen according to the percentage of each size particles in the distribution. To transform a volume distribution into a number distribution the following formula is used; 100V d X (5.12) 3 i i i = 3 Vi di! The simulation of distributed mixtures requires the optimisation algorithm to be more robust in that it must be able to account for a range of particle sizes. It was found that the Hertz-Gravity simulation method was much more adaptable to distributed mixtures than the Lennard-Jones model. From the discussion in section it is apparent that a linear relationship between particle stiffness and particle diameter is required; E i = 0.01 < di < 10 (5.13) d i The distributions simulated were those given in Chapter 4 and were compared to results obtained by the author (lead shot and glass beads) and also results from past literature. The packing efficiency results will be presented in two parts; firstly the results obtained by the author will be presented for which the permeability was also measured (given later in this chapter) and then the results of the past literature will be covered where only packing efficiencies were given. 96

107 The three glass bead mixtures and the five lead shot mixtures were successfully simulated to represent the physical mixtures encountered in the laboratory. Images of the simulations can be seen in Figure 5-11 and Figure Figure 5-11: Simulation of continuous distribution of glass beads (Glass mix 3) 97

108 Figure 5-12: Simulation of discrete lead shot mixture containing five sizes (Lead mix 2) The packing efficiency results for the glass beads and the lead shot are provided in Table 5-4. Table 5-4: Simulation results for the packing efficiency of distributed mixtures Simulated Packing Measured Packing Error (%) Mixture Efficiency Efficiency (Equation B-1) Glass set Glass set Glass set Lead Mix Lead Mix Lead Mix Lead Mix Lead Mix

109 As Table 5-4 shows the simulation results compare well with the actual experimental results. The errors given in the table show a good agreement with the experimental results. There is a relatively small range of packing efficiencies shown in Table 5-4 and this is a result of the narrow particle size distributions of each of the materials. It is a well known fact that the packing efficiency increases with increasing particle size range in the material, and so to compare the results of the simulation algorithm with experimental results for higher packing efficiencies some data from past literature has been used. The data from past literature has been restricted to normal distributions, as the simulation of log-normal distributions presents numerous problems. The main problems associated with log-normal distributions are the large range of particle sizes (with ratios sometimes as large as 1000:1) and the need for huge numbers of particles (in the millions) to accurately represent a given distribution. The authors Sohn and Moreland [71] studied the packing efficiencies of sand mixtures experimentally that were normally distributed with a given standard deviation. Their results along with the simulation results can be seen in Figure Volume Solid Fraction Sohn & Moreland This Work Standard Deviation Figure 5-13: Packing efficiency of normally distributed sand mixtures As the Figure shows, the simulation method once again correlates quite well with the experimental results. The sand particles used in the experimental study were not quite 99

110 spherical as they had a reported sphericity of 0.86, and this may be the cause for the larger discrepancies at the larger standard deviations. 5.6 Radial Distribution Function The radial distribution function (RDF) is another means of comparing simulated results with experimental results. The RDF is defined as the probability of finding one particle centre at a given distance r from the centre of any given particle. It is given by the formula; g( r) n( r) 4" r! r = 2 (5.14) Where n(r) is the number of sphere centres situated between r and r +! r from the centre of a given particle. The RDF was found by averaging a number of RDFs within a central region of each assembly to ensure that no edge effects were apparent. A value of 0.02 times the particle diameter was used for Δr in all of the calculations. A RDF from an assembly created using the Lennard-Jones algorithm can be seen in Figure 5-14, while a RDF from the Hertz-Gravity simulation method can be seen in Figure Average number of centres, NAV/4 r 2 r Radial distance r in diameters Figure 5-14: RDF for Lennard-Jones simulation of mono-size spheres 100

111 2 Average number of centres, NAV/4pr 2 Dr Radial distance r in diameters Figure 5-15: RDF for Hertz-Gravity simulation of mono-size spheres It can be seen that the peaks and troughs in the RDF for the Hertz-Gravity simulation method are not as clear as they are in the Lennard-Jones RDF. This is due to the assembly from the Hertz-Gravity method not being as uniform in the radial direction as in the Lennard-Jones simulations. This result is not unexpected as the particles are optimised in the radial direction for the LJ simulations whereas the particles are optimised in a stack in the HG simulations. Table 5-5 shows a comparison between the locations of the peaks for a RDF between the two simulation methods discussed in this thesis, the results of Finney [28] who calculated the RDF by measuring the exact coordinates of particles in a physical packing, and also results from another simulation method that used DEM [49]. The results presented in Table 5-5 for the LJ and HG simulation methods represent the RDFs for assemblies of particles of different diameters according to Table 5-2 and Table 5-3. Table 5-5: Comparison of peaks for RDFs (location of peaks given in diameters) Lennard-Jones Hertz-Gravity Finney [28] Liu et al [49] 2 nd peak 1.98 ± ± rd peak 2.68 ± ± th peak 3.57 ± ± th peak 4.37 ± ± Over the range of assemblies measured, these two quantities showed a standard deviation of zero 101

112 It is noticeable that the location of the first peak for the Hertz-Gravity method is closer to the particle centre than for the other results. This is a result of the small degree of overlap that is present in the simulated assemblies discussed in the earlier section Also noticeable is that the variation in the results for the HG method is greater than for the LJ method. As mentioned previously this is due to the optimisation method. However, the two methods show a good comparison between the locations of the peaks. The results from the two simulation methods also show good agreement with the experimental results of Finney [28], while the results of Liu et al [49] indicate that their simulated assembly is packed tighter than the other results. In the RDF presented by Finney and Liu et al the second peak is divided so that there is a split peak. The location of the first sub peak is at approximately 1.73 diameters and is of lower magnitude than the second sub peak which is given as the second peak in Table 5-5. The basis for this split peak is discussed in detail in [49] but essentially results from spheres arranged in pairs of equilateral triangles. Figure 5-14 shows a small peak at 1.73 diameters, while in Figure 5-15 it is clear that there is no peak in the location of 1.73 diameters. This absence of a clear split second peak is a result of using a sequential addition method and has been discussed in the past [2]. The results obtained from the two sequential addition methods used here certainly agree with this conclusion. 5.7 Mean free path length With a computer model of an assembly of particles, it allows the study of characteristics that would be very difficult, if not impossible, to measure experimentally. With this in mind, it seems logical to define the assembly by the size of the voids rather than the by size of the particles. The size of the voids can be represented by the mean free path length, X d, which is defined as the average distance from particle surface to particle surface throughout the assembly, which can be found numerically. By using mean free path length instead of particle diameter, it potentially allows any permeability equation to be applied to mixtures that do not have a uniform particle diameter. In terms of finding this variable, the algorithm consisted of selecting a random point that existed within the void space of the assembly, 102

113 and then choosing a random 3D direction. The exact distance to the closest particle in that direction was then calculated, as was the exact distance to the closest particle in the opposite direction. The addition of these two distances from the randomly chosen point gave a distance between two particle surfaces in the assembly. This process was repeated (tens of thousands of times) and it was found the distribution most closely reflected that of a gamma distribution; f ( x; µ 1 " # 1! ",! ) x e " = (5.15)! $ (" ) The gamma distribution is a two parameter (θ, β) continuous distribution, which is shown in Equation (5.15). It could be fitted to the distributions for all assemblies by solving for appropriate values of theta and beta. An example of a distribution is shown in Figure Once a gamma distribution was fitted, the path length with the highest frequency was deemed to be the mean free path length. Figure 5-16: Mean free path for a mono-size assembly of spheres having a diameter of

114 The process of finding the mean free path was carried out for all of the mono-size assemblies from both the LJ and HG simulation algorithms. The results are presented in Table 5-6. Table 5-6: Summary of results for the mean free path length Model Mean Free Path Length (In particle diameters) Lennard-Jones Algorithm ± Hertz-Gravity Algorithm ± The results presented in the above table are the average of a number of assemblies. For the LJ model nine assemblies were used representing three different particle diameters (three assemblies were generated for each particle diameter), while for the HG model 12 different assemblies were used representing four different particle diameters. The important result that this shows is that the average size of a void for a mono-size assembly of particles is a function of the particle diameter. A value of 0.39 times the particle diameter was chosen for the mean free path length as this value falls within the range of both sets of results. The consistency of this result (between particle diameters) allows the following formulation of spherical particles that have been packed to dense random packing; X d = 0. 39d (5.16) The equation listed above is quite a significant result, as it shows that the average size of the mean free path length is a proportion of the particle diameter. Equation (5.16) can be used to replace the need for the particle diameter to appear in any equation that attempts to predict any properties of a material, e.g. pressure loss or permeability. The advantage of Equation (5.16) for simulated assemblies of particles, is that for any given material the particle diameter does not need to be known to predict permeability, only the mean free path length. This is especially useful for hard to define mixtures such as distributed particles and fibres. The inclusion of the mean free path length in formulas to predict 104

115 permeability and pressure loss is discussed in the following section. However, before permeability results are presented the mean free path length of the binary and distributed mixtures will also be presented. Mean Free Path Length (mm) Mass Ratio Figure 5-17: Mean free path length for binary mixtures (Particles of size 2.1 mm and 4.3 mm) Figure 5-17 shows the mean free path length for each of the binary mixtures, with each of the points representing the path length with the highest frequency from the gamma distribution. It can be seen that the mean free path length starts at 0.39 times the small particle diameter (2.1 mm) and increases to a value of 0.39 times the large particle diameter (4.3 mm). The mean free path length increases as there are an increasing number of large particles introduced as the mass ratio increases. This increase in large particles results in the average size of the voids increasing. The results in Figure 5-17 are from the Hertz-Gravity model but it should be noted that the results from the Lennard-Jones model are almost identical. The last set of path lengths found are those from the distributed models of the glass beads and the lead shot. 105

116 Table 5-7: Mean free path length results for distributed mixtures Mixture Mean Free Path Length (mm) Glass set Glass set Glass set Lead Mix Lead Mix Lead Mix Lead Mix Lead Mix The glass bead mixes, which increase in median particle diameter size, show an increase the mean free path size which is consistent with the results of the mono-size particles. The lead shot mixes however show a range of mean free path lengths due to the varying amount of components of each of the individual lead shot sizes. It is anticipated that the experimental permeability results will reflect these mean free path lengths. 5.8 Permeability The results of the previous section can now be used to redefine the permeability equations so that they include the mean free path length of the assembly rather than the particle diameter. As discussed in Chapter 2 of this thesis, the particle diameter for a certain material is not always immediately obvious, and in fact in some cases can be quite troublesome to find. The Ergun equation and the Kozeny-Carman equation can be modified to include the mean free path of the material, with only changes needing to be made to the coefficients. The modified Ergun equation (using equation 5.16) that includes the mean free path length, X d, is; 106

117 2 2 dp 22.82µ f V (1 #! ) 0.68" f V (1 #! ) = + (5.17) 3 3 dx X! X! 2 d d While the modified Kozeny-Carman equation that includes mean free path length is; 2 X d 3! = 27.38µ (1 "! ) k (5.18) f 2 The advantage of the above equations is that only the knowledge of the void ratio and mean free path length of the material must be known, while the disadvantage of the equations is the difficulty in finding the mean free path length for a material without the aid of computer simulations. However, for the materials studied in this thesis this is not a concern. The above equations will now be applied to the set of materials covered so far and their theoretical permeability predicted by Equations (5.17) and (5.18) will be compared to the experimental results presented in the preceding chapter, with the errors being once again calculated using Equation (B-1). The first set of results that these equations will be applied to is the mono-size mixtures of the five sizes of the lead shot. The results can be seen in the table below. 107

118 Table 5-8: Permeability predictions of mono-size particles using modified Equations (5.17) and (5.18) Experimental Ergun K-C Diameter Ergun Error K-C Error Permeability Permeability Permeability (mm) (%) (%) (m 2 /kpas) (m 2 /kpas) (m 2 /kpas) Average Average K-C 7.6 Ergun Error Error 17.8 The table shows that there is a good agreement between the experimental results and the theoretical results predicted by the modified equations. It can be seen that the Ergun equation predicts the permeability slightly better than the Kozeny-Carman equation and this is due to the different coefficient of the equation compared to the coefficient of the viscous term of the Ergun equation. At this stage it is worth noting that the equations that have been applied for the mono-size mixtures are only a different form of the original equations derived by the workers Ergun, Kozeny and Carman. This is because the coefficients of the equations have been varied so that the mean free path appears in the equations rather than the particle diameter. An application of the original equations utilising the particle diameter and the original coefficients would yield the same results as appear above. The real use of the modified Equations (5.17) and (5.18) are for those mixtures where the particle diameter is not readily available, as it is for mono-size mixtures, which can be seen in the following results. The permeability of the binary mixtures was calculated using only data from the simulations, the mean free path length and the packing efficiencies that have been presented earlier. 108

119 Permeability (m 2 /kpas) Experimental Ergun Kozeny-Carman Mass Ratio Figure 5-18: Predicted permeability of binary mixtures As the mean free path length can be found for each mass ratio the modified equations can be used to predict the permeability for each assembly as seen in Figure Once again the modified Ergun equation and the modified Kozeny-Carman equation predict compare well with the experimental permeability results. It can be seen that the Ergun equation compares better with the experimental results for the lower mass ratios while the Kozeny- Carman compares better with the experimental results at the higher mass ratios. The average error for all mass ratios for the Ergun equation though is 6.5% while for the Kozeny-Carman equation it is 12.9%. As the equations are of the same form with only different coefficients the trend in the two theoretical predictions is the same. This binary mixture of varying composition illustrates the use of the modified Equations (5.17) and (5.18) in that they contain the mean free path length, whereas the original equations contain the particle diameter. For the binary mixtures studied here there is no method of calculating an appropriate particle diameter when there are two distinct sizes of particles. Obviously, without the knowledge of the particle diameter the permeability cannot be predicted. The final set of materials that were studied to find permeability results are the continuously distributed mixtures of glass beads and the discrete mixtures of lead shot. Using the mean free path length from each assembly along with the simulated packing efficiency of each 109

