Solutions of a Selection of Partial Differential Equations with Application to Micropore Diffusion and Fixed-bed Adsorption. Paul D.

Transcription

1 Solutions of a Selection of Partial Differential Equations with Application to Micropore Diffusion and Fixed-bed Adsorption by Paul D. Haynes B. App. Sc. (Hons) Supervisors Dr Jorge Aarao Prof. Phil Howlett Mr Basil Benjamin A dissertation submitted to the School of Mathematics and Statistics Division of Information Technology Engineering and the Environment University of South Australia for the degree of Doctor of Philosophy (Mathematics) April 27, 29

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3 Contents Table of Contents List of Figures List of Tables Notation Summary Declaration Acknowledgements i vii xii xv xvii xix xxi 1 Introduction 1 I Modelling the Adsorption Process 5 2 Problem description Activated carbon Physical and chemical adsorption Pore sizes, shapes and distributions Potential energy of physical adsorption Dynamic phase equilibrium Kinetics of Isothermal adsorption: adsorbent particle Fixed-bed adsorption equilibrium considerations i

4 Contents 2.8 Existing models for fixed-bed adsorption Isothermal systems with a linear adsorption rate Isothermal systems with a nonlinear adsorption rate The Thomas and Goldstein models Mathematical model used by Kim et al Effect of pore-size distribution on adsorption Scope of this thesis Modelling isothermal adsorption Particle porosity and bed voidage Conservation of mass Derivation of the differential equation of diffusion Derivation of the fixed-bed mass balance equation Alternative fixed-bed mass balance equation Average concentration inside a spherical particle Fractional approach to equilibrium Proportion of bed unused Time taken by a wave-front to pass through a fixed-bed II Micropore Diffusion in a Particle 51 4 Diffusion in a single microporous particle Diffusion with constant boundary conditions Long-time solution Evaluation of the long-time solution Simplification of the long-time solution Short-time solution Evaluation of the short-time solution Simplification of the short-time solution Short-time solution for non-spherical particles Diffusion in a finite system ii

5 Contents Model for diffusion in a sphere immersed in a finite-volume Long-time solution given by Ruthven Short-time solution given by Crank Finite system: Derivation of long-time solution Using a Laplace transform method Applying the Laplace Inversion Theorem Fractional approach to equilibrium Evaluating the p n in Equation (4.98) Finite system: Derivation of short-time solution Inverse transform of the first terms of series in Laplace domain Short-time solution associated with the n = term only Carman and Haul short-time solution Extending the solution to include the n = 1 term Finite system: Accuracy of the solutions Evaluation of the long-time solution Evaluation of the short-time solutions Finite system: Numerical solution Crank Nicolson scheme with non-dimensional variables Stability of the Crank Nicolson scheme Solving the system of Crank Nicolson equations Fractional approach to equilibrium Conclusion III Adsorption in a Fixed-bed Isothermal adsorption in a fixed-bed Mathematical model Boundary condition for the solid-phase concentration q Alternative boundary conditions Analytic solution iii

6 Contents Transforming the almost-linear system of equations Laplace transformation Inversion of the Laplace solution Obtaining the solutions c and q Numerical solution Algorithm Conservation of mass Experimental results Conclusion Model with a partitioning of the pore-size distribution Mathematical model Saturation concentration q sj for pore-size s j Large slit-shaped pores Small slit-shaped pores Cylindrical pores System of equations for isothermal adsorption in a fixed-bed Matrix form Uniqueness of solution Existence of solution Local existence theorem Global existence Hyperbolic system and characteristics Transformation along characteristics and integral form Evaluating the system of integrals iteratively Weighted pore-size and probability Adsorption and desorption rate constants Numerical solution Algorithm Simplified algorithm iv

7 Contents Conservation of mass Using the experiment parameter values Changes to model parameter-values and the breakthrough curve Breakthrough curves with a bump Explanation for a bump in the breakthrough curve Using the experiment parameter values Conclusion Conclusions and future directions Diffusion in a single microporous particle Isothermal adsorption in a fixed-bed Future directions Bibliography 256 APPENDICES 265 A Error function 267 A.1 Related identities B Verification of solutions to the fixed-bed problem 273 B.1 Solution ψ satisfies the differential equation and BC B.2 Solutions c and q satisfy the differential equations and BC C Parameter values 281 D Numerical solution summary output 283 D.1 Summary output for K = 1/8 with I = D.2 Summary output for K = 1/8 with I = D.3 Summary output for K = 1/8 with I = D.4 Summary output for K = 1/2 with I = D.5 Summary output for K = 1/2 with I = D.6 Summary output for K = 1/2 with I = v

8 Contents D.7 Summary output for K = 1 with I = D.8 Summary output for K = 1 with I = D.9 Summary output for K = 1 with I = D.1 Summary output for K = 2 with I = D.11 Summary output for K = 2 with I = D.12 Summary output for K = 2 with I = D.13 Summary output for K = 8 with I = D.14 Summary output for K = 8 with I = D.15 Summary output for K = 8 with I = D.16 Summary output for K = 8 with I = D.17 Summary output for K = 8 with I = D.18 Summary output for K = 32 with I = D.19 Summary output for K = 128 with I = E Breakthrough curves with bumps 33 Index 37 vi

