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List of Symbols xv -- List of Figures xxi -- List of Tables xxv -- 1 Introduction: Spray Drying and Reactive Drying Processes 1 -- 1.1 Spray Polymerisation. 2 -- 1.2 Single Droplet Models for Spray Drying . 3 -- 1.3 Meshfree Methods, Simulation of Structure Evolution4 -- 1.3.1 Previous Applications of SPH to Drying . 5 -- 2 Theoretical Principles 7 -- 2.1 Transport Equations 7 -- 2.1.1 Transport in a Mass Averaged System 9 -- 2.1.2 Transport in a Molar Averaged System . 13 -- 2.1.3 Reference Velocities and Conversion between Systems . 13 -- 2.1.4 Eulerian and Lagrangian Frames of Reference16 -- 2.2 Diffusion . 17 -- 2.2.1 Fickian Diffusion . 17 -- 2.2.2 Maxwell-Stefan Diffusion 19 -- 2.2.3 Determination of Diffusion Coefficients . 21 -- 2.3 Modelling of Free Radical Polymerisation 21 -- 2.3.1 Reactions in Free Radical Polymerisation 22 -- 2.3.2 Quasi-Steady-State Assumption (QSSA) 27 -- 2.3.3 Method of Moments . 28 -- 2.4 Mixture Thermodynamics 31 -- 2.4.1 Vapour Liquid Equilibrium at the Droplet's Surface . 32 -- 2.4.2 Calculation of Activity Coefficients . 33 -- 2.4.3 The UNIFAC Equations . 34 -- 2.5 Spray Drying: Basic Assumptions 35 -- 2.5.1 Approximate Residence Time in a Spray Dryer . 36 -- 2.5.2 Heat and Mass Transfer . 37 -- 2.5.3 Inner Circulation Inside a Droplet 38 -- 2.5.4 Are Droplets Fully Mixed? 40 -- 3 Modelling of Reactive Droplet Drying and Polymerisation 41 -- 3.1 Transport in a Reaction-Diffusion System 42 -- 3.1.1 Constant Physical Properties. 44 -- 3.1.2 Consideration of Mixture Effects . 46 -- 3.1.3 Diffusion and Reaction Driven Convection at Variable -- Molar Weights47 -- 3.1.4 Transport of Polymer - Quasi-Steady-State Assumption . 50 -- 3.1.5 Transport of Statistical Moments . 52 -- 3.2 Lumped Modelling - 0D approach 56 -- 3.2.1 General Equations for Reactive Spray Drying57 -- 3.2.2 Spray Polymerisation - Quasi-Steady-State Assumption 58 -- 3.2.3 Spray Polymerisation - Method of Moments. 59 -- 3.3 Distributed Modelling - 1D approach. 60 -- 3.3.1 General Equations of the Droplet Continuum60 -- 3.3.2 Boundary Conditions . 62 -- 3.3.3 Spray Polymerisation - QSSA66 -- 3.3.4 Spray Polymerisation - Method of Moments. 68 -- 3.4 Comparison with Existing Models 69 -- 3.5 Implementational Considerations . 70 -- 3.5.1 Implementation of the Moving Boundary Problem . 70 -- 3.5.2 Boundary Conditions . 71 -- 3.5.3 Treatment of Convection Terms . 72 -- 3.5.4 Implementation of Diffusion. 75 -- 3.6 Verification of the Transport Approach76 -- 3.6.1 Diffusion Driven Convection, Constant Properties 76 -- 3.6.2 Diffusion Driven Convection, Variable Molar Weight 79 -- 3.6.3 Diffusion Driven Convection, Excess Volumes . 82 -- 3.6.4 Reaction Induced Convection85 -- 4 Simulation of Spray Polymerisation 87 -- 4.1 Kinetics and Process Conditions . 