Christopher Samuel Handscomb King s College

Transcription

1 SIMULATING DROPLET DRYING AND PARTICLE FORMATION IN SPRAY TOWERS Christopher Samuel Handscomb King s College November, 2008 This dissertation is submitted for the degree of Doctor of Philosophy

2 Summary Title: Simulating Droplet Drying and Particle Formation in Spray Towers Author: Christopher Samuel Handscomb This thesis presents a new modelling framework for the simulation of single droplet drying. Focussing on droplets containing dissolved solids in an ideal binary solution with additional suspended solids, the general model framework tracks the spatially resolved continuous phase composition using volume averaged transport equations and employs a population balance to describe nucleation, movement and crystallisation of the solid particles. Beyond describing the rate of moisture loss from drying droplets, the key achievement of the new model is the ability to describe structural development as influenced by evolving droplet composition and drying conditions and, thereby, the ability to simulate multiple dried-particle morphologies. This is achieved by combining various sub-models with the core droplet description to produce an integrated simulation of structural evolution. The thickening, wet shell, dry shell and slow boiling sub-models are developed, thus allowing the simulation of morphological developments associated with shell and bubble formation and growth. Further, extensive consideration is given to the physical motivation of these sub-models, permitting the development of rational criteria for deciding which sub-model to use at different stages in a droplet s drying history. It is the development of such criteria which allows, for the first time, structural evolution to be simulated as a function of evolving droplet composition and external drying conditions. The new droplet drying model, expressed as a coupled system of ordinary and partial differential equations, is solved using NAG library routines. The computational and numerical issues associated with the model solution are discussed in depth, giving confidence in the results obtained. Several systems of practical interest colloidal silica, sodium sulphate solution and detergent crutcher mix are simulated using the new model and the results obtained are validated against experimental data from the literature. Droplet mass and temperature profiles are accurately reproduced and the ability of the model to simulate structural evolution during drying is demonstrated and investigated.

3 Declaration This dissertation is the result of my own work and includes nothing which is the outcome of work done in collaboration, except where specifically indicated in the text. The work presented was undertaken at the Department of Chemical Engineering at the University of Cambridge, UK, between September 2004 and October Chapters 2 and 3 of this dissertation include work from the dissertation I submitted in August 2005 for a Certificate of Postgraduate Study. No other part of this thesis has been submitted for a degree to this or any other university. This dissertation contains approximately words and 87 figures. Some of the work in this dissertation has been published: 1. C. S. Handscomb, M. Kraft and A. E. Bayly. A new model for the drying of droplets containing suspended solids. Chemical Engineering Science, in press, doi: /j.ces C. S. Handscomb, M. Kraft and A. E. Bayly. A new model for the drying of droplets containing suspended solids following shell formation. Chemical Engineering Science, in press, doi: /j.ces Christopher Samuel Handscomb December 13, 2008 Acknowledgements I would like to acknowledge my supervisor, Dr Markus Kraft along with all members of the Computational Modelling Group. Funding for this project was provided by the EPSRC through the Smith Institute for Industrial Mathematics in collaboration with Procter and Gamble; I should like to thank Dr Andrew Bayly from P&G and Dr Melvin Brown from the Smith Institute for the help and guidance they provided over the course of my PhD. ii

4 Quickly, bring me a beaker of wine, so that I may wet my mind and say something clever. Aristophanes ( BCE) iii

5 Contents 1 Introduction Motivation Novel Elements of This Thesis Structure of the Thesis Background Drying Moisture in Gases and Solids The Driving Force for Drying Stages of Drying Industrial Drying Spray Drying Process Overview Applications of Spray Drying Spray Dryer Modelling Heat and Mass Balances Equilibrium Based Models Rate Based Models using Simplifying Flow Assumptions Rate Based Models Using CFD Single Droplet Drying Models Droplet Drying Behaviour Droplet Morphologies Droplet-Averaged Drying Models Mechanistic Models Scope of this Thesis A New Model for Drying Droplets Prior to Shell Formation Background Descriptions of Moisture Movement in Porous Media Local Volume Averaging Overview of the New Model Model System Notation Continuous Phase Description Defining the Solids Volume Fraction Source Terms Transport Term iv

6 3.3.4 The Continuous Phase Equation Continuous Phase Density The Local Mass-Averaged Velocity Boundary Conditions Discrete Phase Description Population Balance Equation Moments of the Population Balance Equation Size Dependent Diffusion Coefficient Heat and Mass Transfer From The Droplet Evaporation from a Sphere Blowing Effects Thermal Calculation Applications Drying a Droplet of Colloidal Silica Drying a Droplet of Aqueous Sodium Sulphate Solution Conclusions of the Chapter Drying Droplets With an Outer Shell Background Droplet Morphology Formation of a Surface Shell Drying With a Rigid Shell Deciding Which Model to Use Physics of Drying Following Shell Formation Criteria for the Different Models Internal Boiling When Drying in Hot Air Shell Thickening Model Description Shell Growth Boundary Conditions Dry Shell and Slow Boiling Models Model Description Shell Growth Boundary Conditions Wet Shell Model Model Description Shell Growth Boundary Conditions Applications Drying a Detergent Droplet Drying a Droplet of Colloidal Silica Conclusions of the Chapter Solution Methodology Background Equation Classification Solution Methods v

7 5.1.3 Moving Boundary Problems Model Equations and Boundary Conditions Prior to Shell Formation Thickening Regime Wet Shell Regime Dry Shell Regime Solving the Model Equations The NAG Library Routine D03PLF Definition of the System of Partial Differential Equations pdedef Definition of the Coupled ODEs odedef Boundary Conditions and Solution Grid Definition of the Boundary Conditions bndary Grid Resolution Grid Studies The Numerical Flux Function Definition of the Numerical Flux Function numflx Roe s Method for the Numerical Flux Function Numerical Parameter Studies Solution Method Error Tolerances Maximum Time Step Conclusions of the Chapter Conclusion Conclusions of the Thesis Suggestions For Future Work Extending the New Model Experimental Investigations Uses of the New Model Appendices 205 A Abel Transformation 206 A.1 Theory of the Transformation A.2 Testing the Transformation Algorithm B Roe Flux Function 210 B.1 Partial Differential Equation B.2 Linearised Jacobian B.3 Flux Function vi

8 List of Figures 2.1 Illustration of bound and unbound moisture within a drying material Schematic drying rate curves Mechanisms of moisture removal from a porous solid The principle process stages of a generic spray drying process Product discharge from a co-current drying system with: (a) primary separation in the drying tower; and (b) total recovery in the dedicated separation equipment Size and segmentation of the laundry detergent market Schematic showing the structure of modern rate based spray dryer model using CFD and particle sub-models The different stages of drying for a liquid droplet containing solids Schematic showing some of the different dried-particle morphologies that may result when drying droplets containing dissolved or suspended solids Images of hollow spray dried particles Schematic showing a generic drying droplet containing suspended solids and gas pockets Every point, z, within the porous medium has an associated averaging surface, S, containing the volume, V Illustration of the terms used when volume averaging equations in a porous medium consisting of a continuous and discrete phase Schematic showing: (a) the model system; and (b) drying to form a hollow shell Illustration of a droplet with solids represented by a population of spherical particles suspended in a continuous liquid phase Schematic illustrating the fluxes from an evaporating drop of A surrounded by stagnant air Plot of the saturated vapour pressure of water, comparing experimental measurements with fitted equations Comparison of simulated and measured (a) temperature and (b) mass profiles for the evaporation of a pure water droplet Plot of the solids diffusion coefficient used to simulate the drying of a colloidal silica droplet Plot of the three sorption isotherms used in the investigation of colloidal silica droplets drying Plot showing the influence of the sorption isotherm on the predicted temperature and mass profiles when drying a colloidal silica droplet vii

9 3.12 Plot showing the influence of the relative air velocity, v rel, on the predicted temperature and mass profiles when drying a colloidal silica droplet Plot showing the influence of the air relative humidity, R, on the predicted temperature and mass profiles when drying a colloidal silica droplet Simulated solids volume fraction profiles during the drying of a colloidal silica droplet in air at T gas = 101 C Simulated drying of a colloidal silica droplet (lines) compared with experimental results from Nešić and Vodnik (1991) (symbols) at T gas = 178 C with R = 0% and v = 2.5 ms Simulated solids volume fraction profiles during the drying of a colloidal silica droplet in air at T gas = 178 C Plot showing the binary diffusion coefficient, (AB), of Na 2 SO 4 in water at 25 C Water-Sodium Sulphate phase diagram adapted from Wetmore and LeRoy (1951) The effect of initial particle number density and initial seed particle size on the timing of shell formation, t shell The effect of the nucleation rate, Ṅ max, on the timing of shell formation, t shell when drying a droplet initially seeded with 10 nm crystals Plot showing the average solid particle size, L, at r = R at the point of shell formation Plot showing how the maximum solid particle nucleation rate effects the timing of shell formation for a droplet with initial m (0) = m 3 s Plot showing spatially resolved profiles of the mean solid particle size, L, at 2 s intervals from t = 20 s until the point of shell formation, t shell Comparison of the simulated zeroth moment at the outer edge of a drying droplet, (thick line) with that obtained from the first and second moments using a closure approximation of the form given by (3.4.24), (symbols) Simulated drying of a 14 wt% sodium sulphate in water droplet at T gas = 90 C, compared with experimental results from Nešić and Vodnik (1991) Simulated solute mass fraction profiles plotted at 5 s intervals throughout the drying of a droplet of aqueous sodium sulphate solution. In addition, profiles are plotted at the point of shell formation, (dashed line) and solution at the edge of the droplet first becomes supersaturated, (bold line) Simulated normalised moments at the outer edge of the droplet Simulated normalised moments integrated over the entire droplet Simulated solids volume fraction in a drying droplet of aqueous sodium sulphate solution Schematic illustrating the process of shell thickening Illustration showing the drying process in the shell thickening period Schematic demonstrating how the process of shell thickening can be likened to the growth of a filter cake Illustration of the decision process used to select the appropriate drying model to use following shell formation Illustration of the pressure profiles within a drying droplet following formation of a surface shell viii

10 4.6 Illustration of a droplet of radius R drying through a dry shell with internal radius, S Evaporation from a droplet of radius R in the presence of a dry shell Illustration of a droplet of radius, R drying through a wet shell with internal radius, S (a) Initial composition of the crutcher mix droplets as measured by Griffith et al. (2007); and (b) simplified description of the crutcher mix used to model the system in the base case Water sorption isotherm for crutcher mix. The line shows the sorption isotherm, (4.6.6), obtained by fitting to the experimental points obtained by Bayly (2007) Simulated evolution of the moisture mass fraction in a crutcher mix droplet (line) compared with experimentally measured values from Griffith (2008), (symbols) Simulated moisture profiles in a drying detergent droplet (lines) compared with experimental observations (symbols) from Griffith (2008) Simulated evolution of total droplet mass together with the mass of each of the three components in the model droplet Simulated time evolution of the dry shell interface during the drying of a droplet of crutcher mix Simulated solids volume fraction, ɛ, profiles at 10 minute intervals during the drying of a crutcher mix droplet SEM images showing detergent droplets dried in a spray dryer Simulated lye phase sodium sulphate concentration profiles at 10 minute intervals during the drying of a crutcher mix droplet Simulated drying of a colloidal silica droplet (lines) compared with experimental results from Nešić and Vodnik (1991) (symbols) at: (a) T gas = 101 C and R 0 = mm; and (b) T gas = 178 C and R 0 = 0.95 mm Plot illustrating the influence of vapour pressure reduction as predicted by the Kelvin equation on the simulated temperature profiles for a 1.9 mm colloidal silica droplet drying in air at 178 C Plot showing various pressures relating to a droplet of colloidal silica containing 16 nm particles drying in air at T gas = 178 C during the shell thickening regime Simulated morphological evolution of a droplet of colloidal silica containing 16 nm particles drying in air at 178 C Simulated solids volume fraction profiles during the drying of a colloidal silica droplet containing 16 nm particles drying in air at T gas = 178 C Plot showing various pressures relating to a droplet of colloidal silica containing 500 nm particles drying in air at T gas = 178 C during the shell thickening regime Simulated morphological evolution of a droplet of colloidal silica containing 500 nm particles drying in air at 178 C Simulated solids volume fraction profiles during the drying of a colloidal silica droplet containing 500 nm particles drying in air at T gas = 178 C Schematic showing the effect silica particle size on the predicted drying process ix

11 4.27 Plot showing various pressures relating to a droplet of colloidal silica containing 1500 nm particles drying in air at T gas = 178 C during the shell thickening regime Plot showing how the Young s modulus of the shell affects the size and shell thickness of a dried-particle formed from a droplet of colloidal silica containing 500 nm particles Grid points, cells and cell boundaries on a one dimensional domain with uniform grid spacing Following shell formation, the drying droplet may be divided into two adjoining solution domains, both of whose extents vary with time A space time illustration of a uniformly expanding solution domain showing the positions of the grid points, i (t) and demonstrating the origins of virtual flux terms Illustration of the co-ordinate transformation applied to a drying droplet with a shell region. After transforming, the core and shell regions are both fixed on the interval z [0,1] Illustration of the routines called by the time integrator, D03PLF Numerical approximation of a Neumann boundary condition using the two point, one sided difference formula Construction of a solution grid placing grid points using (5.4.9) and taking µ = Illustrating the effect of varying the parameter, µ in (5.4.9). Lower values of µ result in more grid points clustered in the neighbourhood of the right hand boundary An arbitrary distribution of grid points in the neighbourhood of the right hand boundary of the z domain Plot showing the predicted time to shell formation, t shell, when drying a droplet of colloidal silica, illustrating the dependence on the number of cells in the solution grid, npts Plot illustrating the solids volume fraction within a droplet of colloidal silica at the point of shell formation, showing the effect of increasing cell density in a uniform grid Plot illustrating the solids volume fraction within a droplet of colloidal silica at the point of shell formation. The profiles obtained using a uniform grid containing 20 and 500 cells are compared with that returned when solving on a non-uniform grid containing 20 cells Demonstration of grid convergence as measured by the predicted timing of shell formation, t shell, when simulating a drying droplet of sodium sulphate solution Plot illustrating the influence of the finite difference formula used to approximate the boundary gradients on the convergence behaviour of the solution Plot demonstrating the efficiency of the solutions obtained using various grid spacings and finite difference approximations to the boundary gradients Plot illustrating the effect of varying the relative error tolerance, rtol x

12 5.17 Simulated profiles of the solids volume fraction is a droplet of crutcher mix after 50 minutes of drying, illustrating the effect of reducing the maximum integration step size, t max A.1 Numerical transformation of (A.2.1) with profiles showing the effect of varying the number of data points, N A.2 The computation time, t, required to perform a double Abel transform on (A.2.1) is proportional to N 2, where N is the number of data points used in the transform xi

13 List of Tables 2.1 Summary of the major types of continuous driers used in the process industries, (Sinnott et al., 1999) Summary of the major advantages and disadvantages of spray dryers, (Masters, 1992) Sample process specification and results from applying a simple heat and mass balance Sample process results and the sorption isotherm used to calculate the moisture equilibrium at the outlet Values of the l 2 -norm of the deviations, ɛ i, between predicted and measured values of the droplet mass for different drying air velocities, v rel The composition of the detergent slurry investigated by Griffith et al. (2008) Expressions for the various physical properties of water and air used in droplet drying simulations Coefficients of the one-sided n-point gradient approximations written for a right hand boundary Comparison of accuracy and computational times for solutions obtained on a uniform and non-uniform grid A.1 Computational time to perform a double Abel Transform on (A.2.1) xii

14 Chapter 1 Introduction 1.1 Motivation Spray drying is the operation of choice for the production of many commercial products ranging from high value pharmaceuticals to bulk commodities such as dried milk and detergent powders. The needs of these differing applications vary greatly. When producing pharmaceuticals it is essential to maintain a sterile environment, whilst food products must be dried in a way that ensures aromas and nutrients are retained. Detergent powders require tightly controlled physical properties if customer demands concerning flowability and dissolution rate are to be met and, for any bulk drying operation, energy efficiency is likely to be a principle concern. The spray drying operation may be tailored to suit all of these roles and many more. Spray drying works by contacting an atomised feed with drying air in a chamber. The feed composition and process operating parameters are chosen to obtain the correct chemical and physical properties in the final product. In general, these properties will be influenced by the many processes occurring within the spray drying tower: in addition to moisture removal and simultaneous particle formation, these may include particle build-up on the dryer walls or inter-particle interactions such as agglomeration. A comprehensive model of the spray drying operation would simulate all of these processes and such an accurate description is the holy grail of those working with spray dryers. The ideal model would allow accurate control of product properties, prediction of dried-particle composition and morphology, specification of operating conditions for increased energy efficiency and rigourous quantitative design of new towers allowing problems associated with wall deposition to be eliminated. However, despite their ubiquity and substantial research effort over the past twenty years, spray driers continue to present substantial challenges to the modeller and comprehensive dryer models remain elusive. An alternative way of approaching the problem of spray dryer modelling is to focus on simulating the drying behaviour of individual droplets. Simplified droplet drying models are 1

15 1.2. Novel Elements of This Thesis essential components of top-down spray dryer simulations, wherein a computational fluid dynamics description of the gas flow field is coupled with sub-models describing droplet processes. More detailed single droplet drying models describe not only the rate of moisture loss, but can also give valuable insights into the physical evolution of the drying droplets. When droplets contain suspended or dissolved solids, such structural evolution can be complicated; an initial period of ideal shrinkage may be followed by formation of a surface shell, internal bubble growth or inflation. Simulating the evolution of droplets undergoing such processes is complicated, but necessary to predict the morphology and associated physical properties of the final dried-particles. The development and validation of such a model is the subject of this thesis. 1.2 Novel Elements of This Thesis This thesis presents a completely new modelling framework for the simulation of single droplet drying. This framework combines a physically motivated and rigorously derived description of the droplet core, with various sub-models to describe processes following shell formation. A number of these sub-models are developed, implemented and tested within this thesis, but the general framework has the flexibility to allow future researchers to extend the model with ease. Beyond describing the rate of moisture loss from drying droplets, the key objective of the new model is to describe the evolving droplet structure, allowing the simulation of multiple dried-particle morphologies. Focussing on droplets containing dissolved solids in an ideal binary solution with additional suspended solids, the general model framework: tracks the spatially resolved continuous phase composition using transport equations rigorously derived using the theory of volume averaging; employs a population balance to describe the solid phase, allowing solid particles to nucleate, move within the droplet and grow due to crystallisation from the continuous phase. The population balance is solved using a moment method, reducing the description to a series of partial differential equations solved in parallel with that describing the continuous phase. This core droplet model is believed to be the most sophisticated generic description of drying droplets produced to date. Following shell formation, this thesis demonstrates how a number of sub-models may be combined with the core droplet description to produce an integrated simulation of structural evolution. The sub-models investigated are: Shell Thickening a completely new sub-model allowing for continued droplet 2 CSH

16 1.2. Novel Elements of This Thesis shrinkage following shell formation, with evaporation proceeding from the droplet surface; Wet Shell Drying a sub-model from the literature including bubble growth and describing drying in the presence of a rigid shell which is completely wetted by the continuous phase; Dry Shell Drying a common sub-model from the literature describing drying in the presence of a rigid, dry shell where evaporation occurs at a front within the drying droplet; Slow Boiling a new sub-model describing moisture removal from a droplet at temperatures above the boiling point of the continuous phase, without the droplet shattering. The thesis contains a detailed description of how these sub-models can be incorporated within the general framework and validates this by comparing model predictions with experimental data from the literature. Further, for both the new sub-models and those taken from the literature, extensive consideration is given to their physical motivation. This permits the development of rational criteria for deciding which sub-model to use at different stages in a droplet s drying history. It is the development of such criteria which permits, for the first time, structural evolution to be simulated as a function of evolving droplet composition and external drying conditions. In summary, this thesis contains the following novel elements: development of a framework to describe drying droplets based on volume averaged transport equations and a population balance describing suspended solids; development of the shell thickening sub-model; extension of the wet and dry shell sub-models, providing physical motivation and demonstrating compatibility with the new model framework; development of physically motivated criteria to determine the appropriate sub-model to use following shell formation and thus allow simulation of structural evolution influenced by droplet composition and drying conditions; simulation of systems of practical interest and validation of the new droplet drying model against experimental data. 3 CSH

17 1.3. Structure of the Thesis 1.3 Structure of the Thesis After this introductory chapter, Chapter 2 gives background information on the physics of drying before discussing the use and modelling of spray dryers in more detail. The second half of the chapter focusses on the literature relating to the simulation of single droplets, the main subject of this work. Chapter 3 describes the development of the core elements comprising the new droplet drying model in detail. Comparisons with experimental data are presented for droplets containing colloidal silica and dissolved sodium sulphate. Chapter 4 utilises the model framework developed in the previous chapter and shows how this may be augmented with various sub-models to describe droplet drying following formation of a surface shell. The physics of this process is discussed and experimental comparisons are presented for three types of droplet. Chapter 5 discusses the numerical and computational methods used to solve the new droplet drying model and thereby produce the results contained within this thesis. The influence of the solution grid and numerical error tolerances are investigated at length. Finally, Chapter 6 contains conclusions of the work presented and suggests ways in which the novel elements developed herein might be taken forward. 4 CSH

18 Chapter 2 Background In which the background material for the rest of this thesis is introduced. The basic thermodynamics of drying are reviewed and an overview of industrial dryer selection presented. There follows a more detailed discussion of the spray drying process and a review of the literature relating to the simulation of spray drying towers. Single droplet drying models are key to the success of such simulations and the development of a new such model is the focus of this thesis. This chapter concludes with a critical review of existing droplet drying models and indicates how the new model extends the field. 2.1 Drying Drying is the separation of solids and volatile substances most commonly moisture by the application of heat to cause vaporisation 1, (Keey, 1975). The production of most solid materials involves drying at some stage; the goals of such operations range from reducing transport costs to providing specific product properties, such as porosity of a laundry detergent, (Coulson et al., 1996; Sinnott et al., 1999). As well as being a common industrial operation, drying can also be an expensive one due to the inherent heat requirements, (Keey, 1978). A better understanding of the drying process therefore promises both cost savings and enhanced product characteristics. Before going any further, it is worth reviewing the ways in which moisture may be present within solids and the basic thermodynamics of the drying process. Also, since most dryers use hot air to remove moisture from solids, (Coulson et al., 1996), it is important to know how to describe the moisture content of a gas. 1 Squeezing and adsorption are sometimes included in lists of drying processes although these do not involve the use of thermal energy. 5 CSH

19 2.1. Drying Moisture in Gases and Solids Moisture in gases is described in terms of a number of different humidities. The absolute humidity of a gas mixture is defined as the ratio of the mass of water vapour to the mass of vapour-free, bone-dry gas, (Perry and Green, 1997). Assuming the gas to be ideal, this may be written = p A W A (P p A )W g, (2.1.1) where W A and W g are the molar masses of water and dry gas respectively. P is the total pressure and p A is the partial pressure of water in the gas. The saturation humidity, sat, is the humidity of saturated air, i.e., when p A = p sat A. The percentage relative humidity is defined as the partial pressure of water vapour in the air divided by the saturated vapour pressure of water at the same temperature, i.e., R = p A 100%. (2.1.2) p sat A This should not be confused with the percentage absolute humidity, abs, which is defined as the humidity divided by the saturation humidity at the same temperature. Moisture containing solids may be divided into three categories: hygroscopic porous, non-hygroscopic porous and non-porous hygroscopic bodies, (Keey, 1975). Examples of hygroscopic materials are salts, vegetal fibres, most metal oxides and many polymers whilst metal powders and glass granules are examples of non-hygroscopic products. In some materials such as gels moisture behaves as if it was dissolved in the solid and moves by diffusion under ever decreasing concentration gradients. Air cannot penetrate these non-porous hygroscopic bodies. In non-hygroscopic porous materials, moisture is merely trapped in the spaces between solid particles. This unbound moisture exerts an equilibrium partial pressure approximately equal to the partial pressure above pure water at the same temperature. In addition to unbound moisture, hygroscopic porous materials may also contain some bound moisture. This may be in small capillaries 2, adsorbed on to solid surfaces, chemically bound in 2 The Kelvin equation, p A = p sat exp A 2V m,l γ cosθ r M R g T, (2.1.3) may be used to estimate the reduction in vapour pressure for moisture held in capillaries with radius, r M. p A is the vapour pressure above the capillary and p sat is the partial pressure above pure water at the same A temperature, T. V m,l is the molar volume of the liquid and θ is the contact angle. This equation shows that for water at 50 C in a 1 µm capillary and the vapour pressure reduction is less than 0.1%. However, there are circumstances where the correction becomes important, as discussed in Section CSH

20 2.1. Drying 100 Relative Humidity, R / % 0 0 Equilibrium Moisture Bound Moisture Equilibrium Moisture Curve Unbound Moisture Drying air humidity, R,air Free/Active Moisture (moisture removed during evaporation) Dry Mass Moisture Content, u Figure 2.1: Illustration of bound and unbound moisture within a drying material as plotted on a schematic sorption isotherm. Adapted from Masters (1992). the form of a hydroxyl ions or crystalline hydrates or in solutions within the solid, (Masters, 1992). The strength with which the absorbed water is bound to the product depends upon the method of absorption. The atmospheric humidity in equilibrium with a given solid moisture content is given by a sorption isotherm. Such an isotherm is shown schematically in Figure 2.1, which also illustrates some of the other terms defined in this section. The moisture content, u, is given on a dry mass basis, that is, the ratio of the moisture mass to the mass of moisturefree solid material. The complexity arising from the numerous mechanisms for moisture uptake just discussed means that, in practice, sorption isotherms for a given material must be determined experimentally The Driving Force for Drying Water activity reflects the active part of the moisture content. This is the part which, under normal circumstances, can be exchanged between the product and its environment. The water activity is a measure of the vapour pressure generated by the moisture present in a hygroscopic product and is defined a w = p A p sat A. (2.1.4) 7 CSH

21 2.1. Drying Here, p A is the partial pressure of water calculated at the wet bulb temperature and p sat A is the saturation pressure the partial pressure of water vapour above pure water at the product temperature. Bound moisture has a water activity less than unity and therefore exerts an equilibrium vapour pressure lower than the partial pressure above pure water at the product temperature; in contrast, the activity of unbound moisture is unity. Static equilibrium is defined as a set of conditions under which a material does not exchange moisture with its environment. The water activity is defined under equilibrium conditions which, by definition, implies that the partial pressure of water vapour at the surface, p A,sur, is equal to the partial pressure of water vapour in the surrounding bulk gas, p A,. In non-equilibrium situations, moisture exchange between the product and its surroundings, i.e., drying, is driven by a difference in these partial pressures, dm dt = KA p A,sur p A,, (2.1.5) where K is the mass transfer coefficient based on a partial pressure driving forces. M is the total mass of the droplet and A is the area available for mass transfer. Free or active moisture is that in excess of the equilibrium level. This may consist of unbound and some bound moisture. Only free moisture may be removed in a drying operation Stages of Drying Experimental observations of drying porous solids typically record the variation of moisture content with time; a typical such curve is shown schematically in Figure 2.2a. It is clear that the drying rate varies as a function of time, a result arising as a consequence of the various processes influencing moisture removal. It is possible to identify a number of different drying periods, as was first done in the classic works of Sherwood (1929a,b, 1930). These stages of drying are more clearly observed in Figure 2.2b, which is a plot of drying rate against moisture content. 3 As drying commences, there is a brief period (A B in Figure 2.2) during which the solid heats and the drying rate increases. When drying droplets, this stage is typically very short: Oakley (2004) gives an example showing the warming-up period lasts 0.1 s for a 300 µm droplet. The drying rate increases until point B where thermal equilibrium is established between the solid and its surroundings. The line B C represents the continuation of this condition of dynamic equilibrium during which the energy lost by evaporating moisture equals that transferred to the solid from the drying gas. This is the constant-rate drying period where the moisture flux from the surface remains near constant. The surface of the drying material and thus the vapour directly above remains saturated with moisture 3 Note that the plots in Figure 2.2 refer to drying in the context of moisture removal from porous solids. The situation is a little different when moisture is present in solution, as discussed in Section CSH

22 2.1. Drying A B Dry Mass Moisture Content, u C D E 0 0 Time, t (a) Falling-Rate Period C Constant-Rate Drying Period B Drying Rate, ṁ vap Second Falling-Rate Period D First Falling-Rate Period A 0 0 E Dry Mass Moisture Content, u (b) Figure 2.2: Schematic drying rate curves showing: (a) moisture content on a dry basis against time; and (b) drying rate against dry mass moisture content. 9 CSH

23 2.1. Drying Funicular Pendular Flow Diffusion Flow Diffusion Evaporation (a) Figure 2.3: Following the critical point, the drying front recedes within the porous solid. (a) During the first falling-rate period, the liquid is in the funicular state and so flow to the surface is possible. Although there is some diffusion in the vapour phase, most moisture is evaporated at the surface. (b) In the second falling-rate period the fluid near the surface breaks up and enters the pendular regime. Evaporation from the solid is now controlled by vapour diffusion from the funicular pendular boundary. (b) throughout the constant-rate period. Further, when convection is the sole mechanism of heat transfer to the droplet, the solid surface is at the wet bulb temperature, (Perry and Green, 1997). The constant-rate period persists whilst moisture is supplied to the surface at a rate sufficient to maintain saturated conditions. When this is no longer the case the drying rate begins to decrease and the temperature rises, (Scherer, 1990). This is the start of the first falling-rate period, C D, during which air enters the pores and the drying front recedes beneath the solid surface. The transition between these two regimes, point C in Figure 2.2, occurs at the critical moisture content. The critical moisture content is a function of the drying history and since it is an averaged value the mass and geometry of the material being dried. During the first falling-rate period, the liquid remains in the funicular condition and so contiguous pathways to the surface persist, (Figure 2.3a). As a result, the majority of the evaporation continues to occur from the external surface with some moisture evaporating within the unsaturated pore-space and moving to the surface by diffusion, (Schlünder, 2004). It is for this reason that the first falling-rate period is sometimes referred to as the period of unsaturated surface drying, (Perry and Green, 1997). Eventually, the drying front recedes so far that the pathways to the surface break up and the liquid near the exterior enters the pendular condition, (Figure 2.3b). The liquid flow to the surface then stops and moisture is removed solely by vapour diffusion. This is said to be the second falling-rate period, (D E in Figure 2.2), during which the drying rate drops further and the external surface of the solid begins to approach the temperature of the 10 CSH