9 List of Figures 2.1 Activated carbon pore-size distributions for activation temperatures 55 o C, 7 o C, 8 o C and 85 o C Lennard-Jones potential Two-wall enhanced potential for decreasing slit width The Brunauer classification of isotherms Fixed-bed concentration profile Fixed-bed breakthrough curve Effect of shape of isotherm on development of concentration wave-front The irreversible or rectangular isotherm Rectangular parallelepiped element of volume Element of fixed-bed volume Spherical skin r thick at distance r from centre of sphere Cylindrical bed with various types of particles Adsorbent bed and idealised spherical particles Values of the first 1 terms of the series in Equation (4.9) Evaluating fractional uptake with less than 2% error by Equation (4.9) Uptake curve for long-times approximated by Equation (4.27) Fractional uptakes computed by Equations (4.8) and (4.29) Evaluating fractional uptake with less than 2% error by Equation (4.29) Evaluating fractional uptake with less than 2% error by Equations (4.8) and (4.29) vii

10 List of Figures 4.8 Short-time fractional uptakes by linear and quadratic approximations Integrand singularities at imaginary z = iy Integrand singularities at complex z = x + iy Typical intersection of y = tan p n and y = 3p n /(3 + αp 2 n) Evaluating Equation (4.45) with 2% accuracy for α = Evaluating Equation (4.45) with 2% accuracy for α = Theoretical uptake curves calculated by Equation (4.45) for various Λ Finite system: Long-time and short-time solutions for various Λ Finite system: Relative error using short-time solutions for various Λ Concentration profiles inside a spherical particle, relative error 7.2%. R = 1, T = 1, D =.6, α = 1, C =, C = Concentration profiles inside a spherical particle, relative error.5%. R = 1, T = 1, D =.6, α = 1, C =, C = Concentration profiles inside a spherical particle, R = Concentration profiles inside a spherical particle, R = Concentration profiles inside a spherical particle, C =, C = Concentration profiles inside a spherical particle, C =.4, C = Concentration profiles inside a spherical particle, α = Concentration profiles inside a spherical particle, α = Fractional uptakes given by the Crank Nicolson scheme and Equation (4.98) Fractional uptakes given by the Crank Nicolson scheme for α = 1, 1 and 1 and Equation (4.8) Model domain before and after transformations Contour C 1, line L and arc Γ 1 in the complex Laplace plane Schematic diagram of experiment Analytical and numerical breakthrough curves with experimental data Analytical and numerical breakthrough curves with k a = 12.5 sec Analytical liquid-phase concentration distribution inside the column Numerical liquid-phase concentration distribution inside the column viii

11 List of Figures 5.8 Analytical solid-phase concentration distribution inside the column Numerical solid-phase concentration distribution inside the column Pore-size distribution Volume and surface area for slits and cylindrical pores Cross section of a small slit-shaped pore showing distance a Cross section of a cylindrical-shaped pore showing distance a Distance b between circumscribing and circumscribed circles centres Monotonically increasing y = csc(π/x) Smallest cylindrical pore in which sorbate molecules can fit Arc on surface of pore and length a at distance b from pore centre Arc length of curved surface area associated with each length a Filling of a small cylindrical pore Global existence: Claim 3 and Claim Activated carbon pore-size and cumulative pore-size distributions Activated carbon pore-size probability and cumulative probability distributions Numerical breakthrough curve for K = 1, I = 16 and experiment values Numerical breakthrough curve for K = 8, I = 16 and experiment values Numerical breakthrough curve for K = I = 128 and experiment values Liquid-phase concentration distribution inside the column Solid-phase concentration distribution inside the column Breakthrough curves for different interstitial velocities Breakthrough curves for different pore-size distribution partitioning with w a1 = w a2 = Breakthrough curves for different pore-size distribution partitioning with w a1 = 1 and w a2 = Weighted average pore-sizes and associated probabilities versus partitioning bound s Saturation concentration, adsorption rate constant and desorption rate constant versus partitioning bound s ix