88 -- 4.2 Lumped Simulation of Droplet Polymerisation 92 -- 4.2.1 Principle Course of the Process - -- Plain Kinetics, no Monomer Evaporation 92 -- 4.2.2 Effects of Kinetics on the Process 94 -- 4.3 Spatial Effects in Droplet Polymerisation 98 -- 4.3.1 Effect of the Diffusion Coefficient on Concentration Gradients . 99 -- 4.3.2 Inhomogeneities of the Product at Small Diffusion Coefficients, Effect of Moments' Diffusion . 102 -- 4.3.3 Effect of Monomer Evaporation . 106 -- 4.3.4 Pre-polymerisation Before Atomisation . 111 -- 4.3.5 Polymerisation at Elevated Monomer Content in the Drying Gas 116 -- 4.3.6 Influence of Non-Ideality of Activities 122 -- 4.3.7 Interaction with the drying gas125 -- 4.3.8 Applicability of the QSSA model 131 -- 4.4 Summary of Basic Findings on Droplet Polymerisation . 133 -- 4.5 Process Evaluation, Numerical DoEs. 135 -- 4.5.1 DoEs' Setup and Evaluation. 135 -- 4.5.2 Droplet Polymerisation with Solvent in the Feed 138 -- 4.5.3 Bulk Polymerisation within a Droplet 142 -- 4.5.4 Bulk Feed with Pre-Polymerisation before Atomisation . 149 -- 4.6 Discussion and Suggestions for Further Research 153 -- 5 SPH and its Application to Single Droplet Slurry Drying 155 -- 5.1 Mathematical Derivation . 156 -- 5.1.1 SPH Interpolation 156 -- 5.1.2 Integral Approximations . 159 -- 5.1.3 First Derivatives . 160 -- 5.1.4 Laplace-Operator and Divergence of Diffusive Fluxes 162 -- 5.1.5 General Second Derivatives. 165 -- 5.1.6 Choice of Kernel, Smoothing Length and Cut-off Radius 165 -- 5.1.7 Correction of the SPH Approximation 168 -- 5.2 Implementation of Boundary Conditions . 169 -- 5.2.1 Ghost Particles169 -- 5.2.2 Insertion of Boundary Conditions into SPH Equations . 171 -- 5.2.3 Repulsive Forces as Hard Sphere Boundaries172 -- 5.3 Hydrodynamics of an Incompressible Liquid in SPH173 -- 5.3.1 Continuity Equation, Density Evaluation 173 -- 5.3.2 Momentum Balance . 175 -- 5.3.3 Weakly Compressible SPH 178 -- 5.4 Incompressible SPH 179 -- 5.4.1 Boundary Conditions in ISPH182 -- 5.4.2 Boundaries by the Ghost Technique, Wall Boundaries 182 -- 5.4.3 Free Surface Boundaries in ISPH 183 -- 5.4.4 Modifications to ISPH in This Work . 184 -- 5.5 Surface Tension and Wetting . 189 -- 5.5.1 The Interparticle Force Approach 190 -- 5.5.2 The Concept of Surface-Lateral Particle Forces . 195 -- 5.6 Representation of the Solid Phase 200 -- 5.6.1 Primary Particles in the Slurry200 -- 5.6.2 Calculation of Crust Formation . 202 -- 5.7 Modelling of Drying Phenomena in SPH . 205 -- 5.7.1 Heat Conduction . 205 -- 5.7.2 Implementation of Linear Driving Force based Heat and -- Mass Transfer into SPH . 205 -- 5.7.3 Extension to the Second Drying Period . 208 -- 5.7.4 Treatment of Evaporation Concerning Particle Mass and -- Deletion 209 -- 5.7.5 Modelling of Diffusion Driven Drying Involving the Gas -- Phase . 210 -- 5.8 Time Integration 213 -- 5.8.1 Stability Criteria in Explicit Time Stepping. 214 -- 5.8.2 Time Stepping Criteria Employed in This Work and Their -- Reference Length . 216 -- 5.8.3 Implicit Solution of Diffusive Equations . 218 -- 5.8.4 Initialisation of an SPH Calculation . 