120 distribution the permeability was calculated in the same manner as it was for the mono-size and binary mixtures. The results can be seen in Table 5-9. Table 5-9: Permeability predictions for distributed mixtures Mixture Experimental Ergun K-C Ergun Error K-C Error Permeability Permeability Permeability (%) (%) (m 2 /kpas) (m 2 /kpas) (m 2 /kpas) Glass set Glass set Glass set Lead mix Lead mix Lead mix Lead mix Lead mix The theoretical predictions once again compare well with the experimental results, with the errors being relatively low for all of the mixtures. For the glass beads, the average error for the Ergun equation was 6.4%, while for the Kozeny-Carman equation the average error was 11.3%. For the lead shot mixtures, which typically showed a slightly higher packing efficiency, the errors were 21.2% and 10.7% for the Ergun and Kozeny-Carman equations respectively. This result for the errors was in contrast to the other trends in that the Kozeny- Carman equation produced a lower average error in all cases except for the distributed lead shot mixtures. This was because the error is defined in absolute terms and not dependent on whether the equations over or under predict the permeability, and the Kozeny-Carman equation will always be lower in magnitude than the Ergun equation. For the case of the lead shot distributed mixtures the magnitude of the predictions are higher and so the Kozeny-Carman equation, which has for all other cases generally under predicted the permeability, produces results that have gotten closer to the experimental permeability and hence gives a lower average error. 110

121 5.9 Conclusions The simulation of spherical particles is an excellent tool for gathering data on the structure of an assembly. This chapter included two such simulation methods for spherical particles, with both of the methods relying on the optimisation of some form of energy potential. The advantage of the Lennard-Jones model is that due to the spherical growth nature of the assembly it tended to have clearer peaks in the radial distribution function. However, for a more realistic simulation that included the same boundary conditions as in the laboratory the Hertz-Gravity model was used. The results from both models for both the radial distribution function and the packing efficiency were in excellent agreement with the physical results of Finney [28], which are widely recognised as the benchmark for which to compare simulation models. This provided validation of the simulation method. Comparisons between the two simulation models were also excellent although they optimised different forms of energy potential. Furthermore, for a range of particle sizes, the packing efficiency between the models and between particle sizes was excellent. The advantage of using numerical simulation methods was evident through the ability to extract data from the assemblies in the form of the mean free path length. Unlike physical experiments where the gathering of such data would be difficult, if not impossible, calculating the path length between any two particles is relatively straightforward given the particles coordinates and radius. By considering mono-size particles as a starting point it was possible to establish a relationship between the mean free path length and the particle diameter, a relationship that was in fact constant for all particle diameters that had been packed to dense random packing. The establishment of this relationship allowed existing permeability equations to be modified so that they no longer depended on particle diameter, but instead were a function of the mean free path length. These new equations that were developed were then applied successfully to mono-size, binary and distributed mixtures of spherical particles. By developing a permeability equation that depended on void size and void ratio, the need for a parameter describing the particle diameter was eliminated. This shows that spherical 111

122 particle mixtures typical of the ones studied in this chapter can be well characterised by the mean free path length and the void ratio alone. This inclusion of the mean free length in the modified equations eliminates the need for the particle diameter term, and hence eliminates the need for a shape factor for non-spherical particles. As this theory has been successfully applied to spherical particle mixtures, it is now intended to test the validity of the theory with non-spherical particles in the form of fibrous particles. This investigation forms the basis for the following chapters. 112

123 Chapter 6 - PERMEABILITY OF FIBROUS PARTICLES 6.1 Introduction Fibrous materials have not been as widely researched in terms of permeability compared to typical granular media as they are not as commonly encountered. However, a fundamental understanding is no less important, with fibrous materials playing key roles in the design of filters and filter pads, air cleaning elements, and in the recycling industry with the handling of biodegradable materials. The treatment of fibrous materials is also more difficult in that they generally cannot be classed by a single size, as granular media can, as they have both a diameter and a length that can vary. Typical equations that exist to predict the permeability of materials are also generally not applicable to fibrous particles, with the usual procedure being to fit the theoretical equations to experimental data via an empirical shape factor. In an extension of the theory discussed in the previous chapter, fibrous materials will be studied to assess the validity of the theory for particles that are not spherical in shape. Before moving into the realm of computer simulations of fibrous particles, it is first necessary to collect experimental data for a wide range of materials. This chapter presents experimental data for both rigid and non-rigid fibres of varying aspect ratio for the packing efficiency and permeability. Typical factors that affect each of the packing efficiency and permeability are also discussed. The results section of this chapter concludes with results that are presented from past literature that illustrate the dependence of permeability on the packing efficiency and the relationship between packing efficiency and aspect ratio. 6.2 Materials Materials were selected to represent fibrous particles that were both rigid and non-rigid. The requirements of the materials were that they had to be cylindrical in shape (or as close as practically possible) and of a uniform length in order to be similar to the fibres simulated 113

124 using a numerical method. For each of the materials experimental results were collected on the packing efficiency and permeability. Rigid Fibres Rigid fibres are classed as fibres that have an infinite bending stiffness, although in theory no material fits in this category. However, for the purpose of this study, materials were selected that showed no appreciable bending under the conditions of random packing. Two materials that were used to represent uniform length cylindrical fibres were wooden dowel pieces and steel nails. The data for each material is shown in Table 6-1. Table 6-1: Material data for rigid fibres Material Diameter (mm) Length (mm) Aspect ratio Wooden dowel Nails The dowel pieces and the nails were not quite perfect cylinders as the dowel had slightly chamfered ends and the nails obviously a pointed end with a slightly bigger head. A photo showing the particles can be seen in Figure 6-1 and Figure 6-2. Figure 6-1: Photo showing the nails used to represent rigid fibres 114

125 Figure 6-2: Photo showing the wooden dowel used represent rigid fibres Non-Rigid Fibres The material that was used for non-rigid fibres was fishing line. The fishing line was cut into a specific size using an automated guillotine where the fishing line was wound off a large spool. Fishing line was selected due to its ability to bend under load was but also as it still maintains some degree of stiffness. Three aspect ratios were studied for non-rigid fibres. It was intended to cut the fishing line to a uniform length for each aspect ratio, but due to certain factors involved in the physical process the fishing lines pieces had a small distribution within each aspect ratio. These certain factors included friction in the guillotine, variation in the load as the fishing line unwound of the spool, and also variation in the electric motors. However, as Table 6-2 shows, the lengths in the fishing line were still quite uniform. Table 6-2: Material data for non-rigid fibres Material Diameter (mm) Mean Length Standard Deviation Approximate (mm) (mm) Aspect ratio Fishing line Fishing line Fishing line

126 A photo showing the fishing line can be seen in Figure 6-3, while the images of the other two sizes of fishing line can be seen in Appendix A. Figure 6-3: Fishing line pieces 10 mm long representing an aspect ratio of Results The permeability of the fibrous particles were measured using the same test rig as shown in Figure 4-2, however a manometer was used due to the smaller pressure loss for the fibrous materials. A load cell was also installed under the bottom permeable plate to monitor the consolidation pressure to ensure a constant void ratio was maintained Rigid Fibres For each of the two materials representing rigid fibres a range of packing efficiencies was studied. The packing efficiency of the fibres was controlled by varying the rate at which the material was poured into the test chamber. The lowest packing efficiency was achieved by pouring the material into the test chamber as fast as practical, while the highest packing efficiency was achieved by dropping the fibres into the test chamber at a very slow rate. At each packing efficiency the permeability was found. The results for the wooden dowels 116

127 having an aspect ratio of 5 and the steel nails having an aspect ratio of 15 can be seen in the following figure. 9 8 Permeability (m 2 /kpas) Wooden Dowel, AR=5 Steel Nails, AR=15 0 0% 10% 20% 30% 40% 50% 60% 70% Packing Efficiency Figure 6-4: Permeability results for rigid fibres The packing efficiency for the wooden dowel varied over a range 11% and reached as high as 63%, while for the nails the variation in packing efficiency was only 7%. The permeability results for the two materials showed a similar range of values, but this is due to the large differences in the diameter of the fibres, with the dowel having a diameter four times as large as the nails. With both materials the trend was for the permeability to decrease in a linear fashion as the packing efficiency increased over the range of values tested Non-rigid Fibres The packing efficiency of the fishing line pieces was varied by controlling the rate of filling the container, as it was for the rigid fibres, but also by applying a consolidation load to the top of the sample. When using a consolidation load to vary the packing efficiency care had to be taken in order to distribute the load over the entire bed and not just the top section of the sample, with the load cell being used to monitor this. It was found though that the most variation in the sample was obtained by the controlling the rate of flow, with only small changes (1~2%) being seen by adding a consolidation pressure. The following figure shows the permeability results for the three aspect ratios of the fishing line pieces. 117

128 7 Permeability (m 2 /kpas) Fishing line, AR=11 Fishing line, AR=19 Fishing line, AR=38 0 0% 10% 20% 30% 40% 50% Packing Efficiency Figure 6-5: Permeability results for non-rigid fibres The wider range in data in Figure 6-5 reveals an exponential trend in variation of permeability with packing efficiency. The variation that was achieved in the packing efficiency for the three aspect ratios 11, 19 and 38 was 8%, 5% and 5% respectively. This data combined with that of the rigid fibres shows that as the fibres got longer there is less variation that can be achieved in the packing efficiency for fibrous materials. The permeability results for the fishing line all lie on the same curve as the diameter of the fishing line is constant, with the results indicating that the permeability decreases as the packing efficiency increases. Figure 6-5 also shows that the permeability is not directly dependent on the aspect ratio; rather it is a function of the packing efficiency. 6.4 Past Literature Results To further confirm the experimental results, a review of past literature was undertaken for existing data on the permeability of fibrous materials. A study by Jackson and James [37] has collected a vast range of experimental data from many sources and reprinted the results in table form. Results from [37] between the range of 10% and 65% packing efficiency have been used to compare with the experimental results obtained for the wooden dowel, steel nails and the fishing line. To ensure uniformity between all sets of results the 118

129 permeability has been converted into dimensionless quantity, by multiplying by the viscosity of the flowing fluid and dividing by the square of the fibre radius; kµ k d = (6.01) 2 (d 2) The data presented in [37] is for fibres ranging from glass rods, copper wire, varieties of wool, hair, nylon fibres and collagen, and so represents a wide range of materials that are both rigid and non-rigid. The flowing fluid that was used also varied with fluids including glycerol, benzene, water, oil and of course air. The results can be seen in the following figure, along with the experimental results presented earlier for the dowels, nails, and fishing line. 10 Dimensionless Permeability Data from [37] Experimental Data Packing Efficiency Figure 6-6: Experimental permeability compared with existing data from [37] Figure 6-6 shows that the existing data for the experimental permeability for fibrous materials agrees well the experimental data found for this thesis. When plotted on a semilog scale, the figure reveals an exponential relationship between the dimensionless permeability and the packing efficiency. Within the existing data reproduced from [37], there exists a range of values for any given packing efficiency, with the range typically an order of magnitude. This is most likely due to small differences in the method and measuring equipment that each of the authors in [37] have used to find each of their 119

130 experimental results. In comparison, the experimental results for fibrous materials for this thesis show a much lower amount of scatter. The previous discussion has presented permeability results for fibrous particles that depend on the packing efficiency; however in [37] there was no discussion of the length or aspect ratio of these fibres. In some cases, and specifically to compare with a numerical simulation method, it is desirable to know how the packing efficiency varies with the aspect ratio of the fibres. There have been many workers who have gathered experimental data for the packing efficiency of rigid cylindrical fibres. Zou and You [84] used a range of cylindrical wooden rods of diameter 2.5 mm with aspect ratios varying from 1 to 64. These workers studied two packing states, loose and dense random packings, with dense random packing being achieved by tapping the test chamber until no further reduction in volume was observed. From their results the workers developed an empirical equation to predict the packing efficiency from the aspect ratio. Equation (6.02) is for loose random packing, Equation (6.03) is for dense random packing and Equation (6.04) shows the relationship between the particle sphericity and the fibre aspect ratio. The sphericity is defined in the same manner as presented in Chapter 2; the ratio of the surface area of a sphere having the same volume as the particle to the surface area of the particle ln" =! exp[5.89(1 #! )]ln 0.40 (6.02) 6.74 ln" =! exp[8.00(1 #! )]ln 0.36 (6.03) * AR! = (6.04) AR The authors Parkhouse and Kelly [59] used raw spaghetti of diameter 1.8 mm with a varying aspect ratio between 6.8 and 143 to study the effects of aspect ratio on packing 120

131 efficiency. The authors also used experimental data from Milewski [55] and Evans and Gibson [27] to develop an equation to predict the packing efficiency from the aspect ratio; 2ln AR! = (6.05) AR The paper noted that the results amongst different sources was quite consistent and suggested that for a bed of cylindrical fibres of the same diameter but varying length the packing efficiency could be used to determine the average aspect ratio of the material. The study by Rahli et al [63] used copper and bronze fibres cut to uniform lengths to represent aspect ratios varying from 4.5 to The fibres were poured into the test chamber and were not vibrated in any manner. Packing efficiencies were recorded for each aspect ratio, with the authors reporting no noticeable difference in the packing efficiencies for each aspect ratio in three different sized test chambers. These workers also used their results to develop an empirical equation based on the excluded volume theory. The packing efficiency is given by; 11! = (6.06) f V excl Where f V excl is the excluded volume and the value of 11 in Equation (5.06) is chosen from their experimental results. The excluded volume for a fibre is given by; V f excl 2 =! AR (6.07) AR As a comparison of the different equations that have been developed in [84, 59, 63] they have all been plotted on the same graph and appear in Figure 6-7, along with the maximum and minimum values for the packing efficiency achieved with the fibrous materials in this chapter. 121