24 2.2. Spray Drying drying air. The rate of mass transfer is now controlled by the resistance of the solid layer. The drying rate falls to zero once the equilibrium moisture content is reached, (E) Industrial Drying The differing demands made by various industries of their drying equipment has led to a wide variety of dryer types, (Keey, 1978; Perry and Green, 1997). Dryer selection is primarily driven by consideration of the needs of the product. For example, granular detergent manufacturers need to produce a powder with the right dissolution rate, placing great importance on the porosity, density and pore size of the dried product. They also need a drying system that can handle a high throughput of material at a relatively low cost. In contrast, the food industry is interested in flavour retention and may often be working with smaller quantities of high-value, heat sensitive products. All these factors, along with others such as how the product is conveyed and the heating medium used, will play a part in dryer selection, (Coulson et al., 1996). Table 2.1 lists some of the main dryer types used in industry. 2.2 Spray Drying This thesis is concerned with spray drying and, in particular, the development of a new model to describe the drying of individual droplets within a spray drying tower. In this section, the spray drying process and typical industrial applications are discussed in more detail Process Overview At its simplest level, spray drying involves the feed, in liquid or slurry form, being sprayed into a drying medium. This is normally hot air. In more detail, the spray drying unit operation may be considered as being composed of the four stages illustrated in Figure 2.4. The atomisation of a pumpable feed to form a spray is the key characteristic of spray drying. Two principle types of atomiser are used in industry: rotary atomisers making use of centrifugal energy; and pressure nozzles which exploit pressure energy to atomise the spray. Multiple injection levels may be used to handle higher flowrates. Whichever the type of injector chosen, the initial droplet diameter will be in the range µm. The result of the atomisation must be a spray which provides optimum evaporation conditions leading to the desired characteristics in the dried product. The manner in which the spray droplets contact the drying medium determines their subsequent drying behaviour and, in turn, greatly influences the properties of the final product. The form of spray-air contact is determined by the location of the atomiser relative 11 CSH

25 2.2. Spray Drying Table 2.1: Summary of the major types of continuous driers used in the process industries, (Sinnott et al., 1999). Dryer Type Conveyor/ Tunnel Drum Fluidised Bed Pneumatic Rotary Spray Description Solids move along a tunnel on trays or a continuous belt. Drying air is passed over the solids or up through the belt. The temperature profile can be controlled and the thermal efficiency is good, but such driers have long retention times and high set-up and maintenance costs. Used for liquid and dilute slurries which form a film on the surface of a heated, rotating drum. Drum driers have high thermal efficiencies and, due to short residence times, are suitable for heat sensitive materials. Suitable for granular and crystalline solids with a diameter of 1-3 mm. These driers have short retention times, rapid and uniform heat transfer and good temperature control, but power requirements are high. Upward flowing hot gas pneumatically conveys and rapidly dries suspended particles. Such driers have poor thermal efficiency, but are useful for particles which must be dried rapidly but are too fine for a fluidised bed. Short contact times mean large particles cannot be dried. Free-flowing solids move along a rotating, inclined cylinder and are dried by contact with drying air and the cylinder wall. High throughput, high thermal efficiency and relative low capital and running costs are set against difficulty achieving uniform drying. Can handle any pumpable material. The feed is atomized to droplets which then fall through drying air forming particles, which are then collected by cyclones or bag filters. Spray driers have high throughputs, short residence times and good control over product properties. However, energy requirements are high. Feed Atomisation rotary atomizer pressure nozzle Spray-Air Contact co-current countercurrent mixed Powder Separation product discharge from chamber and separation unit total product discharge from separation unit Spray Evaporation Figure 2.4: The principle process stages of a generic spray drying process, adapted from Masters (1992). 12 CSH

26 2.2. Spray Drying to the air inlet. Broadly speaking, the flow may be considered either co- or countercurrent. In the co-current arrangement the product and air pass through the dryer in the same direction. This is by far the most common arrangement, (Zbiciński and Zietara, 2004), and is especially suited to the drying of heat sensitive products. As the wet feed immediately contacts the hottest air, drying is rapid and the drying air cools accordingly. The product temperature remains around the wet bulb temperature throughout the initial drying period. Subsequently, the product is in contact with cooler air and is at no point subject to thermal degradation. Counter-current operation offers greater thermal efficiency as the liquid feed and air enter at opposite ends of the drier. However, this means that the driest material is exposed to the hottest air. Consequently, the set-up is only suitable for products which are non-heat-sensitive. There are also dryer designs which combine co- and counter-current flow patterns and these are termed mixed-flow driers. The choice of how to contact the spray with the drying air is determined by the material being dried and the desired product properties. Co- and counter-current arrangements give different particle morphologies due to the different particle temperature histories. This can lead to counter-current set ups producing a less porous product with higher bulk density. Slower evaporation reduces the tendency to puff, lowering the particle porosity. However, as mentioned above, the configuration may only be used for products which can withstand heat treatment. Conversely, co-current driers feature rapid evaporation preventing high particle temperatures. The downside is that such high drying rates are more likely to cause particle expansion or fracture, producing non-spherical, porous particles. The degree of agglomeration will also be affected by the dryer arrangement and, in turn, will influence product properties such as coarseness. The morphological development of droplets is considered further in Section 2.4. Evaporation occurs as the fluid feed comes into contact with the drying air. As discussed previously, the drying rate will be determined, in part, by the physical layout of the spray drying operation. However, it is also important to understand how the composition of the feed will affect the drying behaviour of the particles in the spray. The unique feature of spray drying is that drying and formation of a powdered product occur simultaneously. Hence an understanding of the physics of droplet drying is essential in order to describe the unit operation. Once the dried product has formed, a final separation stage is necessary. Two principle systems may be identified. In the first, (Figure 2.5a), primary separation occurs in the drying tower itself, with the majority of the product being removed from the base of the tower. The remaining product exits entrained in a separate air discharge stream, which is sent to secondary separation equipment, e.g., cyclones, bag filters or electrostatic precipitators, (Perry and Green, 1997). The second system, (Figure 2.5b) operates with total 13 CSH

27 2.2. Spray Drying Product In Product In Air In Air In Air In Air In ATOMIZER Air Out ATOMIZER Air Out DRYING CHAMBER DRYING CHAMBER Air and entrained powder CYCLONE CYCLONE Primary Product Discharge (a) Secondary Product Discharge (b) Total Product Discharge Figure 2.5: Product discharge from a co-current drying system with: (a) primary separation in the drying tower; and (b) total recovery in the dedicated separation equipment. Diagram adapted from Masters (1992). recovery of the dried product in the separation equipment. This places great importance on the efficiency of the separation system employed and, for obvious reasons, can only be used with a co-current set-up Applications of Spray Drying The range of spray drying technology in current usage reflects the diversity of the industries that dry their products in this way, (Masters, 1992). The ubiquity of this drying technology results from a number of factors. Of the dryer types listed in Table 2.1, the spray and drum dryers are the only two that can handle a pumpable fluid feed. Of these, the spray dryer is the only one which can produce powders of a specific particle size, porosity and moisture content. Spray dryers are also capable of handling a very wide range of feed materials and flow-rates: a pharmaceutical company may use a lab-scale system to dry a few kilograms of high-value product whereas the mining industry might use a much larger drier to continuously process over 100 tonnes of material per hour. The principle advantages and disadvantages of spray dryers are outlined in Table 2.2. The production of laundry detergents is one of the best known applications of spray drying and continues to be a large market, (de Groot et al., 1995). Figure 2.6 shows that spray dried laundry products powder and tablets account for almost 80% of the UK laundry detergent market, (Snapdata, 2005a). In the US, the percentage is less, but this still represented a $1.2 billion market in 2004, (Snapdata, 2005b). Most detergents are formed in counter-current towers with multi-level nozzle atomisation, (Masters, 1992). 14 CSH

28 2.3. Spray Dryer Modelling Table 2.2: Summary of the major advantages and disadvantages of spray dryers, (Masters, 1992). Advantages Specific product properties can be achieved consistently throughout the dryer run. Operation is continuous and easily adaptable to automatic control. Unit can be designed for virtually any capacity required. Can control product density and porosity. Can handle heat-sensitive and heatresistant materials. Can handle flammable, explosive, malodorous and toxic materials and those requiring hygienic conditions. Can dry feedstocks in solution, slurry, paste or melt form, including corrosive and abrasive feeds. Disadvantages High fabrication and installation costs. Large size requires expensive supporting structures. Poor thermal efficiency (except with very high inlet-air temperatures). The product is relatively heat insensitive, allowing drying air temperatures of up to 400 C and, consequently, relatively high thermal efficiencies are achieved. High throughput and good reliability are especially important when producing such a bulk commodity. Spray drying allows this, consistently producing powder with the correct physical properties to yield the desired characteristics in the finished product. 2.3 Spray Dryer Modelling Despite the importance and wide application of spray drying technology discussed in the previous section, the theoretical modelling of spray dryers is still relatively poorly developed. Spray dryers are more difficult to model than other dryer types 4 ; the complex interactions between the swirling gas flow and atomised drying droplets mean that simple scale-up techniques common elsewhere in chemical engineering design cannot be used, (Oakley, 1994). Marshall and Seltzer (1950a,b) considered the basic principles and design aspects of spray drying, reviewing the knowledge and industrial practice at the time. This was, on the whole, based on correlations and qualitative observations of different drying systems. One of the first attempts to model the behaviour of a spray drying tower was by Chaloud 4 Most spray dryer models deal with the first three stages presented in Figure 2.4, i.e., the separation system is not considered. This is the approach adopted in this work. 15 CSH

29 2.3. Spray Dryer Modelling Others 7% Liquid 14% Tablet 36% Powder 43% Liquid 73% Powder 27% Total UK Market Value (2004): 820 million (a) Total US Market Value (2004): $4.5 billion (b) Figure 2.6: Size and segmentation of the laundry detergent market in: (a) the UK; and (b) the US, (Snapdata, 2005a,b). Data refers to the 2004 financial year and segmentation is based on value %. et al. (1957). They demonstrated the logical application of chemical engineering principles to deduce a number of operating lines for spray drying towers. These related the moisture content and bulk density of the dried powder to the drying rate and, beyond this, to independent variables such as slurry moisture content, tower height and drying-air throughput. Such an approach formalised previous intuition-based understanding of tower behaviour, but was still essentially qualitative. To classify more quantitative descriptions of spray dryer behaviour, it is helpful to adopt the hierarchy of modelling levels introduced by Oakley (2004): Level 0 Level 1 Level 2A Level 2B Heat and Mass Balances; Equilibrium Based Models; Rate-based with simplifying assumptions about fluid flow; Rate-based with simulation (CFD) of the continuous gas phase and particle motion. The most appropriate choice of modelling level to use depends on the detail and accuracy required from the solution, that is, on the purpose of the model and problem to be solved. The remainder of this section discusses each of these modelling levels in turn and reviews their application to problems reported in the literature Heat and Mass Balances The simplest modelling layer is to apply appropriate energy and mass balance equations to the system. Such models require no detailed knowledge of the dryer geometry, or the 16 CSH

30 2.3. Spray Dryer Modelling processes occurring therein, (Coulson et al., 1996; Masters, 1992). However, the inlet streams must be specified, along with the exit moisture content. As an example, consider a co-current device drying a product containing 32.8% moisture on a wet basis at a rate of 1800 kghr 1. The required outlet moisture content is 7%, again on a wet basis. There are energy losses of 50 kw and it is assumed that the outlet product is at the temperature of the surrounding gas. The heat and mass balance shows that the air stream specified along with the rest of the problem in Table 2.3, is capable of performing this drying operation: the outlet gas temperature will be 74 C and R = 30%. Whilst the predictive power of heat and mass balances is limited, they are useful to assess the thermodynamic feasibility of an operation at the earliest design stage. There are also some examples of such simple models being used on their own in the literature. For example, Baker and McKenzie (2005) surveyed a number of industrial spray drying installations and used a model based on a simple heat balance to estimate the wasted energy. Goffredi and Crosby (1983) used a set of heat and mass balances to develop simplified models of co- and counter-current driers and investigate their sensitivity to variations in product and drying gas flow-rates. More recently, Montazer-Rahmati and Ghafele-Bashi (2007) modelled a counter-current dryer by applying heat and mass balances over multiple stages within the tower. They applied this approach to simulate a detergent production tower with some success. Table 2.3: Sample process specification and results from applying a simple heat and mass balance. Product Generic Slurry Moisture Water Feed moisture content kgkg 1 wet solids Feed rate (wet solids) 1800 kghr 1 Feed temperature 50 C Specific heat capacity 2 kjkg 1 K 1 Air In Product In Air In Inlet gas Air Absolute humidity 0.05 kgkg 1 bone-dry gas Flow rate (wet) 4500 kghr 1 Temperature 300 C Air Out Product moisture content kgkg 1 wet solids Exit air temperature 74 C Relative humidity at exit 30% Product Out 17 CSH

31 2.3. Spray Dryer Modelling Equilibrium Based Models An alternative to specifying the outlet product moisture content is to use a phase equilibrium relationship to relate the product moisture content to the humidity of the surrounding gas. Assuming that dried-particles at the outlet are in equilibrium with their surroundings allows use of a sorption isotherm to predict the exit product moisture content, ( 2.1.1). Equilibrium based models clearly require that the sorption isotherm is determined, but in return they yield more information about the feasibility of a given drying operation. A sorption isotherm could be used with the example summarised in Table 2.3 to predict the product moisture content at the dryer outlet. Using the generic sorption isotherm shown in Table 2.4 together with the feed conditions specified previously, it is simple to calculate that the product will leave with a wet-basis moisture content of 8%. That is, using the sorption isotherm has shown that the 7% figure guessed previously is thermodynamically unattainable. The air leaving the dryer will be at 82 C with R = 17%. Table 2.4: Sample process results and the sorption isotherm used to calculate the moisture equilibrium at the outlet. Feed specification as in Table 2.3. Product moisture content 8% Exit air temperature 82 C Relative humidity at exit 17% Relative Humidity, R / % Wet Mass Moisture Content, u An approach-to-equilibrium factor may be employed to handle situations where the product outlet stream is not in equilibrium with its surroundings. However, as a user specified parameter this provides little insight, (Oakley, 2004). Hence, the equilibrium models are really only of any use where equilibrium is attained. Intuitively, this is more likely to be valid for small droplets with long residence times in the dryer. Ozmen and Langrish (2003a) provide experimental evidence that this is indeed the case for dried-particles with a final size of 30 µm. 18 CSH

32 2.3. Spray Dryer Modelling Rate Based Models using Simplifying Flow Assumptions When it is not valid to assume equilibrium conditions at the outlet, the rate at which moisture is removed from the sprayed droplets must be considered. Alongside this, it is necessary to model the droplet residence time in the drying unit. Together, these allow the product moisture content at the exit to be predicted. However, achieving both of these objectives presents problems to the modeller. Droplet residence time in a spray drying tower may be modelled in two main ways: the first is by invoking simplifying assumptions about droplet motion whereas the second, discussed in the next section, is to use computational fluid dynamics to simulate the whole flow field. Invoking simplifying assumptions about the droplet motion equates to finding shortcut methods to model droplet residence times. Gauvin and Katta (1976) presented a design methodology whereby laboratory-scale experiments were used to develop correlations for droplet residence times in full-size spray towers. Their analysis allowed both drying rate and residence time to be functions of droplet diameter. Clement et al. (1991) modelled the gas in a spray dryer as a single well-mixed mass, allowing the standard expression for the residence time distribution (RTD) of a perfectly mixed reactor to be used. This was coupled to a shrinking core model to describe the drying of single droplets which were all assumed to be the same size. Birchal and Passos (2005) used the same physical model to simulate the drying of milk emulsions, although they present an improved solution algorithm. Both papers present results which exhibit fair agreement with experimental milk drying data. Later, Birchal et al. (2006) compared this model with a more sophisticated computational fluid dynamics (CFD) simulation, obtaining similar results from both for some key parameters. Palencia et al. (2002) used a similar RTD-based approach, modelling the gas as a series of well mixed vessels in series. Their method reproduced experimental data on milk drying with moderate accuracy. The strengths of using simplified descriptions of the gas phase include the ability to investigate dynamic changes in inlet conditions and flowrates and relatively fast computational times. The main weakness of this approach lies in the assumed residence time distribution. Perfect mixing is nothing more than a first approximation which, for many towers, will prove insufficient. Whilst it is simple to substitute an alternative RTD, this will almost certainly be unique and therefore need to be measured for each unit modelled. Thus, whilst this approach is useful for investigating changes to operating conditions on existing spray drying installations, it is not an adequate tool for detailed design, (Reay, 1988). 19 CSH

33 2.3. Spray Dryer Modelling Rate Based Models Using CFD The most sophisticated level of spray dryer modelling is to extend rate based models by introducing detailed descriptions of the fluid flow and particle processes. The advances in computing power and the rise of computational fluid mechanics has made such detailed descriptions possible, (Oakley, 1994). Today, such approaches represent the most comprehensive simulations of spray drying behaviour, (Fletcher et al., 2003). Crowe (1980) was the first to use computational techniques to simulate a spray drying tower. He used his particle-source-in-cell method, (Crowe et al., 1977) to handle the energy and momentum coupling between the gas and drying droplets. This is a Lagrangian- Eulerian modelling approach, whereby the gas phase is treated as a continuum an Eulerian viewpoint with the droplets being tracked through the flow field in a Lagrangian manner. Such an approach is sensible where one phase occupies a small fraction of the total volume of the solution domain, (Huang et al., 2003), as is the case throughout most of a spray drying tower. It is possible to treat both phases as a continuum the fully Eulerian or two-fluid model as was done by Platzer and Sommerfeld (2003) when modelling the dense spray region around a spray nozzle. The method delivers velocities and volume fractions of both phases as output parameters, but no information about the droplet size distribution. For this reason the approach is rarely used where the spray is dilute and the distribution of droplet sizes is important. A purely Lagrangian approach is also possible, although rare. Salman and Soteriou (2004) presented such a model, which they claim has advantages when modelling high volume-fraction, evaporating spray systems. The vast majority of spray drying simulations use the combined Lagrangian-Eulerian approach. Such a framework facilitates the incorporation of various physically motivated sub-models to describe droplet processes. Fletcher et al. (2003) identified three such droplet processes that merited inclusion in spray dryer simulations: droplet drying; droplet droplet interaction; and droplet wall interaction. A generic Lagrangian-Eulerian spray dryer model incorporating all these elements is illustrated in Figure 2.7. Whilst CFD-based spray drying models are relatively common in the literature, simulations incorporating all of these submodels are rare. There exist a large number of attempts to use CFD to model gas flow patterns in spray dryers, (e.g., Southwell and Langrish, 2000; Fletcher et al., 2003; Oakley, 2004). Langrish and Fletcher (2003) look forward and conclude that commercially available CFD packages are already capable of producing adequate simulations of the gas phase, although there is still considerable discussion regarding the most appropriate turbulence model, (Bayly et al., 2004; Oakley and Bahu, 1991; Zbiciński and Zietara, 2004). More recent papers focus on coupling one or more of the droplet sub-models to the CFD simulation. The description of droplet drying is the most important of the three droplet sub- 20 CSH

34 2.3. Spray Dryer Modelling Full Spray Dryer Model Sub Models Gas Flow Particle-Wall Interaction Particle Drying Particle-Particle Interaction (Agglomeration, Breakage) Figure 2.7: Schematic showing the structure of modern rate based spray dryer model using CFD and particle sub-models. models. In its simplest form, a drying model gives the rate at which moisture is lost and, from this, the average droplet moisture content. More sophisticated models give the spatially distributed moisture content, the temperature profile and perhaps even describe the morphological development of the droplet. Many researchers have included such a sub-model in their CFD simulations of spray dryers. These range from evaporating pure liquid droplets (e.g., Papadakis and King, 1988; Huang et al., 2003; Li and Zbiciński, 2005; Huang and Mujumdar, 2005), through models using a material dependent drying curve (e.g., Langrish and Kockel, 2001; Harvie et al., 2002; Huang et al., 2004; Zbiciński et al., 2005; Zbiciński and Li, 2006), to fully spatially resolved approaches, (Verdurmen et al., 2004). Single droplet drying models are discussed in detail in Section 2.4. Langrish and Fletcher (2003) state that considerable work is needed to model the adhesion and cohesion of particles due to stickiness. This is a pre-requisite to producing physically realistic sub-models for both droplet droplet and, especially, droplet wall interactions. Although stickiness is a common concept, its nature is highly complex and it is still poorly understood, (Kudra, 2003). Consequently, sub-models describing droplet interactions are poorly developed at present. When dealing with droplet wall interactions the typical approach is to assume all droplets stick on collision with a wall, (Langrish and Zbiciński, 1994). While such a methodology can produce useful results, (e.g., Straatsma et al., 1999), it is simplistic and fails to capture any of the complexities found in experimental observations of the modelled systems, (e.g., Kota and Langrish, 2006; Ozmen and Langrish, 2003b). Similarly, agglomeration is normally handled using traditional kernels which only consider droplet size and possibly velocity, (Sommerfeld, 2001; Verdurmen et al., 2004). However, as with droplet wall interactions, other droplet properties in particular the surface moisture content have a strong influence on the adhesion probability. Metzger et al. (2007) identify the development of 21 CSH

35 2.4. Single Droplet Drying Models new kernels to reflect these dependencies as a priority for the future; such descriptions will clearly be dependent on the predictions of the drying sub-model. The most comprehensive simulation of spray dryers to date is the Efficient Design and Control of Agglomeration in Spray Drying Machines (EDECAD) project, (Verdurmen et al., 2004). The principle deliverable of the project was a Design-Tool to relate the dryer geometry, process conditions, product composition and final powder properties. A spatially resolved drying sub-model was used, although evolving droplet morphology was not considered. Droplet droplet collisions are treated using the model of Sommerfeld (2001), with agglomeration treated using the classical model of Brazier-Smith et al. (1972). The project did not explicitly address droplet interaction with the wall. In conclusion, it is clear that rate-based models using CFD coupled with droplet submodels provide the most detailed description of the spray drying process. Whilst considerable progress has been made towards the development of a unified dryer model, there is room for improvement in all three droplet sub-models. The most fundamental of these is that describing droplet drying; a sub-model giving the surface moisture content of drying droplets is a pre-requisite for accurate prediction of particle agglomeration and wall build up. 2.4 Single Droplet Drying Models As discussed in the previous section, a droplet drying submodel is integral to any detailed simulation of the spray drying process, (Langrish and Fletcher, 2003; Oakley, 2004). The droplet moisture content specifically the surface moisture determines the rate of other processes occurring within a spray tower, such as agglomeration and wall deposition, (Fletcher et al., 2003; Blei and Sommerfeld, 2006). Building on the theory presented in Section 2.1.2, this section starts by reviewing the way in which single droplets dry, focussing on those containing suspended solids. The possible morphologies of spray dried particles are then discussed, before the literature relating to single droplet drying models is reviewed Droplet Drying Behaviour The theory introduced in Section 2.1 provides a basis for understanding the drying behaviour of sprayed droplets. However, a number of modifications must be made before such droplets can be modelled successfully. The evaporation of pure liquid droplets has been extensively studied for many years, (e.g., Frössling, 1938; Ranz and Marshall, 1952). Droplets in real spray dryers generally contain dissolved or suspended solids, but understanding the simpler pure liquid system forms the 22 CSH

36 2.4. Single Droplet Drying Models Drying Gas Temperature, T gas F Temperature, T d Boiling Temperature, T boil D E B C Wet-bulb Temperature A 0 0 Time, t Figure 2.8: The different stages of drying for a liquid droplet containing solids. basis for describing more complicated drying mechanisms, (Masters, 1992; Oberman et al., 2004). It is well known that evaporation from droplets containing dissolved solids is slower than that given by (2.1.5) the rate at which mass is lost from pure liquid droplets. Solutes lower the vapour pressure of water, p A,sur, thus lowering the driving force for evaporation, (Masters, 1992). Annamalai et al. (1993) present a neat analysis of evaporation from two-component droplets, considering the case where both components evaporate. Keey (1992) discusses selective evaporation effects which may occur in such circumstances. Whilst dissolved solids reduce the driving force for evaporation, Ranz and Marshall (1952) showed that the presence of suspended, insoluble solids has a negligible vapour pressure lowering effect. Drying during the constant rate period may be treated in the same way as for a pure liquid droplet. However, following formation of a rigid crust, the droplet behaves more like a porous solid and the discussion in Section applies. Figure 2.8 shows a typical temperature history for an individual droplet containing dissolved or suspended solids drying in a spray tower; many of the stages are similar to those outlined for generic solid drying in Section The droplet rapidly heats to the wet-bulb temperature, (A B) and then remains at the wet-bulb temperature and drying proceeds at a near constant rate whilst the surface remains saturated with moisture, (B C). However, the presence of dissolved or suspended solids can alter this behaviour. A high initial solids loading results in the constant rate drying period being brief, if observed at all, (Cheong et al., 1986; Dolinsky, 2001). Any dissolved solids will concentrate as moisture is removed, reducing the moisture vapour pressure and causing the surface temperature to 23 CSH

37 2.4. Single Droplet Drying Models rise above the thermodynamic wet bulb. The falling rate drying period begins when moisture can no longer be supplied to the surface at a rate sufficient to maintain saturated conditions, (C D). The transition between these two regimes occurs at the critical moisture content and, in the presence of suspended solids, may also coincide with the start of crust formation, (Cheong et al., 1986). During the falling rate period the droplet will heat above the wet bulb temperature. Vaporisation will commence if the temperature reaches the boiling point of the solution. Considerable energy is required for vaporisation and so the sensible heating of the droplet halts (D E). This phase is termed the boiling regime and the drying rate is now controlled by external heat transfer to the droplet. The presence of a dissolved solute will typically raise the boiling point of a solution; the droplet temperature therefore increases slowly in the boiling regime as the solution becomes more concentrated. Once all the free moisture has been removed, the temperature will again rise, asymptotically approaching that of the surrounding gas, (E F) Droplet Morphologies The removal of moisture from a spray of droplets involves simultaneous heat and mass transfer and, uniquely in a spray dryer, this process is coupled to concurrent particle formation. When drying droplets containing dissolved or suspended solids, a wealth of different dried-particle morphologies may form. Which type of particle forms depends upon the composition, size, temperature and drying history of the droplet, (Ranz and Marshall, 1952). Further, because droplet drying and particle formation occur simultaneously, the drying mechanism and resultant kinetics are, in turn, strongly dependent on the evolving droplet microstructure, (Huntington, 2004). Figure 2.9 illustrates the main dried-particle morphologies that may result when drying droplets containing dissolved or suspended solids. At very low solids concentrations, droplets continue to evaporate like pure liquid spheres and no particle is formed. However, in any spray drying application of interest one of the other drying routes will be followed. As described in the previous section, droplets initially shrink ideally in the constant rate period. Crust formation commences when the moisture content falls below a critical value at a single preferential surface site, usually the point of maximum mass transfer as determined by the surrounding flow field, (El-Sayed et al., 1990). Once initiated, the crust spreads rapidly over the surface of the droplet, forming a structured solid shell and stabilising the droplet diameter, (Cheong et al., 1986). The behaviour after this point strongly depends on the nature of the shell formed and the drying conditions. Walton and Mumford (1999a) identify three distinct categories of droplet, which behave in morphologically similar ways when dried: crystalline, skin-forming and agglomerate. Within these categories, the major differences in morphology result from the drying air 24 CSH

38 2.4. Single Droplet Drying Models No particle formation Shattered Particle Solid Particle Collapse Low solids concentration, <1%w/w High temperature Re-inflation Dry Shell Wet Shell Initial Droplet Saturated Surface Drying Crust Formation High temperature Internal Bubble Nucleation Uninflated Shell Blistered Particle Shrivelled Particle Inflated Puffed Particle Figure 2.9: Schematic showing some of the different dried-particle morphologies that may result when drying droplets containing dissolved or suspended solids. temperatures and, consequently, the drying rate. At lower temperatures, the mechanisms allowing for droplet shrinkage and deformation are more pronounced, (Alamilla-Beltrán et al., 2005); moisture loss and the rate of shrinkage are slower, allowing more time for structures to deform, shrink and collapse, (Oakley, 1997). A solid dried-particle often forms when the drying gas temperature is below the moisture boiling point, (El-Sayed et al., 1990). Once a rigid crust has formed, such droplets dry somewhat like a porous solid medium with moisture menisci receding into the droplet, ( 2.1.3). However, low temperature drying does not always result in the formation of solid dried-particles, (Walton and Mumford, 1999a,b). With aerated feeds, bubbles or voids can arise as a result of two mechanisms. The droplet can become super-saturated with any dissolved air as a result of increasing solute concentration. Bubbles typically nucleate around the transition from the constant- to falling-rate drying period, (El-Sayed et al., 1990). Greenwald and King (1981, 1982) present results showing internal voidage formed in this way. Alternatively, entrained air pockets can coalesce and expand during drying to produce hollow particles. Verhey (1972) conducted an extensive study on drying milk which demonstrated that the gas vacuoles in this system originated from air entrainment during atomization. At high temperatures, droplets tend to inflate, form crusts and blister or break, (Alamilla- Beltrán et al., 2005). For aqueous solutions, this happens as the droplet temperature ap- 25 CSH

39 2.4. Single Droplet Drying Models (a) Figure 2.10: Images of hollow spray dried particles taken with (a): a scanning electron microscope; and (b) an optical microscope. Courtesy of Cheyne et al. (2002) (b) proaches 100 C, corresponding to the boiling regime discussed in the previous section, (Greenwald and King, 1981, 1982). Inflation results from large partial pressures of water vapour joining inerts in a bubble, (El-Sayed et al., 1990; Oakley, 1997). Subsequent drying behaviour and final dried-particle morphology are determined by the chemical and physical properties of the shell or film regions, (Walton and Mumford, 1999b). For example, the rheological properties of skin-forming materials allow such droplets to undergo multiple inflation-collapse cycles. Such behaviour is observed when drying coffee extract, (Charlesworth and Marshall, 1960; Hecht and King, 2000a) and skim-milk, (Walton and Mumford, 1999a), amongst many others. This behaviour may be contrasted with less pliable crystalline droplets that tend to undergo only partial inflation or form hollow or semihollow dried-particles. As an example, Figure 2.10a shows an image of a dried detergent droplet from Cheyne et al. (2002). The droplet has undergone partial inflation and contains a large central void. This is seen more clearly in Figure 2.10b where a dried-particle has been cut open and imaged following capture in wax. The morphology of the dried-particles produced depends strongly on the nature of the shell formed. The droplet drying models discussed in the next section can say very little about this, thus limiting the scope of the morphological predictions they are able to make; in general, a given drying model is only capable of simulating the morphological evolution towards one type of dried-particle. It would be a great achievement for a single drying model to be capable of simulating multiple dried-particle morphologies, with structural evolution determined by the evolving droplet composition and drying conditions. 26 CSH