220 -- 6 Validation of the SPH Implementation 221 -- 6.1 Implicit Solution of Heat Conduction. 221 -- 6.2 Heat and Mass Transfer by Linear Driving Forces 224 -- 6.2.1 Heat Transfer to a Unilaterally Heated Rod. 224 -- 6.2.2 Coupled Heat and Mass Transfer: Droplet Evaporation . 226 -- 6.3 Diffusion Driven Drying by SPH-Grid Coupling . 228 -- 6.4 SPH Flow Solver . 230 -- 6.4.1 ISPH Solution of a Standing Water Column. 230 -- 6.4.2 Free Surface Flow 231 -- 6.4.3 Surface Tension Approach of Pairwise Forces234 -- 6.4.4 Wetting Phenomena . 237 -- 7 Simulation of Structure Evolution During Drying 243 -- 7.1 Simulation of the First Drying Period243 -- 7.2 Simulation of Crust Formation 245 -- 7.2.1 Simulation of the Second Drying Period without Crust -- Formation . 245 -- 7.2.2 Crust Formation by Caught on First Touch. 247 -- 7.2.3 Crust Formation Determined by the Water Content . 249 -- 7.2.4 Effect of the Density Correction . 250 -- 7.3 Influence of Adjustable Parameters on the Structure. 251 -- 7.4 Effect of the Temperature . 256 -- 7.5 Variation of the Resolution 259 -- 7.6 Drying of a Microporous Structure 261 -- 7.7 Comments on numerical efficiency 267 -- 8 Conclusion 269 -- A Numerical Regression by Gaussian Processes 273 -- B FVM Implementation of the Droplet Polymerisation Model 277 -- C Implementational Aspects of SPH 281 -- C.1 Neighbourhood Search281 -- C.1.1 Linked List 281 -- C.1.2 Verlet List . 282 -- C.2 Performance Aspects, Memory Alignment 282List of Symbols xv -- List of Figures xxi -- List of Tables xxv -- 1 Introduction: Spray Drying and Reactive Drying Processes 1 -- 1.1 Spray Polymerisation. 2 -- 1.2 Single Droplet Models for Spray Drying . 3 -- 1.3 Meshfree Methods, Simulation of Structure Evolution4 -- 1.3.1 Previous Applications of SPH to Drying . 5 -- 2 Theoretical Principles 7 -- 2.1 Transport Equations 7 -- 2.1.1 Transport in a Mass Averaged System 9 -- 2.1.2 Transport in a Molar Averaged System .
13 -- 2.1.3 Reference Velocities and Conversion between Systems . 13 -- 2.1.4 Eulerian and Lagrangian Frames of Reference16 -- 2.2 Diffusion . 17 -- 2.2.1 Fickian Diffusion . 17 -- 2.2.2 Maxwell-Stefan Diffusion 19 -- 2.2.3 Determination of Diffusion Coefficients . 21 -- 2.3 Modelling of Free Radical Polymerisation 21 -- 2.3.1 Reactions in Free Radical Polymerisation 22 -- 2.3.2 Quasi-Steady-State Assumption (QSSA) 27 -- 2.3.3 Method of Moments .
28 -- 2.4 Mixture Thermodynamics 31 -- 2.4.1 Vapour Liquid Equilibrium at the Droplet's Surface . 32 -- 2.4.2 Calculation of Activity Coefficients . 33 -- 2.4.3 The UNIFAC Equations . 34 -- 2.5 Spray Drying: Basic Assumptions 35 -- 2.5.1 Approximate Residence Time in a Spray Dryer . 36 -- 2.5.2 Heat and Mass Transfer . 37 -- 2.5.3 Inner Circulation Inside a Droplet 38 -- 2.5.4 Are Droplets Fully Mixed? 40 -- 3 Modelling of Reactive Droplet Drying and Polymerisation 41 -- 3.1 Transport in a Reaction-Diffusion System 42 -- 3.1.1 Constant Physical Properties. 44 -- 3.1.2 Consideration of Mixture Effects . 