132 Packing Efficiency 70% 60% 50% 40% 30% 20% Zou and Yu (Loose Packing) Zou and Yu (Dense Packing) Parkhouse and Kelly Rahli et al Experimental Data (Max. value) Experimental Data (Min. value) 10% 0% Aspect Ratio Figure 6-7: Packing efficiency results versus aspect ratio from past literature The different equations are in agreement for an aspect ratio of 5, with all three equations (discounting Zou and Yu loose packing) predicting a packing efficiency of approximately 64%. However, for all other aspect ratios the equations show a range of values. This is most likely due to the orientation of the fibres, as discussed in the previous chapter, as it was shown the orientation of the fibres affects the packing efficiency. It can also be seen in Figure 6-7 that there is some variation between the experimental results presented in this chapter and the past literature results. The wooden dowel (AR=5) and the 20 mm fishing line pieces (AR=40) exhibit approximately the same range as past literature. These two sets of results contained a relatively small amount of particles and so achieving a low flow rate to achieve a higher packing efficiency could be achieved in practice. The other 3 results however, required a much larger number of particles to fill the container (test chamber stayed the same size, particles are smaller). This meant that achieving the same low flow rates, in terms of particles per unit time, was much more difficult. For example, there were in the order of 1000 pieces of wooden dowel, while there were more like pieces of 5 mm fishing line. Achieving the same low fill rates for these materials is just not practical. Another factor that may have contributed to these large 122

133 numbers is that there may have been some mechanical interlocking of the fibres which prevented higher packing efficiencies. 6.5 Permeability Predictions As an investigation into the application of existing permeability equations to fibrous materials, the Kozeny-Carman equation and the Ergun equation have been used to predict the permeability for each of the dowels, nails and fishing line. The equations featuring the shape factor can be seen in Equation (6.08) (Kozeny-Carman) and Equation (6.09) (Ergun) " d! k = (6.08) 180 µ! ( 1# ) 2 dp dx µ f V (1 $! ) 1.75# f V (1 $! ) = + (6.09) " d! " d! The permeability was calculated using the above equations, and using Equation (6.04) to determine the shape factor given an aspect ratio, and the results were plotted in dimensionless form along with the experimental permeability to compare. 10 Dimensionless Permeability Experimental Ergun Kozeny-Carman Packing Efficiency Figure 6-8: Permeability predictions for fibrous materials 123

134 As Figure 6-8 shows, the existing equations that predict permeability tend to under predict the permeability, sometimes by as much as two orders of magnitude. It is obvious from these results that the Ergun and Kozeny-Carman equations that were formulated for spherical particles are not applicable to fibrous particles in their current form. The reason for this can be put down to the inability of the shape factor to accurately describe the void space that exists within the particles; the void space between fibrous particles is obviously a lot more complex than the void space between spherical particles. An improvement in the understanding of the void space between particles of varying shape, including fibrous particles, is a key motivator driving the use of computer simulations. The following chapter uses these computer simulations to gain more information into the space that exists between fibrous particles. 6.6 Conclusions Experimental results were found for five aspect ratios of materials covering three distinct fibre diameters. By plotting the dimensionless permeability against packing efficiency good agreement between the aspect ratios and the materials was achieved. The data plotted in Figure 6-6 also showed that there is an exponential trend between the dimensionless permeability and the packing efficiency. This trend in the experimental data was further quantified by existing literature. It was also found that the existing equations to predict the permeability of a material are not applicable to fibrous materials, even with the use of a shape factor. This result highlights the need for further development in the existing equations, particularly in the description of void space. The following chapter expands the simulation method outlined in Chapter 4 to simulate fibrous particles so that vital information regarding the structure of the voids in a fibrous assembly can be extracted. 124

135 Chapter 7 - SIMULATION OF FIBROUS PARTICLES 7.1 Introduction Materials consisting of predominantly fibrous particles are difficult to model using traditional equations for permeability as it is difficult to establish a particle diameter for use in the equations. This thesis has already addressed the use of a shape factor, and it was shown that the shape factor cannot be used accurately to determine an appropriate diameter for use in the equations. As an alternative for fibrous particles, the focus of this chapter is on investigating the application of the theory developed in Chapter 5. This chapter presents two simulation algorithms that have been used to simulate beds of fibrous particles of varying aspect ratio. Through the use of these simulations the assembly structure can be inspected, in a similar manner to the assemblies of spherical particles, to characterise the void space and use this to estimate permeability. Two alternate models are used to simulate fibrous particles, with the first method being used to simulate rigid and non-rigid fibres. In this model, a chain of spheres is modelled by a polynomial in each of the three principal directions. The polynomial is linear for the rigid fibres and cubic for the non-rigid fibres. The fibre is then subjected to gravitational energy; Hertz contact energy, axial strain energy to prevent the particles on each fibre from separating, and also bending strain energy (for non-rigid fibres). The second simulation algorithm discussed in this chapter presents an alternate method for modelling non-rigid fibres. It is based on a method that uses a set of marker points in space to define the radius of curvature between two consecutive particles on the fibre. The advantage of this method is that there is one less variable to optimise and there is a more direct relationship between the variables being optimised and the actual particle coordinates. 125

136 7.2 Rigid Fibres The packing of fibres is more complicated than the packing of spheres, due to each fibres location being defined by not only a set of coordinates but also by its orientation and shape. The Hertz-Gravity model that was defined previously for spheres can be expanded to model fibrous materials with the addition of some energy potentials. Rigid fibres will first be considered and then fibres that are permitted to bend will be discussed Energy Potential The objective of the simulation algorithm is to build a randomly packed assembly of identical cylindrical fibres of diameter d and length L. The position and orientation of the n th fibre can be defined by single order polynomial through space of the form; & # 1 k $ ' a1 kl! k = 0 ~ $ 1! k xn ( l) = $ a kl! $ '! (7.01) 2 k = 0 $ 1! k $ ' a3kl! $ % k = 0!" n Where l! ( 0, L), is the distance measured along the fibre. Each fibre is represented by discrete series of N + 1 points defined by: L l i = i for i = 0, N (7.02) N By optimising the six coefficients and summing the energy potential, an assembly can be made. The energy is calculated by the following equation; U = U + U + U (7.03) G H A The three components are gravitational energy, Hertz contact strain energy and axial strain energy. 126

137 U G Gravitational potential energy This was defined in a similar manner as for the assembly of spherical particles. U G = mgx n2 (7.04) The gravitational energy is summed over each discrete point on each of the fibres. U H Hertz contact strain energy The discrete series of points that define each fibre can be used to calculate the contact strain energy between each fibre. Assuming there exists a spherical particle of diameter d at each discrete point on each fibre; we can calculate the strain energy to be; n 1 =! " U H H ( xn, xm ) (7.05) m= 1 With the Hertzian contact strain being defined as it was for the spherical particles; H ( x, x i j $! 4 E! 15 1& -! ) = #! 0!! " 2, * d xi & x & * j ) ' ' ( 2.5, * + d 4 ) ' ( 0.5 for for x x i i & x & x j j < r % r i i + r + r j j (7.06) With the parameters being defined as they were for spherical particles. Using this approach the fibres are analogous to a string of pearls, however the discrete step length is not necessarily equal to the fibre diameter. It should be noted that the calculation of U H only considers interactions between particles on separate fibres (i.e. the interaction between 127

138 particles on the n th fibre and all other particles on all other fibres in the assembly). This exception is important as spherical particles placed along the individual fibre may be overlapping by a significant amount. U A Axial strain energy The axial stiffness of the fibre is found by adding up the individual axial strain energies of all of the discrete elements of the fibre. U A = E A N) d 8L 2 N! i= 2 ( & ' L % x( li ) " x( li" 1) " # N $ 2 (7.07) In Equation (7.07) E A is the axial stiffness of the fibre rather than of the particles. Once again, a penalty function was applied to prevent the particles from travelling outside of the theoretical container. The container for the fibres was again a cylindrical region with a base at x n2 = 0, and diameter was chosen to contain the fibres. This region provides the following two constraints, which every particle on each fibre is subjected to; x > 0 n2 (7.08) 2 2 n1 x n 3 x + < (7.09) D Simulation Algorithm The optimisation method that was used to implement the algorithm was a Quasi-Newton method with a finite difference gradient. The coefficients of the polynomials were the variables to be optimised rather than the coordinates as used for spherical particles, and so the coordinates for each point on the fibre had to be calculated from the polynomial coefficients. A brief outline of the simulation algorithm can be seen below. 128

139 Initialise parameters - Set the spacing between particles on each fibre, S - Assign a total number of particles to each fibre, POF - Aspect ratio is then, AR = (S x POF) FOR i=1 to number of fibres Assign random coefficients to i th fibre Calculate coordinates using the coefficients The current fibre number is i The coordinates are calculated in the following way, d is the diameter FOR j=1 to POF Coord (i, j, 1) = X(1) + (X(2)*S*d*j) Coord (i, j, 2) = X(3) + (X(4)*S*d*j) Coord (i, j, 3) = X(5) + (X(6)*S*d*j) END FOR Ensure that the fibre is inside the theoretical container Optimise coefficients according to objective function using UMINF Add coordinates of i th fibre to final coordinate array END FOR Output results The array X stores the six coefficients, with the variable d being the fibre diameter. The neighbourhood regime that was discussed for spherical particles in Chapter 4 was used again for fibres. It was used in the same manner with each discrete point on the fibre being placed in a neighbourhood, with each neighbourhood again being one diameter in height. The arrays that were used for the neighbourhood regime were of a different form due to each fibre being made up of a number of particles. 129

140 Neighbourhood (NF, POF) The purpose of this array is to store the neighbourhood number of each particle on each fibre. For example, if each fibre had five particles on it and the first fibre was on the bottom of the container, then; Neighbourhood (1, 1) = 1 Neighbourhood (1, 2) = 1 Neighbourhood (1, 3) = 1 Neighbourhood (1, 4) = 1 Neighbourhood (1, 5) = 1 Neighbourhood List (j, k, l) This array contained a list of the particle numbers in a particular neighbourhood. The first index of the array (j) referenced the neighbourhood number, and the second index (k) held the current number of particles in the j th neighbourhood. The last index (l) was needed to store the fibre number as well as the particle number, and so was of length 2. For the above example with the first fibre being composed of five particles the array would look like; Neighbourhood List (1, 1, 1) = 1; 1 in the third index representing fibre number Neighbourhood List (1, 1, 2) = 1; 2 in the third index representing particle number Neighbourhood List (1, 2, 1) = 1 Neighbourhood List (1, 2, 2) = 2 The length of j was equal to the maximum number of neighbourhoods, the length of k was equal to the maximum theoretical amount of particles that could fit in one neighbourhood and the length of l was equal to one, with only two possible entries. Neighbourhood Count (j) This was the same as before, this array contained the number of particles in each neighbourhood. The length of the array was equal to the number of neighbourhoods. 130

141 With this neighbourhood regime, the code to calculate the objective function which is composed of Hertz contact energy, gravitational energy, axial strain energy and a penalty function is outlined below. i = Number of current fibre being optimised Coord (i, POF, 3) are the coordinates of current fibre U1 = 0; U2 = 0; U3 = 0; U4=0 FOR j=1, POF This part is the Hertz contact energy a = temporary neighbourhood of j th particle on the i th fibre b = Neighbourhood Count (a) FOR j = 1 to b Current fibre number = Neighbourhood List (a, j, 1) Current particle number = Neighbourhood List (a, j, 2) r = distance between j th particle on i th fibre and current particle on current fibre IF r < diameter of fibre U1 = U1 + energy due to Hertz contact energy END IF END FOR The above FOR loop is repeated for the (a-1) and the (a+1) neighbourhoods END FOR (continued next page) 131

142 This part is gravitational energy FOR j=1, POF IF y coordinate of j th particle on i th fibre > 0 U2 = U2 + energy due to gravitational energy ELSE U2 = U2 + much larger energy to prevent particles travelling below zero END IF END FOR This part is the axial strain energy FOR j=2, POF r = distance between j th particle on i th fibre and (j- 1) th particle on i th fibre U3 = U3 + energy due to axial strain energy END FOR This part is the penalty function FOR j=1, POF IF particle is outside container U4 = U4 + energy due to penalty function END IF END FOR U = U1 + U2 + U3 + U4 132

143 7.3 Non-Rigid Fibres The handling of non-rigid fibres involves increased complexity compared to rigid fibres, as there are now more variables to optimise. As discussed in the introduction to this chapter, two methods were used to simulate non-rigid fibres. The need for the second algorithm arose due to limits in the optimisation routine that will be discussed at a later stage. The objective function of the two methods was essentially the same, with the main difference between the two algorithms being the method used to calculate the coordinates of the particles on each fibre. The objective function, or energy potential, is presented first as it is common to both methods followed by a detailed discussion of each simulation method Energy Potential In a similar method used for the rigid fibres, the energy potential can be formed to which the fibres will be subjected to. For a partial assembly of n-1 fibres, the potential of the n th additional fibre is; U = U (7.10) G + U H + U A + U B Where gravitational energy, Hertz contact energy and axial energy are the same as before with the addition of bending strain energy, U B. U B Bending strain energy The stiffness of fibrous particles in the longitudinal direction determines the extent to which the fibre will bend under load. A higher bending stiffness results in the fibre deforming less than if it had a lower bending stiffness. The bending strain energy for each fibre is directly proportional to the degree that the fibre has deformed and can be calculated using the following equation; U B = N! i= 1 E B I 2" A 2 d (7.11) 133

