Droplets and Sprays: Simple Models of Complex Processes (Mathematical Engineering)

This book acts as a guide to simple models that describe some of the complex fluid dynamics, heat/mass transfer and combustion processes in droplets and sprays. Attention is focused mainly on the use of classical hydrodynamics, and a combination of kinetic and hydrodynamic models, to analyse the heating and evaporation of mono- and multi-component droplets. The models were developed for cases when small and large numbers of components are present in droplets. Some of these models are used for the prediction of time to puffing/micro-explosion of composite water/fuel droplets — processes that are widely used in combustion devices to stimulate disintegration of relatively large droplets into smaller ones. The predictions of numerical codes based on these models are validated against experimental results where possible. In most of the models, droplets are assumed to be spherical; some preliminary results of the generalisation of these models to the case of non-spherical droplets, approximating them as spheroids, are presented. Preface Mathematical and Engineering Modelling Scope of the Book Topics and Assumptions References Contents 1 Spray Formation and Penetration 1.1 Spray Formation 1.1.1 Classical WAVE Model 1.1.2 TAB and Stochastic Models 1.1.3 Modified WAVE Models 1.2 Spray Penetration 1.2.1 The Initial Stage 1.2.2 Two-Phase Flow 1.2.3 Effects of Turbulence 1.3 Vortex Ring-Like Structures in Sprays 1.3.1 Conventional Vortex Rings 1.3.2 Turbulent Vortex Rings 1.3.3 Translational Velocities of Vortex Rings-Like Structures 1.3.4 Confined Vortex Rings 1.3.5 Two-Phase Vortex Ring Flows References 2 Heating of Non-evaporating Droplets 2.1 Convective Heating 2.1.1 Stagnant Droplets 2.1.2 Moving Droplets 2.2 Radiative Heating 2.2.1 Basic Equations and Approximations 2.2.2 Mie Theory 2.2.3 Integral Absorption of Radiation in Droplets 2.2.4 Geometric Optics Analysis References 3 Heating and Evaporation of Mono-component Droplets 3.1 Empirical Correlations 3.2 Classical Models 3.2.1 Maxwell and Stefan–Fuchs Models 3.2.2 Abramzon and Sirignano Model 3.2.3 Yao, Abdel-Khalik and Ghiaasiaan Model 3.2.4 Tonini and Cossali Model 3.2.5 Fully Transient Models 3.2.6 d2 and d1.5 Laws 3.3 Effects of Real Gases 3.4 Effects of the Moving Interface 3.4.1 Basic Equations and Approximations 3.4.2 Solution for the Case when Rd(t) is a Linear Function 3.4.3 Solution for the Case of Arbitrary Rd(t) but Td0(R)=const 3.4.4 Solution for Arbitrary Rd(t) and Td0(R) 3.4.5 A Comparison between Model Predictions 3.5 Conventional and Alternative Approaches to Modelling 3.6 Heating and Evaporation of Spheroidal Droplets 3.6.1 Background Research: Non-spherical Droplets 3.6.2 The Tonini and Cossali Model (Spheroidal Droplets) 3.6.3 The Coupled Liquid/Gas Model (Spheroidal Droplets) 3.6.4 Miscellaneous Models 3.7 Effect of Droplet Support 3.8 Modelling Versus Experimental Data 3.8.1 Monodisperse Droplet Stream 3.8.2 Suspended Droplets References 4 Heating and Evaporation of Multi-component Droplets 4.1 Background 4.2 Discrete Component Model 4.2.1 An Analytical Solution for Rd = const 4.2.2 An Analytical Solution for Rd neq const 4.2.3 Bi-component Droplets 4.2.4 Biodiesel Droplets 4.2.5 Kerosene Droplets 4.2.6 Drying Droplets 4.3 Quasi-discrete Model 4.3.1 Description of the Quasi-discrete Model 4.3.2 Application to Diesel and Petrol Fuel Droplets 4.4 Multi-dimensional Quasi-discrete Model 4.4.1 Description of the Model 4.4.2 Application to Diesel Fuel Droplets 4.4.3 Application to Petrol Fuel Droplets 4.4.4 Heating, Evaporation and Ignition of Fuel Droplets 4.4.5 Biodiesel/Diesel/Ethanol/Petrol Droplets 4.4.6 Auto-selection of Quasi-components/Components 4.5 Gas Phase Models for Multi-component Droplets 4.6 Other Approaches to Modelling Multi-component Droplets 4.7 Heating and Evaporation of Multi-component Liquid Films 4.7.1 Mono-component Liquid Film 4.7.2 Multi-component Liquid Film 4.7.3 Solution Algorithm 4.7.4 Validation of the Model 4.7.5 Verification of the Model References 5 Processes in Composite Droplets 5.1 Background 5.2 A Simple Analytical Model 5.2.1 Basic Equations and Approximations 5.2.2 Analysis 5.3 A Simple Numerical Model 5.3.1 Key Equations and Approximations 5.3.2 Preliminary Analysis 5.3.3 Boiling Versus Nucleation Temperature 5.3.4 Times to Puffing/Micro-Explosion 5.4 Puffing/Micro-Explosion in the Presence of Coal Particles 5.4.1 Rapeseed Oil/Water Droplets with Coal Micro-Particles 5.4.2 Modelling Versus Experimental Results 5.5 Puffing/Micro-Explosion in Closely Spaced Droplets 5.5.1 Puffing/Micro-Explosion in Two Droplets in Tandem 5.5.2 Puffing/Micro-Explosion in a String of Three Droplets 5.6 Effects of Thermal Radiation and Support 5.7 Composite Multi-component Droplets 5.7.1 Diffusion of Components 5.7.2 Modelling and Experimental Results 5.8 The Shift Model References 6 Kinetic Modelling of Droplet Heating and Evaporation 6.1 Early Results 6.2 Kinetic Algorithm: Effects of the Heat and Mass Fluxes 6.2.1 Boltzmann Equations for the Kinetic Region 6.2.2 Vapour Density and Temperature at the Boundaries 6.3 Approximations of the Kinetic Results 6.3.1 Approximations for Chosen Gas Temperatures 6.3.2 Approximations for Chosen Initial Droplet Radii 6.4 Effects of Inelastic Collisions 6.4.1 Mathematical Model 6.4.2 Solution Algorithm 6.5 Kinetic Boundary Condition 6.5.1 Molecular Dynamics Simulations (Background) 6.5.2 United Atom Model 6.5.3 Evaporation Coefficient 6.6 Quantum-Chemical Models 6.6.1 Brief Overview of Quantum-Chemical Methods 6.6.2 Evaporation Rate 6.6.3 Interaction between Molecules and Clusters/Nanodroplets 6.6.4 Estimation of the Evaporation Coefficient 6.7 Results of the Kinetic Calculations 6.7.1 Results for βm=1 6.7.2 Results for βm