46 -- 3.1.3 Diffusion and Reaction Driven Convection at Variable -- Molar Weights47 -- 3.1.4 Transport of Polymer - Quasi-Steady-State Assumption . 50 -- 3.1.5 Transport of Statistical Moments . 52 -- 3.2 Lumped Modelling - 0D approach 56 -- 3.2.1 General Equations for Reactive Spray Drying57 -- 3.2.2 Spray Polymerisation - Quasi-Steady-State Assumption 58 -- 3.2.3 Spray Polymerisation - Method of Moments. 59 -- 3.3 Distributed Modelling - 1D approach. 60 -- 3.3.1 General Equations of the Droplet Continuum60 -- 3.3.2 Boundary Conditions . 62 -- 3.3.3 Spray Polymerisation - QSSA66 -- 3.3.4 Spray Polymerisation - Method of Moments. 68 -- 3.4 Comparison with Existing Models 69 -- 3.5 Implementational Considerations . 70 -- 3.5.1 Implementation of the Moving Boundary Problem . 70 -- 3.5.2 Boundary Conditions . 71 -- 3.5.3 Treatment of Convection Terms . 72 -- 3.5.4 Implementation of Diffusion. 75 -- 3.6 Verification of the Transport Approach76 -- 3.6.1 Diffusion Driven Convection, Constant Properties 76 -- 3.6.2 Diffusion Driven Convection, Variable Molar Weight 79 -- 3.6.3 Diffusion Driven Convection, Excess Volumes . 82 -- 3.6.4 Reaction Induced Convection85 -- 4 Simulation of Spray Polymerisation 87 -- 4.1 Kinetics and Process Conditions . 88 -- 4.2 Lumped Simulation of Droplet Polymerisation 92 -- 4.2.1 Principle Course of the Process - -- Plain Kinetics, no Monomer Evaporation 92 -- 4.2.2 Effects of Kinetics on the Process 94 -- 4.3 Spatial Effects in Droplet Polymerisation 98 -- 4.3.1 Effect of the Diffusion Coefficient on Concentration Gradients . 99 -- 4.3.2 Inhomogeneities of the Product at Small Diffusion Coefficients, Effect of Moments' Diffusion . 102 -- 4.3.3 Effect of Monomer Evaporation . 106 -- 4.3.4 Pre-polymerisation Before Atomisation . 111 -- 4.3.5 Polymerisation at Elevated Monomer Content in the Drying Gas 116 -- 4.3.6 Influence of Non-Ideality of Activities 122 -- 4.3.7 Interaction with the drying gas125 -- 4.3.8 Applicability of the QSSA model 131 -- 4.4 Summary of Basic Findings on Droplet Polymerisation . 133 -- 4.5 Process Evaluation, Numerical DoEs. 135 -- 4.5.1 DoEs' Setup and Evaluation. 135 -- 4.5.2 Droplet Polymerisation with Solvent in the Feed 138 -- 4.5.3 Bulk Polymerisation within a Droplet 142 -- 4.5.4 Bulk Feed with Pre-Polymerisation before Atomisation . 149 -- 4.6 Discussion and Suggestions for Further Research 153 -- 5 SPH and its Application to Single Droplet Slurry Drying 155 -- 5.1 Mathematical Derivation . 156 -- 5.1.1 SPH Interpolation 156 -- 5.1.2 Integral Approximations . 159 -- 5.1.3 First Derivatives . 160 -- 5.1.4 Laplace-Operator and Divergence of Diffusive Fluxes 162 -- 5.1.5 General Second Derivatives. 165 -- 5.1.6 Choice of Kernel, Smoothing Length and Cut-off Radius 165 -- 5.1.7 Correction of the SPH Approximation 168 -- 5.2 Implementation of Boundary Conditions . 169 -- 5.2.1 Ghost Particles169 -- 5.2.2 Insertion of Boundary Conditions into SPH Equations . 