144 Where I A is the second moment of area, E B is the bending stiffness of the fibres, and δ is the radius of curvature of the fibre. The radius of curvature is calculated in a different manner for each of the simulation methods and so will be discussed separately Simulation Method 1 The first method is a direct extension of the simulation algorithm used for rigid fibres. It consists of modelling non-rigid fibres by using three cubic polynomials rather than linear polynomials. The position and orientation of the n th fibre can be defined by a cubic parametric curve through space of the form; & # 3 k $ ' a1 kl! k = 0 $ 3! ~ k xn( l) = $ a kl! $ ' 2 (7.12)! k = 0 $ 3! k $ ' a3kl! $ % k = 0!" n Where l! ( 0, L) is the distance measured along the fibre. From this description, each fibre can be represented by a set of 12 coefficients{ a }. Each fibre is represented by a discrete series of N + 1 points defined by; jk { ( )} x~, where n l i L l i = i for i = 0, N (7.13) N The radius of curvature of the fibres is calculated from both the parametric and discrete representations of the orientation. Consider a point moving along the fibre where the parameter l is considered to be time rather than distance. At each discrete position along the fibre, we could calculate the velocity and acceleration of the point from Equation (7.12) as; x v~! =, and! l l i! v! x = (7.14) 2 a~ = 2! l l! l i l i The acceleration of the point would have normal and tangential components, and could be written as; 134

145 2 v a ~ = ate~ t + ane~ n = ate~ t + e~ n (7.15)! Where a t and a n are the tangential and normal components of acceleration, e ~ t and e ~ n are unit vectors in the tangential and normal directions, v = v~ is the magnitude of the velocity and δ is the radius of curvature of the path. The tangential direction corresponds with the direction of the velocity. It therefore follows that the tangential component of acceleration could be written as; a t v~ = a~ o e~ t = a~ o (7.16) v Which would give the normal component of acceleration as; v~ v~ a a~ & n a~ # = ' $ o! (7.17) % v " v So from Equation (7.10), we can calculate the radius of curvature of the path (fibre) as; 2 2 v v " = = an a~ v~ (7.18) a~ o! v~ 2 v Using this method to calculate the radius of curvature, and by optimising the 12 coefficients of each additional fibre with respect to the energy potential it is possible to build an assembly. The algorithm was the same as set out for the rigid fibres, with a slight modification in the calculation of the coordinates of each particle on each fibre due to the cubic polynomial approach. 135

146 d is the fibre diameter FOR j=1 to POF Coord (i,j,1) = X(1) + (X(2)*S*d*j) + (X(3)*(S*d*j) 2 ) + (X(4)*(S*d*j) 3 ) Coord (i,j,2) = X(5) + (X(6)*S*d*j) + (X(7)*(S*d*j) 2 ) + (X(8)*(S*d*j) 3 ) Coord (i,j,3) = X(9) + (X(10)*S*d*j) + (X(11)*(S*d*j) 2 ) + (X(12)*(S*d*j) 3 ) END FOR The same neighbourhood regime was used as described for the rigid fibres. With the addition of the bending stiffness, the fibres were permitted to bend as they were dropped into the container. In theory, with a low enough bending stiffness, the situation of a fibre passing through itself can occur, although this was not observed in practice. The bending stiffness of the fibres is calculated in the following manner; U5=0; FOR j=1 to POF r = radius of curvature between two consecutive particles on fibre U5 = U5 + energy due to bending strain END FOR The result of this packing algorithm was that the packing efficiency was below that of the assemblies of rigid fibres. It was expected that the packing efficiency should at least equal the packing efficiency of the rigid fibres if not be higher. The conclusion that was drawn from this result was that the optimisation routine was not robust enough to deal with this method. This was most likely a result of the relationship between the variables being optimised (12 coefficients of cubic polynomials) and the coordinates of each particle not being closely related. The second method presented used a more direct relationship. 136

147 7.3.3 Simulation Method 2 The process of simulating fibrous particles using this method is similar to the method previously discussed, with the exception of how the coordinates of the fibre are calculated. This model uses a combination of basic dynamics and 3D geometry to calculate the coordinates for each particle on the fibre, rather than using a cubic polynomial to represent each fibre. This method starts by selecting a coordinate in space that represents the location of the first particle on the fibre. Three more points in space are chosen (randomly) that represent the points that the particles on the fibre will rotate around, see Figure 7-1. ~ M 2 ~ r 3 ~ M 3 ~ r 2 ~ M 1 ~ r 1 ~ x 0 Figure 7-1: Illustration showing how the fibre coordinates are calculated As Figure 7-1 shows, once a starting location for the particles on the fibre and the 1 st point of rotation is found, then the rest of the coordinates on the first third can be calculated. The coordinates of the particles on the 2 nd and 3 rd sections of the fibres are found in a similar 137

148 manner. To implement this approach 11 variables are needed, one less than the cubic polynomial approach. Specific details regarding the variables can be found below; 1) ~ x 0 the coordinate of the first particle on the fibre 2) r x, 1 the x component of the centre of rotation for the first third of the fibre 3) r y, 1 the y component of the centre of rotation for the first third of the fibre 4) r x, 2 the x component of the centre of rotation for the second third of the fibre 5) r y, 2 the y component of the centre of rotation for the second third of the fibre 6) r x, 3 the x component of the centre of rotation for the last third of the fibre 7) r y, 3 the y component of the centre of rotation for the last third of the fibre 8)! an angle used in the calculation of starting direction of first particle 9) γ an angle used in the calculation of starting direction of first particle The variables 2 through 9 listed above are used to calculate the coordinates of each particle on the fibre, with the 1 st variable being the coordinate of the 1st particle of the fibre. The method of calculating the coordinates of each particle on the fibre is detailed below. Once again, for illustrative purposes we will use the analogy of a point moving along the fibre. The first step is to calculate the components of the velocity at the 1 st point on the fibre using spherical coordinates. V V x y z = cos( " )sin(! ) = sin( " )sin(! ) V = cos(! ) (7.17) r z,1, representing the z component for the point of rotation for the first third of the fibre, can then be found as it must lie on the same plane as r x, 1 and r y, 1. Using orthogonal vectors and equating the cross product to be zero; 138

149 rx,1 " Vx + ry,1 " Vy rz, 1 = (7.18)! V z The point in space that the first third of the fibre rotates around can now be found since the magnitude and direction from the first particle on the fibre is known; ~ M = ~ r + ~ x (7.19) ~ Where the point M 1 represents the coordinate of the first point of rotation and the point ~ x 0 represents the coordinate of the first particle on the i th fibre. The radius of curvature can then be found for the first section of the fibre as the distance between the first coordinate and the point of rotation is known. The coordinate of the next particle ( j ) on the first section of the fibre, and all subsequent particles on the same section, can then be found in the following manner. The first step is to normalise the velocities of the ( j! 1) th particle ~ ~ V V = (7.20) V Then find the direction from the ( j! 1) th particle to the point of rotation ~ ~ R ~ (7.21) = M! x j! 1 And normalise the directions ~ ~ R R = (7.22) R 139

150 Where R is the radius of curvature for the first section of the fibre. The next step is to find the normal component of the acceleration, and to do this we use the specified magnitude of velocity. Using the analogy of a point moving along a fibre, the specified magnitude of velocity is equal to the distance between consecutive particles along the fibre. The acceleration is then; 2 ( d! spacing between particles) a = (7.23) R The hypothetical velocity at the next point on the fibre can then be calculated; ~ V j ~ ~ = V " R (7.24) j 1 + a! j The coordinates of the next point can then be found by averaging the velocities at the jth and ( j! 1) th points; ~ ~ ~ x ~ x + 0.5( V V ) (7.25) j = j! 1 j + j!1 This calculation is then performed for the remaining particles on the first section of the fibre. Pseudo code showing this calculation is shown on the following two pages. 140

151 FOR j=2,pof/3 Normalise the velocities unit_vec=(vel x (j-1) 2 +Vel y (j-1) 2 +Vel z (j-1) 2 ) 0.5 Vel x (j-1)=(vel x (j-1)/unit_vec)*mag_vel Vel y (j-1)=(vel y (j-1)/unit_vec)*mag_vel Vel z (j-1)=(vel z (j-1)/unit_vec)*mag_vel Find the vectors from coordinate to marker point Rx=M(1,1)-coord(i,j-1,1) Ry=M(1,2)-coord(i,j-1,2) Rz=M(1,3)-coord(i,j-1,3) Find the magnitude of these vectors which is also the radius of curvature mag_r=(rx 2 + Ry 2 + Rz 2 ) 0.5 Normalise the vectors Rx=Rx/mag_r Ry=Ry/mag_r Rz=Rz/mag_r Find the acceleration, mag_vel=spacing between particles times the diameter acc=mag_vel 2 /mag_r Calculate the velocity at the next particle Vel x (j)=vel x (j-1) + acc*rx Vel y (j)=vel y (j-1) + acc*ry Vel z (j)=vel z (j-1) + acc*rz (continued on next page) 141

152 Normalise the velocities unit_vec=(vel x (j) 2 +Vel y (j) 2 +Vel z (j) 2 ) 0.5 Vel x (j)=(vel x (j)/unit_vec)*mag_vel Vel y (j)=(vel y (j)/unit_vec)*mag_vel Vel z (j)=(vel z (j)/unit_vec)*mag_vel Calculate the coordinates of the next particle coord(i,j,1)=coord(i,j-1,1) + 0.5*(Vel x (j)+vel x (j-1)) coord(i,j,2)=coord(i,j-1,2) + 0.5*(Vel y (j)+vel y (j-1)) coord(i,j,3)=coord(i,j-1,3) + 0.5*(Vel z (j)+vel z (j-1)) END FOR The coordinates for the particles on the second and third sections of the fibre can then be calculated in a similar manner. The advantage of this method is that the link between the coordinates of each particle on the fibre and the variables are directly related. For example, the radius of curvature is optimised in this method whereas in the previous method it was calculated from the coefficients which were being optimised. 7.4 Fibre Representation Using a chain of spheres to represent a cylindrical fibre must be treated cautiously, as the representation of a fibre by a chain of spheres spaced one diameter apart does not accurately represent a cylindrical fibre. However, a cylinder can be closely approximated by having the spheres on a particular fibre being spaced less than a particle diameter apart. This is shown in the illustration in Figure

153 Figure 7-2 (a) Fibre with aspect ratio=5, particles 1Ø apart (b) Fibre with aspect ratio=5, particles 0.33 Ø apart As Figure 7-2 shows, by spacing the particles less than a diameter apart then it is possible to closely resemble a cylinder. Also, it must be noted that the fibres in these simulations have rounded ends (semi-spherical) rather than that of a cylinder which has flat ends. The variation in volume due to the rounded ends is very small though and becomes less as the aspect ratio increases. Applying this to the simulation model so that it can be realistically compared with experimental and theoretical results for cylinders the spacing of the particles on each fibre must be chosen accordingly. This means that the volume of the fibre, made up from spheres, should be as close to the volume of a cylinder of equal length as possible. Figure 7-3 shows how this volume percentage increases with a decreasing distance between consecutive particles on a fibre. Percentage of volume of cylinder 100% 90% 80% 70% 60% 50% Particle centres/particle diameter AR=5 AR=40 Figure 7-3: The effect of decreasing spacing between spheres on a fibre on the fibre volume as a percentage of equivalent cylinder volume 143

154 As Figure 7-3 shows, with 10 particle centres per particle diameter the volume of a fibre made from many spheres is very close (99.6%) to the volume of a cylinder of equal length (with semi-spherical ends). Ideally, fibre simulations would then be carried out with this spacing. However, the practical limitations of computing power restrict this. For this reason, a compromise between simulation time and particle centres per diameter was struck. A spacing between particles of 0.33 particles per diameter was chosen, as shown in Figure 7-2b, which tended to be about 96.9% of the volume of a cylinder for an aspect ratio of 5, with this percentage decreasing very slightly as aspect ratio increased. 7.5 Fibre Orientation Particle orientation is not a factor for the spherical particles due to their spherical nature. However, the orientation of fibres is a factor that must be considered in the simulation of fibrous particles. The packing efficiency of fibres is greater when the fibres are orientated in the same direction, either all horizontal or all vertical. In this work the fibres will tend to orientate themselves horizontally, as they would in a physical experiment, due to gravitational energy. As noted earlier in this thesis, the optimisation algorithm employed in this simulation process is relatively simple in that it only finds a local minimum for the solution. This means that each time a particle is introduced into the assembly the final resting position can be one of many positions, largely dependent on the initial coordinates. When the particles are spherical the location of the local minimum each particle resides at has no effect on the final packing efficiency due to the sphericity of the particles. However, as discussed in the following paragraph this was not the case for the fibrous particles. In an effort to increase the packing efficiency of an assembly of fibrous particles, a technique was introduced in which one fibre was introduced a number of times, each with a different starting location, and for each introduction of the same fibre the energy due to the objective function for the optimised position was recorded. The fibre coordinates were then determined by the configuration that resulted in the lowest summation of energy. As 144

155 expected, for each increase in the number of attempts to introduce a new fibre the packing efficiency increased. Figure 7-4 shows the results for straight fibres having an aspect ratio of 5 with the particles being spaced 0.33 diameters apart along the fibre. The number of attempts tested is plotted on the x-axis. The container size that the fibres were simulated in was approximately 4 times wider than the length of the fibre, shown in Figure Packing Efficiency Number of Attempts Figure 7-4: Number of attempts to introduce each new fibre vs packing efficiency Figure 7-5: a) 1 attempt for each new fibre b) 10 attempts for each new fibre 145