12 List of Figures 6.24 Adsorption rate constants versus weights Desorption rate constants versus weights Breakthrough curves for different adsorption rates for different pore-size partitions; using weights w a1 = 1, 3, 6 and 9 with w a2 = Breakthrough curves for different adsorption rates for different pore-size partitions; using weights w a2 = 1, 3, 6 and 9 with w a1 = Breakthrough curves for different desorption rates for different pore-size partitions; using weights w d1 =, 1, 1 and 2 with w d2 = Breakthrough curves for different desorption rates for different pore-size partitions; using weights w d2 =, 1, 1 and 2 with w d1 = Breakthrough curve for weights w a1 = w a2 = w d1 = w d2 = Breakthrough curve for weights w a1 = 1/6, w a2 = w d1 = w d2 = Mass of sorbate at bottom of bed for w a1 = w a2 = w d1 = w d2 = Mass of sorbate at bottom of bed for w a1 = 1/6, w a2 = w d1 = w d2 = Breakthrough curve for weights w a1 = w a2 = w d1 = w d2 = Mass of sorbate at bottom of bed for w a1 = w a2 = w d1 = w d2 = Breakthrough curve for weights w a1 = 2, w a2 = 1, w d1 =, w d2 = Mass of sorbate at bottom of bed for w a1 = 2, w a2 = 1, w d1 =, w d2 = Breakthrough curve for weights w a1 = 1, w a2 = 1, w d1 =, w d2 = Mass of sorbate at bottom of bed for w a1 = 1, w a2 = 1, w d1 =, w d2 = Breakthrough curve for weights w a1 = 3, w a2 = 1, w d1 =, w d2 = Mass of sorbate at bottom of bed for w a1 = 3, w a2 = 1, w d1 =, w d2 = Breakthrough curve for weights w a1 = 1/1, w a2 = 1, w d1 =, w d2 = Mass of sorbate at bottom of bed for w a1 = 1/1, w a2 = 1, w d1 =, w d2 = Breakthrough curve for weights w a1 = 1/2, w a2 = 1, w d1 =, w d2 = Mass of sorbate at bottom of bed for w a1 = 1/2, w a2 = 1, w d1 =, w d2 = x

13 List of Figures 6.46 Breakthrough curve with bump using the experiment parameters, J = Mass of sorbate at bottom of bed for breakthrough curve with bump using the experiment parameters, J = Breakthrough curve with bump using the experiment parameters, J = 4, w a1 = 25, w a2 = 1, w a3 = Mass of sorbate at bottom of bed for breakthrough curve with bump using the experiment parameters, J = 4, w a1 = 25, w a2 = 1, w a3 = Breakthrough curve with bump using the experiment parameters, J = 4, w a1 = 25, w a2 = 1, w a3 = Mass of sorbate at bottom of bed for breakthrough curve with bump using the experiment parameters, J = 4, w a1 = 25, w a2 = 1, w a3 = Breakthrough curve for J = 2, weights w a1 = w d1 = Mass of sorbate at bottom of bed for J = 2, weights w a1 = w d1 = A.1 Swapping order of integration in (A.16), (A.2), (A.22) and (A.25) E.1 Breakthrough curve for weights w a1 =.2, w a2 = w d1 = w d2 = E.2 Mass of sorbate at bottom of bed for w a1 =.2, w a2 = w d1 = w d2 = E.3 Breakthrough curve for weights w a1 =.2, w a2 = 2, w d1 = w d2 = E.4 Mass of sorbate at bottom of bed for w a1 =.2, w a2 = 2, w d1 = w d2 = xi

14 List of Figures xii

15 List of Tables 4.1 Roots of tan p n = 3p n /(3 + αp 2 n) Refinement paths summary results xiii

16 List of Tables xiv

17 Notation Although most symbols are defined as they are used in this thesis, a summary of the most commonly used symbols is given below for quick reference. Symbol Description A p, A pj wall surface area of open-ended pore, for pore-size partition j b adsorption equilibrium constant (b = k a /k d ) c, c liquid-phase (solute) concentration, equilibrium concentration c c F, c in c out D D c, D e, D p initial solute concentration inlet (feed) solution concentration outlet solution concentration diffusivity (ch.4) diffusivity ( 2.6): intraparticle, effective, macropore D L, D b axial dispersion coefficient, D b = ε b D L (see 3.2.3) K Henry constant ( 2.5) k a, k aj adsorption rate constant, for pore-size partition j (ch.6) k d, k dj desorption rate constant, for pore-size partition j (ch.6) L length of bed m t, m mass at time t, mass as time t p j probability for weighted pore-size s j (ch.6) q, q solid-phase (sorbate) concentration and equilibrium concentration q, q initial and boundary sorbate concentration q j q s, q sj q sorbate concentration in pore-size partition j (ch.6) saturation sorbate concentration, ditto for pore-size partition j (ch.6) average sorbate concentration xv

18 List of Tables Symbol Description R distance from centre of pellet ( 2.6, 2.8), sphere radius (ch.3, 4), pore radius (ch.6) R p radius of zeolite pellet ( 2.6, 2.8) r distance from centre of particle, or between molecules ( 2.4) r c radius of zeolite crystal ( 2.6) r m r p radius of sorbate (or solute) molecule radius of pore s, s j pore size (ch.6), weighted pore size for pore-size partition j (ch.6) s t smallest cylindrical pore that can accommodate molecule of size r m time V volume: sphere ( 3.3), liquid (ch.4), particle ( 4.1.7) V a V p, V pj V b, V v v z apparent volume of adsorbent adsorbent pore volume, ditto for pore-size partition j ( 2.3, ch.6) volume of packed bed, volume of voids in bed interstitial fluid velocity distance from bed inlet, z L α ratio of volumes (α = V/(4πR 3 /3) ch.4; α = ε b /(1 ε b ) ch.5, 6 ) ɛ maximum attraction potential ( 2.4) ε a ε p, ε e ε b Λ L [ ] adsorbent porosity pellet porosity ( 2.6), effective porosity (ch.3) voidage of bed non-dimensional concentration (ch.4) Laplace transform σ diameter of molecule ( 2.4) Θ fraction of surface covered by sorbate ( 2.5) τ non-dimensional time (ch.4) or time since wave-front has passed z (ch.5, 6) Υ non-dimensional radius φ potential ( 2.4) χ time wave-front reached location z in bed (ch.5, 6) xvi