40 2.4. Single Droplet Drying Models Droplet-Averaged Drying Models There are many single droplet drying models in the literature. At the highest level, these can be divided into those which only model droplet averaged quantities such as moisture content and temperature and those which are based on mechanistic pictures of droplet drying. This second class of model returns some morphological information such as dried-particle size and may give spatially resolved moisture profiles. The simplest droplet-averaged approach to use is a characteristic drying curve, (Langrish and Kockel, 2001; Chen and Lin, 2004; Huang et al., 2004). Essentially an empirical method, the approach relies upon first identifying an unhindered drying rate and then measuring a drying curve for each material considered. Once normalised, it is assumed that this curve is unique for a given material and independent of external drying conditions, sample size and geometry. If accepted, this implies that drying curves obtained from labbased single droplet studies may be used to describe drying of much smaller droplets in industrial scale equipment. Keey (1992) provides an overview of the experimental tests of the characteristic drying curve. The results show that the method works well for several materials over a modest range of temperatures, air humidities and velocities. However, the concept did not work for large droplets with R 0 > 10 mm. Also, there is no data reported for drying of slurry droplets. Huang et al. (2004) investigated the use of characteristic drying curves in CFD models of spray dryers. They demonstrated that the choice of drying model influences both the droplet mass history and the path it takes through the spray dryer, but presented no comparison with experimental data. The minimal computational expense of this method makes it attractive for use in CFD applications, especially where transient flow patterns are being investigated, (Langrish and Kockel, 2001). However, only droplet averaged properties are returned; no information is obtained about morphological changes that may occur as a result of drying, or even final dried-particle sizes. The reaction engineering approach, introduced by Chen and Xie (1997) maintains the simplicity of implementation associated with the characteristic drying curve whilst claiming a sounder physical basis. The method considers drying as a competitive process between an activation type evaporation reaction and a condensation reaction. A normalised curve of the evaporation activation energy against moisture content is considered to be characteristic of a given material. Chen et al. have successfully applied the reaction engineering approach to a number of spray drying systems, (Chen et al., 2001; Lin and Chen, 2005, 2006, 2007). Chen and Lin (2004) showed the reaction engineering approach gives better predictions than a characteristic drying curve when applied to drying milk droplets. Recently, Woo et al. (2008) conducted a comparison of the two models and again appear to demonstrate the 27 CSH

41 2.4. Single Droplet Drying Models superiority of the reaction engineering approach, especially when using an extension which allows the surface moisture to be predicted. However, wider take up of the method is so far lacking, perhaps because considerable experimental effort is required to obtain the activation energy curve and sorption isotherm for each system. Both the characteristic drying curve and reaction engineering approaches return only droplet averaged properties; they give no spatially resolved or morphological information. Nevertheless, such models are still common in CFD simulations as they are simple to implement and computationally cheap Mechanistic Models Spatially resolved, or mechanistic models are based upon simplified pictures of moisture movement within a drying droplet. A number of such simplified pictures exist, although almost all simulations of solid containing systems are based on just two. These may be loosely termed the effective diffusion and shrinking core approaches. Models containing a bubble may be considered a third class of model which builds on aspects of the first two approaches. This section reviews the many different drying models found in the literature, grouping them into these three broad categories. All the models discussed here assume that the droplets remain spherical throughout their drying history. Effective Diffusion Coefficient Models Effective diffusion coefficient models assume that moisture transport within a drying droplet can be described by Fickian diffusion. It is possible to model binary systems exactly using the convection diffusion equations, (e.g., Hecht and King, 2000a,b), but in most cases the diffusion coefficient matrix is unknown or if there are more than two components the system is simply too complicated. In such cases it is found that an effective diffusion coefficient is required to adequately reproduce experimental results, (Whitaker, 1977). This effective diffusion coefficient is normally a strong function of local moisture concentration and temperature and, as such, needs to be determined experimentally for each system investigated, (Charlesworth and Marshall, 1960; Ferrari et al., 1989; Kentish et al., 2005). There is no explicit formation of a shell region in such models. However, a reduction in the effective diffusion coefficient at low moisture contents can achieve a reduction in mass transfer similar to that caused by shell resistance. Whilst it is generally necessary to determine effective diffusion coefficients experimentally, Whitaker (1977) demonstrated that, under certain assumptions, the same equations can be derived from a rigourous treatment of mass transfer. Van der Lijn (1976) was the first to apply an effective diffusion coefficient approach to individual droplets drying in spray driers, although the general use of such methods goes 28 CSH

42 2.4. Single Droplet Drying Models back much further, (Lewis, 1921; Sherwood, 1929a,b, 1930). Since then, effective diffusion models have been used by a large number of researchers to simulate single droplets drying. Wijlhuizen et al. (1979) based their model on an effective diffusion coefficient approach and applied it to investigate the inactivation of phosphate during spray drying of skim milk. Ferrari et al. (1989) used experimental data to determine an appropriate effective diffusion coefficient for drying milk powder and subsequently obtained close agreement with measured drying curves. Adhikari et al. (2003, 2004) used a model based on that described by Van der Lijn (1976) to investigate parameters affecting surface stickiness during drying. All these studies would be impossible without a spatially resolved model. The model of Sano and Keey (1982) has been used by many researchers to simulate droplets in spray driers. Etzel et al. (1996) and Kentish et al. (2005) both used variants of this to simulate milk drying and a simplified version of the model is used as the droplet drying sub-model for the EDECAD project, (Verdurmen et al., 2004). Similarly, the model of Frey and King (1986) has been used many times, for example, by both El-Sayed et al. (1990) and Wallack et al. (1990) to simulate coffee drying. Recently, Porras et al. (2007) have presented a treatment of a binary liquid in a porous medium using an effective diffusion coefficient approach and volume averaging. The effective diffusion coefficient approach is the most common method used when spatial moisture profiles are desired. However, on its own, the method yields relatively little information about droplet morphological development. For this reason, several models base their description of moisture transport on such an effective diffusion approach which they then combine with a further model for morphological changes. Some such models are discussed below. Shrinking Core Models The shrinking core model consists of two stages. Initially the droplet shrinks ideally, with solids accumulating at the droplet surface. After some time, a crust forms and the subsequent size of the droplet is fixed. The point at which crust formation occurs may be given by a critical moisture fraction averaged across the particle (e.g., Mezhericher et al., 2007), or at the surface (e.g., Liang et al., 2001), or by some critical droplet-averaged porosity, (e.g., Kadja and Bergeles, 2003). After the formation of a surface shell, the model enters the second stage in which the solution-crust interface recedes into the porous particle. Evaporation occurs at this receding interface and water must be transported to the surface by vapour diffusion through the dried shell. The mass transfer resistance of the crust therefore becomes the factor limiting the rate of continued drying. The simplest application of the shrinking core approach assumes that the droplet outside the crust region is well mixed and, consequently, the need to track the spatial distribution 29 CSH

43 2.4. Single Droplet Drying Models of water is removed. Such models have been used to simulate droplets of sodium sulfate decahydrate, (Cheong et al., 1986) and coal slurries, (Kadja and Bergeles, 2003) amongst many others, (Audu and Jeffreys, 1975; Dolinsky, 2001; Nešić and Vodnik, 1991). If spatial information on the droplet moisture content is required, a shrinking core model can be combined with an effective diffusion approach. Prior to shell formation moisture movement within the droplet is described using an effective diffusion coefficient. The trigger for shell formation can then be based on a value at the droplet surface. A dry shell subsequently grows inwards as per the shrinking core model, with an effective diffusion coefficient continuing to describe the wet core. Such an approach is adopted by Dalmaz et al. (2007) to simulate droplets of colloidal silica and skim milk and by Werner et al. to model an amorphous polymer solution, (2008a) and maltodextrin, (2008b; 2008c). Nešić and Vodnik (1991) presented the largest selection of experimental comparisons, investigating colloidal silica, sodium sulphate and skim milk systems. Whilst agreement with experimental measurements is good in all these papers, the comparison is limited to droplet mass and temperature histories. Consequently, the spatial predictions of such models have not been experimentally verified. Elperin and Krasovitov (1995) extend the shrinking core model further, by solving transport equations within the shell region itself. Seydel et al. (2006) present the most advanced model to date: they employ a population balance to describe the suspended solids and couple this to an effective diffusion equation for the droplet moisture. A shrinking core type approach then describes morphological evolution after the formation of a rigid shell. Whilst impressive, the implications of this model are not fully explored and comparison with experimental data is limited to qualitative observations. One of the key postulates of the shrinking core model that evaporation proceeds at the interface between the dried shell and the wet core implies that this interface remains at or around the wet bulb temperature. This results in the prediction of a large temperature gradient across the crust region. Recognising this, Farid (2003) developed a model based on the shrinking core approach in which heat transfer due to both internal conduction and external convection controlled the drying rate. However, the temperature gradients such an approach implies are unphysical and contradict experimental findings, (Schlünder, 2004; Dalmaz et al., 2007). These typically show that their are no major temperature variations within drying particles, (Chen and Peng, 2005). To overcame this objection, Schlünder (2004) introduces a wet surface model which invokes capillary transport through the crust and only allows vaporisation at the particle-air interface. However, to date no work has been done towards implementing such an approach within a droplet drying simulation. Lee and Law (1991) point to a second problem with the shrinking core approach: such models conserve mass only if the porosity of the dried shell is the same as that of the original slurry. This condition is not generally true. Lee and Law overcome this by introducing a 30 CSH

44 2.4. Single Droplet Drying Models continuously expanding vapour-saturated space located at the centre of the particle. Models With a Bubble The third class of single droplet drying model discussed here encompasses those models which include a centrally located bubble. Such models allow the simulation of hollow particles; as discussed in Section 2.4.2, such a dried-particle morphology is common in practice. Wijlhuizen et al. (1979) were among the first to publish a drying model with a bubble. They postulated the presence of a bubble from the start, which can expand and contract as a result of droplet temperature variations. Moisture profiles within the droplet are tracked assuming Fickian diffusion of moisture. However, their model does not simulate drying into the boiling regime, i.e., inflation. Sano and Keey (1982) presented a model that can simulate inflation but is otherwise similar to that of Wijlhuizen et al.. Results are compared with experimental measurements of milk drying and good agreement is obtained. As the model is incapable of simulating inflation-deflation cycles, a key question raised is how to set the maximum particle size. Without some attempt to do this, the inflating droplet would continue expanding until it was unrealistically large. This same problem is faced by Hecht and King (2000b) who present a further model containing a bubble. They introduce a check which considers the surface tension of the shell and prevents expansion if the pressure within the bubble is insufficient to over come this. They also impose an arbitrary maximum particle size to prevent excessive expansion. Frey and King (1986) presented a model capable of simulating a droplet with multiple small internal bubbles. This worked by homogenising the entire droplet and dealing with the equivalent binary, ideal homogeneous mixture. El-Sayed et al. (1990) made experimental measurements of sucrose and maltodextrin solutions, coffee extract and skim milk drying and applied the model of Frey and King to make satisfactory predictions prior to droplet boiling. Minoshima et al. (2001, 2002) developed a relatively simple model to predict the formation of hollow granules. Their model uses the structural properties of the crust to the predict dried-particle size and shell thickness. Tsapis et al. (2005) considered the electrostatic stabilisation forces between colloidal silica particles to produce impressive simulations of shell formation and buckling. Neither of these models simulated spatial moisture profiles, but are interesting because they demonstrate how the physical properties of droplets might be used inform the structural and morphological simulation. 31 CSH

45 2.5. Scope of this Thesis 2.5 Scope of this Thesis There have been numerous attempts to simulate droplets drying, as reviewed in Section 2.4. Whilst these previous models contain several good ideas, there are a number of obvious deficiencies. Droplet moisture content is often resolved spatially, but variations relating to the solids within the droplet beyond inclusion of an explicit shell region are normally ignored. Morphological development may be simulated by means of the shrinking core or bubble-based approaches, but no existing model is capable of simulating multiple dried-particle morphologies; once the use of a particular drying model has been specified, the type of dried-particle is determined. Related to this inability to predict dried-particle morphologies is the lack of any clear rational for choosing a particular model for morphological development. Given the importance of dried-particle morphology in determining the properties of the dried-powder, the combination of these shortcomings amounts to a serious weakness in existing models. The new model presented in this thesis attempts to address the major weaknesses identified in existing descriptions of droplet drying. In particular, the new droplet drying model will: simulate spatially resolved moisture profiles and, further, be capable of modelling the local concentration of a dissolved solute; utilise a population balance approach, based on that developed by Seydel et al. (2006), to simulate the nucleation, growth and transport of suspended solids; provide for the inclusion of a centrally located bubble and explicit shell region; incorporate existing and develop new sub-models to describe morphological development following shell formation; provide a physical basis for the use of different sub-models following shell formation and develop coherent criteria for choosing the appropriate description to use. The combination of these features produces a model that, for the first time, is capable of describing structural development as influenced by evolving droplet composition and drying conditions and is thereby able to simulate multiple dried-particle morphologies. The more sophisticated, stand-alone, droplet drying model developed here gives useful physical insights into the droplet drying process. These elucidate the relationship between droplet composition, drying rate and dried-particle structure. Eventually, it is intended that the model may be incorporated within a comprehensive computational fluid dynamics simulation of a spray drying tower, or may help with the formulation of improved reduced models for this purpose. 32 CSH

46 Chapter 3 A New Model for Drying Droplets Prior to Shell Formation In which a new droplet drying model is introduced. This novel drying model incorporates a population balance to describe the suspended solids and uses volume averaging to obtain appropriate transport equations for the continuous phase. After demonstrating the formulation and discussing the benefits of this new model, drying droplets of colloidal silica and sodium sulphate are simulated up until the point of shell formation. The results obtained are compared with experimental data from the literature. The material presented in this chapter is based upon work published in Handscomb et al. (2008a). 3.1 Background Spray towers require a pumpable feed, ( 2.2.1) which typically contains suspended solids or dissolved materials which crystallise during drying, (Masters, 1992). Droplets may also contain air present in the feed or entrained in the atomiser, ( 2.4.2) and will therefore, in general, constitute a three phase system as illustrated in Figure 3.1. The literature contains a number of methods for describing moisture transport and drying in such systems. Before proceeding with a detailed exposition of the new droplet drying model, this section briefly discusses these various approaches. The reasons for selecting a volume averaged description are expounded and the key results underpinning such an approach are introduced Descriptions of Moisture Movement in Porous Media The drying of individual droplets can be described in a number of different ways. It is perhaps convenient to consider the various length-scales on which such drying could be simulated, (Kohout et al., 2006b). At the largest scale, whole droplets can be considered 33

47 3.1. Background Vapour and Inert Gas Phase Solid Phase Liquid Phase Figure 3.1: Schematic showing a generic drying droplet containing suspended solids and gas pockets. with the drying rate prescribed as some function of droplet size and droplet-averaged quantities such as moisture concentration or temperature. These average quantities are, in turn, evolved yielding droplet temperature and moisture histories. The characteristic drying curve and reaction engineering approaches discussed in Section are examples of modelling at this scale. Although simple to implement, such models provide very limited information. At the other extreme, drying can be modelled at scale of individual pores. The traditional approach is to use pore network models, of which Prat (2002) and, more recently, Metzger et al. (2007) have written thorough reviews. The pore networks in such models consist of regularly or randomly located pores connected by throats. The geometry of both pores and throats can be chosen to simulate different aspects of real porous media. Assuming that capillarity is the process controlling moisture transport, drying can be simulated as an invasion-percolation process, (Yiotis et al., 2005, 2006). The objective of such models is to predict the dependence of parameters such as the permeability, capillary pressure and effective diffusion coefficient on moisture concentration. The effect of the pore size distribution on drying rate can also be investigated, although typically only in a qualitative manner, (Bray and Prat, 1999). This is due to both computational limits on the size of networks that can be simulated and experimental limitations analysing the nano-scale structure of porous materials. 34 CSH

48 3.1. Background A more sophisticated development of pore network models is to work with more realistic geometries and use a volume of fluid approach 5 to track the fluid menisci, (Orr et al., 1977). Such reconstructed medium methods are capable of simulating three phase, solid liquid gas systems. For example, Štěpánek et al. (2001) investigated the effect of porous structure on the evolution of gas pockets trapped within a granular sludge. The realistic geometries used are typically assemblies of randomly packed spheres, (e.g., Bryant and Johnson, 2003; Mayer and Stowe, 2006), but recent developments have seen the move towards more physically accurate descriptions. Kohout et al. (2006b) digitally encoded a porous structure recorded by X-ray micro-tomography, thus running simulations on a geometry identical to that used for comparative experiments. The most detailed approach to modelling porous media is to perform actual hydrodynamic simulations at the pore-scale. The appropriate laws of physics are applied directly to determine the moisture distribution as a function of time, (Whitaker, 1977). Each phase in the system must obey the overall continuity equation, ρ t + ρv = 0, (3.1.1) and, where a phase consists of more than one component, the continuity equation for species j, ρ j t + ρ j v j = r j. (3.1.2) Here ρ is the total mass density and v is the mass averaged velocity. ρ j is the mass density of species j and v j is the velocity of species j with respect to a stationary frame of reference. The volumetric mass rate of production of species j is represented by r j. The third important physical law is the requirement that linear momentum is conserved. This is expressed in the equation of motion, ρ v t + v v = ρg + Π, (3.1.3) where g is the acceleration due to gravity and Π is the total stress tensor. This later quantity depends on the fluid, with the Navier-Stokes equation being obtained when the Newtonian constitutive relation is employed. The equations above are sufficient to describe mass transfer in a porous medium, but only in conjunction with a complete geometric configuration of the phase interfaces. Even when such information is available, two problems face workers using such an approach: 5 The volume of fluid method is a numerical technique to track the movement and development of free liquid surfaces, (Hirt and Nichols, 1981). 35 CSH

49 3.1. Background constructing a suitable solution grid and limited computer power, (Benzi, 2003). To a varying degree, the second of these problems is an issue facing all numerical methods for hydrodynamic simulation. The difficulty of meshing an intricate pore-space is a major issue for traditional CFD techniques, but can be overcome in the lattice Boltzmann method. In this approach, the fluid is represented by an array of computational particles and their associated number density functions. These particles move across a regular lattice with a discrete set of velocities, according to rules ensuring that the required conservation laws are satisfied, (Benzi et al., 1992; Succi, 2001). Macroscopic quantities are obtained by averaging over this particle ensemble, with the Navier-Stokes equation being recovered in the hydrodynamic limit. 6 Because a regular lattice is used irrespective of the geometric complexity of the flow domain, this approach is considerably more robust than traditional CFD approaches in applications like the simulation of pore-scale flow in porous media, (Chen et al., 2003; Sullivan et al., 2006). However, Vogel et al. (2005) note that some degree of lattice refinement may be necessary to capture the narrow films of moisture which play an important role when drying in the pendular regime. In practice, the detailed geometric description required for hydrodynamic simulation of porous media is rarely available and may not even be measurable. Moreover, it also likely to prove unnecessary, as, for most applications, the detailed velocity distribution within each pore is not required. Rather, it is more important to know the variation of the average velocity over distances which are large compared with a typical pore diameter. Certainly for the present application where radial moisture profiles are sought, the coarse fidelity of such an effective medium length scale is sufficient. To this end, it is chosen to model moisture movement within the drying droplets at this intermediate scale through the use of volume averaging. The fundamental continuity equations describing transport in a porous medium, ( ) are point equations they are true everywhere in the porous medium since mass and momentum must be conserved at each point. The idea behind the volume averaging technique is to average these fundamental continuity equations over some suitably chosen region, (Whitaker, 1977, 1980; Slattery, 1999). The resulting description is that of a homogenised- or effective-medium approximating the true porous structure. Volume averaging smears out information about flow at the pore-scale and such homogenised models cannot be expected to predict phenomena resulting from large-scale heterogeneities in the liquid phase distribution, such as surface dry spots, (Laurindo and Prat, 1996). However, spatial concentration profiles for each species in the system can be obtained and, for many purposes, this is sufficient. As with most averaging techniques, a number of terms requiring closure are introduced 6 The hydrodynamic limit is that in which ɛ 0, (Benzi et al., 1992). Here, ɛ = λ F, where λ is the F mean free path and F is a general macroscopic field. The parameter, ɛ, can be likened to a local Knudsen number, i.e., a ratio of molecular kinetic to hydrodynamic length scales. 36 CSH

50 3.1. Background when volume averaging the mass transport equations for a porous medium. Whitaker (1977) describes how these can be treated in a rigourous way and this approach has been used to model textile dyeing, (de Souza and Whitaker, 2003a), catalytic reactions in a packed bed reactor, (de Souza and Whitaker, 2003b) and drying of porous sandstone, (Wei et al., 1985a,b). However, in practice, most researchers group the terms requiring closure into a number of effective transport coefficients such as an effective thermal conductivity, effective vapour-phase diffusivity, and effective liquid-phase permeability, (Erriguible et al., 2005; Kohout et al., 2006a,b). These parameters which are typically strong functions of local microstructure and moisture content can then be measured from well designed experiments, (Perré and Turner, 1999; Metzger et al., 2007) or obtained by simulations at the pore space length-scale, (Bray and Prat, 1999; Kohout et al., 2004). As droplets dry in a spray tower, they form particles and any of the methods discussed in this section can be applied to model their continued drying. However, a different approach is required whilst the droplets are more accurately described as droplets containing suspended solids rather than porous particles containing moisture. In this case, the pore network and reconstructed medium methods are not applicable as they require the identification of a porous structure. Methods based on hydrodynamic simulation can still be used, but the additional need to describe solid phase growth renders them even less attractive from a computational perspective. Consequently, the new model introduced in this thesis uses the idea of volume averaging to describe moisture transport in a drying droplet. Such an approach can handle droplets containing suspended solids and moisture transport in a porous medium with equal ease. The method is therefore ideally suited to modelling droplets in spray dryers, where simultaneous drying and particle formation is the defining feature of the process Local Volume Averaging As discussed in the previous section, the model introduced in this thesis uses local volume averaging to describe transport within a drying droplet. This section gives an overview of the basic mathematical concepts underpinning the approach and which are used later in developing the new model. The theory of volume averaging as applied to porous media is discussed in detail in several sources, (Slattery, 1967, 1969, 1970, 1999; Whitaker, 1969, 1977, 1980). The reader is referred to these works for more detail on any of the methods or theoretical results introduced in this section and, consequently, references for individual results are not supplied. 37 CSH

51 3.1. Background V z S Figure 3.2: Every point, z, within the porous medium has an associated averaging surface, S, containing the volume, V. Definitions Every point, z, within the porous medium whether in the solid, fluid or gas has, associated with it, an averaging surface, S. A spherical averaging surface is illustrated in Figure 3.2, but any shape can be used, provided that the dimensions and orientation are invariant. The minimum acceptable size for S is, roughly speaking, such that the average of the averages taken within S is equal to the average taken at the central point. The total volume enclosed by S is denoted V, with V (i) denoting the volume occupied by phase i within the averaging surface. Consider a quantity B associated with the phase i, where B may be a scalar, vector or tensor valued quantity. The average of B within the volume enclosed by S may then be defined in three ways. Firstly, the superficial volume average for phase i of B is defined B (i) 1 V V (i) B d. (3.1.4) Note that, where appropriate, the phase with which a given averaged quantity is associated is written in superscripted parentheses. The superficial volume average may therefore be understood as the mean value of B (i) in V. This notation is used throughout the remainder of the thesis. The intrinsic volume average for phase i of B the mean value of B (i) in V (i) is defined B (i) 1 V (i) V (i) B d. (3.1.5) 38 CSH

52 3.1. Background From their definitions, it is clear that the superficial and intrinsic volume averages are related according to B (i) = V (i) V B (i). (3.1.6) Finally, the total volume average of B over all phases present the mean value of B in V is given by B 1 B d V V = B (i). i=1 (3.1.7) These three methods of volume averaging will be used frequently in the remainder of this thesis. Having defined superficial and intrinsic volume averages, the minimum acceptable size of the averaging volume enclosed by S can be specified more precisely. If the averaging surface is characterised by a dimension, L 0, then the minimum acceptable size of S is such that B (i) is nearly independent of position over distances of order L 0. As a consequence of this, the following two relations are true: B (i) (i) = B (i) ; (3.1.8a) and B (i) (i) = B (i). (3.1.8b) Averaging Theorem The key result which enables local volume averages to be taken of interesting equations 7 is the theorem for the local volume average of a gradient. Consider again the quantity B (i) and take the superficial volume average of its gradient, 8 B (i) 1 1 B d = B d + 1 B ˆndA V V (i) V V (i) V S w = B (i) + 1 B ˆndA. (3.1.9) V S w 7 Here, interesting equations is taken to mean equations of relevance to transport in porous media. 8 Here B = B i j...k...m x k, and B ˆn is evaluated appropriately. 39 CSH

53 3.1. Background S V (c) z S e S w Figure 3.3: Illustration of a porous medium consisting of a continuous and discrete phase. S represents the boundary of the averaging volume associated with the point, z. The fraction of this volume occupied by the continuous phase, V (c), is bounded by walls and exits. Walls are denoted S w and represent those portions of the bounding region defined by phase interfaces; exits comprise the remainder of the bounding surface and are those areas where the boundary of V (c) coincides with S. A special case of this theorem is divb (i) 1 divb d V V (i) = divb (i) + 1 B ˆndA, V S w (3.1.10) where B is to be interpreted as a spatial vector field or second-order tensor field. In these equations, S w represents the portions of the surface bounding V (i) comprised of phase interfaces, termed walls. The remaining regions, i.e., those regions coincident with S, are termed exits and denoted S e. Figure 3.3 illustrates these terms for a porous system comprising a continuous and discrete phase. Volume Averaged Differential Mass Balance Having defined the volume averages and stated the averaging theorem, it is now possible to proceed with averaging the conservation equations. The key equation for the purposes of the drying model developed below is the continuity equation for an individual species, (3.1.2). Written for a specific component, A, this reads ρ A t + ρ A v A ra = 0, (3.1.11) 40 CSH

54 3.1. Background and is also known as the differential mass balance for species A. As a point equation, the differential mass balance is applicable in the solid, liquid and gas phases, although it may take on simpler forms in some of these. For example, if the solids are assumed to be at rest and not growing, the differential mass balance for this phase becomes redundant. Consider a system comprising a continuous liquid and discrete solid phase. It is sought to associate a local volume average of (3.1.11) with each point in the continuous phase. Taking the superficial volume average of this equation gives, 1 V ρa + ρ t A v A ra d = 0, (3.1.12) V (c) (t) where V (c) (t) is volume of the continuous phase within the local averaging surface. Although the solids are presently assumed not to move, V (c) may still change as a function of time due to solid phase growth through crystallisation. Denoting the velocity of this growing interface by w, (3.1.12) becomes d dt 1 V 1 V V (c) (t) ρ A d S w ρ A w ˆn da+ 1 V V (c) (t) ρ A v A d 1 V V (c) (t) r A d = 0. (3.1.13) Using the definition of a superficial volume average, (3.1.4), it is possible to introduce r (c) A 1 V V (c) (t) r (c) d, (3.1.14) A which is the homogeneous rate of production of A by, for example, chemical reaction. Recognising the first term in (3.1.13) as the time derivative of ρ (c) (c), and substituting for r A A gives 9 ρ (c) A t 1 V Sw ρ A w ˆn d A+ ρ A v A (c) r (c) A = 0. (3.1.15) 1 V 9 Here d dt fixed point in space. V (c) (t) ρ A d = dρ(c) A dt = ρ (c) A t, because the time derivative of an average is associated with a 41 CSH

55 3.1. Background Using the theorem for the volume average of a gradient, (3.1.9), on the third term gives ρ (c) A t 1 V ρ A w ˆn d A+ ρ A v A(c) + 1 S w V ρ (c) A + ρ t A v (c) A 1 V S w ρa v A ˆn da r (c) A = 0 S w ρa (w v A ) ˆn da r (c) A = 0. (3.1.16) The third term on the left hand side may be written 1 ρa (w v A A ) ˆn da (3.1.17a) V S w = 1 r (σ) V d A, (3.1.17b) A S w r where r (σ) A denotes the rate of production of species A per unit area of interface by heterogeneous chemical reactions. r A therefore denotes the volume averaged rate at which species A is produced per unit area of the fluid-solid interface. This gives the final locally averaged mass balance for species A as ρ (c) A t Alternatively, defining + ρ A v A (c) r A r (c) A = 0. (3.1.18) n A = ρ A v A, (3.1.19) as the mass flux of species A with respect to a stationary frame of reference, (3.1.18) may be written ρ (c) A t + n (c) A r A r (c) A = 0. (3.1.20) This equation concludes the introduction of required background material. In the following sections, this knowledge is applied to the development of a new model for single droplet drying. 42 CSH

56 3.2. Overview of the New Model 3.2 Overview of the New Model Model System Spray drying is unique in that it combines moisture removal with particle formation; any model for the drying of such droplets must be able to describe both these concurrent process. The model must therefore simulate the liquid phase containing most of the moisture and the suspended or dissolved solids. In general, drying droplets may also contain a dispersed gas phase, but the present model does not consider this possibility further. As the feed to the spray dryer must be pumpable, it is reasonable to assume that the liquid phase is initially continuous with discrete suspended solids. This will change as a solid dried-particle structure forms, but throughout this work the liquid and solids are referred to as the continuous and discrete phases respectively. Continuous Phase Both the liquid and the solid phases may be complex mixtures, e.g., the crutcher mix feed to a tower producing detergent powder contain may contain more than ten components, (Appel, 2000). Simulating such a complicated system in detail would be a daunting task and it is not sought to do so here. Rather, the new droplet drying model introduced in this thesis approximates the continuous phase as an ideal binary solution consisting of a solvent normally water and single solute and considers the solids phase to be comprised of a single component. The decision to model the continuous phase as a binary solution permits the description of initially liquid droplets where solids crystallise as drying progresses. Whilst clearly not capturing the great complexity possible in many real systems, this approach does allow modelling of some key morphological developments associated with dried-particle formation. Further, the ideal binary assumption means that the continuous phase composition can be described by a single equation, derived from the volume-averaged differential mass balance for one of the species. This derivation is presented in Section 3.3. Discrete Phase The description of the discrete solid phase is key to modelling the morphological evolution of the droplet. As stated above this, along with predicting the drying rate, is one of the prime deliverables from a droplet drying model. In reality, droplet structure and moisture transport are strongly interdependent and, consequently, the solid phase description also influences the predicted drying rate and the continuous phase description affects the morphological simulation. In the present framework, the solid phase is modelled by a population of discrete solid particles, assumed spherical, which evolve according to a population 43 CSH