156 Figure 7-4 shows a large variation in packing efficiency, with approximately a 10% difference between introducing a fibre once and ten times. This difference in packing efficiency is reflected in Figure 7-5 where it can be seen that the fibres are all orientated in a similar manner in image (b) which has a higher packing efficiency compared with the image in (a) where there is a greater inhomogeneity with regards to orientation. As a further measure of this effect Table 7-1 shows the results of an investigation into the average height difference between the 1 st and last particles on each particular fibre. Table 7-1: Average change in height for each fibre Change in height per fibre Number of Attempts (in fibre diameters) The table shows that for the higher number of attempts to introduce each new fibre the fibres are indeed more horizontal, indicated by the lower average change in height between the first and last particles on each fibre. This means that for each aspect ratio for straight fibres there is a range of packing efficiencies that can be attained by changing the orientation of the fibres. The physical explanation for this result is discussed in the previous chapter. If a bed of fibrous particles is made by pouring the particles very slowly into the container, then the packing efficiency will be higher than if the particles were poured into the container at a faster rate. This is due to the interlocking of the particles that occurs when the particles are dropped into the container at a faster rate. Since sequential addition was used in the simulation method this physical trait could not be reproduced directly, but it can be seen that this technique is a good substitute. 146

157 7.6 Packing efficiency The calculation of the packing efficiency for an assembly of fibrous particles was done in the same manner as described in Chapter 5. It was based on a probability method in which a point with a randomly chosen length and direction was calculated relative to a randomly chosen particle in the assembly, and then it was tested to see if this point intersected with another existing fibre in the assembly. After carrying out this procedure a number of times the packing efficiency of the assembly can be found Rigid Fibres Assemblies of rigid fibres were created for aspect ratios of 5-40, in increments of 5. For each aspect ratio, the technique discussed in section 7.5 was used to study the effects of orientation. The size of the theoretical container for each aspect ratio, and hence the number of fibres to fill the container, was varied for each aspect ratio. However, due to time constraints and computational power the ratio between fibre length and container size could not be kept constant. The following table lists the details of the simulation conditions for each aspect ratio. Aspect Ratio Container Diameter (Fibre lengths) Table 7-2: Simulations conditions for each aspect ratio Simulated Fibres Number of particles on a fibre Number of particles simulated The reason that the same ratio between the fibre length and container diameter could not be kept constant is because the computational time would have been too large. As Table 7-2 shows, the number of particles simulated increases at a fast rate depending on the aspect 147

158 ratio and on the container diameter. If the same ratio of 4.2 for the container diameter to fibre length had been kept for all of the aspect ratios, this would have meant the number of particles needed to be simulated for an aspect ratio of 40 would have been in the order of 2.6 million particles. For the simulation algorithm used in this study this was just not feasible. Regardless of the difference in size of the containers, the assemblies still contained sufficient information so that meaningful data could be extracted. The packing efficiency results for each of the aspect ratios described in Table 7-2 can be seen in the following figure. Also plotted in the figure are the results from past literature [84, 59, 63] for rigid fibres that were first presented in chapter 5. Packing Efficiency 70% 60% 50% 40% 30% 20% 10% Simulation Model (Max. value) Simulation Model (Min. value) Zou and Yu (Loose Packing) Zou and Yu (Dense Packing) Parkhouse and Kelly Rahli et al 0% Aspect Ratio Figure 7-6: Packing efficiency results for rigid fibres The two sets of data presented in Figure 7-6 are for the minimum and maximum values due to orientation for the packing efficiency for each aspect ratio. As the figure shows, the two sets of data encompass the results of previous authors. This leads to the conclusion that the differences in packing efficiency for each of the results for rigid fibres presented by the different authors are a result of the orientation of the fibrous particles. The figure also shows that the difference in packing efficiency for each aspect ratio is approximately constant. Packing efficiencies for two aspect ratios were obtained in the laboratory, using 148

159 the materials wooden dowel and steel nails, and if we compare the results to the simulation results there are slight differences. The range of values obtained for the minimum and maximum packing efficiency can be seen in Table 7-3. Table 7-3: Range of values for minimum and maximum packing efficiency Laboratory Simulation Minimum Maximum Minimum Maximum AR=5 (dowel) 51.7% 63.1% 52.4% 61.8% AR=15 (nails) 28.7% 35.2% 30.8% 45.0% For the aspect ratio 5 wooden dowels, the minimum packing efficiency in the experimental situation and the simulation method are very similar. For the maximum value however, it was found that the packing efficiency in the laboratory was a little higher than the simulation method produced. This is most likely due to a higher level of alignment in the fibrous particles in the experimental procedure. For the steel nails, both the minimum and maximum packing efficiency in the laboratory situation were below the value achieved in the simulation method. The difference is most notable at the maximum packing efficiency where the difference is 10%. The reason that the steel nails pack below the simulated packing efficiencies is because in reality the packing experiences some mechanical interlocking that prevents the nails from reaching a higher packing efficiency. This mechanical interlocking is experienced for the nails and not the dowel due to the larger aspect ratio of the fibres. The simulation method introduces the fibres at the slowest rate possible, one at a time, and so the mechanical interlocking experienced in the laboratory does not exist in the simulations. Sample images of the simulations of fibrous particles can be seen in the following figures. All of the simulations were carried out with a particle spacing of 0.33 particle diameters but have been shown at a lower resolution due to the complexity of rendering spherical particles. 149

160 Figure 7-7: Simulation image of aspect ratio 10 fibres, packing efficiency 48.7% Figure 7-8: Simulation image of aspect ratio fibres 20, packing efficiency 37.3% 150

161 Figure 7-9: Simulation image of aspect ratio fibres 30, packing efficiency 17.5% Figure 7-10: Simulation image of aspect ratio fibres 40, packing efficiency 12.7% 151

162 7.6.2 Non-Rigid Fibres The cubic polynomial approach that was presented in section was first implemented to simulate beds of non-rigid fibrous particles. This method was successful for the lower aspect ratios, with the figure below illustrating a bed of fibres with an aspect ratio of five. Figure 7-11: Aspect ratio 5 non-rigid fibres simulated using cubic polynomial model The simulation shown in Figure 7-11 has a packing efficiency of 62.9%, representing an increase of approximately 1% from the rigid fibres under the same simulation conditions. This increase is achieved due to the fibres being able to bend as they are optimised to lower the gravitational energy in the optimisation routine. However, as the aspect ratio increased the cubic polynomial method failed to increase the packing efficiency compared to the rigid fibres. This effect is seen immediately as fibres of aspect ratio 10 have a packing efficiency that is 1% lower than that of the rigid fibres. The reason for this is most likely a result of the indirect relationship between the polynomial coefficients and the fibre coordinates from which the objective function is calculated. As the fibres get longer and the relationship between the fibre length and container diameter gets smaller, the increase in the variables to 152

163 optimise (compared to the rigid fibres) becomes too much for the optimisation routine to process. This deficiency in the cubic polynomial model necessitated the need to implement the alternative model which has a more direct relationship between the fibre coordinates and the variables being optimised. This simulation method used the same objective function and the same optimisation routine but different method of calculating the coordinates as detailed in section An example can be seen in the following image. Figure 7-12: Aspect ratio 5 non-rigid fibres simulated using alternate model The main difference between the two models, shown in Figure 7-11 and Figure 7-12, is that the fibres shown using the cubic polynomial method have a continuous curvature whereas the fibres using the alternate model have a discrete curvature along the fibre. This ability of the fibres to have a varying curvature is another reason why this model is successful for the larger aspect ratios while the cubic polynomial model was not. 153

164 Using this method the packing efficiency results showed similar results to the rigid fibres, with some aspect ratios showing a marginal increase for the non-rigid fibres. This can be seen in the following graph, where only the maximum packing efficiencies have been shown. 70% Packing Efficiency 60% 50% 40% 30% 20% 10% 0% Rigid Fibres (Max value) Non-Rigid Fibres (Max Value) Aspect Ratio Figure 7-13: Comparison between rigid and non-rigid fibres This lack of improvement for the packing efficiency across the entire range of aspect ratios was an interesting result. However, it is one that is not entirely unexpected. Firstly, the maximum packing efficiency of the rigid fibres is quite high, as it has been achieved by aligning almost all of the fibres in the horizontal direction, which from the images (e.g. Figure 7-5b) show that the fibres are very efficiently packed. It is expected that the maximum packing efficiencies found for the rigid fibres for the given simulation conditions are very close to the maximum packing efficiency that can be found for any stiffness of fibre under any type of packing conditions. The second reason is that it is possible that the advantage that non-rigid fibres can offer in terms of increasing packing efficiency can only be truly realised by using a global rearrangement approach. Using a global rearrangement method all of the fibres would be moved at once which would essentially simulate each fibre being deformed under load. However, the optimisation method would need to be highly advanced to handle such a problem considering that for each fibre there are 12 variables (using cubic polynomial model) which would mean optimising variables at once for a simulation of 2000 fibres. 154

165 7.7 Mean Free Path Length The mean free path length of the fibrous particle assemblies was found in the same way that was used for the assemblies of spherical particles. For all of the assemblies, it was once again found that the distributions most closely reflected that of a gamma distribution. Two examples of such distributions for the fibrous particles can be seen in the following figures. Figure 7-14: Distribution of mean free path lengths for aspect ratio of 5, packing efficiency 52.4% Figure 7-15: Distribution of mean free path lengths for aspect ratio of 40, packing efficiency 12.7% 155

166 The distributions shown in Figure 7-14 and Figure 7-15 were found for each assembly of fibrous particles and the mean free path length was chosen as the value with the highest frequency from the gamma distribution. 10 Mean Free Path Length (Fibre Diameters) Mono-size Spheres AR=5 AR=10 AR=15 AR=20 AR=25 AR=30 AR=35 AR=40 0 0% 10% 20% 30% 40% 50% 60% 70% Packing Efficiency Figure 7-16: Mean free path length for the assemblies of rigid fibres As Figure 7-16 shows there is a clear relationship between the mean free path length and the packing efficiency of the fibrous assemblies. This relationship depends only on the packing efficiency of the assembly and not on the aspect ratio of the fibres, although in Chapter 6 it was shown that the packing efficiency of fibrous particles depends to a large degree on the aspect ratio of the fibres. The single point on the graph for the mono-size spheres represents the mean free path of an assembly of spheres having a spherical particle diameter the same as the fibre diameter. This point shows the agreement between the relationship of packing efficiency mean free path length for both the spheres and fibres. The curve shown in Figure 7-16 is tending towards the value of 0.39 times the particle diameter for the mean free path length at a packing efficiency of 64%. The mean free path length of the non-rigid fibres followed the same trend as for the rigid fibres, as seen in Figure This figure once again confirms the excellent relationship between the mean free path length and packing efficiency for the fibrous materials. 156

167 Mean Free Path Length (Fibre Diameters) % 10% 20% 30% 40% 50% 60% 70% Packing Efficiency Rigid Fibres Non-Rigid Fibres Figure 7-17: Comparison between mean free path length for rigid and non-rigid fibres 7.8 Permeability Through the use of the simulation algorithms assemblies of fibrous particles of varying aspect ratio have been successfully modelled. The data that was then extracted from these models, in the form of packing efficiency and mean free path length, can then be used to predict the permeability of these assemblies using the equations derived in Chapter 5 of this thesis. The two equations are; 2 2 dp 22.82µ f V (1 #! ) 0.68" f V (1 #! ) = + (5.17) 3 3 dx X! X! 2 d d 2 X d 3! = 27.38µ (1 "! ) k (5.18) f 2 157

168 Dimensionless Permeability Experimental Modified Ergun Modified Kozeny-Carman % 10% 20% 30% 40% 50% 60% 70% Packing Efficiency Figure 7-18: Permeability predictions using the modified equations including mean free path length Figure 7-18 shows that the permeability predictions using the modified equations, which are based on the mean free path of the assembly, are a poor estimate of the actual permeability of fibrous materials. The predictions shown in Figure 7-18 are actually worse than if a simple shape factor was used, as shown in Figure 6-8 in Chapter 6, with the predictions shown in Figure 7-18 at the low end of packing efficiency over estimating the permeability by as much as 3 orders of magnitude. This is in fact a general trend for the modified equations in that they over estimate the permeability of the fibrous materials. The result of this is that the assemblies of fibrous particles in an experimental situation actually offer a greater resistance to flow than what the models predict. The reasons contributing to the poor predictions by the equations including mean free path length are not immediately obvious, but a factor that has not been fully explored is the connectivity of the voids throughout the material. The connectivity of the voids can be expressed by the term tortuosity, defined as the length the flowing fluid actually takes through the bed normalised by the bed height. The original equation developed by Kozeny [41] and extended by Carman [8] included a value for the parameter representing the materials' tortuosity. As a result of a lack of better data, the Kozeny-Carman equation and 158

169 the modified equation, appearing in Equation (5.18), uses a value for the tortuosity for a bed of mono-sized spherical particles. The tortuosity is most likely not the same for a bed of fibrous particles as it is for a bed of spherical particles, and a higher value for the tortuosity will result in a decrease in the permeability. This is obviously due to the fact that if the flowing fluid has to take a longer and more curved path, then the pressure drop through the material will also be greater. This concept of tortuosity is explored in further detail in the following chapter. 7.9 Conclusions The numerical method was successful in the simulation of assemblies of fibrous particles. Assemblies of rigid fibres of varying aspect ratio were simulated using the optimisation algorithm, and using a computational technique it was possible to achieve a range of packing efficiencies for each aspect ratio. The range of packing efficiencies that could be achieved for each aspect ratio was found to encompass most of the existing experimental results for rigid fibres which shows that the differences between those sets of experimental results is due to the orientation of the fibres. The simulation of non-rigid fibres required the use of an alternative algorithm to the one used for rigid fibres. This was due to the optimisation routine not being able to successfully simulate the non-rigid fibres using the cubic polynomial model as a result of the indirect relationship between the variables being optimised and the fibre coordinates. The alternate model that was used, that utilised a combination of 3D geometry and basic dynamics, was able to simulate non-rigid fibres over the same aspect ratio as the rigid fibres but was not able to show any real improvement in the packing efficiency. This lack of increase in the packing efficiency is due to the already high packing efficiency of the rigid fibres and also the constraints of the sequential addition simulation method that was used. Through the simulation of fibrous particles the same data could be extracted that was discussed for the assemblies of spherical particles. By finding the mean free path length for each of the fibrous assemblies a relationship was established that showed the dependence of the mean free path length on the packing efficiency. This dependence indicates that the 159