19 Summary The process of adsorption is one in which one or more solutes are removed from a fluid passing through an insoluble porous solid, called an adsorbent, and accumulate on the walls of the adsorbent pores. Commercially viable adsorbents have a large ratio of surface area to volume where the walls to the pores provide the surface area. Activated carbon, having a low affinity for water, is the most important adsorbent used in water treatment. Activated carbons have different pore structures depending upon the raw material used, activation conditions and activation techniques employed. When some theoretical results did not appear to agree with experimental data using activated carbon adsorbents we were asked for our help with the mathematics. The research for this thesis involved collaboration between mathematicians and chemists. Although the pore-size distributions for adsorbents are known, existing mathematical models for the adsorption process do not consider that the rate of adsorption may be different in pores of different sizes. We have developed a mathematical model for adsorption in a fixed-bed of activated carbon that incorporates a partitioning of the pore-size distribution. In Part I of my thesis I have provided an overview on the modelling of adsorption processes with an emphasis on physical adsorption by activated carbon adsorbents. Part II looks at two models pertaining to micropore diffusion inside a particle. The first model assumes constant concentration of the adsorbed solute at the surface. In the second model the adsorbed solute concentration changes at the surface. After describing each model, we derive the short-time and long-time solutions given by others for the second model. The short-time solution for the second model was given without derivation by Carman and Haul [11]. We have published our derivation in Haynes and xvii

20 Lucas [42]. After assessing the accuracy of the short-time and long-time solutions for both models, we have given recommendations regarding their use. Ruthven [59, 1984] gave a linear simplification for the first case short-time solution. However, I discovered that Ruthven s simplification is of limited use when accurate results are required. In this thesis, I have extended this idea by giving a quadratic simplification that, with a little extra effort, provides a much better approximation. Modelling the adsorption process in a cylindrical fixed-bed is looked at in Part III. I began with a model that made no assumptions about the pore-size distribution. After finding the analytical solution and a numerical solution to the model, and using experimental data to calibrate the model, I extended the model to incorporate a partitioning of the adsorbent pore-size distribution. The extended model can be used for activated carbon adsorbents with their slit-shaped pores, or for other adsorbents with cylindrical pores. We have used the extended model to show that the results can differ if the adsorption rate (or the desorption rate) is different for different pore-sizes, and to explain a bump in the breakthrough curve. xviii

21 Declaration I declare that this thesis presents work carried out by myself and does not incorporate without acknowledgement any material previously submitted for a degree or diploma in any university; that to the best of my knowledge it does not contain any materials previously published or written by another person except where due reference is made in the text; and all substantive contributions by others to the work presented, including jointly authored publications, is clearly acknowledged. Paul D. Haynes B. App. Sc. (Hons) xix

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23 Acknowledgements Firstly I thank Stephen Lucas, my original principal supervisor, who suggested the research problem and was my supervisor for the first 18 months when the background reading was undertaken. Yalçin Kaya is thanked for his moral support and discussions following Stephen s abrupt departure for the USA. Thanks are also due to Phil Howlett who stepped into the principal supervisor role and became actively involved 15 months after Stephen s departure, despite my project with its numerical content not quite falling into any of his many areas of interest or expertise. I thank Jorge Aarao who provided supervision whilst Phil was away for 6 months then continued as my principal supervisor until the end of the project, and who suggested we build a model that incorporates the pore-size distribution. I also appreciate Jorge s knowledge of wave equations. Special thanks go to Basil Benjamin who became my associate supervisor 15 months after Stephen s departure. Basil has devoted numerous hours of his retirement time travelling to Mawson Lakes for weekly meetings, reading my thesis and spending many late nights discussing my work. His enthusiasm, willingness to be involved, constant support and encouragement are greatly appreciated. Thanks to Phillip Pendleton and Jung-Hee Kim for their discussions regarding the adsorption process and for providing the experimental data and parameters used to validate the fixed-bed models. The examiners are thanked for their constructive comments and suggested amendments, in particularly that I include the von Neumann stability analysis. Lee White is also thanked for his guidance and advice, regarding amendments to my thesis, while Jorge Aarao was overseas on sabbatical for six months. In addition, I thank the administrative staff of the School of Mathematics and Statistics and the Centre for Industrial and Applied Mathematics for their assistance xxi

24 and support. I also thank my fellow PhD students who have supported and encouraged me over the years, particularly during the 15 months following Stephen s departure. Last, but certainly not least, I thank my wife Aileen for her love, understanding and support. xxii