57 3.2. Overview of the New Model Ideal Shrinkage T (t) R(t) b(t) Ideal Binary Solution Vapour Bubble Shell Slurry Discrete Solid Phase Hollow Shell (a) Vapour Bubble (b) Figure 3.4: Schematic showing: (a) the model system; and (b) drying to form a hollow shell. balance equation. The volume fraction of the solid phase, along with other quantities of interest, can be extracted and used to inform the predicted morphological development. The approach used to model the solid phase is discussed in detail in Section 3.4. Shell and Bubble In addition to the description of the suspended solids, two additional features are included in the model enabling simulation of a greater range of experimentally observed dried droplet morphologies, ( 2.4.2). The first of these is provision for an explicit shell region, within which the equations describing the continuous and discrete phases can be modified. As almost all spray dried droplets form some sort of solid structure, provision for a structured shell region is essential. Secondly, it is observed that many experimentally observed morphologies are based on hollow particles. Whilst a dispersed gas phase is not included in the present model, there is provision for a single centrally located bubble. This bubble can expand (and contract) with time, thus allowing the simulation of hollow droplets. Depending on the nature of the droplet when the bubble expands, inflated shells or uninflated hollow spheres can be simulated. Figure 3.4a shows a schematic of the droplet system outlined so far in this section, highlighting the ideal binary continuous phase, discrete solid phase, external shell region and central bubble. Figure 3.4b illustrates just one morphological history capable of being simulated by the new model drying to an inflated hollow shell. 44 CSH

58 3.2. Overview of the New Model Droplet Temperature In general, drying models do not attempt to model the spatially resolved temperature distribution within a droplet. This approximation is generally considered valid provided that the Biot number is less than 0.1, (Incropera and DeWitt, 2002). The Biot number a ratio of internal and external heat transfer resistances may be defined Bi = hl λ drop, (3.2.1) where h is the heat transfer coefficient of the mass transfer film and λ drop is the thermal conductivity of the droplet. Alternatively, Bi may be given as a function of the Nusselt number, Bi = Nu λ, λ drop (3.2.2) where λ is the thermal conductivity of the film. In this way, the effect of flowing drying air can be accounted for through correlations for Nu. Farid (2003) gives an example of a 200 µm milk droplet drying in an air stream at 90 C with a relative velocity of 1 ms 1. Here, the Biot number is initially 0.15, and so the temperature distribution within the droplet can be ignored. Provided the thermal conductivity of the droplet doesn t change, the Biot number will only decrease from this initial value as the droplet shrinks. This is confirmed in the simulations of pure water droplets conducted by Oberman et al. (2004). Here, the temperature profiles start relatively flat, with any variation across the droplet getting increasingly small with time. However, Farid (2003) points to some systems, such as skimmed milk, where the thermal conductivity of the droplet decreases by as much as an order of magnitude as moisture is removed. In such cases, Farid argues that the internal temperature distribution becomes important as the droplet dries and presents a model based upon this hypothesis. The simulation results show surface temperatures up to 10 C greater than those averaged over the relatively large, 2 mm, droplets at low drying air temperatures. However, as the author himself notes, it is difficult to validate these predictions against experiments. Indeed, the measured experimental temperatures, which are droplet averages, fit the predicted surface and averaged temperatures equally well. The effect of decreasing droplet thermal conductivity may, to some extent, be offset by the effect of surface evaporation. This idea is explored by Chen and Peng (2005), who conclude that the Bi < 0.1 criterion for uniform droplet temperature may be relaxed for evaporating particles. 45 CSH

59 3.2. Overview of the New Model For the new model presented in this thesis, it was decided not to calculate the internal droplet temperature distribution. As predicted temperature variations where they are significant at all are very hard to verify experimentally, it was felt that there was little to gain from spatially resolving the temperature profiles. Clearly it is important to model the averaged droplet temperature and the details of how this is done are to be found in Section Notation Throughout this work, quantities relating to the solvent, solute and solid components are given the subscripts A, B and D respectively. Quantities relating to the continuous and discrete phases are indicated with the superscripts (c) and (d) respectively. The volume fraction of the discrete phase is given the symbol, ɛ and, assuming volume additivity, this means the volume fraction occupied by the continuous phase is (1 ɛ). The term droplet always refers to the overall droplet being dried, even when physically this entity might more accurately be described as a particle. This is to enable the term particle to be used solely in reference to those solid particles suspended within the drying droplet and described by the population balance equation. Finally, dried-particle is used to describe the droplet at the end of drying. The liquid phase mass fraction of component j is denoted ω j. These mass fractions are related to the corresponding liquid phase mole fractions by ω A = x W A A x W j =1 j j, (3.2.3) where W j is the relative molecular mass of species j and is the number of species in the phase. For a binary system, this reads ω A = x A W A x A W A + x B W B. (3.2.4) Similarly, mass fractions may converted to mole fractions using x A = ω A /W A j =1 ω j /W j, (3.2.5) which, for a binary solution reads x A = ω A /W A ω A /W A + ω B /W B. (3.2.6) 46 CSH

60 3.2. Overview of the New Model Mass and mole fractions of component j in the gas phase are denoted w j and y j respectively. The mass density of a particular component in phase i is denoted ρ (i). The mass density A of the associated phase is then given by ρ (i) = j =1 ρ (i) j, (3.2.7) where is again the total number of components in phase i. Mass fractions are related to mass densities by ω (i) A = ρ(i) A ρ. (i) (3.2.8) The material density of a given component, A that is, the density of the pure substance is denoted ρ 0 A. The molar flux of component A with respect to a fixed co-ordinate frame is notated N A. The corresponding mass flux is n A, and the two are related by N A = n A W A. (3.2.9) The velocity of this component with respect to the same fixed frame of reference is then denoted v A, and the mass-averaged velocity of the entire phase is v = ω j v j. j =1 (3.2.10) The mass flux of the component is then given by n A = ρ A v A, (3.2.11) and the total mass flux of the phase by n = n = ρv. (3.2.12) j =1 Having defined the core notation used, the following three sections detail the mathematical development of the new droplet drying model outlined above. This chapter focusses on the core features of the new model: the continuous and discrete phase descriptions, together with an analysis of heat and mass exchange with the bulk. The features of the new model concerned with morphological development after shell formation the shell region 47 CSH

61 3.3. Continuous Phase Description and central bubble are introduced in Chapter Continuous Phase Description The composition of a binary solution is uniquely specified once the mass fraction of one of the components is known. Therefore, it is only necessary to solve the differential mass balance for the solvent or the solute. The equation for the solute is chosen as this enables the crystallisation process to be described more easily. Recalling (3.1.20), the differential mass balance for the solute can be written ρ (c) B t + n (c) B r B r (c) B = 0. (3.3.1) This equation is completely general and not specific to the present problem of simulating drying droplets. To be implemented in the droplet drying model, the reaction source terms r B and r (c) B need to be specified, together with an expression for the averaged mass transfer flux, n (c) B to its equivalent in terms of intrinsic volume averaged quantities.. It is also useful to transform (3.3.1) written in terms of superficial volume averages Defining the Solids Volume Fraction Before proceeding with manipulations of (3.3.1), it is first necessary to carefully define ɛ, the volume fraction occupied by the discrete phase. This is done by introducing the function 0 if z lies in the continuous phase, α(z) = 1 if z lies in the discrete phase, (3.3.2) and defining the solids volume fraction at the point z to be the total volume average of this function i.e., ɛ α. Recalling (3.1.6), the following relations are immediately obvious ɛ = V (d) V = 1 V (c) V, (3.3.3) where V (d) and V (c) denote the volume of the discrete and continuous phase within an appropriately defined averaging surface enclosing total volume, V. From this, it follows that B (c) = (1 ɛ) B (c), B (d) = ɛ B (d), (3.3.4a) (3.3.4b) 48 CSH

62 3.3. Continuous Phase Description where B is any scalar, spatial vector or tensor valued function associated with the system. Using these results in (3.3.1) gives t (1 ɛ) ρb (c) + (1 ɛ) n B (c) r B r (c) B = 0. (3.3.5) Source Terms As there are no chemical reactions, the homogeneous rate of production of solute, r (c) B, is zero. However, the volume average interfacial production rate is non-zero due to the possibility of a crystallisation process. Relating the mass rate of crystallisation of the solute, B, to the changing mass of the solid phase gives r B = ρ0 D ɛ, (3.3.6) t crys where ρ 0 is the material density of the solid phase. The local solids volume fraction may D evolve as a result of crystallisation from the continuous phase or spatial transport of existing solids from elsewhere in the droplet. The volume average interfacial production rate only contributes to the former process, as indicated by the subscript on the derivative in (3.3.6). A more detailed expression is given for r B in Section 3.4.2, following discussion of the discrete phase description. It has been assumed in writing (3.3.6) that there is no change in the total volume on crystallisation. This is equivalent to saying that the material density of the solute, ρ 0 is B equal to that of the crystallised solid, ρ 0. Such an assumption is not valid in general, but D is often invoked in the modelling of crystallisation, (Gerstlauer et al., 2002) and is a good approximation in certain systems. Substituting into (3.3.5) gives t (1 ɛ) ρb (c) + (1 ɛ) n B (c) + ρ 0 D Transport Term ɛ = 0. (3.3.7) t crys The next task is to introduce an expression for the intrinsic solute mass flux, n B (c). Fick s first law can be used to describe binary diffusion of a gas within a porous medium and it is current common practice to always use Fick s first law when analysing binary diffusion in liquids, (Slattery, 1999). Fick s first law may be written n B = n A + n B ωb (AB) ρ ω B, (3.3.8) 49 CSH

63 3.3. Continuous Phase Description where (AB) is a binary diffusion coefficient. Recognising na + n B = n = ρv, (3.3.9) and taking the intrinsic volume average with respect to the continuous phase gives n B (c) = ρ (c) ω B v (c) (AB) ρ (c) ω B (c), where ρ (c) is the density of the continuous phase. Whilst (AB) may vary appreciably within the drying droplet, it seems reasonable to neglect variations within the averaging volume. This allows the previous equation to be written n B (c) = ρ (c) ω B v (c) (AB) ρ (c) ω B (c). (3.3.10) The relation given in (3.3.4a) can be used to cast the averaging theorem, (3.1.9), in terms of intrinsic volume averages, (1 ɛ) B (i) = (1 ɛ) B (i) + 1 V which allows (3.3.10) to be written S w B ˆndA, (3.3.11) n B (c) = ρ (c) ω B (c) v (c) ρ (c) (AB) (1 ɛ) ω 1 ɛ B (c) δ (B). (3.3.12) where the mass tortuosity vector is given by δ (B) = ρ (c) ω B (c) v (c) (AB) ρ (c) ω B (c) ρ (c) ω B v (c) + (AB) ρ (c) ω B (c) + ρ (c) (AB) V (1 ɛ) S w ω B ˆn da. (3.3.13) It is often helpful to think of the mass flux in terms of an effective diffusivity tensor, i.e., n B (c) = ρ (c) ω B (c) v (c) ρ (c) 1 ɛ D(e) (AB) (1 ɛ) ω B (c), (3.3.14) and this is the approach adopted for the present model. Here, the terms on the right hand side represent convective and diffusive mass transport respectively. The material presented in this section is well known in the field and the interested reader is referred to Slattery (1999) for a thorough discussion. 50 CSH

64 3.3. Continuous Phase Description The Continuous Phase Equation Having obtained an expression for the solute mass flux, it is now possible to write the equation describing the composition of the continuous phase. Substitution of (3.3.14) into (3.3.7) gives t (1 ɛ) ρ (c) ω B (c) + (1 ɛ) ρ (c) ω B (c) v (c) ρ (c) D (e) (1 ɛ) ω (AB) B (c) ɛ + ρ 0 = 0. (3.3.15) D t crys Assuming spherical symmetry allows this equation to be simplified to t (1 ɛ) ρ (c) ω B (c) + 1 r 2 r + ρ 0 D r 2 (1 ɛ) v r (c) ρ (c) ω B (c) r 2 }{{} eff ρ (c) (1 ɛ) ωb (c) } r {{} ADVECTION DIFFUSION ɛ t crys }{{} CRYSTALLISATION = 0. (3.3.16) This is the equation used to describe the continuous phase in the new droplet drying model. In assuming spherical symmetry, the effective diffusion tensor, D (e) has been reduced to a (AB) scalar effective diffusion coefficient, eff. In reality, the only way to obtain an expression for this diffusion coefficient is to use an empirical form measured from experiments. It still remains to consider appropriate expressions for v (c), the volume-averaged, massaveraged velocity in the radial direction the only non-zero component assuming spherical r symmetry. Also, an appropriately volume averaged expression for the continuous phase density, ρ (c), is required. These two tasks are addressed in the following sections Continuous Phase Density The density of the continuous phase may be expressed in terms of the material densities of the solvent and solute ρ 0 and A ρ0 and the mass fractions of each. It is relatively trivial B to show that ρ 0 A ρ0 B ρ (c) = ρ 0 ω + B A ρ0 ω A B. (3.3.17) 51 CSH

65 3.3. Continuous Phase Description Making use of the fact that mass densities must sum to one, and defining the density ratio gives ρ0 B Λ B = ρ 0, ρ 0 A ρ0 A ρ0, (3.3.18) B B ρ (c) = ρ0 A Λ B ω B + Λ B. (3.3.19) It is necessary to take the intrinsic volume average of this expression for use in (3.3.16). This is done by first expressing the point solute mass fraction as ω B = ω B (c) + ω (c) B, (3.3.20) where ω (c) is the deviation from the intrinsic averaged mass fraction within the averaging B volume. In general, it is expected that ω (c) ω B B (c). Substituting for ω B from (3.3.20) in (3.3.19) and taking the intrinsic volume average gives ρ (c) = ρ 0 A Λ B ωb (c) (c) (c) 1 + Λ B + ω. (3.3.21) B Expanding the term in square brackets and making use of (3.1.8b) gives ρ (c) = ρ 0 A Λ B 1 ω B (c) + Λ B 1 ω (c) B + ω (c)2 (c) ω B (c) A + Λ B ρ 0 A Λ B ρ0 Λ A B ω B (c) ω B (c) + Λ B ωb (c) 2, (3.3.22) + Λ B where terms of the order ω (c)2 have been dropped. Further, it may be argued that the A term involving ω B (c) may also be ignored since ω B (c) ω B (c) < 1. Alternatively, the correction could be incorporated into the effective diffusion tensor in (3.3.14). 10 Either way, the expression used for ρ (c) in the remainder of this work is ρ 0 ρ (c) A = Λ B. (3.3.23) ω B (c) + Λ B 10 It is possible to expand (3.3.10) in terms of intrinsic means and variations in a similar fashion to that demonstrated here. The equation can then be re-arranged to develop an expression for the mass tortuosity vector in terms of these fluctuations. This provides some insight regarding the likely magnitude of the various terms in δ (B) but does not remove the eventual need for empirical closures. More information on this approach can be found in Whitaker (1977). 52 CSH

66 3.3. Continuous Phase Description The Local Mass-Averaged Velocity The advective transport term in (3.3.16) is seen to arise naturally from the assumption of Fickian diffusion. It reflects the continuity requirement for the continuous phase, (3.1.1), and it is seen from (3.3.9) that this term only vanishes in the case of equi-mass counter diffusion where n A + n B = 0. For all situations in which the material density of the solute is not equal to that of the solvent, the local mass-averaged fluid velocity, v (c), is non-zero and needs to be calculated. In general, this requires that the equation of motion, (3.1.3), be solved for the fluid. However, with the assumption of spherical symmetry it is possible to make use of an overall continuity equation to determine the mass-averaged radial velocity, v (c) the sole remaining non-zero component of v (c). r The simplest way to obtain an expression for the mass-averaged radial velocity is to consider conservation of continuous phase volume within an averaging surface, S. Recalling the terms illustrated in Figure 3.3, it is clear that material may only enter or exit the volume occupied by the continuous phase, V (c), through the surfaces denoted S e. Assuming volume additivity in the ideal binary liquid, a volume balance may thus be written S e na ρ 0 A + n B ρ 0 B ˆndA = 0, (3.3.24) where it is assumed that V (c) doesn t change with time. Since the fluxes through the surfaces S w are both zero, 11 it is possible to write S na ρ 0 A + n B ρ 0 B ˆndA = 0, (3.3.25) where the integral is now over a closed surface. The divergence theorem then gives V (c) div na ρ 0 A + n B ρ 0 B d = 0. (3.3.26) Since the averaging surface, S, was associated with an arbitrary point, z, the integrand in (3.3.26) must be identically equal to zero. Further, assuming spherical symmetry and reducing to one dimension gives 1 r 2 nar r 2 r ρ 0 A + n B r ρ 0 B = 0, 11 The flux through S w will be non-zero due to crystallisation, but it is reasonable to ignore this for the present discussion since the material involved doesn t cross the averaging surface, S. 53 CSH

67 3.3. Continuous Phase Description or, upon integrating, n Ar ρ 0 A + n B r ρ 0 B = f (t) r 2. (3.3.27) Taking the intrinsic volume average of this equation and using the 1-D form of (3.3.14) to substitute for n Ar (c) and n B r (c) gives 1 ρ 0 A v r (c) ρ (c) ω A (c) eff ρ (c) (1 ɛ) ωa (c) 1 ɛ r + 1 v ρ 0 r (c) ρ (c) ω B (c) eff ρ (c) (1 ɛ) ωb (c) = f (t). 1 ɛ r r 2 B Multiplying through by ρ0 A ρ0 B (1 ɛ), and collecting like terms yields ρ (c) (3.3.28) v r (c) eff (1 ɛ) ρ 0 ω B A (c) + ρ 0 ω A B (c) r (1 ɛ) ρ 0 B ω A (c) + ρ 0 A ω B (c) = (1 ɛ) f (t) r 2. (3.3.29) Recognising that ω A (c) = 1 ω B (c), and using the density ratio ρ0 B Λ B = ρ 0 A ρ0 B, (3.3.30) gives the required velocity as 1 ω v r (c) B = (c) eff 1 ɛ + f (t). (3.3.31) Λ B + ω B (c) r 1 ɛ r r 2 The function, f (t), is determined by the boundary conditions on the droplet. In the absence of a central bubble, the symmetry condition at r = 0 immediately gives f (t) = 0. However, this is not always the case. For example, consider the case of a droplet containing a centrally located bubble of radius b, which is expanding at a given rate. Further, assuming that the spatial gradients of the solute mass and solids volume fractions are zero at the bubble interface, (3.3.31) gives db dt = f (t) b 2 f (t) = b 2 db dt. 54 CSH

68 3.3. Continuous Phase Description Such a situation is discussed in Section 4.5. Substituting (3.3.31) into the 1-D form of (3.3.14) gives, on taking f (t) = 0, n B r (c) = eff ρ ω (c) B (c) Λ B. (3.3.32) r Λ B + ω B (c) Comparison with Fick s law demonstrates that this expression has the standard form for a diffusive flux. The factor in square brackets modifies the flux to take account of density differences between the species. As ρ 0 A ρ0 B, Λ B, Λ B / Λ B + ω B (c) 1 and Fick s law is recovered Boundary Conditions Having derived the equation describing the continuous phase, (3.3.16), and given expressions for all the relevant quantities involved, it only remains to specify the boundary conditions. In the absence of a central bubble, symmetry considerations require that the gradient of the solute profile is zero at the centre of the droplet, i.e., ω B (c) = 0. (3.3.33) r r =0 A full discussion of the boundary conditions once a bubble has formed is deferred to Section Basic drying theory, ( 2.1.2) explains how water will leave a drying droplet at the surface due to a higher water activity adjacent to the droplet surface than in the bulk. In contrast, it is assumed that the solute does not leave the droplet at any time, i.e., the solute mass flux following the receding interface is zero, n B r (c) r =R(t) = n B r (c) r =R ρ (c) ω B (c) (1 ɛ) dr dt = 0. (3.3.34) Substituting for the solute mass flux from (3.3.32) and re-arranging gives the solute boundary condition at the droplet surface, ω B (c) = ω B (c)ṁ. (3.3.35) r r =R ρ (c) eff Here, ṁ is the solvent mass flux from the droplet surface and is related to the rate of 55 CSH

69 3.4. Discrete Phase Description shrinkage by dr dt = ṁ ρ 0 A. (3.3.36) Discussion of how ṁ is calculated is deferred until Section Discrete Phase Description Population Balance Equation The discrete solid phase is modelled by a population of particles. These are described by a particle number density, N, which evolves in time according to a population balance equation. Ramkrishna (2000) gives the most general form of such a population balance equation as N t + x ẊN + r }{{ ṘN = } x D x x D T x + r D r r D T +h. (3.4.1) r }{{} ADVECTION DIFFUSION In this equation, terms containing an x refer to the internal co-ordinate, and those containing an r to the external co-ordinate. Internal co-ordinates refer to those properties such as size, shape or even colour which are intrinsic to the particles. In contrast, the external co-ordinates merely denote the location of the particles in physical space. The population balance equation allows the particle number density function to evolve in the multi-dimensional state-space of internal and external co-ordinates via both advective and random, diffusive, processes. Birth and death terms used to model processes such as coagulation can be included via the source term, h. Model Equations The particles considered in the model are assumed spherical, allowing a single internal coordinate to be used to classify particle size. The particle diameter, L, is chosen as a suitable internal co-ordinate, with corresponding state space, Ω x = [L min, ). Here L min is the minimum stable crystal size a parameter which must be obtained from experimental measurements. Assuming spherical symmetry of the drying droplet allows a single external co-ordinate the radial position of its centroid to completely specify the spatial location of a solid particle. This external co-ordinate, r, has the state space Ω r = b + L min, R L min 2 2 where b is the radius of the central bubble and R is the external radius of the droplet. Figure 3.5 illustrates a drying droplet as described by this population balance approach: the solid 56 CSH

70 3.4. Discrete Phase Description Solid Particles N (L, r, t) L Continuous Phase r Figure 3.5: Illustration of a droplet with solids represented by a population of spherical particles suspended in a continuous liquid phase. The number density function shown schematically on the right characterises the particle population, describing the distributions of particle size and radial location within the drying droplet. phase is represented by a population of spherical particles and characterised by the evolving number density function, N (L, r, t). The velocity of particle motion through the state space of internal co-ordinates is denoted by G. The most appropriate form for this linear growth rate depends on the system being simulated and is obtained from experimental measurements see Section for an example. The only assumption made at this stage is that the linear growth rate is independent of crystal size, i.e., G f (L). It is valid to approximate the linear growth rate as being independent of both particle size and the continuous phase flow if the suspended particles are smaller than 50 µm, (Nagata, 1975). When describing the evolution of the number density function in the space of internal co-ordinates, it is assumed that there are no particle sources or sinks, i.e., h = 0 in (3.4.1). The solid phase may grow through particle growth or nucleation of new particles at the lower boundary of Ω x, but not by agglomeration or breakage. Allowing such processes is certainly possible within the framework of the model developed here, but further discussion of this is beyond the scope of the present thesis. The solid phase may evolve in physical space through a convective or diffusive process, or a combination of the two. Thus, combined with the assumptions discussed above, the general population balance equation, (3.4.1), reduces to the following form used in this work, t N + L (GN) + 1 r 2 r r 2 v (d) N 1 r r 2 r r 2 D N = 0. (3.4.2) r 57 CSH

71 3.4. Discrete Phase Description It is assumed that, prior to shell formation, the solid particles only move by diffusion, i.e., = 0. Furthermore it is postulated that once a rigid shell has formed around the droplet, v (d) r the solid particles are no longer free to move at all within this shell region. That is, particle growth is the sole mechanism operating within a solid shell. However, the advective term is used when modelling shell thickening, as discussed in Section 4.3. Boundary Conditions New particles are permitted to form through nucleation. In general the rate of nucleation of new particles per unit volume per unit time, Ṅ 0, may be linked to the boundary condition at the lower bound of the space of internal co-ordinates by Ṅ 0 = Ẋ ˆn x N (x,r, t), x Ωx, (3.4.3) where Ẋ is the velocity through the space of internal coordinates and ˆn x is a local outwards unit normal to Ω x at the boundary, (Ramkrishna, 2000). Reduced to one dimension for crystal nucleation and growth the form required for the present model this becomes Ṅ 0 = GN (L min ). (3.4.4) This nucleation condition is applied at the lower end of Ω x ; it may be understood by recognising that the nucleation rate must give the flux of particles across this boundary. The regularity condition is imposed at the upper boundary of Ω x, that is, N 0 as L, (Ramkrishna, 2000). It is assumed that no solids leave the droplet. Instead prior to shell formation the receding continuous phase menisci between particles drive the solids towards the centre of the shrinking droplet, (Liang et al., 2001). In the population balance, this is captured by means of a birth term at the outer boundary, r = R(t). Again recalling the assumption that the solids move via a purely diffusive process prior to shell formation, the appropriate boundary condition is obtained by equating the volume flux from (3.4.2) with the flux caused by the receding interface, i.e., D N r = dr dt N = ṁ N, (3.4.5) giving rise to the condition N = r r =R ṁ Dρ 0 A ρ 0 A N. (3.4.6) Here, ṁ is the evaporative flux of solvent, A, from the droplet. 58 CSH

72 3.4. Discrete Phase Description Symmetry implies zero gradient boundary conditions should be employed at the centre of the droplet until the formation of a central bubble Moments of the Population Balance Equation Full information about the form of the number density at all times is not required; rather it is sufficient to know how a small number of the moments of the population evolve. The a th integer moment of the internal co-ordinate is defined by m a (r, t) = L a N (r, L, t) d L, L min (3.4.7) and several quantities of interest are related to the lower such moments, (Hounslow et al., 1988). The zeroth moment, m 0, gives the total number of particles per unit volume, whilst the first moment, m 1, is related to the total length per unit volume. The average solid particle size at a given spatial location is therefore L = m 1 m 0. (3.4.8) Similarly, the total surface area per unit volume is related to the second moment of the particle number density, with a scaling factor derived from the assumption that all particles are spherical, S V = πm 2. (3.4.9) The key variable of interest when describing the solid phase is the local solids volume fraction, ɛ(r ), which appears in the equation describing the continuous phase, (3.3.16). The solids volume fraction can be calculated from the particle number density by integrating over all particle sizes, ɛ = π 6 L min N L 3 d L = π 6 m 3, (3.4.10) where it is seen that ɛ is directly related to the third moment of the number density function. As discussed in Section 3.3.1, ɛ is an inherently averaged quantity and so it is appropriate to use the solids volume fraction from the moments of the number density function directly in the volume averaged continuous phase equation. The evolution of the moments can be obtained directly without the need to solve the whole population balance equation. Substituting from (3.4.2) into (3.4.7) and differentiat- 59 CSH

73 3.4. Discrete Phase Description ing gives m a t = L N a L min t d L (3.4.11) = L a L min L (GN) + 1 r 2 v (d) N 1 r 2 r r r 2 r = L min L (La GN) al a 1 GN d L 1 r 2 v (d) L a N d L r 2 D L a N d L r 2 r r L min r L min = L a GN + agm L min a 1 1 r 2 v (d) m r 2 r a r 2 D m a, r r r 2 D N r d L where it has been assumed that the particle growth rate, advection velocity and diffusion coefficient are independent of particle size. The first of these assumptions is fairly common, (Rosenblatt et al., 1984), and a relaxation of the later is discussed in Section below. Applying the nucleation and regularity boundary conditions at the lower and upper boundaries respectively gives m a t = L a Ṅ min 0 + agm a 1 1 r 2 r 1 r 2 v (d) m r a + r 2 r D m a r. (3.4.12) For some drying droplets, particularly those with high initial solids loading, the particle inception rate, Ṅ 0, will be equal to, or close to, zero. Consequently in these systems the number density of the smallest particles will vanish and the total number of particles in the system will not change with time. Recall that an expression for the time evolution of the solids volume fraction is sought. From (3.4.10) it is clear that the evolution of ɛ is related to that of the third moment by ɛ t = π 6 m 3 t. (3.4.13) The general moment evolution equation, (3.4.12), then shows that a hierarchy of moment equations is required in order to obtain this. Namely, it is necessary to solve the following 60 CSH

74 3.4. Discrete Phase Description closed set of four partial differential equations, ɛ t = π 6 L3 Ṅ min 0 + π 2 Gm 2 1 r 2 r = L 2 Ṅ min 0 + 2Gm 1 1 r 2 m 2 t m 1 t m 0 t r = L min Ṅ 0 + Gm 0 1 r 2 = Ṅ0 1 r 2 r r 1 r 2 v (d) m r 0 + r 2 r 2 v (d) ɛ + 1 r r 2 1 r 2 v (d) m r 2 + r 2 1 r 2 v (d) m r 1 + r 2 D m 0 r r D ɛ r r D m 2 r r r D m 1 r (3.4.14a) (3.4.14b) (3.4.14c). (3.4.14d) In Section 3.3.2, the solute evolution equation, (3.3.7) was shown to contain a source term describing the rate at which solute crystallises to the solid phase. Having derived the relevant moment equations, it is now possible to revisit (3.3.6) and express this source term as r B = ρ0 D = ρ 0 D ɛ t π crys 6 L3 min Ṅ 0 + π 2 Gm 2. (3.4.15) Here, ɛ/ t crys, the rate at which the solids volume fraction evolves due to crystallisation, is identified with the first two terms on the right hand side of (3.4.14a). The general moment evolution equation, (3.4.12), allows the calculation of more than these first four moments of the population balance indeed, any number of moments could theoretically be evaluated. Whilst this would give a more detailed representation of the number density function, N, the information provided by the limited set stated above is sufficient for the present application. For example, one of the main uses of this information will be in predicting the structural properties of the shells formed. It is often difficult to measure the functional dependence of such properties on solids fraction alone and so, in such cases, further knowledge of the size distribution of particles constituting the shell would be of limited practical interest. The boundary conditions for the moments are obtained by inserting (3.4.6) into (3.4.7) and integrating. At the outer edge of the droplet, the following conditions are then easily obtained, m i = ṁ r r =R ρ 0 D m i for i = 0,1,2, (3.4.16) A ɛ and = π ṁ r r =R 6 ρ 0 D m 3 = ṁ. (3.4.17) ρ 0 Dɛ A A 61 CSH