170 mean free path length is not a function of the aspect ratio but only the packing efficiency, although the results indicated that the packing efficiency is obviously determined to an extent by the aspect ratio. The use of the mean free path length in the modified permeability equations that were developed for spherical particles resulted in poor predictions for the fibrous particles. The equations were successful for the spherical particles, but over estimated the permeability of the fibrous assemblies by as much as three orders of magnitude. The result of this is that the equations do not predict enough of a resistance to flow when compared to experimental results, and one possible factor to account for this is the tortuosity of the assembly. The modified equations that appear in this chapter use a value for the tortuosity determined for assemblies of spherical particles, and so if the tortuosity for fibrous particles is higher than for spherical particles the permeability equations would predict a higher resistance to flow. An investigation has been undertaken into the tortuosity of assemblies of particles which is the direction for the following chapter. 160

171 Chapter 8 - TORTUOSITY 8.1 Introduction It has been clearly established throughout this thesis that the permeability of porous materials is dependant upon both the amount and structure of the voids within the material. The amount of voidage can be expressed using well defined measures such as the void ratio, or volume solid fraction, while the structure tends to be defined in a more equivocal way. Previous chapters have shown that the particle (or fibre) diameter is not always adequate in describing the size of the voids, and so the parameter mean free path length was introduced. However, it was seen in the previous chapter that this single parameter is not enough to predict the resistance of the material to flow. One reason for this lies in the descriptor for the connectivity of the voids. One measure that is often used to describe the connectivity of the voids is the tortuosity of the pore structure. This parameter was first introduced by Carman [9], and is defined by the ratio of the distance travelled by a fluid particle permeating through the material to the overall distance travelled in the direction of the pressure gradient. Obviously this is an important (although difficult to determine) parameter that has an influence on the permeability of a material. Of the two most widely known equations for predicting the permeability of granular materials (Ergun [26], and Kozeny-Carman [8]), only the Kozeny- Carman equation explicitly uses this measure. This would imply that the Ergun equation implicitly assumes that all granular materials have a constant tortuosity, or that the tortuosity is uniquely determined (and included in the correlation) from the void ratio, or particle size. While the tortuosity is explicitly used in the Kozeny-Carman equation, very little guidance is given as to suitable values. From his experimental work, Carman observed that the path a fluid takes in a random packing of mono-size spherical particles is approximately at an angle of 45 degrees to the superficial flow direction. This tends to suggest that a value of 161

172 2 is acceptable for mono-size spherical assemblies. The lack of other relevant data has tended to mean that this value is used almost as a default for any granular material regardless of its morphology or size distribution. While this may be accurate enough for many granular materials, care must be taken when the equation is applied to highly irregular particles such as fibrous particles. An approach that has been used in the past to investigate the tortuosity of spherical particles is the use of Delaunay empty spheres [66]. The Delaunay empty sphere [19] is defined as a sphere that fills a void and is tangent to at least four particles without overlapping any other particle in the assembly. The method used in [66] consists of finding all of the Delaunay empty spheres within the assembly, and then linking a series of the Delaunay empty spheres to find a pathway through the assembly. Each two of the Delaunay empty spheres that are linked are connected by two truncated cones, with the radius decreasing to a minimum value and then increasing again to the radius of the second empty sphere. When conducting this process the authors in [66] used the most direct path between any two Delaunay empty spheres to characterise the tortuosity. The authors studied both disordered and partly ordered packings, finding that the tortuosity for the disordered packings was approximately 1.44 for a packing efficiency of 57% and decreased to 1.38 for dense random packing (64%). This value is in reasonably good agreement with the value of predicted by Carman [9]. The direction for this chapter is to examine the void structure within the assemblies of spherical and fibrous particles, using a similar method to the one used in [66], but altered slightly to form continuous chains of voids through the assemblies. In this investigation, the tortuosity (as defined by Carman) will be used as a measure which is particularly relevant to the permeability of these materials. The tortuosity will be principally investigated in the direction of the global pressure gradient experienced in the laboratory (axial direction for a cylindrical container of particles) but the tortuosity in the radial direction will also be considered. This is principally a numerical investigation used as a means to compare the connectivity of the voids for the different assemblies presented in earlier stages of this thesis. 162

173 8.2 Numerical Method The main focus of this study is to calculate the tortuosity for each assembly in the direction of the global pressure gradient. However, also of interest is finding the tortuosity for each assembly in a direction perpendicular to the global pressure gradient. By doing this it will give an indication of the degree of isotropy of the assembly. The assemblies that will be considered in this study are only those ones created using the Hertz-Gravity simulation method, and as such are contained in a cylindrical container. To best match experimental conditions the global pressure gradient is defined to be in the axial direction while the direction perpendicular to this is the radial direction Axial Direction The algorithm used to find the tortuosity of each assembly of particles was based on finding a continuous chain of voids through the assembly. The main difference between the algorithm used here and the one presented in [66] is that this algorithm places a sphere in a void, which is in contact with four particles, and then the next one is placed in a void but must stay in contact with the void before hence making a chain of voids. The method used in [66] approximated the void space between two voids that were not in contact by two truncated cones. An explanation of the algorithm used in this thesis is discussed below. 1. A random location was chosen in the void space at the bottom of the assembly, typically in the region between the floor of the container and one particle (or fibre) diameter in height. This location serves as the starting position for the first void in the chain. 2. Using a small step size, typically one thousandth of a particle diameter, the radius of this void was increased until it made contact with a particle in the assembly. 3. The diameter of this void is further increased while moving its centre to ensure it maintains (tangential) contact with the particle. This process continues until the void makes contact with a second particle. 4. Once the void had made contact with a second particle, its size is further increased while moving its centre to ensure it maintains contact with the two particles. This 163

174 process is slightly more difficult than maintaining contact with a single particle, as the direction in which to move is more difficult to determine. The direction is approximated by the vector addition of unit vectors defined by the positions of the centres of the particles and the void (shown in Figure 8-1). In the case of mono-size spheres this direction exactly defines the direction to move the centre of the void. If the two particles have different diameters, the direction is an approximation and the location of the void is further adjusted to maintain contact with both particles (after each diameter increase). u 1 u +u 1 2 u 2 Figure 8-1: The direction in which the centre of a void (small circle) must move in order to maintain tangential contact with two particles (larger circles) as its diameter increases. This direction is approximated by adding together the unit vectors that point towards the centre of the void from the centre of the two particles. Note that, while this is demonstrated in two dimensions, an analogous procedure can be defined in three dimensions and for simultaneous contact with three particles. 5. The process is repeated until the void makes contact with a third particle. In the two-dimensional case (eg, Figure 8-1), once contact had been made with a third particle the location and diameter of the void is fixed. 6. In three dimensions the void can continue to expand (using a similar process) until it contacts a fourth particle. This represents the maximum size of the void that is possible from the given starting point (note that the final void may not contain the original starting point). 164

175 7. A number of starting locations for the next void are then chosen randomly on the surface of the upper hemisphere of the previous void. For each starting location, a void is expanded into the void space above according to steps 3-6 listed above. The new void must maintain contact with the previous void, and so the process ends when the new void contacts three particles. 8. From the various starting positions of step 7, two to three possibilities for the next void in the sequence are usually presented. The void with the highest coordinate in the vertical direction (defined by the gravitational potential) is selected. 9. This process of finding successive voids continues until the path had passed though the entire assembly of particles. A two dimensional representation of the results of this process is shown in Figure 8-2. Figure 8-2: A two dimensional void chain (white circles) passing through an assembly of particles (grey circles). The void chain begins at the bottom of the assembly. With a focus on the permeability of granular materials, these void chains can be used to interrogate the structure of the voidage in these materials. The quantitative parameters that are of interest include; the tortuosity of the path, defined by the sum of the diameters of the 165

176 particles divided by the height of the bed, and the average diameter of the chain of voids. Qualitative understanding can also be gained from examining the appearance of the paths Radial Direction To find the tortuosity in the radial direction the chain of voids is created in the same manner as in the above discussion. The only change is of course the direction in which the chain of voids is moving. For each assembly of particles a square cross-section was taken through the height of the container and void chains were created from one side to the other. Figure 8-3: Cross-sectional view of cylindrical container indicating radial direction The process of finding chains of voids in the radial direction is described below. 1. An arbitrary square cross-section was defined for the assembly of particles 2. A number of void chains were then found, using the process described in the previous section, from one side (A) to the other (B). Each chain of voids was found in the horizontal plane, representing a fluid flowing from side A to side B. The height of each void chain, in the vertical direction, was random and could vary between the top and bottom of the container. 3. A second square cross-section was then found that was rotated at an angle of 10 degrees about the centre of the cross-section. 4. Step 2 immediately above was then carried out again, with this process being repeated a total of ten times until a square cross-section was located at angle of 90 degrees from the original square cross-section. By rotating from 0 to 90 degrees ensured all radial directions were considered. 166

177 Using this method of finding a chain of voids in different radial directions allowed for a more complete understanding of the connectivity of the voids in each assembly. This knowledge also indicated the degree of isotropy of the bed of particles. 8.3 Tortuosity The tortuosity of an assembly of materials can be found using the previously described algorithm. The tortuosity results will first be presented for the assemblies of spherical particles presented in Chapter 5 and then for the assemblies of fibrous particles presented in Chapter 7. The images that appear in the following sections that show the typical tortuous paths for each of the materials have been created by using conical surfaces to blend between the individual voids that define these paths Spherical Particles Assemblies The assemblies of particles studied include mono-size particles, binary mixtures of spherical particles and distributed mixtures of spherical particles. The assemblies that will be studied will be limited to the Hertz-Gravity models as they represent the physical packings studied in the laboratory compared to the spherical assemblies produced from the Lenard-Jones algorithm. A number of paths were found through the mono-size assemblies for the diameters 0.1, 1.0 and 10.0 as presented in Chapter 4. Typical examples of the paths can be seen in Figure 8-4 where each of the paths has been normalised to allow for a direct comparison. 167

178 Figure 8-4: Examples of the tortuous paths through assemblies of mono-sized particles of diameter 0.1, 1.0 and 10.0 (left to right) As Figure 8-4 shows, there is no discernible difference in the shape or length of the tortuous path through the assemblies of different sized diameters. The only difference in the paths of the assemblies of varying sized diameters was the path diameter, which is completely expected as the voids surrounding the particles increase in size as the particle diameter also increases. By considering all of the void diameters on each of the paths found through the assemblies the distribution of void diameters can be found. An example of the distribution can be seen in Figure 8-5, where the void diameters have been normalised by the particle diameter. The figure shows that the distribution most closely resembles a gamma distribution as was the case with the mean free path lengths of the assemblies. The average void diameter was then chosen as the void diameter that had the highest frequency. A summary of the results for the tortuosity and the average void diameter for each of the mono-size assemblies can be seen in Table

179 Figure 8-5: Distribution of void diameters on the tortuous paths Table 8-1: Tortuosity results for mono-size assemblies Particle Diameter (Program units) Tortuosity Average Path Diameter (mm) The results in the table show a good consistency between simulations of different sized particles for both the tortuosity and the average path diameter. It can be concluded from these results that the tortuosity is about 1.23 and the average path diameter is about 22% of the particle diameter. The method of finding Delaunay empty spheres used in [66] found the average diameter of the voids to be 38% of the particle diameter for dense random packing. The value of the mean free path found in Chapter 5 of this thesis was 39% of the 169

180 particle diameter, which shows a good agreement with the results of [73]. The good agreement between the mean free path method and the Delaunay empty spheres method can be attributed to the fact that the void sizes in packings of spherical particles are regular and all of a similar size. The reason for the lower ratio from the method presented in this chapter is that the voids are constrained to pass through the capillaries between the voids as well as the voids themselves. As a result of this constraint the size representation using this method is not a measure of the Delaunay void sizes. The method of finding the tortuosity and average path length of mono-size assemblies can be applied in the same manner to the other spherical particle assemblies presented in Chapter 5. These assemblies were binary mixtures of ratio 1:2.05 and also distributed mixtures of glass beads, lead shot and sand having a sphericity of Some images of the tortuous paths through the binary assemblies are presented in Figure 8-6. Figure 8-6: Tortuous paths for binary assemblies; mass ratio (L to R) 0.17, 0.37, 0.47, 0.68,

181 60% 1.50 Average Path Diameter (percentage of small particle diameter) 50% 40% 30% 20% 10% Average Path Diameter Tortuosity 1.25 Tortuosity 0% Mass Ratio Figure 8-7: Tortuosity and the average path diameters for binary mixtures of spherical particles. Figure 8-7 shows all of the results for the binary assemblies for the tortuosity and average path diameter. It was found that the tortuosity remained constant through all of the binary assemblies and that the value of the tortuosity, 1.23, was the same as the value found for the mono-size assemblies. The average diameter shown in Figure 8-7 was found by fitting a gamma distribution, and is shown in the graph as a percentage of the small particle diameter. As with the mono-size particles, the path diameter at zero mass ratio (mono-size particles) was found to be 22% of the particle diameter. As the mass ratio increased, and the number of large particles in the assembly increased, the average path diameter also increased. This follows the same trend as the mean free path length found in Chapter 5 and is due to the size of the voids increasing. The final sets of spherical particle assemblies studied were of a distributed nature. This included the simulations with the same particle size distributions as the three sets of normally distributed glass spheres and the five lead shot mixtures that were discretely distributed with a limited amount of sizes in each mix. The distributions for each assembly were given in Chapter 4. Images of the tortuous paths can be seen in Figure 8-8, where the lengths of the paths have once gain been normalised to view them at the same scale. The results for each of the assemblies can be seen in Table