25 Chapter 1 Introduction The process of adsorption is one in which one or more solutes 1 are removed from a fluid passing through an insoluble porous solid, called an adsorbent, and accumulate on the walls of the adsorbent pores. There are instances where a solute attached to a surface detaches and returns to the fluid. This is desorption. Commercial adsorbents include silica gel, zeolites, carbons, polymers, resins and clays ([15, p.747]). According to Sontheimer et al. ([63, p.1]), activated carbon is the most important adsorbent employed in water treatment. Phillip Pendleton 2 found that certain theoretical results did not appear to agree with data from adsorption experiments involving activated carbon adsorbents, and sought our help with the mathematics. We have collaborated with Phillip Pendleton and Jung-Hee Kim 3 on this problem. To model and analyze the kinetics of adsorption by a single particle, we consider an idealized isotropic spherical particle so that the mass transport inside the particle may 1 According to Hiller Jr. and Herber [44, p.441], a solution is composed of two distinct parts: the dispersing medium, called the solvent, and the dissolved material, called the solute. If a teaspoon of sugar is added to hot water, then the sugar is the solute and the water the solvent. If 1 ml alcohol is added to 99 ml of water, then the alcohol is the solute and the water the solvent. However, if 1 ml of water is added to 99 ml of alcohol, then the water is the solute and the alcohol the solvent. 2 Center for Molecular and Materials Sciences, University of South Australia, Mawson Lakes, SA, Department of Chemical Engineering, Pikyong National University, Busan , Korea. 1

26 1. Introduction be described by the diffusion equation written in spherical coordinates ([59, ch.6]). In fixed-bed adsorption, where adsorbent particles are packed into a cylindrical bed, the flow of concentrated solute through the bed may be represented by an axially dispersed plug flow model coupled with an adsorption rate expression ([59, ch.8]). The adsorption rate expression may be implicit and could be realized as a system of equations. There are many existing models that use different combinations and different forms of the above equations with various boundary conditions. We have developed a mathematical model that incorporates a partitioning of the pore-size distribution for an activated carbon. We have used this new model to investigate how the breakthrough curve 4 may differ from those given by other models if the rates of adsorption and desorption are different for different pore-sizes. Part I of this thesis provides an overview on modelling adsorption. In Chapter 2, I define the terminology used in this thesis, describe the adsorption problem and discuss common adsorbents, including information about the shapes of pores and pore-size distributions, with an emphasis on activated carbons. Following an explanation of the differences between chemical and physical adsorption, I discuss the two-wall enhancement of the potential energy of physical adsorption pertaining to the attraction between a molecule and a surface being enhanced when the molecule is between two closely separated surfaces. I also discuss the dynamic phase equilibrium between the solid-phase concentration and the liquid-phase concentration during the adsorption process, and provide a summary of adsorption isotherm models. The kinetics of adsorption by a single adsorbent particle and the adsorption equilibrium in fixed-beds are discussed before describing some existing models for fixed-bed adsorption. The scope of this thesis is given at the end of Chapter 2. In Chapter 3, we provide some pre-requisite general theory on modelling isothermal adsorption, including definitions for particle porosity and bed voidage. After providing derivations for the differential equation of diffusion and the fixed-bed mass balance equation, we derive the formula for the average concentration inside a spherical particle, explain the fractional approach to equilibrium and 4 The breakthrough curve is a plot of the solute concentration output by the bed as a function of time see page 23 for further details. 2

27 the proportion of bed unused, and discuss the time taken for a rectangular wave-front to pass through a bed. In Part II, Chapter 4 looks at diffusion inside microporous particles immersed in a bulk liquid of infinite volume and in a bulk liquid of finite volume. After describing each model, we derive the short-time and long-time solutions given by others for the finite-volume problem. Carman and Haul [11] gave a solution to the micropore diffusion equation in a finite system without providing further details of the derivation. Subsequently, we have published our derivation in Haynes and Lucas [42]. After discussing the accuracy of the short-time and long-time solutions for both the infinite volume and finite volume models, we make recommendations as to which simplifications of the solutions should be used to evaluate the fractional uptake with less than 2% error 5. We also use the Crank Nicolson scheme to obtain a numerical solution to the finite-volume problem. The numerical solution is compared with the long-time solutions to both the finite-volume problem and the infinite-volume problem. Part III pertains to the isothermal adsorption in a fixed-bed with a Langmuir adsorption rate. In Chapter 5, we look at a model for plug flow, without axial dispersion and with constant velocity, coupled with a non-linear adsorption rate expression that makes no assumption about the adsorbent pore-distribution and derive an analytical solution to the model. Since the solution is a ratio involving exponentials and power series that is not easily evaluated, we use a finite difference scheme to obtain a numerical solution. The model is calibrated using experimental data. Chapter 6 extends the model to incorporate a partitioning of adsorbent pore-size distribution. I begin by discussing the partitioning of the adsorbent pore-size distribution; derive formulas for the sorbent saturation concentration associated with each pore-size partition where the pores may be large slits, small slits or cylindrical in shape; and present my model as a non-linear system of equations. It is shown that a unique solution exists. After finding the characteristics for the hyperbolic system, the model is written in integral form, and we discuss iteratively evaluating the system of integrals simultaneously. I then calculate the weighted pore-size and associated probability for each pore-size 5 Our analysis was based on a 2% error since this is the value used by Ruthven [59, p.168]. 3