75 3.4. Discrete Phase Description Symmetry implies that the spatial gradients of all the moments are zero in the centre of the droplet, prior to the formation of a bubble. It is often useful to express moment related quantities in terms of their value integrated over the whole droplet, rather than on a volumetric basis. Such quantities are easily obtained by integrating the moments over the full range of the external co-ordinate, M i = = m i V d R b d 4πr 2 m i (r ) d r, (3.4.18) where M i is the i th moment of the internal co-ordinate summed over the entire droplet volume, V d. For example, the integral of the zeroth moment over the entire droplet gives the total number of solid particles. The rate at which the number of particles changes is obtained by differentiating (3.4.18) and substituting from (3.4.14d). Assuming no advective motion or bubble growth, dm 0 dt R(t) = 4πr 2 m 0 (r ) d r t b = 4π R 2 m 0 (R) dr R dt + b = 4π R 2 m 0 (R) dr R dt + b = 4π R 2 m 0 (R) dr R dt + b r 2 m 0 t d r r 2 Ṅ 0 1 r 2 r 2 Ṅ 0 d r r 2 D m 0 d r r r r 2 D m 0 r R b, which, on substituting the boundary conditions on m 0 from (3.4.16) gives dm 0 dt R = 4π r 2 Ṅ 0 d r, (3.4.19) b that is, the change in the total number of solid particles is as expected the local volumetric nucleation rate integrated over the entire droplet volume Size Dependent Diffusion Coefficient In some applications, it may not be acceptable to assume the solids diffusion coefficient is independent of particle size. Instead, the Stokes-Einstein equation might be used to give a 62 CSH

76 3.4. Discrete Phase Description relationship between particle diameter, L and diffusion coefficient, D = kt d 3πµL. (3.4.20) Here, k is Boltzmann s constant, with T d and µ representing the temperature and viscosity of the continuous phase respectively. Substituting this expression for D into (3.4.2) gives N t + L (GN) + 1 r 2 r r 2 v (d) N 1 r r 2 r r 2 kt d N = 0. (3.4.21) 3πµL r Taking the discrete phase advection velocity, v (d), to be zero as is assumed to be the case r prior to shell formation gives, on substituting into (3.4.11) and proceeding as in the previous subsection, m a t = L a Ṅ min 0 + agm a kt d r 2 3πµ r r m 2 a 1 r. (3.4.22) Again, the linear growth rate has been assumed independent of particle size and the assumption of a uniform droplet temperature allows T d to be taken outside the spatial derivative. The hierarchy of moment equations required to solve for the solids volume fraction, ɛ, is therefore ɛ t = π 6 L3 Ṅ min 0 + π 2 Gm 2 + kt d 1 r m 2 2 (3.4.23a) 18µ r r m 2 t m 1 t m 0 t = L 2 Ṅ min 0 + 2Gm 1 + kt d 18µ = L min Ṅ 0 + Gm 0 + kt d 18µ = Ṅ0 + kt d 1 18µ r r 2 r 2 r 2 1 r m 2 1 r 2 r r 1 r m 2 0 r r r 2 m 1 r (3.4.23b) (3.4.23c). (3.4.23d) Using the Stokes-Einstein equation to give the diffusion coefficient a dependence on the internal co-ordinate results in an unclosed equation set (c.f. (3.4.14)a-d). The final term in (3.4.14d) means that the evolution equation for the zeroth moment depends upon the unknown value m 1 ; it is necessary to approximate m 1 to close these equations. A possible solution to this closure problem is to use an extrapolative closure. One such scheme is based on simple linear extrapolation of the moments when plotted on a logarithmic scale. This gives m 1 = m2 0 m 1, (3.4.24) 63 CSH

77 3.5. Heat and Mass Transfer From The Droplet which, combined with (3.4.22) for a {0,1,2,3}, gives a closed system of four partial differential equations for the discrete phase. This interpolative approach has been shown to be accurate in a number of different applications, (Frenklach, 2002), and it is believed to be a good approximation in the present case. Nevertheless, a full numerical study would be needed to prove this conclusively. The boundary conditions on the moments in the case of a size dependent diffusion coefficient are m i = ṁ 3πµ m r r =R ρ 0 i+1 for i = 0,1,2, (3.4.25) kt A d ɛ and = π ṁ 3πµ m2 3 = ṁ 18µ ɛ 2, (3.4.26) r r =R 6 kt d m 2 kt d m 2 ρ 0 A ρ 0 A where the extrapolation m 4 = m 2/m 3 2 has been used in the last equation to close the system. As before, symmetry boundary conditions are applied at the centre of the droplet. The example presented in Section uses the closure introduced in this section to model the drying of a droplet of sodium sulphate solution. As discussed, this should be done with care. The quality of the closure will likely be system dependent and vary depending on the relative magnitudes of the nucleation, growth and spatial diffusion rates. The likely quality of the closure for the initially saturated sodium sulphate system is discussed in Section Heat and Mass Transfer From The Droplet The previous two sections introduced the equations describing transport of the continuous phase and suspended solids within a drying droplet. It was seen that, in obtaining suitable boundary conditions for these equations, the solvent mass flux, ṁ, or total rate of evaporation from the droplet, ṁ vap, were required. In this section, expressions for these quantities are derived Evaporation from a Sphere When considering mass transfer in mixtures, it is traditional to start from the Maxwell- Stefan equations for multi-component mass transfer, y i = n j =1, j i y i N j y j N i C t i j. (3.5.1) 64 CSH

78 3.5. Heat and Mass Transfer From The Droplet Stagnant Air r N air N A 2 R Figure 3.6: Schematic illustrating the fluxes from an evaporating drop of A surrounded by stagnant air. Here the equation is written in terms of mole fractions, y i, molar fluxes, N i and the total molar concentration, C t. Assuming a binary mixture a fair approximations in a spray drying tower where the air behaves like a single component this equation reduces to y A = y A N air y air N A C t A,air N A = N A + N air ya C t A,air y A, (3.5.2) which is the familiar mass transport equation for a binary mixture. For the present model, it is more convenient to recast (3.5.2) in terms of mass fractions. Substituting from (3.2.4) and (3.2.9) gives, after re-arrangement, n A = n A + n air wa ρ A,air w A. (3.5.3) A number of assumptions are now made concerning the drying droplets. Firstly, it is reemphasised that the droplets are assumed to remain spherical at all times. This is important as shape appreciably affects the rate of droplet evaporation, (Michaelides, 2006). The drying system assuming spherical symmetry is shown schematically in Figure 3.6. The separation between the droplets is assumed large compared with their diameter; the evaporation characteristics from single droplets differ from those within a spray, (Masters, 1992). Finally, it is assumed that vapour is transported away from the droplet by pure Stefan flow, i.e., n air = 0. This allows (3.5.3) to be written n A = ρ A,air 1 w A w A. (3.5.4) 65 CSH

79 3.5. Heat and Mass Transfer From The Droplet The assumption of spherical symmetry leads to the continuity requirement d dr 4πr 2 n A = 0, (3.5.5) which, on substituting for the mass flux from the 1-D form of (3.5.4), integrates to give 4πr 2 ρ A,air 1 w A dw A dr = ṁ vap, (3.5.6) where ṁ vap is the total rate of solvent mass lost from the droplet. It is now sought to integrate this expression from the surface of the droplet out to the bulk. Under isothermal conditions, the product ρ A,air is constant and may be taken outside the integral. However, Frank-Kamenetskii (1969) demonstrated that, even when the temperature varies, it is still possible to write T A,air R = R 1 r 2 T A,air d r, (3.5.7) where A,air is the averaged diffusion coefficient from A,air = R R T 1 r 2 T d r, (3.5.8) A,air and T is suitably chosen to represent the average temperature in the gas film surrounding the evaporating droplet; Yuen and Chen (1976) recommend using T = T d T T d. (3.5.9) The density in (3.5.4) can be similarly averaged and this is denoted ρ. Integrating then gives d r ṁ vap r = 4πρ 2 A,air R wa, w A,sur d w A 1 w A ṁ vap = 4πRρ A,air log(1 + B M ), (3.5.10) 66 CSH

80 3.5. Heat and Mass Transfer From The Droplet where B M is the Spalding mass transfer coefficient, 12 defined as B M = w A,sur w A, 1 w A,sur. (3.5.11) This coefficient is a dimensionless expression of the driving force for mass transport, (Spalding, 1963), so that it is always possible to write n A = k ω B M, (3.5.12) where k ω is a mass transfer coefficient based upon a mass fraction driving force. Recognising that ṁ vap = 1 4πR n, (3.5.13) 2 A r =R and substituting from (3.5.10), it is seen by inspection that k ω = ρ A,air R log(1 + B M ) B M, (3.5.14) i.e., the mass transfer coefficient depends upon the driving force, B M a well known result when dealing with high rates of mass transfer, (Bird et al., 1960). In the limit of a low driving force, i.e., when B M 1, the expression for the total mass vaporisation rate, (3.5.10), reduces to ṁ vap = 4πRρ A,air B M. (3.5.15) Comparison with (3.5.12) using (3.5.13) allows the identification of k, the mass transfer ω coefficient in the limit of a low driving force, k ω = ρ A,air R, (3.5.16) which, it is seen, does not depend on B M. It is often more convenient to express the mass transfer coefficient, k ω, in terms of the Sherwood number. 13 Using transfer coefficients based on mass fraction driving forces, the 12 The Spalding mass transfer coefficient is sometimes referred to as the Blowing coefficient, (Michaelides, 2006). 13 The Sherwood number is a dimensionless number representing the ratio of convective to diffusive mass transport rates. 67 CSH

81 3.5. Heat and Mass Transfer From The Droplet Sherwood number may be defined Sh = (2R) k ω ρ A,air, (3.5.17) which, in the limit of a low driving force, is Sh 0 = (2R) k ω ρ A,air = 2. (3.5.18) With a low driving force, the mass vaporisation rate, (3.5.10), may therefore be written as ṁ vap = 2πRSh 0 ρ A,air log(1 + B M ). (3.5.19) Modifications to account for the effect of bulk gas flow and high mass transfer rates are discussed in the next section Blowing Effects As is well known, the rate of droplet evaporation is greater when drying in moving air than when the surrounding gas is stagnant. This observation can be explained by thinking of mass transfer from the droplet in terms of film theory. Film theory is based upon the hypothesis that the resistance to mass transfer between a surface and the bulk gas may be analysed through the introduction of a gas film adjacent to the surface, (Bird et al., 1960; Slattery, 1999). In stagnant gas, this film thickness, δ M is constant. However, if the droplet is moving in the gas stream, the film narrows and the resistance to mass transfer decreases. There exist a number of common correlations to describe this effect, (e.g., Frössling, 1938; Ranz and Marshall, 1952), giving Sh = f (Re,Sc). In this work, the correlation of Clift et al. (1978) is used, Sh = 1 + (1 + ReSc) 1 3 f (Re), (3.5.20) where 1 Re 1, f (Re) = Re Re 400. (3.5.21) The thickness of the mass transfer film can also be influenced by the evaporation process itself. At high rates of mass transfer, the radial velocity field associated with the flow of vapour from the droplet to the bulk sometimes termed Stefan convection acts to 68 CSH

82 3.5. Heat and Mass Transfer From The Droplet thicken the boundary layer, (Schlichting, 1979). This is the so-called blowing effect at high evaporation rates. Abramzon and Sirignano (1989) propose that this boundary layer thickening can be described by a correction factor of the form F m = δ M δ M0 = (1 + B M ) 0.7 log(1 + B M ) B M. (3.5.22) A similar correction factor is introduced for the heat transfer film, with B M replaced by B T, the Spalding heat transfer coefficient defined as T T d B T = C p, A, (3.5.23) H vap + Q ṁ vap where C p, A is the average vapour specific heat in the film and Q is the total heat penetrating the droplet. A modified Sherwood number, taking into account blowing effects, is then defined by (Sh 2) Sh = 2 +, (3.5.24) F M where (3.5.20) gives the dependence of Sh on the velocity relative to the drying gas. Replacing Sh 0 with Sh in (3.5.19) then gives the expression used for the evaporation rate in this work, ṁ vap = 2πρ A,air Sh Rlog(1 + B M ). (3.5.25) Thermal Calculation Although the current model does not seek the temperature distribution within a drying droplet, ( 3.2.1), the evolving droplet temperature, T d, is very important. This section outlines the theory used to evaluate the heat transfer to the droplet from the drying gas and concludes by giving the algorithm used to obtain both the mass vaporisation rate and the heat penetrating the drying droplet. In the previous sections, an expression for the mass vaporisation rate was derived by considering mass transfer driving forces. This problem could instead be approached by considering a heat balance on the droplet; the material evaporated has required an quantity of heat expended per unit mass, (Michaelides, 2006). Following a method similar to that 69 CSH

83 3.5. Heat and Mass Transfer From The Droplet detailed in Section 3.5.1, the following expression is obtained, ṁ vap = 2π λ air C p, A Nu 0 R log(1 + B T ), (3.5.26) where λ air is the thermal conductivity of the gas mixture in the film, C p,a is the average specific heat capacity of the vapour in the film and B T is the Spalding heat transfer coefficient defined in (3.5.23). Re-arranging this definition, it is seen that the heat penetrating the droplet, Q, can be obtained from C p,a T T d Q = ṁ vap B T H vap T d. (3.5.27) As expected from the heat and mass transfer analogy, the discussion in Section on the effect of blowing maps across to the present problem of heat transfer to the droplet. The effect of movement relative to the drying gas is described by, (c.f. (3.5.20)) Nu = 1 + (1 + RePr) 1 3, (3.5.28) where f (Re) is again given by (3.5.21). The modified Nussult number is then given by, (c.f. (3.5.24)) (Nu 2) Nu = 2 +, (3.5.29) F T where F T is given by an equation analogous to (3.5.22). This means (3.5.30) can be rewritten ṁ vap = 2π λ air C p, A Nu R log(1 + B T ), (3.5.30) which, comparing with (3.5.25), shows that the Spalding heat and mass transfer coefficients are related by where B T = (1 + B M ) φ 1, (3.5.31) φ = C p, A C p Sh 1 Nu Le. (3.5.32) 70 CSH

84 3.5. Heat and Mass Transfer From The Droplet This relationship between the heat and mass transfer numbers reflects that the two transport processes are coupled in the droplet drying system. The equation system can be solved iteratively to calculate the evaporation rate, and the heat penetrating the droplet. Algorithm 3.1, based on that given by Abramzon and Sirignano (1989), shows the way in which these equations are solved in the current model. To calculate the rate of mass transfer from the droplet, it is necessary to first obtain the saturated vapour pressure of water at the droplet surface, (line [3.1]-2) and in the bulk, (line [3.1]-3). Together with the sorption isotherm and the bulk relative humidity, this then enables calculation of w A,sur and w A,, which are used in the definition of B M, (3.5.11). The Goff-Gratch equation, (Goff and Gratch, 1946; Goff, 1957) is recommended for use when determining the saturation vapour pressure of water, (WMO, 1988a,b). Although fitted to values measured over a flat surface, it is assumed to be a fair approximation for spherical droplets. The equation is p sat A log 10 = T st T log 10 2 T T st T 1 T st T st T 1, (3.5.33) where the temperature T must be supplied in Kelvin and T st = This equation is valid from 50 to 102 C, (Gibbins, 1990); above this temperature range a polynomial is fitted to data reported by Lide (1992), giving p sat A 10 3 = T T T T T T 6, (3.5.34) where the temperature in this equation must be supplied in degrees Celsius. Figure 3.7 shows both functions plotted with experimental data up to a temperature of 300 C. The polynomial in (3.5.34) gives a better fit to the data at high temperatures and, consequently, the routine PSAT_CALC uses this expression to obtain p sat A (T ) for T 102 C. Validation The heat and mass transfer algorithm used for the new model, (3.1) was tested by simulating the drying of a pure water droplet and comparing against data from the literature. Experimental droplet mass and temperature profiles from Werner et al. (2008c) were used as the authors give a thorough description of their drying apparatus. In particular, the dimensions of the filament used to suspend the droplet and the thermocouple used to record 71 CSH

85 3.5. Heat and Mass Transfer From The Droplet Algorithm 3.1 The procedure used to calculate the heat and mass transfer between a drying droplet and the bulk, (Abramzon and Sirignano, 1989). 1: procedure Q_CALC(T d, T, P, w A,, R, B old) T 2: p sat PSAT_CALC T A,sur d 3: p sat PSAT_CALC(T A, ) 4: p A,sur SORPTION_CALC ω B (c),ɛ r =R, p sat A,sur 5: p A, R p sat A, 6: y A,sur p A,sur /P 7: w A,sur y A,sur W A y A W A +y air W air from (2.1.2) 8: calculate ρ, C p, A, C p,λ, µ, A,air, Le,Pr,Sc Evaluated at T given by (3.5.9) 9: Re ρ air v rel R /µ air 10: if Re 1 then 11: f (Re) = 1 12: else 13: f (Re) = Re : end if 15: Nu 1 + (1 + RePr) 1 3 f (Re) 16: Sh 1 + (1 + ReSc) 1 3 f (Re) 17: calculate B M Using (3.5.11) 18: F M (1 + B M ) 0.7 log(1 + B M )/B M 19: Sh 2 + (Sh 2)/F M 20: calculate ṁ vap Using (3.5.10) 21: B T B old T 22: repeat 23: B old B T T 24: F T B old T log 1 + B old T /B old T 25: Nu 2 + (Nu 2)/F T C 26: φ p,a Sh C p Nu 1 Le 27: B T (1 + B M ) φ 1 28: until B T B old < ɛ 29: B old B T T 30: C p, A T T d Q ṁ vap 31: return ṁ vap, Q 32: end procedure T B T H vap T d 72 CSH

86 3.5. Heat and Mass Transfer From The Droplet Pressure / MPa P / kpa T / o C 2 1 Goff Gratch Polynomial Temperature / o C Figure 3.7: Plot of the saturated vapour pressure of water. Symbols show experimental data from Lide (1992) and lines are predictions from the Goff-Gratch equation, (3.5.33) and a polynomial fit, (3.5.34). The inset shows the vapour pressure at lower temperatures on an expanded scale. the temperature are given. Towards the end of drying, as much as 35% of the total heat penetrating the droplet is supplied by these items, (Werner et al., 2008c; Nešić and Vodnik, 1991). To account for this, the filament and thermocouple were modelled as semi-infinite rods, for which Incropera and DeWitt (2002) give the expression Q therm/fil = π 2 L λl T gas T d, (3.5.35) where h and λ are respectively the heat transfer coefficient to and thermal conductivity of the rod. For cylindrical rods, the appropriate characteristic dimension, L, is the diameter. In the comparative experiments, the diameters of the filament and thermocouple were 0.31 mm and 70 µm respectively; the thermal conductivities of the filament and thermocouple were 1.14 Wm 2 K 1 and 91.7 Wm 2 K 1. The thermocouple comprised two wires entering the droplet, each of which conducted heat. The heat transfer coefficients to the filament and thermocouples will depend on the precise geometry and flow which are unknown. Consequently, it is assumed that these heat transfer coefficients will be the same as calculated for the droplet, i.e., having obtained Nu from line 25 in Algorithm 3.1, the heat 73 CSH

87 3.6. Applications transfer coefficient for use in (3.5.35) is given by h = Nu λ gas 2R. (3.5.36) Figure 3.8 shows a comparison of simulated and measured temperature and mass profiles for a 6 µl pure water droplet. The droplet was suspended from a filament for all experimental runs, but the thermocouple was only in place for those runs used to obtain the temperature profile. The data from Werner et al. (2008c) refers to drying in air at T gas = 40 C, flowing with a velocity of 0.3 ms 1 and with R = (3.75 ± 2)%. Since the temperature profile in particular is quite strongly dependent on the ambient relative humidity, simulations were performed across the range covered by the experimental uncertainty. Given likely experimental uncertainties in the thermal readings (not reported), the fits between simulated and experimental temperature and mass profiles are good. Having validated this part of the code, the remaining sections in this chapter present simulations of physical systems prior to the formation of a surface shell. 3.6 Applications Having introduced the new droplet drying model, this section presents two applications which validate the model formulation and demonstrate some of its core features. The simulations presented in this chapter are run up until the point at which a shell is predicted to form. This is deemed to happen once the solids volume fraction at the surface of the droplet exceeds a certain critical value, ɛ crit. The rationale behind this choice of trigger for shell formation is discussed in detail in Section 4.1. In general, ɛ crit is a system-dependent parameter, but it is worth considering the range of values it might take. For uniformly sized spheres, the highest theoretically possible packing density is 74%, corresponding to close-packing in space, (Hales, 1992). Particles in drying droplets are unlikely to adopt this most efficient arrangement; rather it is more realistic to assume random packing of spheres. Random close packing of spheres in three dimensions is not a well defined arrangement, with possible packing densities covering the range 0.06 to 0.65, (Jaeger and Nagel, 1992; Torquato et al., 2000). However, capillary consolidation during drying makes figures towards the top of this range are more likely. In most systems of practical interest, the suspended solids will not be mono-sized spheres. For non-spherical particles, higher packing fractions are achievable, similarly when there is a range of particle sizes. In such cases it is necessary to resort to experimental observations to chose a suitable value for ɛ crit. 74 CSH

88 3.6. Applications Droplet Temperature / o C Experiment RH=1.75% RH=3.75% RH=5.75% Time / s (a) 6 5 Experiment RH=1.75% RH=3.75% RH=5.75% Droplet Mass / mg Time / s (b) Figure 3.8: Comparison of simulated and measured (a) temperature and (b) mass profiles for the evaporation of a pure water droplet. Experimental data is taken from Werner et al. (2008c) and refers to a 6 µl droplet drying in air at T gas = 40 C and flowing with a velocity of 0.3 ms 1. The experimental relative humidity was reported as R = (3.75 ± 2)% and the plots show results from simulations run across this range. 75 CSH

89 3.6. Applications Drying a Droplet of Colloidal Silica The first test case simulated is the drying of a droplet of water-borne colloidal silica. Colloidal silica is a dispersed system in which silica constitutes the dispersed phase. The silica in question is in the colloidal size range, i.e., less than 1 µm so that the particles are unaffected by gravitational forces, but larger than 1 nm so that the dispersion differs from a true solution, (Bergna, 2006). Such a stable dispersion of solid colloid particles in a liquid is called a sol. Gelation occurs as the concentration of colloidal silica is increased. A gelling sol first becomes viscous before developing rigidity, (Bergna, 2006). There is therefore no volume change on gelation, nor is there an increase in the silica concentration in any macroscopic region, (Iler, 1979). The mass fraction at which a gel forms is dependent on the size of the suspended particles, (Roberts, 2006); the smaller the particles, the lower the gelling concentration. In the system simulated here the droplets contain 16 nm particles which gel at mass fractions above 40%. Nešić and Vodnik (1991) report that the solids diffusion coefficient in this gel phase is D = exp ω A ω A where ω A is the mass fraction of water, given by, (3.6.1) (1 ɛ)ρ 0 A ω A = 1 ω D = ɛρ 0 + (1 D ɛ)ρ0 A. (3.6.2) The material density of the suspended silica particles, ρ 0 D, is 2250 kgm 3. This value, taken from Roberts (2006), represents the average density of silica calculated from a number of commercially available silica sols; the density variation between sols from different manufacturers is found to be small, i.e., ±40 kgm 3. The droplet simulated had an initial silica mass fraction of 30%, corresponding to an initial solids volume fraction, ɛ 0, of Prior to gel formation that is, at solids volume fractions below 0.23 experimental observations suggest that internal convection currents keep the drying droplets well mixed, (Nešić and Vodnik, 1991). To simulate this, the solids diffusion coefficient is set to 10 7 m 2 s 1 a relatively large value at low solids concentrations. The discontinuity at the point of gel formation was found to cause problems for the numerical solution routine and so was smoothed using a hyperbolic tangent weighting function. The expression used for the solids diffusion coefficient is therefore D ω D = f ωd f ω D exp 1 ωd ω D, (3.6.3) 76 CSH

90 3.6. Applications 10 6 Solid Diffusion Coefficient / m 2 s Solid Mass Fraction Figure 3.9: Plot of the solids diffusion coefficient used to simulate the drying of a colloidal silica droplet. where f 1 ω D = 1 tanh 20π ωd (3.6.4) 2 A plot of the solids diffusion coefficient is shown in Figure 3.9. To simulate the pure water continuous phase using the current model wherein the continuous phase is assumed to be an ideal binary solution the solute mass fraction is initialised to a very small number, There is no particle nucleation or growth in this example and so there is only one particle size at all times. This considerably simplifies the solids description and makes this an ideal initial test case. The simulation results prior to shell formation are compared with experimental results from Nešić and Vodnik (1991) at two different drying air temperatures. For dispersed colloidal silica, (i.e., not aggregated using flocculants), Guo and Lewis (1999) conducted experiments demonstrating that gravity driven settling produced beds with a solids volume fraction around 0.6. Capillary driven consolidation was found to increase the packing fraction towards the random packing limit. The consolidated packing fraction was found to have little dependence on the initial solids concentration. For the present example, the critical solids volume fraction representing the formation of surface shell is therefore taken as ɛ crit = The heat conducted to the particle via the suspending filament is modelled as described 77 CSH

91 3.6. Applications in Section 3.5.3, using (3.5.35): Q fil = π 2 L λl T gas T d. (3.6.5) The glass filament balance used by Nešić and Vodnik (1991) was 0.3 mm in diameter and is here assumed to have a thermal conductivity of 1.14 Wm 2 K 1. Effect of the Sorption Isotherm As described in Section 3.5, the moisture vapour pressure above the surface is required to calculate the mass vaporisation rate from the droplet. Since suspended insoluble solids have a negligible vapour pressure lowering effect, (Ranz and Marshall, 1952), it might be expected that the mass vaporisation rate in the present example could be calculated by assuming p A,sur = p sat. However, once a gel has formed, it is perhaps more accurate to view A the moisture as dissolved in the network of branched chains formed by the suspended silica particles, (Keey, 1975; Bergna, 2006). As such, the increasing solids concentration will be expected to decrease the equilibrium moisture vapour pressure above the surface, as described by the associated sorption isotherm, ( 2.1.1). Roškar and Kmetec (2005) measured the water sorption isotherm for one type of colloidal silica, 14 with the data reproduced in Figure Least squares regression was used to fit an exponential law to the sorption data, giving p A,sur = p sat 1 exp 5ω 1 2 A A, (3.6.6) where ω A is the mass fraction of moisture at the droplet surface. This expression is also shown in Figure 3.10, along with the isotherm that results from assuming insoluble particles. Comparison shows that, as expected, the measured sorption profile exhibits minimal vapour pressure reduction prior to gel formation, i.e., whilst ω A > 0.6. The effect of the sorption isotherm used is illustrated in Figure This shows the predicted temperature and mass profiles for a droplet of a colloidal silica with an initial diameter of mm drying in air at 101 C, with experimental data from Nešić and Vodnik (1991) shown as symbols. The choice of sorption isotherm is seen to have virtually no effect on the simulated mass profile. In contrast, the effect on the temperature profile is relatively pronounced as the surface solids concentration increases. The figure shows the profile resulting when the particles are assumed insoluble, i.e., no vapour pressure reduction la- 14 The colloidal silica investigated by Roškar and Kmetec (2005) was Aerosil 200F from Degussa, now Evonik Industries. This is a hydrophilic fused silica with a specific surface area of m 2 kg 1, (Aerosil, 2008) 15 Roškar and Kmetec (2005) originally presented their data using a dry-mass moisture content. This has been converted to a wet-mass basis for use in Figure CSH

92 3.6. Applications Relative Humidity / [ ] (a) Insoluble Particles (b) Fit to Sorption Data (c) Fit to Temperature Profile Sorption Data Water Mass Fraction / [ ] Figure 3.10: Plot of the three sorption isotherms used in the investigation of colloidal silica droplets drying. The three isotherms are obtained by: (a) assuming insoluble solids; (b) fitting experimental data shown in the figure from Roškar and Kmetec (2005); and (c) fitting an experimentally determined temperature profile for a droplet drying in air at 101 C. belled (a) and that obtained when using (3.6.6), isotherm (b). Whilst a difference is discernable, neither predicts the measured values as shell formation is approached and the temperature rises above the wet-bulb. Indeed, it is not clear that the colloidal silica for which Roškar and Kmetec (2005) measured their sorption isotherm is a good match for that used by Nešić and Vodnik (1991) in their experiments; Gun ko et al. (2006) demonstrate that whilst different types of fumed silica exhibit qualitatively similar sorption behaviour, the precise sorption isotherms vary appreciably. Consequently, a new sorption isotherm was developed by adjusting the constants in the exponential law sorption isotherm to give the best fit to the experimental temperature data at 101 C. The resulting isotherm is p A,sur = p sat A 1 exp 5ωA, (3.6.7) which is the isotherm labelled (c) in Figure The new sorption isotherm continues to predict limited vapour pressure reduction before gelation. Once a gel has formed, the partial pressure of moisture above the drying surface is lower than that predicted by the previous two isotherms. Figure 3.11 shows that the temperature profile predicted using this isotherm agrees well with the experimental data as is to be expected given this is how it was derived. This last sorption isotherm is the one now used when investigating the influence of drying air relative velocity and humidity. 79 CSH

93 3.6. Applications Droplet Mass / mg (a) Insoluble Particles (b) Fit to Sorption Data (c): Fit to Temperature Profile Droplet Temperature / C Isotherm t shell / s (a) 55.8 (b) 56.4 (c) Time / s Figure 3.11: Simulated drying of a colloidal silica droplet (lines) compared with experimental results from Nešić and Vodnik (1991) (symbols) at T gas = 101 C with R = 0% and v rel = 2.5 ms 1. The figure illustrates how the sorption isotherm influences the predicted temperature and mass profiles Droplet Mass / mg Droplet Temperature / C v rel / ms 1 t shell / s 0 > ms 1 1 ms 1 2 ms 1 3 ms Time / s Figure 3.12: Simulated drying of a colloidal silica droplet (lines) compared with experimental results from Nešić and Vodnik (1991) (symbols) at T gas = 101 C with R = 0%. The figure illustrates the effect of relative air velocity, v rel, on the predicted mass and temperature profiles. The time at which a shell forms, t shell, is given in the table for each air velocity. 80 CSH

94 3.6. Applications Effect of Relative Air Velocity Figure 3.12 illustrates how the velocity of the air relative to the drying droplet, v rel, influences the predicted temperature and mass profiles. The influence on the mass profile is seen to be considerably greater than that on the predicted droplet temperature; as expected, air blowing over the droplet accelerates drying considerably. The relative velocity also strongly affects the timing of shell formation, t shell. Also plotted in the figure is the experimental data from Nešić and Vodnik (1991) relating to colloidal silica drying in air at 101 C. It is claimed that the relative air velocity in the experiment was 1.78 ms 1, with no error estimate provided. Whilst such a relative velocity is not completely unrealistic, it is not the most likely value according to the simulation. Table 3.1 shows the values of the l 2 -norm 16 of the deviations between the predicted and experimentally measured values of the droplet mass for a range of drying air relative velocities. From this it is seen that a velocity of 2.5 ms 1 gives the best agreement between experiment and simulation. Given the likely uncertainties in both experiment and model, this is not an unreasonable value. Table 3.1: Values of the l 2 -norm of the deviations, ɛ i, between predicted and measured values of the droplet mass for different drying air velocities, v rel. v rel ε Effect of Drying Air Humidity Figure 3.13 illustrates how the drying air humidity influences the simulated temperature and mass profiles. The profiles in the figure were obtained using (3.6.7) for the sorption isotherm and with v rel = 2.5 ms 1. In contrast to the effect of the relative air velocity, it is seen that the bulk humidity has a significant influence on the temperature profile and has comparatively little effect on the timing of shell formation. Drying in more moist air results in a greater droplet temperature a result that can be readily deduced from a psychometric chart. Nešić and Vodnik (1991) do not give the air humidity when drying their droplet of colloidal silica but, from the Figure 3.13, it is clear that the air must have had a relative humidity close to zero. 16 The l 2 -norm of a vector, ε n, is the Euclidian norm, ε 2 = n i=1 ɛ i (3.6.8) 81 CSH