171 -- 5.2.3 Repulsive Forces as Hard Sphere Boundaries172 -- 5.3 Hydrodynamics of an Incompressible Liquid in SPH173 -- 5.3.1 Continuity Equation, Density Evaluation 173 -- 5.3.2 Momentum Balance . 175 -- 5.3.3 Weakly Compressible SPH 178 -- 5.4 Incompressible SPH 179 -- 5.4.1 Boundary Conditions in ISPH182 -- 5.4.2 Boundaries by the Ghost Technique, Wall Boundaries 182 -- 5.4.3 Free Surface Boundaries in ISPH 183 -- 5.4.4 Modifications to ISPH in This Work . 184 -- 5.5 Surface Tension and Wetting . 189 -- 5.5.1 The Interparticle Force Approach 190 -- 5.5.2 The Concept of Surface-Lateral Particle Forces . 195 -- 5.6 Representation of the Solid Phase 200 -- 5.6.1 Primary Particles in the Slurry200 -- 5.6.2 Calculation of Crust Formation . 202 -- 5.7 Modelling of Drying Phenomena in SPH . 205 -- 5.7.1 Heat Conduction . 205 -- 5.7.2 Implementation of Linear Driving Force based Heat and -- Mass Transfer into SPH . 205 -- 5.7.3 Extension to the Second Drying Period . 208 -- 5.7.4 Treatment of Evaporation Concerning Particle Mass and -- Deletion 209 -- 5.7.5 Modelling of Diffusion Driven Drying Involving the Gas -- Phase . 210 -- 5.8 Time Integration 213 -- 5.8.1 Stability Criteria in Explicit Time Stepping. 214 -- 5.8.2 Time Stepping Criteria Employed in This Work and Their -- Reference Length . 216 -- 5.8.3 Implicit Solution of Diffusive Equations . 218 -- 5.8.4 Initialisation of an SPH Calculation . 220 -- 6 Validation of the SPH Implementation 221 -- 6.1 Implicit Solution of Heat Conduction. 221 -- 6.2 Heat and Mass Transfer by Linear Driving Forces 224 -- 6.2.1 Heat Transfer to a Unilaterally Heated Rod. 224 -- 6.2.2 Coupled Heat and Mass Transfer: Droplet Evaporation . 226 -- 6.3 Diffusion Driven Drying by SPH-Grid Coupling . 228 -- 6.4 SPH Flow Solver . 230 -- 6.4.1 ISPH Solution of a Standing Water Column. 230 -- 6.4.2 Free Surface Flow 231 -- 6.4.3 Surface Tension Approach of Pairwise Forces234 -- 6.4.4 Wetting Phenomena . 237 -- 7 Simulation of Structure Evolution During Drying 243 -- 7.1 Simulation of the First Drying Period243 -- 7.2 Simulation of Crust Formation 245 -- 7.2.1 Simulation of the Second Drying Period without Crust -- Formation . 245 -- 7.2.2 Crust Formation by Caught on First Touch. 247 -- 7.2.3 Crust Formation Determined by the Water Content . 249 -- 7.2.4 Effect of the Density Correction . 250 -- 7.3 Influence of Adjustable Parameters on the Structure. 251 -- 7.4 Effect of the Temperature . 256 -- 7.5 Variation of the Resolution 259 -- 7.6 Drying of a Microporous Structure 261 -- 7.7 Comments on numerical efficiency 267 -- 8 Conclusion 269 -- A Numerical Regression by Gaussian Processes 273 -- B FVM Implementation of the Droplet Polymerisation Model 277 -- C Implementational Aspects of SPH 281 -- C.1 Neighbourhood Search281 -- C.1.1 Linked List 281 -- C.1.2 Verlet List . 282 -- C.2 Performance Aspects, Memory Alignment 282.
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Computational fluid dynamics simulation of spray dryers : an engineer's guide / / Meng Wai Woo