28 1. Introduction partition by interpolating experimental pore-size data, and give formulas for assigning adsorption and desorption rate constants to each pore-size partition. Using a finite difference scheme that incorporates Newton s method, we obtain numerical solutions to the equations and discuss the stability of the numerical scheme. We then use the numerical solutions for different adsorption and desorption rate constants associated with each pore-size partition to demonstrate the effect of different rates of adsorption and desorption on the shape of the breakthrough curve and to explain a bump in the breakthrough curve. Chapter 7 summarizes the conclusions and proposes some future directions for research. 4

29 Part I Modelling the Adsorption Process 5

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31 Chapter 2 Problem description In this thesis we study problems that relate to the process by which a solute being carried in a liquid through a porous solid leaves the liquid and accumulates on the walls of the pores. We use the following definitions given by Atkins [4, p.874] in this thesis. Adsorption is the accumulation of particles on a surface 1. The porous solid is called the adsorbent and the substance that adsorbs, or accumulates on the surface, is called the sorbate. We also refer to sorbate as adsorbate or adsorbed solute. Desorption is the process where sorbate detaches from the surface and returns to the liquid. The greater the surface area of the porous solid, the greater the quantity of sorbate that can be adsorbed. The surface area can be greatly increased by increasing the number of pores or by increasing the surface area of the existing pores. The adsorption process is different to the absorption process where the solute accumulates within the pores themselves so that the greater the volume of the pores, the greater the quantity absorbed. Although the reversible ability of porous solids to adsorb large volumes of vapour was recognized in the eighteenth century, the practical use of this property in industrial separation and purification processes is relatively recent ([59, p.1]). For example, Gregg and Sing [37, p.1] point out that in 1777, Scheele found that when heated, charcoal expelled a volume of air that was eight times the volume of the charcoal, and that this 1 Broadly, adsorption is the accumulation of material at the interface between two phases: liquidliquid, liquid-solid, gas-liquid, gas-solid ([54, p.1]). 7

32 2. Problem description air returned to the charcoal when the charcoal was cooled. Although adsorption from liquids had been less well understood than adsorption from gases according to Coulson et al. [15, p.766] there has been considerable recent research on adsorption from liquids. See for example [7, 14, 2, 47, 48, 5, 56, 55, 57, 62]. Seader and Henley [61, p.782] state, Most solids are able to adsorb species from gases and liquids. However, only a few have sufficient selectivity and capacity to make them serious candidates for commercial adsorbents. Ruthven [59, p.3] also includes life as a primary property for a commercially viable adsorbent. Thus, the adsorbent should have sufficiently high selectivity to enable sharp separation of the sorbate from the gas or liquid, a large surface area to volume ratio to maximise the quantity of sorbate that can be adsorbed, high resistance to fouling and the capacity for reuse at the end of the process all for a relatively low cost. Seader and Henley [61, pp ] list other properties for an adsorbent to be suitable for commercial applications. Coulson et al. [15, p.747] give a list of commercial adsorbents and their typical applications. For example, silica gel is used inside packaging and double glazing to adsorb moisture; zeolite is used for the purification of hydrogen and the separation of oxygen and argon; carbons are used for the removal of odours from gases and the recovery of solvent vapours; polymers and resins are used in water purification; and clays are used in the treatment of edible oils. According to Seader and Henley [61, p.787], activated and molecular-sieve 2 carbons are the most widely used commercial adsorbents followed by molecular-sieve zeolites, silica gel and activated alumina. Surfactants 3 are not only used as soaps and detergents, but are also used as surfaceactive agents in many industrial processes that cause environmental pollution. Zeolites, silica, alumina, polymers, natural and synthetic fibres, and activated carbons have been investigated as adsorbents to remove surfactants ([47, 69]). However, activated carbons are being used increasingly as economic and stable mass separation agents ([47]). Activated carbon is also used as the adsorbent in many purification processes 2 Using special procedures, carbon adsorbents can be activated with a very narrow distribution of micropore size and therefore behave as molecular sieves, Ruthven [59, p.8]. 3 Surfactants are surface-active agents that accumulate at the interface between two phases and modify the surface tension ([4, p.711], [69]). 8