95 3.6. Applications Droplet Mass / mg RH=0% RH=1% RH=2% RH=3% RH=4% Time / s Droplet Temperature / C R / % t shell / s Figure 3.13: Simulated drying of a colloidal silica droplet (lines) compared with experimental results from Nešić and Vodnik (1991) (symbols) at T gas = 101 C with v rel = 2.5 ms 1. The figure illustrates the effect of air relative humidity, R, on the predicted mass and temperature profiles. The time at which a shell forms, t shell, is given in the table for each humidity. Predicted Solids Profiles Figure 3.14 shows the predicted evolution of the solids volume fraction within the colloidal silica droplet drying at T gas = 101 C. The results are obtained from simulations of a droplet drying in air with R = 0% and v rel = 2.5 ms 1. Profiles are plotted at 5 s intervals. Initially the profiles are flat across the droplet due to internal circulation prior to the formation of the gel phase. After formation of a gel at ɛ = 0.23, the profiles develop a pronounced curvature as particles build up at the receding interface. In the current model, shell formation is predicted when the surface solids volume fraction reaches 0.65, which is seen to occur at t = 58 s. Figure 3.15 shows a comparison between experimental data and the simulated temperature and mass profiles for a 0.95 mm droplet drying in air at T gas = 178 C. The data is taken from Nešić and Vodnik (1991) as before. The simulation results presented assume, as discussed above, that the relative humidity of the drying air is 0%. The reported air velocity for these data is 1.4 ms 1 but, after investigation, a velocity of 2.5 ms 1 was again found to give the best fit to the experimental data. The results presented in Figure 3.15 demonstrate good agreement with the data, but the range over which the comparison can be made is limited since shell formation is predicted after 24.4 s. Figure 3.16 shows the corresponding solids volume fraction profiles, plotted at 5 s intervals up until the point of shell formation. A higher drying air temperature clearly 82 CSH

96 3.6. Applications Solids Volume Fraction [ ] Shell formed at solids volume fraction = 0.65 t=58s Radial Position / mm x 10 4 Figure 3.14: Simulated solids volume fraction profiles during the drying of a colloidal silica droplet in air at T gas = 101 C. Profiles plotted at 5 s intervals, with shell formation predicted at t = 58 s, (highlighted) Droplet Mass / mg Shell formed at t=24s Droplet Temperature / C Time / s Figure 3.15: Simulated drying of a colloidal silica droplet (lines) compared with experimental results from Nešić and Vodnik (1991) (symbols) at T gas = 178 C with R = 0% and v = 2.5 ms CSH

97 3.6. Applications Solids Volume Fraction [ ] Shell formed at solids volume fraction = 0.65 t=24s Radial Position / mm x 10 4 Figure 3.16: Simulated solids volume fraction profiles during the drying of a colloidal silica droplet in air at T gas = 178 C. Profiles plotted at 5 s intervals, with shell formation predicted at t = 24 s, (highlighted). results in faster drying and correspondingly earlier shell formation. Further, the droplets dried more quickly form a surface shell after having lost less mass than those drying in air at lower temperatures: the droplet where T gas = 178 C has lost 70% of its initial moisture content at the point of shell formation, compared with a 79% loss for the droplet where T gas = 101 C. Correspondingly, droplets drying at higher temperatures are larger at the point of shell formation. This is an important observation if shrinkage is assumed to cease with the appearance of a surface crust. Conclusions Simulating droplets of colloidal silica allowed the new droplet drying model to be tested on a simple application where solid nucleation and growth can be ignored. The model has been successfully validated against experimental measurements and the sensitivity to drying conditions investigated. This example is revisited in Section where, after having discussed models for use following shell formation, the drying droplets simulated in this section are modelled beyond t shell Drying a Droplet of Aqueous Sodium Sulphate Solution The second set of results presented relate to the drying of a droplet of sodium sulphate solution. The droplet simulated has an initial solute content of 14 wt%, which is just below saturation at the initial droplet temperature of 20 C, (Rosenblatt et al., 1984). Raoult s law 84 CSH

98 3.6. Applications is used to obtain the moisture vapour pressure above the surface of the drying droplet, i.e., p A,sur = p sat A ω A (c). (3.6.9) Sodium Sulphate Diffusion Coefficient The effective diffusion coefficient, eff, is the key physical parameter in the continuous phase equation, (3.3.16). This can be estimated using correlations, of which perhaps the best known is that published by Wilke and Chang (1955), 17 (AB) = (φ A W A ) 0.5 T µv 0.6 m,b. (3.6.10) Here, φ A is an association factor to account for hydrogen bonding in the solvent, W A is the relative molecular mass of the solvent, V m,b is the molar volume of the solute at its normal boiling point and µ A is the solvent viscosity. Applied to the sodium sulphate water system, it is appropriate to take φ A = 2.26 for water and estimate V m,b = m 3 kmol 1 for sodium sulphate. 18 Using these values gives, at 25 C, an estimate of (AB) = m 2 s 1. Without experimental data, using such a correlation is the only way to obtain the binary diffusion coefficient. However, there are severe limitations to the values obtained; for example, the Wilke-Chang correlation is only valid for dilute solutions and, even then, is typically only accurate to within 10%, (Perry and Green, 1997). It is therefore always preferable to obtain the diffusion coefficient from experimental data, where this is available. Rard and Miller (1979) report the binary diffusion coefficient for sodium sulphate in water at 25 C as a function of solute concentration. Data is reported for concentrations from zero to saturation. After converting the measured molarities to mass fractions, the measured data is shown in Figure It is interesting to note that the diffusion coefficient measured in the limit of an infinitely dilute solution agrees very well better than could be reasonably expected with the value obtained from the Wilke-Chang correlation. Least-squares minimisation was used to fit a second-order polynomial in ω 1 2 to the data, B 17 The form of the correlation given here has been converted for use with SI units. 18 The molar volume of sodium sulphate is estimated using the structural contributions of each atom, taken from Sinnott et al. (1999): Element V m m 3 kmol 1 Na S O (in union with S) CSH

99 3.6. Applications 1.2 x 10 9 Diffusion Coefficient / m 2 s Na SO Mass Fraction / [ ] 2 4 Figure 3.17: Plot showing the binary diffusion coefficient, (AB), of Na 2 SO 4 in water at 25 C. Symbols show experimental data from Rard and Miller (1979), covering the range from infinitely dilute through to saturation; the line shows a polynomial fit to the data, (3.6.11), which is extrapolated to supersaturated values. giving (AB) = ω B ω m 2 s 1. (3.6.11) B This fitted equation is also plotted in Figure 3.17, where it has been extrapolated beyond the range of the experimental data to cover supersaturated solute mass fractions. Whilst extrapolation of this kind is dangerous, the general monotonic-decreasing form of the resulting curve agrees with that reported by Horvath (1985) for the system. The experimental data and associated fitted curve are only applicable at 25 C; the binary diffusion coefficient at other temperatures may be obtained using (AB) (T ) = (AB) (T = 298K) T 298. (3.6.12) To obtain the effective diffusion coefficient, eff, for use in (3.3.16), it is necessary to consider the effect of the suspended solids on the diffusion rate. As before, experimental data is preferable when available but, in its absence, the influence of suspended solids may be modelled (Silva et al., 2000). Assuming randomly distributed spherical solid particles, the effective diffusion coefficient through the medium is given by the Clausius-Mossotti 86 CSH

100 3.6. Applications equation, 19 3D p Ds D p ɛ eff = D p + D s + 2D p. (3.6.13) D s D p ɛ Here D p is the solute diffusion coefficient through the pore space, D s is that through the solids and ɛ is the solids volume fraction. In the present model, it is assumed that the solute cannot pass through the solids, i.e., D s = 0, giving 2(1 ɛ) eff = (2 + ɛ) (AB). (3.6.14) For the sodium sulphate example, the final functional form for the effective diffusion coefficient is therefore eff ωb,ɛ, T 2(1 ɛ) T = (2 + ɛ) 298 Crystallisation Kinetics ω B ω 1 2 B (3.6.15) The phase diagram for the sodium sulphate water system, Figure 3.18, shows that an aqueous sodium sulphate solution will crystallise directly to solid sodium sulphate at temperatures above 30 C without forming a hydrated phase. Since the drying droplets in this example rapidly achieve a wet bulb temperature in excess of 30 C, the kinetics used are those for direct crystallisation to the solid. For such conditions, Rosenblatt et al. (1984) gives the equation dl dt = G max exp E Ci A 1.5 C eqm, (3.6.16) RT where E A = 57.4 kj mol 1 is the activation energy for the growth process, G max = ms 1 is the maximum growth rate 20 and R g is the gas constant. C i and C eqm are the local and saturated solute concentrations respectively in kmol m 3. These are related to the solute mass fraction by C i = ρ (c) W B ω B (c), (3.6.17) where ρ (c) is the continuous phase density, ( 3.3.5), and W B = 142 kg kmol 1 is the relative molecular mass of sodium sulphate. The dependence of the continuous phase den- 19 See Barker (1973) for a derivation of this equation. 20 Note that (3.6.16), the expression for the linear growth rate from Rosenblatt et al. (1984), is an empirical equation and, consequently G max is best thought of as a fitted coefficient with no physical meaning. 87 CSH

101 3.6. Applications SOLUTION Na 2 SO 4 + SOLUTION Temperature / ºC Na 2 SO 4-10H 2 O + SOLUTION Na 2 SO 4 + Na 2 SO 4-10H 2 O Weight Percent Na 2 SO 4 Figure 3.18: Water-Sodium Sulphate phase diagram adapted from Wetmore and LeRoy (1951). sity on the continuous phase composition, as given by (3.3.23), must be considered when evaluating (3.6.16). In terms of the density ratio, Λ B, solute mass fraction, ω B (c) and equilibrium solute mass fraction, ω B (c), the driving force for crystallisation is eqm C i C eqm = ρ 0 A Λ2 ω B B (c) ω B (c) eqm. (3.6.18) W B ωb (c) + Λ B ω B (c) + Λ eqm B The equilibrium solute mass fraction, ω B (c), is obtained from the phase diagram shown eqm in Figure The crystal growth expression given by (3.6.16) corresponds to the linear growth rate, G/ms 1, in the population balance equation (3.4.2). The saturation ratio, S, is defined as the ratio of the solute molar concentration and the equilibrium solubility, i.e., S = C i C eqm. (3.6.19) Substituting from (3.6.17), and using (3.3.23) to write the continuous phase densities in 88 CSH

102 3.6. Applications terms of the density ratio and mass fractions, the saturation ratio may be expressed S = 1 + Λ B ω B (c) eqm. (3.6.20) 1 + Λ B ω B (c) For the sodium sulphate system, the saturation ratio is less than five throughout drying, so heterogeneous nucleation dominates, (Dirksen and Ring, 1991). The following standard functional form was used for this heterogeneous nucleation rate, log 10 Ṅ0 Ṅ max = 2 log 10 2 (S), (3.6.21) where Ṅmax is a constant representing the maximum nucleation rate. No nucleation kinetics for the aqueous sodium sulphate system could be found in the literature and values for Ṅmax were found to vary considerably between similar systems, (Dirksen and Ring, 1991). The sensitivity of the results to the nucleation rate and initial seeding conditions are investigated below. However, it is worth explicitly noting that the lack of experimental data relating to crystallisation kinetics is the key factor preventing accurate simulation of the sodium sulphate system. Effect of Initial Seeding When solving the droplet drying model, it is necessary to provide initial conditions for all variables. Assuming a well mixed system implies the initial spatial profiles are uniform across the droplet. The initial concentration of sodium sulphate is 14 wt%, but the appropriate initial conditions for the moment system are less clear. Since the solute is at the saturation concentration, it seems reasonable to assume that there are some crystallites initially present in droplet, although their number is unknown. This is unfortunate since seeding that is, the initial number of crystallites is known to have a large effect on the subsequent crystallisation process, (e.g., Dirksen and Ring, 1991; van Drunen et al., 1996). This effect can be investigated using the present model. Figure 3.19 demonstrates the effect of the initial particle number density on the timing of shell formation. Recall that the particle number density is given by the zeroth moment of the particle size distribution. Shell formation as plotted in this figure is taken as the point at which the solids volume fraction at the droplet surface reaches As expected, a greater number of seed particles results in earlier shell formation. Whilst the initial particle number density gives the initial condition of the zeroth moment of the number density function, m (0), it remains necessary to specify the initial conditions for the three remaining moments. The droplet drying model requires that the 0 solid 89 CSH

103 3.6. Applications L 0 =2 nm L 0 =5 nm L 0 =10 nm t shell / s Initial Particle Number Density / # m 3 Figure 3.19: The effect of initial particle number density and initial seed particle size on the timing of shell formation, t shell. The maximum nucleation rate, Ṅ max, is set equal to m 3 s 1. particles are assumed spherical. If it is further assumed that the initial crystallites are the same size then the second and third moments may be calculated once the particle diameter or the first moment of the particle size distribution has been specified. It seems sensible to take the initial particle diameter, L 0, equal to the minimum stable crystal size which, for sodium sulphate, is around 10 nm, (Shi and Rousseau, 2001). However, as seen from Figure 3.19, the choice of initial particle size does not significantly affect the timing of shell formation. This result is due to the fact that the particle growth rate, (3.6.16) is assumed independent of particle size. Since all sizes of seed crystal investigated are small compared with the particle sizes at shell formation, the choice of L 0 has minimal effect on t shell. Effect of Nucleation Rate No information relating to the nucleation rate for aqueous sodium sulphate could be found in the literature. Assuming that the dependence of the nucleation rate on the saturation ratio follows the standard expression in (3.6.21), the new droplet drying model was used to study the effect of varying Ṅmax. The results are plotted in Figure 3.20, wherein the timing of shell formation is shown as a function of initial particle number density for four different maximum nucleation rates. The curve relating to the lowest maximum nucleation rate, Ṅ max = m 3 s 1, reproduces the results already shown in Figure From the present plot it is clear that 90 CSH

104 3.6. Applications t shell / s N max =10 10 m 3 s 1 N max =10 20 m 3 s 1 N =10 30 m 3 s 1 max N =10 40 m 3 s 1 max Initial Particle Number Density / # m 3 Figure 3.20: The effect of the nucleation rate, Ṅmax, on the timing of shell formation, t shell when drying a droplet initially seeded with 10 nm crystals. new particle nucleation has little effect on the timing of shell formation in this case. The situation is quite different when the maximum nucleation rate is increased, as seen by the form of the corresponding curves in Figure Below a certain limit, the timing of shell formation is seen to become virtually independent of the initial number density. In this regime, the formation of the shell is dominated by the nucleation of new particles which fill space and trigger the condition that ɛ > ɛ crit at the droplet surface. The dominance of the nucleation process renders the initial condition with respect to the particle number density inconsequential. In contrast, outside this regime, the formation of a surface shell is dominated by the growth of the seed particles and, consequently, the timing of t shell remains dependent on m (0) in this region. The size of the seed particles is found to have no influence 0 in either regime. The two regimes outlined above are expected to influence the mean size of the particles constituting the newly formed surface shell. This is confirmed in Figure 3.21 wherein the average particle size in the shell, L is plotted against maximum nucleation rate, Ṅ max for three different initial particle number densities, m (0). Mean particle sizes are obtained from 0 the first two moments of the particle number density function using (3.4.8). The ability to predict the average particle size in the shell along with an idea of the width of the particle size distribution using the second moment is a key feature of the new model. Looking first at the points corresponding to the highest initial particle number density, it is seen that the predicted mean particle size in the shell is around 3.5 µm and largely independent of nucleation rate. This corresponds to the second regime discussed above 91 CSH

105 3.6. Applications 10 4 Average Particle Size / m m (0) = m 3 0 (0) 10 3 m 0 = m (0) 12 3 m 0 = m Maximum Nucleation Rate / m 3 s 1 Figure 3.21: Plot showing the average solid particle size, L, at r = R at the point of shell formation. The dependence on the maximum nucleation rate is illustrated for three different initial particle number densities. where shell formation occurs as a result of seed growth. Since new particle nucleation plays a limited role in this regime the particle sizes in the shells that result would not be expected to depend on Ṅmax. In contrast, a dependence on the maximum nucleation rate is observed for lower initial particle number densities. Higher values of Ṅmax result in more particles being nucleated and, consequently, smaller mean particle sizes are observed. For low maximum nucleation rates the formation of a shell is again dominated by seed growth and the mean size of particles in the shell is once more independent of Ṅmax. Figure 3.22 shows the influence of the maximum nucleation rate, Ṅ max on the timing of shell formation for a droplet with initial particle number density, m (0) = m 3. 0 As expected, t shell is seen to be constant for low values of Ṅ max, where nucleation plays almost no role. Interestingly, the initial effect of increasing the maximum nucleation rate is to delay the onset of shell formation. This effect can be explained by understanding that nucleation creates small particles at the expense of existing particle growth. These small particles are able to diffuse away from the droplet interface far more quickly than the, relatively immobile, large particles. Consequently, the effect of nucleating new particles at the expense of growing existing ones is to delay the onset of shell formation. At still higher maximum nucleation rates, the production of new particles is so fast that this initial effect is negated and a shell is formed increasingly quickly. Figure 3.23 plots the average particle size within a drying droplet as a function of radial position and time. The profiles shown relate to a droplet where the initial particle number 92 CSH

106 3.6. Applications t shell / s Maximum Nucleation Rate / m 3 s 1 Figure 3.22: Plot showing how the maximum solid particle nucleation rate effects the timing of shell formation for a droplet with initial m (0) = m 3 s 1. 0 density is m 3 throughout the droplet and the maximum nucleation rate is m 3 s 1. The timing of shell formation for such a droplet is expected to be dominated by new particle nucleation. Solid particle growth adjacent to the receding droplet interface is easily identified from the figure. New particle nucleation acts to lower the mean particle size, since all newly nucleated particles have the minimum size, L min. Close inspection of the profiles in Figure 3.23 shows that this behaviour is observed at all radial locations following the onset of nucleation around t = 42 s. The mean particle size at the very edge of the droplet is noticeably greater that that even a short way in and consequently, following the onset of nucleation, a local minimum develops in the radial profiles. The larger mean size of particles at the edge of the droplet is understandable this is where the solute concentration is greatest, resulting in the highest rates of growth. Further, larger solids diffuse into the bulk more slowly, with the effect that the size distribution at the boundary becomes skewed towards bigger particles. Solids Diffusion Coefficient and Moment Closure From the figures and discussion above it is clear that crystal nucleation and growth mean that the solid particles in the sodium sulphate system vary greatly in size. This is in contrast to mono-sized particles in the droplet of colloidal silica studied in Section and, for this reason, a size dependent solids diffusion coefficient is necessary here. The modifications required to the moment system to handle a size dependent diffusion 93 CSH

107 3.6. Applications Particle Size / m Time / s Radial Location / m 1 0 x 10 4 Figure 3.23: Plot showing spatially resolved profiles of the mean solid particle size, L, at 2 s intervals from t = 20 s until the point of shell formation, t shell. The profile at t shell is shown in bold. coefficient have been discussed in Section In particular, the closure used is that given by (3.4.24), based on simple linear extrapolation of the moments when plotted on a logarithmic scale. Frenklach (2002) showed such a closure to be suitable in a number of similar applications, but a full numerical study would be required to establish its applicability here beyond doubt. However, preliminary investigations provide encouragement. For example, the closure approximation may be applied to the first and second moments to approximate the zeroth moment. Since the zeroth moment is explicitly calculated by the model, the approximated values using the closure can be compared with the simulated values. Such a comparison is shown in Figure 3.24 for two different initial particle number densities, m (0) 0. The moment values plotted are those at the outer edge of the receding drying droplet. The closure is initially very good indeed, it should be exact for a mono-sized distribution of particle sizes and remains fair up until the point of shell formation. Experimental Comparison The validity of any experimental comparison is clearly limited by the uncertainty associated with the crystallisation kinetics. This uncertainly most obviously manifests itself around the time of shell formation. Further, as has been demonstrated above, the timing of shell formation is itself strongly dependent on the initial seeding and subsequent nucleation kinetics. 94 CSH

108 3.6. Applications Normalised Moments / [ ] Zeroth Moment First Moment Second Moment Approximated Zeroth Moment Normalised Moments / [ ] Zeroth Moment First Moment Second Moment Approximated Zeroth Moment Time / s (a) Time / s Figure 3.24: Comparison of the simulated zeroth moment at the outer edge of a drying droplet, (thick line) with that obtained from the first and second moments using a closure approximation of the form given by (3.4.24), (symbols). The maximum nucleation rate is Ṅ max = m 3 s 1 and the initial particle number density is: (a) m (0) 0 = m 3 s 1 ; and (b) m (0) 0 (b) = m 3 s Droplet Mass / mg Shell formed at t=47s Droplet Temperature / C Time / s Figure 3.25: Simulated drying of a 14 wt% sodium sulphate in water droplet at T gas = 90 C, compared with experimental results from Nešić and Vodnik (1991). For this simulation, the maximum nucleation rate was set to Ṅmax = 1020 m 3 s 1 and the initial particle number density was m (0) = m 3. 0 Nevertheless, it is instructive to compare predictions with experimentally determined temperature and mass profiles, if only to validate the model in the initial drying period where the solids remain relatively unimportant. To this end, comparisons with experimental measurements from Nešić and Vodnik (1991) of a sodium sulphate droplet drying in air at 90 C are presented in Figure The size of the model droplet has been fixed such that the initial mass corresponds 95 CSH

109 3.6. Applications Solute Mass Fraction / [ ] t=47 s t=18 s Radial Position / m x 10 4 Figure 3.26: Simulated solute mass fraction profiles plotted at 5 s intervals throughout the drying of a droplet of aqueous sodium sulphate solution. In addition, profiles are plotted at the point of shell formation, (dashed line) and solution at the edge of the droplet first becomes supersaturated, (bold line). with the experimental observations, thus giving a diameter of 1.78 mm. For this and all subsequent figures in this section, the simulation was run with Ṅmax = 1020 m 3 s 1 and with an initial solid particle number density of m (0) 0 = m 3. Taking the trigger for shell formation to be a surface solids volume fraction of 0.65, it is seen that this choice of parameters corresponds to the appearance of a rigid crust after 47 s. The match between the simulated and measured results up until this point is seen to be excellent, demonstrating the validity of using Raoult s law to model the solvent partial pressure above the droplet surface. Solute Profiles and Moment Evolution Figure 3.26 plots the solute mass fraction profiles at 5 s intervals through the drying of the droplet. The dashed curve is the solute profile at the point of shell formation and the lower bold curve indicates the first profile when the solution becomes supersaturated. The saturated mass fraction is obtained from the phase diagram, Figure 3.18, where it is seen that, above 30 C, this is only a weak function of temperature and approximately equal to 33 wt%. Naturally, the first appearance of a supersaturated solution occurs at the outer boundary of the drying droplet. The evolution of the moments at the moving external boundary of the droplet are plotted in Figure In this figure, the three moments are all normalised against their initial values. Initially all the normalised moments are coincident, indicating that no growth is taking place. The moment values increase in this period simply because the particle number 96 CSH

110 3.6. Applications Normalised Moments / [ ] Zeroth Moment First Moment Second Moment Time / s Figure 3.27: Simulated normalised moments at the outer edge of the droplet. density at the droplet surface rises as a result of shrinkage. Figure 3.26 shows that the continuous phase at the surface of the droplet becomes saturated with the solute at around 18 s. This corresponds to the point at which growth starts to be observed in the moment system, as would be expected from inspecting (3.6.16), the expression for linear crystal growth. The zeroth moment, representing the particle number density, is seen to increase rapidly after about 42 s. This point marks the time at which new particle nucleation becomes appreciable and, as has been discussed at length above, is determined by the choice of Ṅmax. The decreasing first and second moments reflect the fact that the newly nucleated particles are smaller than those particles which have grown from the seeds present at the start of drying. Figure 3.27 may be contrasted with Figure 3.28 which again shows the normalised moments, but now integrated over the entire droplet. This figure more clearly indicates the onset of new particle nucleation around t = 42 s. The change in the rate of increase of the zeroth moment around t = 18 s in Figure 3.27 is a consequence of the size dependent diffusion coefficient. As the particles at the droplet edge grow, their mobility decreases markedly. This reduces the rate at which they diffuse into the droplet bulk and therefore increases the rate of accumulation at the droplet surface. In contrast, the zeroth moment integrated over the entire droplet, as shown in Figure 3.28, shows no change in its second derivative prior to the onset of nucleation. This confirms that the change observed at the droplet edge is due to a local change in particle mobility rather than a droplet-wide effect. Finally, Figure 3.29 shows the predicted evolution of the solids volume fraction in the 97 CSH

111 3.6. Applications Integrated Normalised Moments / [ ] Zeroth Moment First Moment Second Moment Time / s Figure 3.28: Simulated normalised moments integrated over the entire droplet. drying droplet. The volume fraction of solids is negligible until just before shell formation. The relative immobility of the particles forming the shell means that the profiles representing the solids volume fraction are extremely steep. This suggests a narrow region in which crystals form, surrounding a more dilute core. Qualitative experimental evidence supports this picture, (Seydel et al., 2006), but no quantitative data could be found in the literature. The shell forms rapidly at a time that, as shown above, is highly dependent on the nucleation kinetics employed. Good knowledge of these kinetics is required for accurate prediction of the time to shell formation, however such data is currently unavailable. Conclusions The simulation of a saturated droplet of aqueous sodium sulphate demonstrates the full capabilities of the new model. Drying rates are successfully calculated, as validated against experimental data, and spatially resolved moisture and dissolved solute profiles are returned. In addition, the equations describing the evolving moments of the solid particle population are solved using a size dependent diffusion coefficient, enabling prediction of particle number densities, mean particle sizes and the solids volume fraction as a function of radial position within the droplet. The work presented here has demonstrated the number of physical and kinetic parameters required to successfully simulate this drying system. Some of these parameters such as the sodium sulphate diffusion coefficient and crystal growth rate are readily available, whereas for other required information such as the particle nucleation rate no rele- 98 CSH

112 3.7. Conclusions of the Chapter Solids Volume Fraction / [ ] Radial Position / m x 10 4 Figure 3.29: Simulated solids volume fraction in a drying droplet of aqueous sodium sulphate solution. vant data could be found in the literature. This lack of key experimental data hampers the ability of the model to make quantitative predictions. However, the new model may still be used to investigate likely sensitivities to these unknown parameters; such studies have been conducted to investigate the influence of the maximum particle nucleation rate and initial particle number density. The population balance describing the solid phase enables structural quantities of interest to be predicted, for example, the average size of the solid particles forming the initial surface shell. As discussed in the next chapter, this type of information is essential if droplet drying and morphological development following shell formation are to be simulated successfully. 3.7 Conclusions of the Chapter The aim of this chapter was to develop a new droplet drying model capable of simulating droplets containing both dissolved and suspended solids. The continuous phase was modelled as an ideal binary solution and, after reviewing various approaches for describing transport in the presence of solids, volume averaging was identified as the most promising method for deriving appropriate transport equations. The suspended solids were described by means of a population balance equation, allowing for particle nucleation, growth and transport in physical space. Solving for the moments of this equation allowed properties of interest relating to the solids such as particle number density, mean particle size and 99 CSH

113 3.7. Conclusions of the Chapter solids volume fraction to be spatially resolved. Equations for the discrete and continuous phases were derived, along with a detailed discussion of the appropriate boundary conditions. Taken together, this represents a comprehensive framework for describing drying droplets. The later sections of this chapter dealt with model validation. Simulations of a colloidal silica droplet reproduced experimentally observed mass and temperature profiles. The ability of the new model to investigate sensitivity to drying conditions was also demonstrated. The more complicated aqueous sodium sulphate system was then investigated, illustrating the full capabilities of the model. The influence of particle nucleation rate and initial seeding on the timing of shell formation and mean solid particle size was investigated, producing several interesting qualitative results: for example, increasing the maximum nucleation rate was initially found to delay shell formation. Experimental data relating to droplet mass and temperature histories were reproduced, although uncertainty associated with key kinetic parameters prevented further quantitative validation. In summary, this chapter contained the following novel elements: a framework to describe drying droplets based on volume averaged transport equations and a population balance describing suspended solids was developed; the core droplet drying model was used to simulate systems of practical interest and validated against experimental data. 100 CSH

114 Chapter 4 Drying Droplets With an Outer Shell In which the novel droplet drying model introduced in Chapter 3 is extended to droplets drying in the presence of a surface shell and a centrally located bubble. Following the initial appearance of an outer shell, a number of sub-models may be used to described subsequent drying and structural development. These sub-models include a novel shell thickening description, as well as the wet and dry shell models from the literature. Physically motivated criteria are developed that allow the appropriate sub-model to be chosen at each stage in the droplet drying history. Comparisons between model predictions and experimental data from the literature are presented for two systems of interest: detergent crutcher mix and colloidal silica. These simulations demonstrate the extended model is capable of simulating multiple dried-particle morphologies together with other properties of interest such as moisture profiles and moments of the particle size distribution. The material presented in this chapter includes work published in Handscomb et al. (2008b). 4.1 Background Spray drying removes moisture and simultaneously produces particulate products. Chapter 3 introduced the core features of a new model capable of simulating the removal of moisture from droplets. However, the discussion in that chapter was limited to droplet drying prior to shell formation; the simulations in Section 3.6 were explicitly terminated on the appearance of an external crust. Whilst such a model provides some important information, a drying model incapable of simulating dried-particle formation is clearly limited. The extended model developed in this chapter allows prediction of moisture removal rates following shell formation but, equally importantly, simulates the evolving morphology of the drying droplet. 101