182 Figure 8-8: Tortuous paths for distributed assemblies; (L to R) Glass beads set 1, 2, 3, Lead shot mixtures 1, 2, 3, 4, 5 Table 8-2: Tortuosity results for distributed mixtures of spherical particles Mixture Tortuosity Average Path Diameter (mm) Glass set Glass set Glass set Lead Mix Lead Mix Lead Mix Lead Mix Lead Mix Figure 8-8 shows that the paths through the assemblies of distributed spherical particles are once again of the same shape as for the mono-size and distributed mixtures, with Table 8-2 confirming the magnitude of the tortuosity. The average path diameter in Table 8-2 is given 172

183 in units of millimetres, and represents the average diameter of the paths for the assemblies of spherical particles with dimensions the same as those given in Chapter 4. The average path diameters were once again found by fitting a gamma distribution to the data. These results for distributed mixtures confirm that the tortuosity for an assembly of spherical particles, regardless of the distribution of particle sizes, is constant. The average path diameter is a function of the particle diameter and reflects the fact that as the particles increase in size so do the voids. Finally the tortuosity of a bed of particles in the radial direction was also found. For all of the assemblies of spherical particles it was found that the assemblies were isotropic, meaning that the tortuosity was the same in the radial direction as in the axial direction. The following figure illustrates the tortuosity in the radial direction, as found by the method described previously, for varying angles of rotation in the horizontal plane (for an assembly of mono-size particles). Tortuosity Angle of rotation (degrees) Figure 8-9: Tortuosity in the radial direction for a packed bed of spherical particles As Figure 8-9 shows the tortuosity is constant through the varying angles in the horizontal plane, with the magnitude of the tortuosity being equal to the tortuosity found in the axial direction. As the particles are spherical in shape this result is expected as the orientation of the particles is irrelevant. It was also found that the void diameter in the radial direction was the same as in the axial direction. 173

184 This method of finding tortuous paths through the assemblies of particles results in an alternate method for classifying the size of the void space that exists in the assembly. The average size of the voids for a mono-size assembly was 22% of the particle diameter using the method described in this chapter, whereas using the mean free path length method discussed in Chapter 5 it was 39% of the particle diameter. A comparison between both methods was carried out for all of the spherical particle assemblies and the results are shown in Figure Mean free path length (mm) Mono-size Binary Distributed Average path diameter of the tortuous paths (mm) Figure 8-10: Comparison between mean free path length (chapter four) and the average diameter of the tortuous path (this chapter). The figure shows that the relationship between the two methods is the same for all of the assemblies, as the data reflects a straight line through the origin. This comparison serves as a good indication of the reliability of the two algorithms as the same ratio of magnitudes is found over a wide range of packing efficiencies Fibrous Particle Assemblies The same method that was used for the spherical particles was used for assemblies of fibrous particles. The fibrous assemblies studied consisted of only the rigid fibres as the same packing efficiencies were achieved with the non-rigid fibres. It was found using this method that the tortuosity for assemblies of fibrous particles was higher than for assemblies of spherical particles. Examples of the paths can be seen in Figure 8-11 (aspect ratios 5-20) and Figure 8-13 (aspect ratios 25-40). 174

185 Figure 8-11: Tortuous paths for assemblies of fibrous particles (L to R) Aspect ratio 5, 10, 15, 20. It is immediately apparent that the path lengths through the fibrous assemblies are much longer than for the assemblies of spherical particles. The main reason for this is that the fibres tend to orientate themselves horizontally and in the same direction. In these assemblies it is often found that three fibres can essentially form a passage through which there is an opening at each end only (Figure 8-12). When three fibres aligned themselves in this manner the path passing through the assembly has long straight sections as seen in Figure 8-11, most notably in (b). As the fibres increase in length the likelihood of them aligning decreases, which leads to a decrease in tortuosity. The average lengths of the tortuous paths for each aspect ratio are presented in Figure 8-14, where the packing efficiencies are as presented in Chapter

186 Figure 8-12: Illustration of fibres aligning (grey), and the voids (red) travelling through passageway Figure 8-13: Tortuous paths for assemblies of fibrous particles (L to R) Aspect ratio 25, 30, 35,

187 Tortuosity AR=1 AR=10 AR=5 AR=15 AR=20 AR= AR=30 AR=35 AR=40 0 0% 10% 20% 30% 40% 50% 60% 70% Packing Efficiency Figure 8-14: Tortuosities for varying aspect ratios of varying packing efficiency As Figure 8-15 shows, as the aspect ratio increases the tortuosity also increases until some length where the orientation of the fibres became random enough so that few passages form. This occurs roughly at an aspect ratio of 20. For assemblies of fibres longer than this the tortuosity decreases, although it remained at all times greater than the tortuosity for spherical particle assemblies. The void diameters in the fibrous assemblies tended to follow a gamma distribution as initially observed in the spherical packings. The most frequent path diameter (normalised by the fibre diameter) is shown as a function of the packing efficiency in Figure 8-15 (for all of the assemblies examined). 177

188 2.5 Path Diameter (fibre diameters) AR=5 AR=15 AR=25 AR=35 AR=10 AR=20 AR=30 AR=40 0 0% 10% 20% 30% 40% 50% 60% 70% Packing Efficiency Figure 8-15: Variation in the path diameter for varying aspect ratios It is apparent from Figure 8-15 that there is a large variation in the path diameter with the packing efficiency. As Figure 8-11 and Figure 8-13 show, it is clearly visible that the path diameter is larger for higher aspect ratio fibres (mostly due to the increase in void ratio). For the fibres with an aspect ratio of 40, at the lowest packing efficiency the path diameter is approximately 2 fibre diameters. While this may seem large, it should be noted that the assembly has a void ratio of 91.2%. This set of results indicates that the tortuosity of the fibrous materials depends on both the packing efficiency and the aspect ratio, while the mean path diameter seems to depend mainly on the packing efficiency of the assembly. The tortuosity in the radial direction for the fibrous particles showed that the assemblies were clearly anisotropic. An example of the tortuosity in the radial direction is shown in Figure 8-16 which is for an assembly of aspect ratio 15 fibres which have a packing efficiency of 42.6%. 178

189 Tortuosity Angle of Rotation (degrees) Figure 8-16: Radial tortuosity for aspect ratio 15 fibres with a packing efficiency of 42.6% As Figure 8-16 shows the tortuosity is dependent on the direction in the horizontal plane. At the angles of zero and 90 degrees, the direction of flow is perpendicular to the majority of the fibres. In contrast to this, at the angle of 50 degrees the direction of flow is parallel to the fibres which means the fluid can travel straight through the passageways formed by the aligned fibres. The following images illustrate this. Figure 8-17: Tortuous paths in the radial direction that are perpendicular to the majority of the fibres 179

190 Figure 8-18: Tortuous paths in the radial direction that are parallel to the majority of the fibres Figure 8-17 shows an example of two paths that are in the radial direction that are travelling in a direction that is perpendicular to the majority of the fibres. When this is the case the tortuosity tends to be greater than when travelling in a direction parallel to the direction of the fibres, as shown in Figure The tortuosity tends to be greater due to the fluid having to wind its way over and under the fibres, whereas when it is travelling in the same direction as the fibres the fluid is able to travel in the passageways formed by aligned fibres (seen as long straight sections in Figure 8-18). As the above images show, the tortuosity in the radial direction varies according to the direction of the fibres. Furthermore, there is also a difference between the tortuosity in the radial direction and tortuosity in the axial direction. The tortuosity in the axial direction for the assembly studied in Figure 8-16 was 2.68, whereas in the radial direction it varied between 1.1 and This comparison highlights the fact that the assembly of particles is anisotropic, which is caused by the orientation of the fibres. All of the assemblies of fibrous particles that were studied displayed some degree of anisotropy. 8.4 Permeability The tortuosity results presented in this chapter can be used to modify the existing Kozeny- Carman equation to predict permeability. The Kozeny-Carman equation as presented in its original form is; 2 3 d " k = (2.23) b0 (1 # " )! 180

191 Where the parameters have been explained previously in earlier chapters. When Carman originally developed the equation it was noted that b 0 was usually grouped together with the square of the tortuosity and a value of 5 was found experimentally by fitting the above equation to actual data. The above equation can be modified by using the data for assemblies of mono-size particles. The modifications proposed to the equation are; Replace the particle diameter with the average path diameter as the relationship between the two for spherical particles is known Find a new coefficient to replace the value of 36b 0 using the value of tortuosity for spherical particles In the modification of the equation, the numerator and denominator will be treated separately to simplify the process. The parameter d p is used to denote the average path diameter through the assemblies of particles. From the assemblies of mono-size particles the relationship between the particle diameter and the average path diameter is; d = 4. 55d p (8.01) Which means the numerator of (2.23) can be rewritten as 2 p d!. Using the value of the tortuosity for spherical particles, 1.23, found using the method described in this chapter, a new coefficient of the denominator of (2.23) must be found such that; K d! 2 = 180 (8.02) Where the value of b 0 has been absorbed into the coefficient of the denominator, K d. Substituting the value of the tortuosity into this equation yields a value of 119 for K d. The form of the modified Kozeny-Carman equation is now; 181

192 2 3 d p " (1 # " )! k = (8.03) Equation (8.03) allows the permeability of a material to be predicted with the knowledge of the average path diameter through the material, the tortuosity of the material (as defined by the method presented here) and the void ratio. The equation will now be applied to the distributed mixtures of spherical particles simulated in Chapter 5 and also the assemblies of fibrous particles simulated in Chapter Spherical Particles The modified equation presented in Equation (8.03) was used to predict the permeability of a range of materials, starting with mono-size and binary mixtures of spherical particles. The results for mono-size (two sizes, mass ratio 0 and 1) and binary mixtures can be seen in Figure Permeability (m 2 /kpas) Experimental Modified Kozeny-Carman Mass Ratio Figure 8-19: Permeability predictions for mono-size and binary mixtures of spherical particles As stated earlier, the mass ratios between the experimental and simulated assemblies were not quite the same due to the nature of the simulation algorithm. However, the results still show a good agreement between the simulations and the experimental results. Although it is a little hard to calculate the exact error due to the permeability values not being at the same mass ratios, an approximate estimate can be made by interpolating between 182

193 successive data points and comparing values. Using this method and the error as defined in Equation (B-1) resulted in an average error of 18%. The mean free path method presented in Chapter 5 returned an average error of 13%, although the method used in this chapter is preferred as it uses all of the parameters from the specific assembly rather than any empirical factors. The same equation can be applied to the sets of distributed mixtures for the lead shot and glass beads. The permeability predictions are shown in Table 8-3. Table 8-3: Permeability predictions for distributed mixtures Mixture Experimental K-C K-C Error Permeability Permeability (%) (m 2 /kpas) (m 2 /kpas) Glass set Glass set Glass set Lead mix Lead mix Lead mix Lead mix Lead mix Once again, good agreement between simulations and experimental data was achieved, with the average error being about 11%. In comparison with the mean free path method that predicted permeability the magnitude of the errors was very similar, although once again this method is preferred as it uses actual values of the tortuosity found from the assemblies of particles. The comparisons between this method that includes the tortuosity and the mean free path method will always be similar for assemblies of spherical particles as the tortuosity for mixtures of spherical particles is constant. However, the difference between the two methods becomes more evident in the following section on fibrous materials. 183

194 8.4.2 Fibrous Particles The real advantage of the method presented in this chapter lies in the prediction of permeability for assemblies of fibrous particles. The inclusion of the tortuosity helps to make the permeability prediction more accurate as it was found that assemblies of fibrous particles have values for tortuosity vastly different from assemblies of spherical particles. The permeability predictions for the aspect ratios simulated in Chapter 6 can be seen in the following figure. Permeability (Dimensionless) Experimental AR= AR= AR=15 AR= AR=25 AR= AR= AR=40 Mean free path method % 10% 20% 30% 40% 50% 60% 70% Volume Solid Fraction Figure 8-20: Permeability predictions from varying aspect ratio fibrous particles Figure 8-20 shows that at the lower aspect ratios the agreement between the experimental results and the predicted ones are quite good. At the higher aspect ratios there is a discrepancy between the two with the worst case scenario being two orders of magnitude in difference. However, this is still an improvement on the predictions made using the mean free path method presented in Chapter 7. It can also be seen that at the higher aspect ratios (lower packing efficiencies) there is a much wider range in permeabilities predicted from the equation using the tortuosity method. This large variation in the permeability is due to large variations in the path diameter and to a lesser extent variations in the tortuosity for the higher aspect ratios. 184

195 To further investigate the large differences between the predictions using the tortuosity and the experimental results the radial tortuosity was plotted for each aspect ratio. The values that appear in the following figure represent the average tortuosity for each assembly in the radial direction Radial Tortuosity AR=1 AR=5 AR= AR=15 AR=20 AR= AR=30 AR=35 AR=40 0 0% 10% 20% 30% 40% 50% 60% 70% Volume Solid Fraction Figure 8-21: Radial tortuosities for varying aspect ratios of varying packing efficiency Careful inspection of Figure 8-21 in conjunction with Figure 8-20 reveals that the assemblies for which the permeability is grossly over predicted are those ones that have a much higher value for the radial tortuosity than the assemblies of spherical particles. This is obviously a factor that affects the permeability of the assembly of particles but has not been taken into account in the modified Kozeny-Carman equation. Another point to consider regarding the tortuosity is the relationship between the axial and radial tortuosity. A graph showing the ratio of the axial tortuosity to the radial tortuosity is shown in the following figure. 185