33 2.1. Activated carbon including gas separation, solvent recovery and drinking water purification ([14]). I describe activated carbon and some of its properties in Section 2.1, then explain the differences between physical and chemical adsorption in Section 2.2. In Section 2.3 we discuss pore sizes, pore shapes and pore-size distributions; and in Section 2.4, I explain the potential energy of physical adsorption and how the potential is enhanced in small pores. We then discuss the dynamic isothermal equilibrium between the solute concentration (fluid-phase) and the sorbate adsorbed (solid-phase) in Section 2.5; including the classification of isotherms and a description of some special isotherms. In Section 2.6 we look at the resistances to mass transfer and the mathematical modelling of the mass transfer inside a single adsorbent particle during isothermal adsorption. In Section 2.7, we explain what a fixed-bed is, then discuss the adsorption equilibrium considerations in fixed-beds. A summary of existing models for fixed-bed adsorption is given in Section 2.8. We look at the effect of the pore-size distribution on adsorption in Section 2.9. The chapter is concluded by giving the scope of this thesis in Section Activated carbon According to Coulson et al. [15, p.752] and Ruthven 4 ([59, p.7]), activated carbon is obtained from carbonaceous materials such as coal, wood, coconut shells and bones. These materials are decomposed in an inert atmosphere heated to about 8 K to create a carbonised material without pores. A further activation process is required to remove tarry carbonization products and open the pores. The activation may involve chemically treating the raw material or selectively oxidising the carbonised material at temperatures in excess of 1 K. Activated carbon may be used in granular form or as a powder. Granular carbon is normally regenerated after use. Powdered carbon is generally used when it is not economical to regenerate the carbon. Powdered activated carbons have faster adsorption rates than granular activated carbons, but they compact and resist flow. A relatively new Aqualen activated carbon fibre does not compact 4 Ruthven is highly cited Google searches on 14th November 28 and 23rd February 29 respectively show Ruthven had been cited by 1834 and by

34 2. Problem description under flow, has a higher adsorption rate and has a greater adsorption capacity than the powdered and granular forms ([62]). Due to its low affinity for water, activated carbon is suited to adsorbing sorbate from aqueous solutions or from moist gases ([15, p.752]). For example, activated carbon is used in the treatment of polluted surface and groundwater for drinking and the treatment of wastewater ([63, p.1]). The coconut-based steam activated carbon used in the investigation of the adsorption of surfactants by Kim et al. [47] had 15 µm mean particle size,.73 cm 3 /g total pore volume, density.66 g/cm 2, and surface area 1369 m 2 /g. That is, the surface area associated with each gram of activated carbon is larger than the area of a 3 m 4 m residential block of land. The wood-based phosphoric activated and coal-based steam activated carbons used by Kim et al. [48] respectively had surface areas 1149 m 2 /g and 1738 m 2 /g. 2.2 Physical and chemical adsorption There are two types of adsorption with contrasting properties outlined by Alberty and Silbey [2, 24.1], Atkins [4, 29.3], Noll et al. [54, pp21-22] and Ruthven [59, pp.29-3]. Physical adsorption, or physisorption, does not involve sharing or transfer of electrons but involves only relatively weak intermolecular van der Waals forces 5 and electrostatic interactions 6. Physical adsorption is not site specific. Thus the sorbate molecules are free to cover the entire surface. A monolayer (single layer) or multilayer of sorbate may accumulate at a surface. The energy (heat) released by physical adsorption is low. Physical adsorption is rapid and readily reversible but decreases at higher temperatures. Chemical adsorption, or chemisorption, involves site specific chemical bonds with only a monolayer at the surface. Generally, chemisorption is exothermic, releasing a high heat energy. The process may be slow and is irreversible. 5 Always present van der Waals forces are dispersion-repulsion forces between uncharged molecules ([44, p.9], [59, p.3]). 6 Electrostatic interactions involve polarization, dipole and quadrupole interactions and are only significant in adsorbents that have an ionic structure, such as zeolites ([59, p.3]). In these cases, the physical adsorption may appear to be a slow activated chemisorption with a high heat of adsorption. 1

35 2.3. Pore sizes, shapes and distributions Chemisorption may be strongly inhibited at low temperatures if the reaction has an activation temperature where the rate of adsorption increases rapidly as the temperature increases. 2.3 Pore sizes, shapes and distributions The surface of a porous particle has indentations and cracks or fissures. It is important to distinguish between the external and internal surface when discussing the surface properties of an adsorbent ([37, pp.23-25]). The accepted simple demarcation is that any prominence or crack that is wider than it is deep forms the external surface whereas the surfaces of the walls of all cracks, pores and cavities which are deeper than they are wide form the internal surfaces. Pores are classified by their shape: cylindrical, slitshaped, ink-bottle shaped or funnel shaped ([58]). Pores are also classified according to their size 7 where the size of cylindrical pores is measured by their diameter and the size of slit-shaped pores is measured by the distance between the opposite sides ([22, p.2], [37, pp.25-26], [58]). Pores that are sufficiently small, with width less than 2 Å (2 nm), for the attraction potential to be enhanced are called micropores (see Section 2.4). Mesopores pores, where capillary condensation occurs, are between 2 and 5 Å wide. The large pores, greater than 5 Å (5 nm) wide, are called macropores. According to Satya Sai and Krishnaiah [6], Micropores are mainly responsible for adsorption, with macro- and mesopores acting as channels and conduits for the passage of the adsorbate. According to Rouquerol et al. [58], pore-size distributions are represented by the derivatives da p /dr p or dv p /dr p where respectively A p, V p and r p are the pore wall area, volume and radius. However, Satya Sai and Krishnaiah [6] use the more appropriate notation V p / r p, to reflect a discrete change in volume over a discrete interval of pore sizes see Figure 2.1 taken from Satya Sai and Krishnaiah. If the pores are slit-shaped then r p is the pore width. Satya Sai and Krishnaiah [6] found that the pore-size distribution, pore volume and average pore diameter for coconut shell based 7 Pore sizes measures. Angstom: 1 Å = 1 1 m. Nanometre: 1 nm = 1 9 m. 11