115 4.1. Background Droplet Morphology The morphologies observed when droplets are spray dried were discussed at length in Section 2.4.2, with the major particle types illustrated in Figure 2.9. In principle, a drying model could be formulated to simulate a droplet drying to any single one of these morphologies. Of more interest would be a model which could simulate the production of multiple dried-particle structures and, crucially, be capable of deciding which of these structures will result when a given initial droplet is subjected to different drying histories. The model developed in this chapter seeks to do this, making use of the information about the solid phase given by the moments of the population balance ( 3.4) to simulate different structural developments. However, before proceeding to develop such a model, it is necessary to narrow the field of potential dried-particle morphologies. A core assumption of the model developed in the previous chapter was spherical symmetry of the drying droplets. This immediately limits the range of dried-particle morphologies capable of being simulated by the extended model. Three possibilities remain: solid particles; particles with a centrally located void; and inflated shells. All three of these particle types are observed at the outlet of real spray drying installations, (Marshall and Seltzer, 1950a). Processes that are symmetry breaking notably particle bursting or collapse cannot be explicitly simulated within such a framework. However, if the conditions leading to these processes are identified, the new model might be used to flag when the associated dried-particle morphologies are likely to be observed. The formation of any dried-particle commences with the appearance of a surface shell. The present discussion proceeds by considering what is meant by this term and discussing the physical processes associated with the appearance of a shell Formation of a Surface Shell There is no unique meaning of the term surface shell. Rather, the definition depends on the context, i.e., the particular model or experiment. Whilst a structural feature associated with the term might be clearly identifiable in the later stages of drying, the precise point at which such a shell forms is not obvious, leading to the various definitions. First consider the processes preceding shell formation however that may be defined. Depending on the droplet composition, suspended solids may be present initially or may crystallise out of solution as moisture is evaporated. If present initially, the solids are assumed uniformly distributed within the droplet as illustrated in Figure 4.1a. As drying proceeds, the solid particles are drawn inwards by surface tension forces, causing the solids volume fraction at the receding surface to increase. Where the solids are initially in solution, crystallisation will first occur in the regions of highest solute supersaturation, i.e., at the droplet surface, (Farber et al., 2003). Once formed, these crystals are drawn inwards 102 CSH

116 4.1. Background behind the receding interface as described previously. The speed with which the surface solids concentration increases depends on the relative rates of moisture evaporation and internal solids mixing. That is, the ratio between the interface recession rate and the rate at which solids can diffuse back into the bulk. Tsapis et al. (2005) introduced characteristic mixing and drying times to illustrate this idea; the characteristic mixing time in a droplet of radius R is related to the solids diffusion coefficient by τ mix R2 D, (4.1.1) so that a shell forms when τ mix τ dry. One way to identify the point at which a surface shell forms is to specify some associated critical solids concentration at the interface. This is a common approach in droplet drying models, (e.g., Elperin and Krasovitov, 1995; Kadja and Bergeles, 2003; Seydel et al., 2006), and is indirectly used in those experiments which take a visual change as the criterion for shell formation. For example, Walton and Mumford (1999a) noted changes in droplet opacity and Eslamian and Ashgriz (2007) used changes in the amount of reflected light from droplets viewed under an optical microscope. In the new droplet drying model, a shell is deemed to have formed once the volume fraction of particles at the droplet surface rises above a certain critical value, ɛ crit, (Figure 4.1b). A number of researchers (e.g., Brouwers, 2006; Masuda et al., 2006) have discussed how the solid particle size distribution may be used to predict the maximum packing fraction and it is imagined that the information from the population balance could be used to inform the point at which shell formation is deemed to occur. However, for the systems simulated in this thesis, ɛ crit is taken to be a pre-defined parameter. Using a critical surface solids concentration to identify the timing of shell formation says nothing about the subsequent drying behaviour, nor about the nature of the growing shell. Drying models make numerous assumptions on these two issues, but it is important to remember that these are features of the model rather than a reflection of the physics. For example, most models in the literature assume the droplet ceases shrinking as soon as a shell is formed, (e.g., Mezhericher et al., 2007; Dalmaz et al., 2007; Kadja and Bergeles, 2003), but this need not be the case. Of the three morphological categories identified by Walton and Mumford (1999a), shell formation arrested shrinkage only for droplets forming dried-particles with a crystalline structure. Skin-forming droplets and those forming dried-particles with an agglomerated structure both continued to decrease in size after the appearance of a surface shell. It is this observation which motivates the development of the shell thickening model, introduced in Section 4.3. This thickening regime allows for continued shrinkage of the droplet whilst the shell grows to a critical shell thickness, T crit. At this point, the shell is is capable of supporting itself and prevents the droplet shrinking 103 CSH

117 4.1. Background ɛ ɛ ɛ ɛ crit ɛ crit ɛ crit T crit T min r r r R 0 S R S R R 0 (a) (b) (c) Figure 4.1: Schematic illustrating the process of shell thickening: (a) initially the droplet is homogenous and drying proceeds ideally; (b) a shell forms when the maximum packing fraction, ɛ crit, is reached at the surface of the droplet; (c) the solid particles continue to redistribute and the droplet continues to shrink until a critical shell thickness, T crit, is attained. further, (Figure 4.1c) Drying With a Rigid Shell Most drying droplets stop shrinking at some point. 21 Some researchers define this to be the point of shell formation, but in the current work the stabilisation of the droplet radius is associated with the formation of a rigid shell. Such a rigid shell fixes the volume of the drying droplet; further moisture removal must therefore lead to an expanding vapour-saturated space somewhere within the droplet. This vapour space may be dispersed throughout the droplet, in which case detailed methods for simulating moisture transport through porous media containing a gas phase, as discussed in Section 3.1.1, are required. However, for the reasons explained in that section, such detailed simulation is not attempted here. Rather, following Lee and Law (1991), two limiting cases are identified upon which shell drying models are developed. In the dry shell model, the vapour space is located in the shell itself and an evaporative front recedes through the droplet. This is a variant of the classic shrinking core approach used by many researchers to model droplets drying and discussed in detail in Section This approach is a fair approximation for several drying systems and an implementation for 21 Exceptions to this statement are droplets with very low initial solid (suspended or dissolved) concentrations, or those droplets which inflate before shrinkage is arrested. 104 CSH

118 4.2. Deciding Which Model to Use use with the new model developed in this thesis is introduced in Section 4.4. However, a number of problems are evident. Firstly, most implementations assume that the porosity and other properties of the shell are uniform, (e.g.,cheong et al., 1986; Dalmaz et al., 2007). This is not necessarily the case, especially if solute is crystallising from the continuous phase within the still-wet core. The dry shell model described in Section 4.4 overcomes this objection by allowing the shell properties to vary. A more difficult objection to overcome is that, on its own, the dry shell model can only predict solid dried-particles. Furthermore, these dried-particles will have a solids volume fraction similar to that of the droplet when it stops shrinking. The simulated dried-particle densities are, consequently, often too low. A further challenge to the dry shell model comes from Walton and Mumford (1999a), who observed that droplets which form agglomerated dried-particles that is, droplets containing suspended rather than dissolved solids exhibit saturated surface drying throughout the shrinking period and for a short time thereafter. Similar behaviour is found with skin forming droplets, which remain wetted at the surface during skin thickening even though the surface film can present a significant obstacle to mass transfer. Such behaviour is in stark contrast to that predicted by the dry shell model, based as it is upon the idea of evaporation occurring at an interface receding into the droplet bulk. These problems are addressed by the wet shell model. The wet shell model assumes that the shell region is, at all times, wetted by the continuous phase; the vapour now lies in a single, centrally-located, bubble. Evaporation proceeds from the wetted external surface, as observed in those experiments discussed above. Furthermore, the model predicts the formation of hollow dried-particles and, with suitable modifications, can also be used to simulate inflating droplets. The development of the wet shell model, along with details of its implementation, is given in Section 4.5. The wet and dry shell models in combination are capable of simulating the three droplet morphologies identified at the start of this section. After giving details of each model in isolation, it is shown how the evolving droplet properties can be used to select the appropriate model to use. The chapter closes with two applications of the new model showing comparison with experimental results from the literature. 4.2 Deciding Which Model to Use Before proceeding to discuss the structure and implementation of the thickening, wet and dry shell models in more detail, this section considers which of these models should be used following shell formation. As mentioned above, the dried-particle formation model is traditionally specified a priori. In contrast, the new droplet drying model developed in this thesis allows the choice of drying model and consequently the generic morphology of the final particle to be decided during the course of the simulation. In order to explain 105 CSH

119 4.2. Deciding Which Model to Use the manner in which this is achieved, it is first necessary to take a closer look at the physics of moisture removal following shell formation Physics of Drying Following Shell Formation Pressure Drop Across Surface Menisci Initially, all the solid particles are completely wetted by the continuous phase as illustrated in Figure 4.2a. Once a shell has formed, the continuous phase recedes to the pore mouths and menisci form between the particles at the surface. These menisci support a pressure drop of P = 2γ r M, (4.2.1) where γ is the surface tension of the liquid phase and r M is the radius of curvature of the meniscus. The pressure gradient within the droplet resulting from this capillary pressure leads to transport of the continuous phase towards the outer surface. At the same time, the tension from these menisci drives the solid particles past each other towards the droplet centre, (Tsapis et al., 2005). These two processes occur simultaneously and provided the capillary forces are strong enough result in continued droplet shrinkage after the formation of a surface shell. This continued, capillary driven shrinkage is termed shell thickening. During this thickening period the shell remains quite weak and the pore network shrinks in response to the capillary tension, (Brinker, 2006). As a result, the radius of curvature, r M, remains large, as shown in Figure 4.2b. In some types of gel, it is possible for the pore structure to shrink to as little as one tenth of the original volume during the thickening period, (Scherer, 1990). However, the network will stiffen as it shrinks, increasing the capillary pressure. At some point, the radii of curvature of the surface menisci will become so small that they can fit into the pores, as shown in Figure 4.2c. This point corresponds with the start of the first falling-rate period discussed in Section and also represents the point with the maximum capillary pressure in the liquid. Simple geometry shows that, if the pores are assumed cylindrical, the critical radius of curvature for a meniscus entering a pore is r M = a cosθ, (4.2.2) where a is the pore radius and θ is the contact angle, (Scherer, 1990). Clearly there will be some characteristic pore radius, r c, corresponding to the peak capillary pressure and this will likely be related to the size of the narrowest pores. Several workers have shown 106 CSH

120 4.2. Deciding Which Model to Use r M r H (a) (b) (c) Figure 4.2: Illustration showing the drying process in the shell thickening period. (a) Prior to shell formation, the suspended solids are completely wetted by the liquid phase. (b) On shell formation, tension develops in the liquid as surface menisci form with radii of curvature, r M. Initially, the solid network yields easily requiring little stress to deform and, as a result, the radii of the menisci are large. (c) As the porous network stiffens, the stress required for deformation increases and the menisci radii decrease. The limiting radius corresponding to the maximum capillary pressure occurs when the menisci recede into the pores. that this characteristic pore size is closely tracked by the hydraulic radius of the porous medium, (Scherer, 1994; Smith et al., 1995). If the total surface area per unit volume is given by S V, the hydraulic radius is r H = 2(1 ɛ) S V. (4.2.3) Taking a in (4.2.2) as r H and substituting into (4.2.1) gives the maximum pressure drop across the surface menisci as P max = γ cosθ (1 ɛ) S V, (4.2.4) where the need to assume cylindrical pores has been removed. Instead, the present model allows the volumetric solids surface area to be obtained form the second moment of the particle number density using (3.4.9), that is S V = πm 2. (4.2.5) If the suspended particles are assumed to be mono-disperse with diameter L, then (4.2.3) becomes r H = (1 ɛ) 3ɛ L, (4.2.6) and it is seen that the characteristic pore size is inversely proportional to the solids volume 107 CSH

121 4.2. Deciding Which Model to Use fraction. Equation (4.2.4) shows that it is theoretically possible to develop substantial pressure drops across surface menisci. As an example, consider the drying of a water based gel where the solids volume fraction, ɛ, is 0.65 and S V = m 2 m 3. Water has a surface tension of Nm 1 and the contact angle is taken as θ = 0 rad for convenience. Under such, fairly typical, circumstances the maximum pressure drop is greater than 40 MPa. In general, the pressure drops predicted by (4.2.4) cause a negative absolute pressure within the fluid. This implies that the drying fluid is under tension, (Imre, 2000). Water the fluid of interest in the present thesis is capable of withstanding very large negative pressures; that is, water has a high tensile strength. Theoretical values as high as 3000 bar have been derived for the tensile strength of water at room temperature, (Tas et al., 2003), with negative pressures as high as 1400 bar at 42 C having been observed experimentally, (Zheng et al., 1991). Cavitation occurs when the negative pressure exceeds the tensile strength of the fluid, or earlier in the presence of impurities capable of acting as cavitation nuclei, (Briggs, 1950; Tas et al., 2003). Cavitation is the result of the energy barrier to bubble nucleation being overcome; the result is the spontaneous appearance of a bubble within the stretched liquid, (Maris and Balibar, 2000). Pressure Drop Across a Thickening Shell During most of the thickening period the capillary pressure is significantly less than the maximum value given by (4.2.4) and, consequently, the meniscus radius is greater than the hydraulic radius, r H. It is therefore not possible to use (4.2.1) to directly calculate P in this regime. Instead, the capillary pressure, P cap, is obtained by likening the movement of the continuous phase through the shell to filtration through a porous filter with the same thickness as the growing crust, (Minoshima et al., 2001). The pressure drop across the shell is then assumed equal to the pressure drop across the air continuous phase interface. This analogy is illustrated schematically in Figure 4.3. Given knowledge of the liquid flow rate, Darcy s law may be used to obtain the corresponding pressure drop across a porous medium. Darcy s Law may be written q = K µ P, (4.2.7) where K is the permeability tensor for the porous body and µ is the viscosity of the flowing fluid. Making use of the assumed spherical symmetry in the present problem, this reduces to q = κ µ P r, (4.2.8) 108 CSH

122 4.2. Deciding Which Model to Use ṁ ρ 0 A P e ṁ ρ 0 A P e ṁ ρ 0 A P e T min r r r T (a) (b) (c) Figure 4.3: Schematic demonstrating how the process of shell thickening can be likened to the growth of a filter cake: (a) Prior to shell formation; (b) the point of shell formation corresponds with the appearance of a filter cake one particle thick; (c) the filter cake thickness, T continues to increase as the shell grows. The volumetric flow through the filter cake is simply related to the mass flux from the droplet, ṁ, and results in a pseudo-pressure acting on the shell, P e. where q is the volume flux of liquid through the shell. At the external surface, the volume flux is simply related to the mass vaporisation flux, ṁ, by q = ṁ ρ 0 A = dr dt. (4.2.9) Assuming that the problem may be analysed as a pseudo-steady state and, further, that the total volumetric fluid flow through the shell is not a function of radius, allows (4.2.8) to be integrated, giving P = P R P S = µ κ = µ κ R S ṁ ρ 0 A ṁ vap 4πr 2 ρ 0 A R d r S T, (4.2.10) where T = R S is the shell thickness. In deriving this expression, it has also been assumed that the permeability and viscosity are constant across the thickening shell. This assumption seems fairly reasonable and is used again in the next section to calculate the shell growth rate. The permeability, κ, in (4.2.10) is estimated using the Carmen-Kozeny relation, (Coulson et al., 1996), κ = 1 (1 ɛ) 3 L 2. (4.2.11) 180 ɛ 2 In the above expression, L is the diameter of the solid particles assumed spherical and monodisperse making up the porous medium. To account for a distribution of particle 109 CSH

123 4.2. Deciding Which Model to Use sizes, (4.2.11) may be replaced by κ = 1 5 (1 ɛ) 3 S 2 V, (4.2.12) where S V is the solids surface area per unit volume, discussed above, (Scherer, 1990). Strength of a Surface Shell So far, this section has demonstrated a method for calculating the pressure drop that exists across the surface menisci of a droplet drying in the presence of a surface shell. This pressure drop is important as it can cause continued shrinkage of the droplet following shell formation. In order to say when the droplet stops shrinking, it is necessary to determine when the thickening shell becomes structurally capable of supporting itself. The continued deformation of the shell is hypothesised to occur through a series of mini-buckling events driven by the capillary pressure of the receding continuous phase. Timoshenko (1936) showed that a spherical shell of radius R and thickness T will buckle when subjected to a uniform external pressure, P buck, given by P buck = T R T 2 2E, (4.2.13) 3 1 ν 2 where E is the Young s modulus of the material and ν is its Poisson s ratio. It is hypothesised that the newly formed shell will continue to experience buckling events and therefore continue to thicken so long as the capillary pressure is greater than the buckling pressure, P buck Criteria for the Different Models Having introduced the necessary theory, the criteria for applying the different sub-models following shell formation are now presented. These criteria are based on the relative magnitudes of the capillary pressure in the surface pores, the strength of the growing shell and the pressure drop across it quantities which are continuously tracked in the course of the simulation. The decision process applied to decide the appropriate model to use is summarised in Figure 4.4; this section explores that decision process in more detail. Thickening Regime Upon formation of a surface shell, all drying droplets are assumed to enter the thickening regime. The initial shell is set to have a certain minimum thickness. Whilst this is an essentially free parameter, the diameter of the smallest solid particle sets a physical 110 CSH

124 4.2. Deciding Which Model to Use P shell > P buck YES NO P buck < P max YES P buck P shell > P crit YES NO NO DRY SHELL WET SHELL THICKENING Figure 4.4: Illustration of the decision process used to select the appropriate drying model to use following shell formation. lower bound on possible shell thicknesses. The pressure drop across this shell is calculated using (4.2.10) and this will be equal to the pressure drop across the air continuous phase interface. Further, the pressure required to buckle the shell, P buck, is calculated using (4.2.13). As explained above, the capillary pressure in the surface pores can be thought of as exerting an external pseudo-pressure, P e, on the thickening shell. The magnitude of this pseudo-pressure is equal to the capillary pressure at the surface which, in turn, is equal to the pressure drop across the shell. The shell will continue to thicken so long as this pseudo-pressure is greater than the pressure required to cause buckling of the shell. That is, thickening continues whilst P shell > P buck, (4.2.14) or, on substituting from (4.2.10) and (4.2.13) and re-arranging, whilst R(R T ) ṁ > κ 2Eρ 0 A T µ. (4.2.15) 3 1 ν 2 The structural properties of the growing shell are seen to directly influence the duration of the thickening regime, as is the drying rate, ṁ. The thickening period extends whilst the inequality in (4.2.15) is satisfied. However, this condition is sufficient but not necessary for thickening, i.e., thickening does not necessarily cease once (4.2.15) is no longer satisfied. The reason for this is that the surface menisci between suspended solid particles are potentially capable of supporting a greater 111 CSH

125 4.2. Deciding Which Model to Use R R S S P P P shell 0 r 0 r P shell P buck (a) Figure 4.5: Illustration of the pressure profiles within a drying droplet following formation of a surface shell: (a) the pressure drop across the liquid air interface is equal to the pressure drop across the shell, P shell when this is greater than the buckling pressure, P buck ; (b) when P buck > P shell, the pressure drop across the interface is equal to the buckling pressure and the pressure in the droplet core falls below ambient. (b) pressure drop than that across the thickening shell. Once (4.2.15) is no longer satisfied, the menisci will retreat into the surface pores, reducing their radius of curvature and thus increasing the associated capillary pressure. This retreat will continue until the pressure is sufficient to cause further buckling, or until the capillary pressure exceeds the maximum possible value as given by (4.2.4). Thickening will cease once the capillary pressure exceeds the maximum possible value, P max. At this point, the menisci retreat beneath the surface and the dry shell regime commences. However, it is possible for the thickening regime to end earlier by progression to the wet shell regime. Wet Shell Regime When drying droplets with a surface shell, the surface menisci support a pressure drop across the air-continuous phase interface. As explained above, it is initially assumed that this pressure drop is equal in magnitude to the pressure drop across the thickening shell and, consequently, the pressure profile in the drying droplet is similar to that sketched in Figure 4.5a. Once the menisci begin retreating into the surface pores, the pressure drop across the menisci grows larger than that across the shell. As a result, the fluid pressure in the core region falls below atmospheric, as illustrated in Figure 4.5b. The wet shell model, as described above, postulates the growth of a centrally located vapour-saturated bubble within the drying droplet. This approach is clearly a gross simplification of the physics occurring within a real drying droplet, but it is possible to motivate the idea of a centrally located bubble in a number of ways. Firstly, dissolved gases present in 112 CSH

126 4.2. Deciding Which Model to Use the initial droplet may come out of solution to form bubbles, or small bubbles may even be present in the initial feed. As the pressure in the droplet falls, these bubbles will expand. In the absence of such features, cavitation will occur when the pressure falls below the vapour pressure of the liquid by an amount dictated by the tensile strength, ( 4.2.1). However the bubble forms in a particular system, the present drying model postulates that a droplet will cease thickening and enter the wet shell regime once P buck P shell < P crit, where P crit is some critical pressure. The complete process for deciding the relevant drying regime is illustrated in Figure 4.4. Wet Shell to Dry Shell Switch As mentioned above, the dry shell model can directly follow the thickening regime when P buck > P max. This is likely to occur for droplets containing large suspended solids, e.g., the detergent crutcher mix droplet simulated in Section However, it is also possible that the dry shell regime might be entered following a period of wet shell drying. The wet shell model includes an expanding central vapour space with a slowly growing outer shell. It is clear that if this regime persist for a sufficiently long time, the growing bubble will meet the retreating inner shell surface. At this point, the bubble can expand no more and further moisture removal necessitates drying of the shell itself. The new model therefore executes a switch from wet shell to dry shell drying when b = S. An example of this behaviour is presented in Section In general, it is possible that the switch between the wet and dry shell drying regimes might occur before b = S. Such an eventuality leaves three spatial domains: a region adjacent to the bubble, formerly the wet core; a region corresponding to the previous wet shell; and the new, growing, dry shell region. Such a situation may also be handled in the current model formulation and an example is shown in Section Internal Boiling When Drying in Hot Air When the drying air has a temperature greater than the boiling point of the continuous phase, T boil, the possibility exists that the drying droplet will boil. Since the new drying model developed in this thesis assumes a uniform temperature across the droplet, ( 3.2.1) the condition for boiling of the continuous phase is simply T d > T boil. The behaviour following the start of boiling depends on the morphology of the droplet at that point. Whilst the details of this dependence are beyond the scope of the current thesis, it is possible for a number of observations to be made concerning likely drying behaviour. 113 CSH

127 4.2. Deciding Which Model to Use Boiling with a Dry Shell The assumption that there are no internal temperature gradients is approximately true, but only holds for regions wetted by the continuous phase; in the dry shell model, the dried shell is be expected to be hotter than the droplet average. Consequently, it seems reasonable to assume boiling will commence at the evaporative front marking the edge of the hotter dry shell region when the droplet average temperature, T d, rises above the boiling temperature, T boil. Making this assumption allows boiling in the presence of a dry shell to be simply incorporated into the existing model framework provided that the rate of boiling is slow enough to enable the vaporised moisture to escape through the porous shell. Assuming that this condition is satisfied, the slow boiling model may be applied, as discussed in Section 4.4. Boiling causes the partial pressure of water vapour at the evaporative interface to rise. The initial effect of this rise is to enhance the vapour flow away from the interface towards the external surface. Such conditions can be modelled using the slow boiling model. However, if the pressure rise is sufficiently high then the droplet is expected to crack or shatter. Obtaining a qualitative measure corresponding to this sufficiently high criterion requires that the vapour flow through the shell be modelled. Such a calculation could, in principle, be incorporated within the new model framework. Structural information about the shell provided by the moments of the solid particle population balance could then be used to predict cracking or shattering of the droplet. However, this is beyond the scope of the present thesis. Boiling with No Shell or a Wet Shell If the boiling condition is satisfied prior to shell formation, or during the wet shell or thickening regimes, vaporisation is expected to commence within the drying droplet. This results in bubble formation and droplet inflation or puffing. Such behaviour has been observed by Walton and Mumford (1999b) who report that, at high drying air temperatures, nearly all skin forming particles undergo inflation as a result of internal bubble nucleation. Although the temperature is assumed uniform throughout the droplet, the temperature at which the continuous phase boils is not necessarily constant. Rather, T boil will be a function of the continuous phase composition, with more dilute regions boiling at lower temperatures. Consequently boiling is expected to commence in the most dilute regions which, for a drying droplet, will be at the centre. 22 This motivates the expansion of a centrally located bubble in those drying models which attempt to simulate puffing, (e.g., Sano and Keey, 1982; Hecht and King, 2000b). 22 Whilst the continuous phase is most dilute at the droplet centre in most systems, this is not necessarily the case for droplets which are already saturated at the start of drying. 114 CSH

128 4.3. Shell Thickening Implementation of a sub-model to simulate drying during inflation is beyond the scope of this thesis. However, the model presented here can indicate when such inflation is likely to occur. Furthermore, it is noted that the new model framework developed in the present work lends itself to future incorporation of such a sub-model; the infrastructure to handle an expanding, centrally located bubble has been incorporated during implementation of the wet shell model. 4.3 Shell Thickening The previous section demonstrated the criteria for deciding which of the sub-models thickening, dry shell, wet shell or slow boiling to use following formation of a surface shell. Here and in the following two sections, each of these drying regimes is discussed in more detail. Particular attention is paid to specifying how each sub-models is implemented within the overall framework of the new droplet drying description introduced in this thesis. A complete listing of all the equations in the present model in the precise form in which they are solved is given in the following chapter. This section focuses on the thickening regime which all droplets are hypothesized to enter upon shell formation. As discussed in Section 4.1.2, there is experimental evidence showing that some drying droplets continue to shrink following the appearance of a surface shell; contraction only ceases when the structural strength of the shell is sufficient to withstand those collapsing forces acting on the droplet arising from continued evaporation, ( 4.2.2). The thickening model describes the droplet drying behaviour between the initial appearance of a shell and the formation of rigid crust Model Description The continuous phase continues to wet the shell during the thickening regime and consequently the equation which describes the behaviour of this phase, (3.3.16), is unchanged: (1 ɛ) ρ (c) ω B (c) t + 1 r 2 (1 ɛ) v r 2 r (c) ρ (c) ω B (c) r 2 eff ρ (c) (1 ɛ) ωb (c) r r ɛ + ρ 0 D t = 0. (4.3.1) 115 CSH

129 4.3. Shell Thickening Under ordinary circumstances it is assumed that there is no central bubble present during the thickening regime and, consequently, f (t) = 0 in (3.3.31), giving 1 ω v r (c) B = (c) eff 1 ɛ. (4.3.2) Λ B + ω B (c) r 1 ɛ r Similarly, the equation describing the solids behaviour in the core region, (3.4.2) is unchanged, with v (d) = 0, giving r t N + L (GN) 1 r 2 r r 2 D N = 0. (4.3.3) r In contrast, the solids within the thickening shell now behave differently. During the thickening regime, the particles in the shell are re-arranging themselves as the crust deforms. The model allows for continued solids growth during this period, but does not seek to model the precise nature of the spatial re-arrangement. Rather it is assumed that the solid particles are no longer free to diffuse but are subjected to an imposed velocity, v (d) r R 2 dr = r dt (4.3.4) arising from the deforming shell. This expression is derived from a volume balance on the solids over an element in the shell, assuming that there is no accumulation as a result of buckling. Equation (3.4.2) therefore reduces to t N + R 2 L (GN) + dr N r dt r = 0, (4.3.5) in the thickening shell Shell Growth The continuous phase is assumed to continue wetting the surface particles although not necessarily completely during the thickening regime and so evaporation proceeds at the external surface of the droplet. Consequently, the external radius of the particle continues to decrease according to (3.3.36), i.e., dr dt = ṁvap ρ 0 A. (4.3.6) Algorithm 3.1, used to calculate the mass vaporisation rate ṁ vap in this expression, is modified to account for the vapour pressure reduction as the menisci recede between the solid 116 CSH

130 4.3. Shell Thickening particles. The reduced partial pressure of moisture at the droplet surface, p, is related to A that above a solids-free surface at the same temperature, T, by the Kelvin equation, p = p A A exp 2V m,l γ cosθ. (4.3.7) r M R g T Here, r M is the radius of curvature of the surface menisci, as introduced in the previous section, V m,l is the molar volume of the continuous phase and θ is the contact angle. This correction becomes significant as the radii of the surface menisci approaches the hydraulic radius of the porous shell, r H. The rate at which the thickening shell grows is calculated by considering a number balance on the solid particles in the region [S (t), R(t)]. Assuming that nucleation of new particles does not occur within the shell region, the total number of particles may only change as a result of flux across the interface at r = S (t), i.e., S ds 2 dt m 0 S = 1 4π = d dt d dt R(t) S(t) V shell m 0 d r 2 m 0 d r = R dr 2 dt m 0 ds R S2 dt m 0 S + + R r m 2 0 d r, (4.3.8) t S } {{ } =0 where the Leibniz integral rule has been used to write the last line. In these equations, S = S δ r represents a radial location just inside the thickening shell, and S + = S + δ r a location just outside. Making the further assumption that the number density of particles within the growing shell is spatially uniform and unchanging with time allows the final term in (4.3.8) to be set equal to zero. The shell growth rate is then given by ds dt = m 0 R m 0 S + m 0 S R 2 dr S dt, m 0 S m 0 S +. (4.3.9) It is also possible for the thickening shell to grow through the process of sink diffusion, where the inner boundary of the shell is viewed as a sink for randomly diffusing particles. However, the contribution made by this process will be significantly less than the term on the right hand side of (4.3.9), and further discussion is deferred until Section Assuming that the solid particle number density is temporally constant within the shell implies that it remains equal to its initial value. The zeroth moment of the particle number density function throughout the shell therefore equals the value at droplet edge at the point 117 CSH

131 4.3. Shell Thickening of shell formation, i.e., m 0 t tshell, r [S (t), R(t)] = m 0 r = R t = tshell. (4.3.10) The rational behind assuming a constant particle number density to define the shell growth rate as opposed to, say, a constant solids volume fraction, is that no new particles are allowed to nucleate in the shell region. In contrast, growth of existing particles is still permitted and the solids volume fraction in the shell region is expected to continue increasing. In situations where only one particle size is present, such as the colloidal silica example discussed in Section 4.6.2, all moments are constant in the thickening shell region Boundary Conditions At the point of shell formation, that is, when ɛ = ɛ crit at r = R, the shell thickness, T min, is set equal to the diameter of the smallest of the suspended particles, (Figure 4.1b). This sets up two spatial domains, termed the central core region, [b, S], and the wet- or thickeningshell, [S, R]. The five partial differential equations describing the time evolution of the solute mass fraction and moments of the solid particle number density are solved in both of these domains. In effect, following shell formation, there are therefore ten coupled partial differential equations to be solved, each requiring two boundary conditions. Continuous Phase Equation In the thickening regime, the solute boundary condition at the external droplet surface, (3.3.35), remains unchanged: ω B (c) = ω B (c)ṁ. (4.3.11) r r =R ρ (c) eff However, the appropriate boundary conditions to be implemented at the internal surface of the thickening shell are more complicated. Considering the superficial solute flux at r = S gives n (c) B r = (1 ɛ) n B r (c) r = (1 ɛ) v r (c) ρ (c) ω B (c) r =S =S eff ρ (c) r = (1 ɛ) n B r (c) r =S(t) (1 ɛ) ωb (c) (4.3.12a) ds + ρ (c) (c) (1 ɛ) ω B dt, (4.3.12b) 118 CSH