196 Ratio of axial tortuosity to radial tortuosity Ratio of Axial tortuosity to Radial tortuosity Predicted permeability (using axial tortuosity) Experimental results 0 0% 10% 20% 30% 40% 50% 60% 70% Volume Solid Fraction Dimensionless permeability Figure 8-22: Comparison between axial and radial tortuosity and the relevance to the permeability predictions In Figure 8-22 the solid dots represent the ratio of the axial tortuosity to the radial tortuosity. A value greater than one (which covers most assemblies) indicates that the axial tortuosity is greater than the radial tortuosity while the reverse is true for a value less than one. The permeability predictions along with the experimental results are also plotted in the figure corresponding to the secondary axis. It is noticeable that the assemblies that have a greater radial tortuosity than axial tortuosity are the ones that correspond to large differences between the predicted and experimental tortuosity. At this stage it is not known how to quantify these results but qualitatively it can be said that there is a higher level of inhomogeneity amongst the void structure which leads to an increased resistance to flow (compared to what the modified equation predicts). One final point specifically related to the permeability of porous materials is that it may not be sufficient to represent the tortuous nature of the path taken by the length of the path alone. Another factor that is closely related to the previous discussion may be the frequency and magnitude of the changes in direction of the path. The impact of this on viscous flows may not be significant, however at higher Reynolds numbers (inertial flows) this feature of the structure is possibly one of the main factors influencing the pressure drop. 186

197 The magnitude and frequency of direction changes for the paths in the axial direction was examined directly in the assemblies discussed earlier. This was characterised by the angles at which the path changes at every void along a chain. Practically, the angle change was determined from the dot product of successive vectors joining the centres of voids along the path. The results of this process (for both the spherical and fibrous materials) are shown in Figure For the spherical and low aspect ratio assemblies, the change in angle is on average between degrees and did not vary greatly with the packing efficiency. As the fibres get longer (and the tortuosity increases), the average change in angle shows a lot of variation. The highest values of the average angle change are for high aspect ratios, and low packing efficiencies. It was earlier noted that the tortuosity of these materials (calculated using our approach) starts to decrease as the packing becomes less efficient. In other words, the total length of the path is decreasing with the void ratio, however it is changing direction more severely (and more often). The overall effect of this on the permeability is unknown at this stage. 60 Average Angle change (degrees) AR=1 AR=10 AR=20 AR=30 AR=40 AR=5 AR=15 AR=25 AR=35 0% 10% 20% 30% 40% 50% 60% 70% Volume Solid Fraction Figure 8-23: Average angle change between consecutive points on the void paths. 187

198 8.5 Conclusions The tortuosity of assemblies of spherical particles, including mono-size, binary and normally distributed assemblies, was found to be constant. The constant value was found to be approximately 1.23 which is slightly smaller than the experimental value found by Carman [8] of The main reason for the difference between the two results is that the path found by the algorithm in this study does not necessarily represent the actual path taken by the flowing fluid. However, the algorithm does provide an effective means for comparisons between different assemblies. The average path diameter of the tortuous paths was found to vary with particle size, as would be expected. The application of the same algorithm to the assemblies of fibrous particles led to an interesting result. It was found that the tortuosity of the fibrous particle assemblies was greater than for assemblies of spherical particles, with some assemblies showing an increase of 250%. For some of the aspect ratios, the combination of the container size and fibre length resulted in groups of fibres aligning, as illustrated in Figure When this occurred it was found that this is when the tortuosity increased the most compared to the tortuosity of spherical particles, due to the path taking a winding and less direct route through the material. As the fibres got larger in aspect ratio it was found that the packing of the particles was random and dilute enough so that there was little chance of any fibres aligning with each other, which meant that the tortuosity was smaller than for the lower aspect ratios when there was a high chance of fibres aligning. By also considering the radial tortuosity of the assemblies it was found that while the spherical particle assemblies were isotropic the assemblies of fibrous particles were not. The average void diameter of the tortuous paths generally varied with aspect ratio the larger the aspect ratio the larger the average void diameter. Modifying the Kozeny-Carman equation based on the results for spherical particles and including tortuosity provided moderate success for the assemblies of fibrous particles. At the higher end of the packing efficiencies the agreement between theory and experiments was quite good, much better than the mean free path method. It was only at the packing 188

199 efficiencies less than about 15% did the theory start to break down, with the modified Kozeny-Carman equation over predicting the permeability by as much as 2 orders of magnitude. However, the results were still better than the mean free path method. The discussion also highlighted two possible reasons for the failing of the theory at the lower packing efficiencies. These reasons included the larger than average values for the radial tortuosities and larger angle changes between consecutive voids on each void chain. It is clear that the connectivity of the voids in different assemblies is quite a complex subject and a more in depth study must be undertaken to obtain a greater understanding of this area. 189

200 Chapter 9 - CONCLUSIONS This thesis has addressed the theme of permeability and its dependence on parameters such as the void ratio of the material as well as the average size of the voids. The knowledge of the permeability of any given material is important in many fields and must be considered in the design of filters, pneumatic conveying systems and silos to name a few. General equations for the prediction of permeability, or the underlying property determining permeability, the pressure gradient, have been in existence since the 1950s. The most well known and used equation to determine permeability is the Kozeny-Carman equation while the Ergun equation considers an inertial component as well as a viscous component to determine the pressure gradient. Both of these equations have been used with some success over the years in a diverse range of fields. These equations are based on a conduit flow type assumption that assumes the fluid flows through cylindrical channels within the material. The key properties in these types of equations are the void ratio, which indicates the total amount of void space, and the particle diameter of the material, which implies the average size of the voids. In Chapter 3 of this thesis a number of problems were highlighted that indicate specific applications of these types of equations where the void ratio and particle diameter are not adequate in determining the permeability. The failure of these equations in these situations can be attributed to the poor characterisation of the average void space by the particle diameter term. The underlying reason for the particle diameter being a poor estimate is due to it being not very well defined for materials that differ from the ideal case of mono-size spherical particles. A shape factor has been used in the past with moderate success, but as a tool to predict permeability it is useless, as it was shown that the shape factor calculated by the equation is very different from the shape factor found by empirically fitting the equations to experimental results. 190

201 As a tool to further study permeability and its relationship with average void space numerical simulations were undertaken. These simulations started with the ideal case of mono-size spherical particles and were based on the Lennard-Jones energy potential which determines the attractive and repulsive force between any two particles. To simulate a packed bed of particles an optimisation method was used in which the Lennard-Jones energy potential was summed for each particle in the assembly and optimised accordingly. The particles were introduced sequentially into the assembly with the final result leading to a packing efficiency that was equal to dense random packing of mono-size spherical particles. As well as using a comparison to dense random packing to validate the simulation method, the radial distribution function was also considered. A comparison between the radial distribution function from the simulations and the experimental results achieved by Finney [28] indicated that the simulated assembly was a very good representation for an actual physical bed of particles. However, due to the nature of the energy potential the final assemblies of particles resulting from the Lennard-Jones energy potential were a spherical shape. To more closely resemble the structure studied in the laboratory a simulation method was used in which particles were dropped into a container and the energy potential was made up of gravitational energy and Hertz contact strain energy for any two particles in contact. The container dimensions were defined by imposing constraints on the coordinates of each particle. Simulations for this method once again showed good agreement with dense random packing and the radial distribution function. The validation of the numerical simulation method for mono-sized spherical particles allowed other spherical particles to be simulated including binary mixtures and mixtures of varying distributions. The Hertz-Gravity simulation method was also used to simulate packed beds of fibrous particles. By representing a fibrous particle as a chain of mono-size spheres and with the inclusion of axial strain energy and bending strain energy an assembly of fibres could be formed. The first optimisation method that was considered was to represent each fibre by a set of polynomials in each of the three principal directions, with linear polynomials being used to simulate rigid fibres. This method for rigid fibres proved very successful for aspect ratios ranging from 5 to 40 with orientation effects also being studied. To simulate nonrigid fibres cubic polynomials were used, however this method proved unsuccessful with 191

202 the fibres not being optimised very well. An alternate method for simulating non-rigid fibres that utilised a combination of 3D geometry and basic dynamics was able to achieve packing efficiencies equal to the rigid fibres, but no improvement was made in the packing efficiency compared to the rigid fibres. This lack of improvement was attributed to the already high packing efficiency of the rigid fibres and the possibility of needing a global rearrangement simulation method to realise the advantages of non-rigid fibres. Simulating assemblies of particles was a very useful tool as the exact location of every particle within the assembly was known. By initially considering the assemblies of monosize spherical particles the mean free path length of the voids within the assembly could be extracted, and it was found that the relationship between particle diameter and mean free path length was constant. By using this mean free path length the permeability equations could be modified to include the mean free path length rather than the particle diameter. The obvious advantage of this is that the average void space is no longer implied by the particle diameter; rather it is explicitly defined by a parameter directly representing the void size. Using these modified equations that included the mean free path length for assemblies of spherical particles including mono-size, binary and distributed mixtures resulted in an excellent comparison between theoretical and experimental results. This same method was then applied to the assemblies of fibrous particles with the results being very poor. In most cases the permeability was over predicted by up to three orders of magnitude. At this stage it was necessary to consider the connectivity between the voids in the assembly. To study the connection of the voids the parameter tortuosity was used, which represented the ratio of the actual length a fluid particle travelled to the length in the direction of the global pressure gradient. An algorithm was written that found chains of voids through any given assembly from which the tortuosity could be determined. For all of the assemblies of spherical particles it was found that the tortuosity was constant, not dependent on particle size or particle composition. However, for the varying aspect ratios of the fibrous materials the tortuosity was varied and was larger than for the assemblies of spherical particles. This fact explains the success of the equations containing the mean free path length for spherical particles (the voids are connected in the same manner) and the 192

203 failure for the fibrous particles (the voids are connected differently compared to spherical particles). Using only the Kozeny-Carman equation this time, as the Ergun equation has no parameter representing the tortuosity of the material, the results for the tortuosity and average diameter of a tortuous path can be used to modify the equation. Using this method the results for the spherical particles were once again in excellent comparison to the experimental results, but this time the results for the fibrous particles were actually quite good compared to experimental results. Only at the lower end of the packing efficiencies (<15%) does the equation tend to fail. At these lower packing efficiencies there were two factors that went some way towards accounting for the discrepancy between the theory and the experimental results. The first factor was the increase in radial tortuosity for the assemblies with a low packing efficiency which was caused by a difference in the orientation of the fibres. The second factor was the average angle change between consecutive voids on the chain of voids. At these lower packing efficiencies the average angle chain is considerably larger than for spherical particles indicating a higher degree of randomness within the material. While these two factors are likely to be involved as part of the reason in the assemblies being less permeable than the modified Kozeny-Carman equation predicts, at this stage it is not known how to quantify these into tangible parameters. The end result for the modified Kozeny-Carman equation is quite a robust equation that predicts the permeability of a material which is dependent on the total void space, the average void size and the inter-connectivity of these voids. This equation is now based on the properties of the void space rather than material and no longer implies either the void size or void structure. With the current progression towards simulations that is rapidly increasing in all areas of research the theory developed in this thesis has the potential to become more relevant as a wider range of particle shapes can be simulated. The advantage this theory has over the more traditional equations such as the Ergun and Kozeny-Carman equations is that with the help of simulated assemblies it can include real parameters for the void space and structure. 193

204 Further work There are currently a number of areas that can be addressed as future areas of work directly related to the subject matter in this thesis. They have been listed below in point form. It is clear that more study is needed on the inter-connectivity of voids in different assemblies of particles for different shapes. What is needed is to determine the actual path that a permeating fluid takes through a bed of particles and how each of the axial and radial tortuosity affects this. It also needs to be investigated as to whether or not the magnitude and frequency of the angle change between successive voids on a chain of voids has an effect on the permeability. The theory developed using the tortuosity has only been tested on spherical particles and fibrous particles. The validity of this theory on other shapes of particles needs to be tested. The simulation of non-rigid fibres was only briefly addressed in this thesis due to problems encountered with the optimisation method and the simulation method of sequential addition. A global rearrangement method could be used in which all of the fibres are optimised simultaneously to study the effects of particle rigidity on the packing efficiency. As mentioned earlier in this thesis though this would need a high level optimisation routine. The simulation method used here could be extended to simulate other particle shapes. For example, the fibres could be simulated with a width of a number of particles to simulate flat platelets. 194

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214 APPENDIX A: MATERIAL PHOTOS Figure A-1: Bagasse Figure A-2: Lead Shot 2.1 mm Figure A-3: Lead Shot 2.4 mm Figure A-4: Lead Shot 3.3 mm Figure A-5: Lead Shot 3.6 mm Figure A-6: Lead Shot 4.3 mm Figure A-7: Glass Beads (mean 1.65 mm, standard deviation 0.10 mm) Figure A-8: Glass Beads (mean 2.05 mm, standard deviation 0.15 mm) Figure A-9: Glass Beads (mean 2.86 mm, standard deviation 0.20 mm) Figure A-10: Steel Nails (30mm x 2mm) Figure A-11: Wooden Dowel (40mm x 8mm) Figure A-12: Fishing line pieces 5 mm Figure A-13: Fishing line pieces 10 mm Figure A-14: Fishing line pieces 20 mm

215 Figure A-1: Bagasse Figure A-2: Lead Shot 2.1 mm 205

216 Figure A-3: Lead Shot 2.4 mm Figure A-4: Lead Shot 3.3 mm 206

217 Figure A-5: Lead Shot 3.6 mm Figure A-6: Lead Shot 4.3 mm 207

218 Figure A-7: Glass Beads (mean 1.65 mm, standard deviation 0.10 mm) Figure A-8: Glass Beads (mean 2.05 mm, standard deviation 0.15 mm) 208

219 Figure A-9: Glass Beads (mean 2.86 mm, standard deviation 0.20 mm) Figure A-10: Steel Nails (30mm x 2mm) 209

220 Figure A-11: Wooden Dowel (40mm x 8mm) Figure A-12: Fishing line pieces 5 mm 210

221 Figure A-13: Fishing line pieces 10 mm Figure A-14: Fishing line pieces 20 mm 211