36 2. Problem description activated carbons are dependent on activation process parameters. For example, using steam instead of CO 2 produces smaller pores. Figure 2.1 taken from Satya Sai and Krishnaiah shows the pore-size distributions for different activation temperatures where V p is the pore volume per gram of adsorbent. Gun ko and Do [4] give the pore-size distributions for a variety of activated carbons with different structural characteristics. Activated carbon consists of microcrystallites of graphite only a few layers thick and less than 1Å in width stacked together in an arbitrary orientation, where the spaces between the crystals form the micropores, and the pore size distribution is typically trimodal ([54, pp.25-27], [59, p.7]). According to Do [22, pp.4-6,149-51] and Sontheimer et al. [63, pp.73 74], the activated carbon micropores are slit-shaped and provide most of the adsorption capacity. Adsorption in the micropores is by volume filling due the force field encompassing the entire volume of micropores see Section 2.4. The adsorption capacity of macropores is not significant as the surface area of the macropores is negligible compared to the total area of the micropores. The macropores act as transport pores, allowing sorbate molecules to diffuse from the bulk liquid into the interior of the adsorbent particles. Mesopores also act as transport pores or conduits for the transfer of sorbate molecules. 2.4 Potential energy of physical adsorption According to Atkins [4, p ], when molecules are squeezed together during physical adsorption, the repulsive forces between the nuclei and the electrons begin to dominate the attractive forces and the repulsive forces increase steeply with decreasing separation, as depicted in Figure 2.2. In a simple model, where molecules are taken to be spherical, the potential energy φ is given by the Lennard-Jones potential function 8, [ (σ ) 12 ( σ ) ] 6 φ = 4ɛ (2.1) r r 8 The discovery of carbon nanotubes and C 6 fullerenes may lead to new nanomechanical devices and nanocarriers for drug delivery. Researchers are using the Lennard-Jones potential to calculate the van der Waals interaction between the fullerenes and carbon nanotubes ([5], [6], [17], [18], [64]). 12

37 2.4. Potential energy of physical adsorption Figure OmiTTED from electronic copy Figure 2.1: Fitted curves V p / r p against r p for activated carbon pore-size distributions at activation temperatures 55 o C, 7 o C, 8 o C and 85 o C. where r is the distance between the centres of the interacting molecules, 1/r 6 represents the attraction, 1/r 12 represents the repulsive potential, σ is the diameter of the molecules and ɛ is the maximum attraction potential see Figure 2.2 taken from Ruthven [59, p.31]. If the interaction is between two different molecules, then Ruthven [59, p.31] uses, σ = σ 1 + σ 2, ɛ = ɛ 1 ɛ 2. (2.2) 2 Figure 2.2 also represents the interaction potential of a single molecule with an exposed plane surface of an adsorbent. In the following discussion, φ denotes the minimum potential between a single molecule and an exposed plane surface of the adsorbent. Gregg and Sing [37, pp.27-9] explain that if an adsorbate molecule enters a slit-shaped micropore, the combined interaction potential of the molecule with the two walls is larger than that for the single plane surface. The size of the two-wall enhancement is related to the ratio of the pore-size to that of the adsorbate molecule, 13

38 2. Problem description Figure OmiTTED from electronic copy Figure 2.2: Lennard-Jones potential showing the dependence of the interaction potential on the distance of a single molecule from an exposed plane surface. d/r, where d is the half-width of the slit and r is the collision radius 9 of the molecule. Given an adsorbate molecule of radius r, Figure 2.3 depicts the two-wall potential, φ, for different slit widths. When d = 2r, that is the slit is two molecular diameters wide, there are two minima with little enhancement as shown by curve (a). As the slit-width decreases, as shown by curves (b) and (c), the minima merge to a single minimum with an enhanced attraction potential. When d = r the enhanced potential is 1.6φ. Everett and Powl [26] have calculated (to the accuracy shown) the maximum enhanced potential as 2φ when d/r =.858 for slit-shaped pores, and 3.388φ when R/r = 1.86 for cylindrical pores of radius R. 2.5 Dynamic phase equilibrium In adsorption, a dynamic phase equilibrium is established for the distribution of the solute between the fluid and the solid surface., Seader and Henley [61, p.794]. The fluid-phase is expressed in terms of concentration, c, of the solute in the fluid, and the solid-phase is expressed in terms of the sorbate loading, q, on the adsorbent. The 9 Two molecules are considered to have collided when their centres are within the collision diameter apart ([4, p.729]). For hard spherical molecules of the same size, the collision diameter is the diameter of the molecules, σ. If the molecules are of different size, the collision diameter is given by Equation (2.2) ([2, p.64]. 14