132 4.3. Shell Thickening which, on substituting for v r (c) from (4.3.2) may be re-arranged to give ω B (c) = ρ0 A ds n (c) ω r eff ρ (c) B dt + B r (c) ρ (c) r =S(t). (4.3.13) Note that the intrinsic solute mass fraction and, consequently, the continuous phase density must be the same either side of the inner shell wall at r = S. Further, the solute mass flux across the growing shell boundary must be continuous, that is, it is required that ω B (c) r =S + = ω B (c) r =S (4.3.14a) and n (c) B r r =S (t) = n (c) B r r =S + (t) n B r (c) r 1 ɛ = n =S + (t) 1 ɛ + B r (c) r =S (t). (4.3.14b) In these last two equations, S = S δ r represents a radial location just inside the thickening shell, and S + = S + δ r a location just outside; ɛ and ɛ + denote the solids volume fraction at S and S + respectively. Equation (4.3.13) may now be evaluated at both S and S + which, on substitution into (4.3.14b), can then be re-arranged to give a relationship between the solute mass fraction gradients on either side of the growing shell interface, ω B (c) r r =S = + eff eff 1 ɛ + ωb (c) 1 ɛ r r =S + ρ0 A ω B (c) eff ρ (c) ɛ ɛ + 1 ɛ ds dt. (4.3.15) This, along with (4.3.14a), gives the two boundary conditions required for the solute equation at the shell interface. The boundary condition at the centre of the droplet, assuming no bubble growth in the thickening regime, remains unchanged, i.e., a zero-gradient symmetry condition is applied. Discrete Phase Equations The appropriate boundary conditions for the moment system in the shell region are those of zero spatial gradient at both ends. This arises from the assumption that spatial transport of solids in the shell is purely advective, (4.3.5). The moment boundary conditions in the central core are also those of zero spatial gradient at both ends, although inclusion of sink diffusion at the shell interface modifies this. As mentioned previously, a thorough discussion of the moment boundary conditions in the central region is deferred until Section CSH

133 4.4. Dry Shell and Slow Boiling Models 4.4 Dry Shell and Slow Boiling Models Model Description The dry shell model is, in its simplest form, a classical shrinking core type analysis whereby a central core of wetted material contracts as moisture evaporates from a receding interface, (Audu and Jeffreys, 1975; Cheong et al., 1986). The dry shell is defined to be the region beyond this wetted core. As suggested by the schematic of dried-particle morphologies in Figure 2.9, droplets drying via the dry shell route might be expected to form solid particles or, at high temperatures, they might shatter. Although the initiation of shell formation may be signalled by some critical solids volume fraction at the droplet surface, it is noted that the dry shell model does not explicitly place a requirement on the solids volume fraction within the growing shell region. The porosity of dry shell particles might therefore be expected to vary perhaps considerably with position. In contrast to several previous implementations which assumed constant dry crust properties, (e.g., Cheong et al., 1986; Dalmaz et al., 2007), porosity variations in the dry shell are tracked in the present model. In the present implementation of the dry shell model, the shell is assumed to be completely dry, that is, there is no solvent or solute remaining in the crust region. Further, it is assumed that there is no vapour in the wet core and, consequently, the continuous phase remains funicular throughout drying, as illustrated in Figure 4.6. Within the wet core, the equations to be solved for the solute mass fraction and the moments of the population balance are the same as outlined in Chapter 3. As demonstrated in the Section 4.6.1, continuing to solve these equations in the wet region allows the properties of the growing dry crust to be predicted. If the temperature of the droplet rises above the boiling point of the continuous phase during the dry shell regime then, as discussed in Section 4.2.3, the droplet is assumed to undergo slow boiling. Two key assumptions underpin this sub-model: first, the pressure increase arising from continuous phase boiling is assumed to be lower than that which would cause the droplet to shatter; secondly, the boiling rate is assumed to be heat transfer limited, that is, all energy supplied to the droplet is assumed to go towards vaporising the continuous phase. Improving upon these assumptions would require vapour transport through the dry shell to be modelled and is beyond the scope of this thesis Shell Growth The dry shell is assumed to grow according to ds R 2 ṁ dt =, (4.4.1) S 1 ɛ S ρ 0 A 120 CSH

134 4.4. Dry Shell and Slow Boiling Models wet core S r wet core dry shell region R Figure 4.6: Illustration of a droplet of radius R drying through a dry shell with internal radius, S. The shell region is completely dry, containing no solvent or solute. The wet core remains free of vapour and thus the continuous phase remains funicular throughout drying. R ṁ vap S w w A A,S w A,sur w A, r S R Figure 4.7: Evaporation from a droplet of radius R in the presence of a dry shell. Evaporation proceeds from a front located at r = S(t), which is receding towards the centre of the droplet. The graph below the diagram shows illustrative radial profiles of the water mass fraction, w A, in the vapour-filled pores and surrounding gas. where the solids volume fraction is evaluated at the receding interface, r = S (t). The ratio of radii accounts for the fact that ṁ still refers to the evaporative solvent flux at the external surface whereas the moisture actually evaporates from a front within the drying droplet. Under such circumstances it is necessary to consider the increased resistance to mass transfer resulting from the presence of a dry shell, (Incropera and DeWitt, 2002). This situation is illustrated in Figure 4.7, where it is seen that the moisture mass fraction adjacent to the evaporative interface, w A,S, falls to w A,sur at the surface of the droplet before dropping to w A, in the bulk gas. The additional mass transfer resistance from an internal evaporative front may be char- 121 CSH

135 4.4. Dry Shell and Slow Boiling Models acterised by the effective diffusion coefficient of solvent vapour through the dried shell region, D eff. This may be related to the binary diffusion coefficient of water in air, A,air, via D eff = (1 ɛ) A,air σ, (4.4.2) where σ is the tortuosity of the porous shell, (Cussler, 1997). Assuming a pseudo-steady shell thickness, (3.5.4) can be integrated over the dry region to give 1 ṁ vap = 4πρD eff R wa,sur log, (4.4.3) S 1 w A,S where w A,sur is unknown. The transport from the droplet surface to the bulk can be analysed as in Section to obtain 1 wa, ṁ vap = 2πρ A,air Sh Rlog, 1 w A,sur which is the same as (3.5.10). Using this to substitute for w A,sur in (4.4.3) gives, after re-arrangement, ṁ vap = 4πR 2 ṁ = 2πρ A,air R A,air S R + 1 2S Sh D eff 1 log(1 + B M ). (4.4.4) As expected, this expression reduces to (3.5.10) when S = R, i.e., when there is no dry shell surrounding the droplet. During the slow boiling regime, all energy supplied to the droplet is assumed to go towards vaporising the continuous phase. The expression for the mass vaporisation rate (4.4.4), is therefore modified to read ṁ vap = 4πR 2 ṁ = Q, (4.4.5) vap H where Q is the heat penetrating into the droplet and H vap is the latent heat of vaporisation of the continuous phase. The dry shell growth rate in the boiling regime is still obtained from (4.4.1) with ṁ now given by (4.4.5). An example of this behaviour is presented in Section CSH

136 4.5. Wet Shell Model Boundary Conditions The symmetry boundary conditions in the centre of the droplet remain unchanged once a dry shell has formed. However, the external boundary conditions, now at r = S (t), require modification. The solute boundary condition at this outer edge of the wet central core becomes ω B (c) ω B (c) ṁ =, (4.4.6) r r =S ρ (c) eff 1 ɛ S where ṁ is given by (4.4.4) or (4.4.5) in the dry shell and slow boiling regimes respectively. Comparison with (3.3.35) shows that the only difference between this expression and the external solute boundary condition prior to shell formation is the inclusion of the 1 ɛ S 1 term. The outer boundary condition on the moment system is now N = 0, r r =R (4.4.7) as no solids are allowed to enter or leave the dried region. This completes the description of the dry shell and slow boiling models. 4.5 Wet Shell Model Model Description The key assumption of the wet shell model is that the solids in the shell region remain wetted by the continuous phase and, consequently, the evaporative front remains at the droplet surface. Continued solvent evaporation from an unshrinking droplet requires the presence of a growing vapour space. In the wet shell model, this takes the form of a single, centrally located bubble, as illustrated in Figure 4.8. As the continuous phase wets all the solid particles in this model, the shell itself cannot be identified in terms of a dry region as was done in the dry shell approach. Instead the shell is defined as the region which has a solids volume fraction higher than some critical value. This value is normally taken to be the same as the critical solids volume fraction triggering shell formation, ɛ crit, although it is noted that this need not be the case. Figure 4.8 demonstrates how the droplet is divided into the core and wet shell regions, separated by the internal edge of the growing shell at r = S (t). Volume conservation requires that the rate of growth of the central bubble must be 123 CSH

137 4.5. Wet Shell Model central bubble core S S S + wet shell region r b S R Figure 4.8: Illustration of a droplet of radius, R drying through a wet shell with internal radius, S. The solid particles in the wet shell remain wetted by the continuous phase and a central bubble grows as evaporation proceeds. r related to the evaporative moisture flux according to db dt = R2 ṁ, b > 0. (4.5.1) b 2 ρ 0 A From this expression, it is clear that a seed bubble of finite size is required to avoid an infinite initial growth rate. Physically, this is likely to be an air bubble present in the feed material or introduced during spraying. The central bubble is filled with vapour-saturated drying air, not pure solvent vapour. That is, the partial pressure of solvent in the bubble will be in equilibrium with that in the adjacent droplet. As the bubble grows, it is assumed that the mass of solvent in the bubble remains small compared with the amount in the surrounding droplet. Consequently, the moisture flux to the bubble is negligible and has no effect on the surrounding concentration gradients. The growing bubble is modelled as imposing an outward advective velocity on the droplet. The bulk continuous-phase velocity, (3.3.31), is modified to read v (c) r 1 ω B = (c) eff 1 ɛ Λ B + ω B (c) r 1 ɛ r + b 2 r 2 db dt, (4.5.2) where the final term is the additional advective component due to the bubble. With this modification to v (c), the continuous phase equation, (3.3.16), can be applied without further alteration in both the shell and core regions. r The evolution equation for the population of solid particles, (3.4.2), has the same additional advective component as a result of the growing bubble. Therefore, the equation 124 CSH

138 4.5. Wet Shell Model describing the solids in the core region now reads t N + L (GN) + b 2 db N r 2 dt r 1 r 2 r r 2 D N = 0. (4.5.3) r Within the shell region, the solid particles are no longer free to move at all; the particle population evolves purely as a result of crystallisation from the continuous phase and (3.4.2) reduces to t N + (GN) = 0. (4.5.4) L Nucleation of new particles within the wet shell is possible, but this is thermodynamically unlikely in a region where, by definition, the volume fraction of solids is already high. Nucleation is therefore neglected Shell Growth The shell grows as a result of solid particles depositing on the inside wall at r = S. From (4.5.3) it is seen that the particle number density flux in the space of the external co-ordinate is given by Ṅ r = b 2 r 2 db dt N D N r. (4.5.5) If the advective flux of solids is assumed to dominate the deposition process, then Ṅ r b 2 r 2 db dt N, (4.5.6) and so, following the growing shell interface at r = S(t), it is possible to write N b 2 db S 2 dt = N N + ds dt. (4.5.7) Here N and N + represent the particle number density at S and S + respectively. Rearranged, this gives the shell growth rate as ds dt = b 2 db N S 2 dt N + N. (4.5.8) 125 CSH

139 4.5. Wet Shell Model Note that this expression implies that the gradient of the particle number density is zero at the shell interface, i.e., N r r =S = 0. (4.5.9) This might be viewed as unrealistic as the growing boundary also acts as a sink for diffusing solid particles. This effect can be captured through the incorporation of a sink diffusion term (Hansson, 2003; Won et al., 2001). It is assumed that the rate of sink diffusion may be described by the simple model Ṅ sink = k sink N, (4.5.10) which, included in (4.5.6), gives a new solids flux to the growing wall, Ṅ r = b 2 S 2 db dt N + k sink N. (4.5.11) The wet shell growth rate is then modified to read ds dt = N N + N b 2 db S 2 dt + k sink which, in terms of the moments, (3.4.22), is ds dt = m a m + a m a b 2 db S 2 dt + k sink Boundary Conditions, (4.5.12). (4.5.13) The existence of two physical domains following the formation of a wet shell the central core and the wet shell region requires that four boundary conditions be specified for each of the equations, one at either end of both domains. Continuous Phase Equation The solute boundary condition at the external droplet surface remains unchanged from the base case considered in Chapter 3, i.e., (3.3.35) continues to be used once the moisture evaporation rate, ṁ vap has been suitably modified using (4.3.7) to account for the vapour pressure reduction caused by the wet shell. The appropriate boundary conditions to be implemented at the internal surface of the wet shell are similar to those in the shell thickening sub-model, as discussed in Section Substituting the revised expression for the 126 CSH

140 4.5. Wet Shell Model continuous phase velocity, (4.5.2), in to the superficial solute flux balance at r = S (t) gives (1 ɛ) n B r (c) r = (1 ɛ) v (c) ρ (c) ω =S r B (c) eff ρ (c) (1 ɛ) ωb (c) r = (1 ɛ) n B r (c) r ds + ρ (c) (c) (1 ɛ) ω B =S(t) dt, (4.5.14) and so (4.3.13) becomes ω B (c) r = ρ0 A ds eff ρ (c) ω B (c) dt b 2 db n B r (c) r =S(t) + S 2 dt ρ }{{} (c) v s. (4.5.15) Imposing continuity of the solute mass flux across the growing shell and re-arranging as in Section gives the boundary condition where ω B (c) r r =S = + eff eff 1 ɛ + ωb (c) 1 ɛ r r =S + ρ0 A ω B (c) eff ρ (c) ɛ ɛ + v 1 ɛ s, (4.5.16a) v s = ds dt b 2 S 2 db dt. (4.5.16b) Solute mass fraction continuity across the shell interface gives the second boundary condition required at r = S, ωb (c) = ωb (c)+. (4.5.16c) The fourth boundary condition is that applied at the bubble interface at r = b. The moisture content of the gas in the bubble is assumed to be negligible compared with that in the adjacent wet core, (Sano and Keey, 1982) The flux of moisture from the droplet to the bubble is negligible and, consequently, the presence of the expanding bubble is assumed to have no effect on the concentration gradients at r = b. That is, the solute boundary condition at this interface is unchanged from the symmetry condition used previously, ω B (c) = 0. (4.5.17) r r =b 127 CSH

141 4.6. Applications Discrete Phase Equations The solid particles in the wet-centre of the droplet remain free to move once the outer shell has formed. When these particles contact the inner wall of the shell through either advection or the sink diffusion process discussed above they aggregate and contribute to shell growth. This is modelled by introducing a death-rate boundary condition at r = S, which is obtained directly from the sink diffusion condition for the number density function, (4.5.10). The relevant equations for the moments therefore read ɛ = k r sink ɛ, (4.5.18a) r =S m a = k r sink m a {0,1,2}. (4.5.18b) a r =S From the definition of the wet shell region the region where the solids volume fraction exceeds the critical value, ɛ crit it is clear that the appropriate boundary condition for the solids volume fraction at r = S + is ɛ + = ɛ crit. (4.5.19a) Re-arranging (4.5.13) then gives the boundary conditions for the remaining moments at r = S + as b 2 m + = db ds 1 a m 1 a S 2 dt + k sink, a {0,1,2}, (4.5.19b) dt where the shell growth rate is ds dt = ɛ ɛ crit ɛ b 2 db S 2 dt + k sink. (4.5.20) The specification of the wet shell model is completed by applying zero gradient boundary conditions to all moments of the solids number density at the both the droplet external radius and at the bubble interface. 4.6 Applications Two physical systems are simulated using the new model and the results obtained compared with experimental data from the literature. Both examples presented focus on aspects of the model relating to shell formation and the subsequent drying behaviour. 128 CSH

142 4.6. Applications Drying a Detergent Droplet The first test case is the simulation of a droplet of detergent slurry drying. Such crutcher mix is typically a complex mixture of around 10 components, with precise formulations being closely guarded commercial secrets, (de Groot et al., 1995). Griffith et al. (2008) have recently conducted a series of experiments investigating the drying of a generic crutcher mix formulation, the composition of which is shown in Table 4.1. Griffith et al. report that the droplets they observed showed no shrinkage during drying which indicates that this system might be a suitable choice to demonstrate the dry shell model. Table 4.1: The composition of the detergent slurry investigated by Griffith et al. (2008). Component Initial Mass Fraction LAS 0.09 Water 0.29 Acusol Polymer 0.03 Sodium Sulphate 0.35 Sodium Aluminosilicate 0.24 The methodology described in Section can be applied to further motivate the use of the dry shell approach for this example. Assume that the solid particle population in the crutcher mix is mono-disperse with a diameter of 10 µm. This represents the base case described below. Further, assume that shell formation is triggered at a critical solids volume fraction, ɛ crit = The hydraulic radius of the capillaries in this freshly formed shell can be calculated from (4.2.6), giving r H = 1.79 µm. According to (4.2.4), the maximum pressure drop capable of being supported by the menisci in such capillaries is P max = 80 kpa. Bortolotti et al. (1992) reports the Young s moduli for a number of different zeolite containing detergent powders; these are all of the order of 10 MPa. Using this value along with ν = 0.3 in (4.2.13) gives, on setting P buck = P max and assuming that T is small compared with the droplet radius, T crit R. (4.6.1) The result suggests that the assumption regarding the relative magnitudes of droplet radius and critical shell thickness is justified. For a 1.5 mm droplet, the critical shell thickness is less than six solid particle diameters. Therefore, following the methodology outlined in Section 4.2.2, the thickening regime is predicted to be negligibly long. In such situations, there are computational advantages associated with dictating that the simulation initiates the dry shell model immediately on shell formation namely, this avoids the need for creating the second co-ordinate system. This is the approach taken for the simulations presented in this section. 129 CSH

143 4.6. Applications CONTINUOUS PHASE SOLID PHASE CONTINUOUS PHASE SOLID PHASE 1.0 neat lye 1.0 neat lye mass fraction LAS water polymer zeolite Na 2SO 4 mass fraction LAS LAS water zeolite Na 2SO Na 2SO Na 2SO mass fraction (a) mass fraction Figure 4.9: (a) Initial composition of the crutcher mix droplets as measured by Griffith et al. (2007); and (b) simplified description of the crutcher mix used to model the system in the base case. (b) The description of the crutcher mix system needs to be further simplified before the present model can be used. Griffith et al. (2007) report that the continuous phase actually comprised individual neat and lye phases which were rich in LAS and sodium sulphate respectively. This is illustrated in Figure 4.9a. To simulate this system within the present framework, the lye phase was modelled as an aqueous sodium sulphate solution. Similarly, the small amount of sodium sulphate in the neat phase is ignored and this is considered to be a LAS water binary. This further simplified system is illustrated Figure 4.9b, and represents the initial composition used for the simulations presented in this section. In terms of the present model, the solid phase, D, comprises the zeolite sodium aluminosilicate and some crystallised sodium sulphate and the water in the system is the solvent, A. The LAS and dissolved sodium sulphate must be described as component B. For the remainder of this section, these two components will collectively be referred to as the solute. However, since only sodium sulphate crystallises to form new solid, it is important to track the mass fraction of the combined solute that is Na 2 SO 4, i.e., ξ = mass of Sodium Sulphate mass of Sodium Sulphate + mass of LAS. (4.6.2) Because of the simplifying assumptions made above, ξ also represents the mass fraction of the solute in the lye phase. Assuming that this fraction is uniform throughout the droplet allows ξ to be simply calculated from the initial masses and knowledge of the mass of Na 2 SO 4 that has crystallised during the drying period. This assumption is justifiable as the rate of moisture removal is slow when drying at T gas = 60 C. For the droplets observed 130 CSH

144 4.6. Applications by Griffith et al. (2008), ξ (t = 0) = NMR results obtained by Griffith et al. (2007) demonstrate that when drying crutcher mix, water is initially lost from the lye phase. Assuming that the ratio of LAS to water in the neat phase remains unchanged until the lye phase has dried allows the mass concentration of sodium sulphate in the lye phase, ω (c), to be determined from ξ : Na ω (c) Na = ξ ω B (c) ω B (c) (1 ξ ). (4.6.3) Initially, ω (c) Na = 0.31, which is the saturated mass fraction at 60 C, (Wetmore and LeRoy, 1951). This is to be expected as the aqueous and crystallised sodium sulphate are initially in equilibrium. Sodium sulphate crystallises out of solution as the drying proceeds. In the absence of any data relating to the model crutcher mix system, the kinetics used to describe this process are those reported by Rosenblatt et al. (1984) for direct crystallisation to the solid in a sodium sulphate water mixture. First introduced in Section 3.6.2, the relevant expression is dl dt = exp 57.4 Ci 1.5 C eqm, (4.6.4) RT which corresponds to the linear growth rate, G, in the population balance equation (3.4.2). C i and C eqm are the local and saturated sodium sulphate concentrations respectively in kmol m 3. As discussed in Section 3.4, the moment system describing the solid phase allows nucleation of new particles to be modelled. However, the presence of a considerable amount of crystallised sodium sulphate at the outset of drying in the detergent system makes nucleation of new crystals thermodynamically unlikely. Therefore, the nucleation rate, Ṅ 0, is set to zero in this simulation. In turn, this avoids the difficulties discussed at length in Section associated with determining a suitable value for Ṅmax. The solids contained in the crutcher mix droplets are relatively large typically between 2 and 50 µm in diameter, (Bayly, 2008) and, consequently, are relatively immobile. For simplicity, the solid particles are here assumed initially mono-disperse with a size of 10 µm. A single, non-size dependent, solids diffusion coefficient is then used, D = m 2 s 1. Griffith et al. (2008) determined an effective diffusion coefficient appropriate for modelling the movement of water in crutcher mix, u eff = exp u, (4.6.5) which, as expected, is a strong function of the dry mass basis moisture content, u. The 131 CSH

145 4.6. Applications sorption isotherm is required to obtain the rate of moisture evaporation from the drying droplet. 23 The isotherm was obtained by fitting a standard sorption isotherm to measurements on the crutcher mix obtained by Bayly (2007). The resulting equation is R = p A p = f ω 0.8ω A sat A + 1 f ωa 3.92 ω ω A exp, (4.6.6) A 0.15 where f 1 ω A = 1 tanh 100π ωa 0.25 (4.6.7) 2 is a blending function required to join the different functional forms fitted at high and low moisture contents. The isotherm is plotted in Figure 4.10 along with the associated experimental data. A crutcher mix droplet with an initial diameter of 1.5 mm was simulated drying in dehumidified air at 60 C with a relative velocity of 1.68 ms 1. Figure 4.11 shows the simulated moisture mass fraction compared with experimental measurements of a droplet drying under these conditions from Griffith (2008). The simulated moisture content matches the experimental results very well, although the errors associated with the measured data were Relative Humidity / [ ] Water Mass Fraction / [ ] Figure 4.10: Water sorption isotherm for crutcher mix. The line shows the sorption isotherm, (4.6.6), obtained by fitting to the experimental points obtained by Bayly (2007). 23 The sorption isotherm is evaluated by the function SORPTION_CALC which is called by Algorithm 3.1 in the course of determining the moisture evaporation rate. 132 CSH

146 4.6. Applications Moisture Mass Fraction / [ ] Simulated Moisture Mass Fraction Experimental Moisture Mass Fraction Time / s Figure 4.11: Simulated evolution of the moisture mass fraction in a crutcher mix droplet (line) compared with experimentally measured values from Griffith (2008), (symbols). not reported. The use of NMR techniques to follow the droplet drying by Griffith (2008) allows for further validation of the current model. Figure 4.12 shows a comparison between experimentally measured moisture profiles and those extracted from the simulation. To render the model results suitable for comparison with the experimental data, it is necessary to apply two Abel transforms, (Bracewell, 1990). Details of the Abel transformation, the routine used to perform the projection and its associated accuracy are contained in Appendix A. The NMR data returns intensity readings in arbitrary units. Therefore, assuming an initially homogeneous spherical particle, an appropriate scaling factor was determined using least squares minimisation and applied to all the simulated profiles. It is clear that the experimental droplet was not a perfect sphere, but the error associated with this approximation is small. The model fit to the measured profiles is fair. The intensity maxima at the centre of the droplet are very well predicted for the 10 and 20 minute profiles, but the predicted profile at 30 minutes does not match the data so well. Between 20 and 40 minutes, the experimental data shows faster drying than predicted by the model as is demonstrated by the observation that the 30 minutes data coincides with the 40 minutes predicted profile. It is possible that this discrepancy is a result of the experimental data which appears to show particularly fast drying between 20 and 40 minutes. However it could also be that the dry shell assumption used in the model is not entirely compatible with this system, or that water associated with the polymer not considered in the simplified model becomes important in the later 133 CSH

147 4.6. Applications Water Signal / [a.u.] min 10 min 20 min 30 min 40 min 50 min 0 min 10 min 20 min 30 min 40 min 50 min Radial Position / m x 10 3 Figure 4.12: Simulated moisture profiles in a drying detergent droplet (lines) compared with experimental observations (symbols) from Griffith (2008). Measured and simulated profiles are displayed at 10 minute intervals. stages of drying. Figure 4.12 clearly shows the influence of the dry shell assumption on the model results. A dry shell contains no moisture and consequently no NMR signal would be observed from such a region. The simulated profiles reflect this, showing the diameter of the wet core shrinking with time. Such an effect is, however, not clear in the experimental data, suggesting that an extension to the dry shell approach may be required for this system. One idea is to allow for a damp shell, whereby a certain fraction of the pore-volume in the shell remains filled by the continuous phase. The new droplet drying model gives additional information about the drying droplet not immediately apparent from the experimental data. For example, Figure 4.13 shows the time evolution of the individual component masses. The mass of solids increases as sodium sulphate crystalises out of solution. This is also reflected in the decreasing solute mass, which tends to a constant value which equals the mass of LAS in the system. Figure 4.14 shows the position of the receding dry shell interface as drying proceeds. As demonstrated by (4.4.4), the growing dry shell acts as a resistance to further moisture evaporation. In the simulation, the tortuosity of the shell is taken as σ = 10, which is representative of the values measured by Griffith et al. (2007). This value is used in (4.4.2) to give the effective diffusion coefficient of moisture through the dried shell. The shell solids volume fraction used in this expression is the average value of ɛ in the shell region. Figure 4.15 shows the solids volume fraction profiles through the droplet drying history. 134 CSH

148 4.6. Applications Solvent Mass Solute Mass Solids Mass Total Mass Mass / mg Time / s Figure 4.13: Simulated evolution of total droplet mass together with the mass of each of the three components in the model droplet x 10 3 Radial Distance / m Dry Shell Interface Outer Droplet Radius Time / s Figure 4.14: Simulated time evolution of the dry shell interface during the drying of a droplet of crutcher mix. 135 CSH

149 4.6. Applications ε / [ ] mins S (t = 0 mins) ε / [ ] mins S (t = 10 mins) Radial Position / mm Radial Position / mm ε / [ ] mins S (t = 20 mins) ε / [ ] mins S (t = 30 mins) Radial Position / mm mins Radial Position / mm mins ε / [ ] 0.5 ε / [ ] S (t = 40 mins) 0.4 S (t = 50 mins) Radial Position / mm Radial Position / mm Figure 4.15: Simulated solids volume fraction, ɛ, profiles at 10 minute intervals during the drying of a crutcher mix droplet. The diamond marker in each subplot shows the position of the inner edge of the dry shell at the corresponding time. Initially, the crutcher mix droplet is homogeneous with a uniform solids volume fraction of As the solid particles are relatively immobile, the surface solids volume fraction quickly reaches 0.65 and shell formation is triggered almost immediately. The rigid shell prevents further shrinkage of the droplet and, following the discussion at the start of this section, the dry shell model is immediately applied. The solids volume fraction behind this initial shell is still quite low. The dry shell is therefore predicted to have a loosely packed region immediately behind the external skin; recall that the dry shell model in contrast to the wet shell model does not place an explicit requirement on the solids volume fraction within the growing shell region, ( 4.4.1). As more water is removed, crystallisation causes the solids volume fraction to rise, as is reflected in later profiles. This in turn causes the packing of the dry shell to increase once more. Each subplot in Figure 4.15 contains a diamond indicating the position of the evaporative front at the relevant time. The solids volume fraction is, as expected, continuous across this dry shell interface. This is a further difference in expected behaviour between the wet and dry shell models. 136 CSH

150 4.6. Applications Figure 4.16: SEM images showing detergent droplets dried in a spray dryer. The dried-particles are seen to have a thin surface skin covering a core region in which the solids are loosely packed. Courtesy of Cheyne et al. (2002) By the plot at 40 minutes, all of the sodium sulphate has crystallised out of solution. This corresponds with the removal of all the water from the lye phase. At this point, there is no further mechanism within the model to increase the solids volume fraction and the dried-particle is therefore predicted to have a loosely packed centre. The pore space in this central region is completely filled by the neat phase, i.e., LAS and its associated water. Continued evaporation removes the water from this neat phase and, in practice, some of the LAS will solidify as it concentrates. Nevertheless, it still seems likely based on the model predictions that the centre of the dried-particles will be less dense than their outer regions. There is some evidence that the particle morphology predicted above is indeed observed in real dried-particles. Figure 4.16 shows SEM images of two detergent droplets similar to those simulated in the present study. The imaged droplets were dried at a higher temperature and therefore more quickly than those simulated, but they did not undergo inflation. The particles have a fairly smooth surface skin. However, this skin is thin and encloses a large central region where the solids are far less densely packed. Qualitatively, this agrees well with the predictions from the simulation. Tracking the sodium sulphate mass fraction in the solute, ξ, allows profiles of the concentration of sodium sulphate in the lye phase to be reconstructed. Figure 4.17 shows some such profiles plotted at 10 minute intervals through the drying of a crutcher mix droplet. It is seen that the concentration stays relatively constant around 30wt% until around 40 minutes. At this point it falls quickly to zero across the entire droplet. As discussed above, this leads to the cessation of solid crystallisation at the same point in the drying history. The observation of a relatively constant sodium sulphate concentration agrees with experimental results for similar systems. Griffith (2008) used 23 Na and 1 H NMR to observe the drying of a vial of crutcher mix over an extended period. 23 Na NMR can track the amount of sodium sulphate in solution as the vial dried. This was found to decrease in step with the free water 137 CSH