Charge injection systems : phycical principles, experimental and theoretical work

C Specific heat at constant pressure p D Displacement field D Diffusion coefficient d D Orifice diameter E Electric field E Electron charge F Force G Acceleration due to gravity I Current J Current flux K Conductivity k Boltzmann constant B L Atomizer geometry: length from electrode tip to orifice plane i L Atomizer geometry : length of orifice channel o P Polarization Q Flow rate/Heat flux Q Charge r Atomizer geometry : electrode tip radius p T Time T Temperature U Velocity V Voltage W Energy X Distance Nomenclature (Greek) Thermal expansion coefficient ? Permittivity ? Permutation operator ? ijk Ion mobility ? VI Nomenclature Debye length ? D ? Dynamic viscosity ? Mass density Surface tension ? T Electrical conductivity ? ? Timescale ? Vorticity Nomenclature (Subscripts) Reference state ? o Cartesian tensor notation ? ijk Volume density (? per unit volume) ? v Surface density (? per unit area) ? s Linear density (? per unit length) ? l 'critical' state ? c Bulk mean injection ? inj Nomenclature (Superscripts) Time or ensemble averaged ? Contents Contents 1 Introduction................................................................... 1 1.1 Introduction and Scope.................................................. 1 1.2 Organization.............................................................. 3 2 Electrostatics, Electrohydrodynamic Flow, Coupling and Instability.................................................................. 5 2.1 Electrostatics.............................................................. 5 2.1.1 The Coulomb Force............................................. 5 2.1.2 Permittivity...................................................... 6 2.1.3 Conductors, Insulators, Dielectrics and Polarization........ 6 2.1.4 Gauss'sLaw...................................................... 8 2.2 Mobility and Charge Transport........................................ 10 2.2.1 Introduction...................................................... 10

1 Introduction	p. 1
1.1 Introduction and Scope	p. 1
1.2 Organization	p. 3
2 Electrostatics, Electrohydrodynamic Flow, Coupling and Instability	p. 5
2.1 Electrostatics	p. 5
2.1.1 The Coulomb Force	p. 5
2.1.2 Permittivity	p. 6
2.1.3 Conductors, Insulators, Dielectrics and Polarization	p. 6
2.1.4 Gauss's Law	p. 8
2.2 Mobility and Charge Transport	p. 10
2.2.1 Introduction	p. 10
2.2.2 Convective Transport by Fluid Motion	p. 10
2.2.3 Mobility and the Drift Term	p. 10
2.2.4 Diffusion and the Debye Length	p. 11
2.2.5 Charge Conservation	p. 12
2.3 Momentum and Energy	p. 13
2.3.1 Introduction	p. 13
2.3.2 Electrical Forces	p. 13
2.3.3 Momentum Conservation	p. 14
2.3.4 Energy Conservation	p. 14
2.4 Electrical Timescales	p. 15
2.4.1 Introduction	p. 15
2.4.2 Ohmic-Charge Relaxation	p. 15
2.4.3 Space-Charge Relaxation	p. 15
2.4.4 Ionic Diffusion Timescale	p. 16
2.4.5 Ionic Transit Timescale	p. 16
2.4.6 Electro-viscous Timescale	p. 17
2.4.7 Electro-inertial Timescale	p. 17
2.5 Non-dimensional Transport Equations	p. 17
2.5.1 Introduction	p. 17
2.5.2 Momentum Conservation: Free Flow	p. 18
2.5.3 Momentum Conservation: Forced Flow	p. 19
2.5.4 Non-dimensional Parameters	p. 19
2.6 Electrohydrodynamics	p. 21
2.6.1 Introduction	p. 21
2.6.2 Fundamentals	p. 22
2.6.3 Instability	p. 22
2.6.4 Plumes	p. 26
2.6.5 Transition to Turbulence	p. 29
2.7 Electrohydrodynamic Turbulence in a Propagating Flow Front	p. 30
2.7.1 EHD Vorticity	p. 30
2.7.2 EHD RANS in a Propagating Flow Front	p. 31
2.7.3 Transient Turbulence	p. 32
2.7.4 AC Turbulence	p. 33
2.7.5 Current and Voltage	p. 33
2.8 Chapter Summary	p. 35
3 Charge Injection into a Quiescent Dielectric Liquid	p. 37
3.1 Charge and Field Distribution	p. 37
3.1.1 Field Emission and Ionization	p. 37
3.1.2 Electrochemical	p. 38
3.1.3 Ohmic Conduction	p. 39
3.1.4 Space-Charge	p. 39
3.1.5 Point Sharpness	p. 39
3.1.6 Hyperbolic Field Expression	p. 40
3.2 IV Characteristics of Point-Plane Systems	p. 40
3.2.1 Steady-State Behavior	p. 40
3.2.2 Current Instabilities	p. 44
3.3 Vapor Bubble Creation and Pressure Dependence in Liquids	p. 49
3.3.1 Vapor Bubble Formation	p. 49
3.3.2 Vapor Bubble Growth: Pulsed Voltage Operation	p. 51
3.3.3 Vapor Bubble Growth: Constant Voltage	p. 52
3.3.4 Vapor Bubble Evolution	p. 58
3.4 Chapter Summary	p. 60
4 Single Charged Drop Stability, Evaporation and Combustion	p. 61
4.1 Maximum Spherical Drop Charge	p. 61
4.2 Maximum Spheroidal Drop Charge	p. 69
4.3 Spheroidal Deformation of Non-stationary Charged Drops	p. 70
4.4 Models for Products of Charged Drop Disruption	p. 72
4.5 Combustion of Single Drops	p. 76
4.6 Summary	p. 77
5 Charge Injection Atomizers: Design and Electrical Performance	p. 79
5.1 Overview: Electrostatic Atomization for Electrically Semi-conducting Liquids	p. 79
5.2 Overview: Electrostatic Atomization for Electrically Insulating Liquids	p. 81
5.3 Atomizer Construction	p. 82
5.4 Nozzle Design	p. 84
5.5 Rig Design	p. 85
5.6 Liquids Used	p. 86
5.7 Breakdown Limits and Typical Current-Voltage Response	p. 87
5.7.1 Sub-critical Breakdown	p. 87
5.7.2 Super-critical Breakdown	p. 91
5.7.3 Overview of the Breakdown Regimes	p. 94
5.8 Total Current Versus Voltage: Observations	p. 94
5.9 Total Current Versus Voltage: Comparison to Quiescent Fluid Data	p. 96
5.10 Effect of Flow-Rate/Injection Velocity	p. 100
5.11 Specific Charge Regimes	p. 101
5.12 Specific Charge: Summary	p. 106
5.13 Variation of Electrode Gap Ratio (L i /d), L 0 /d=2, d=500¿m, Version 1 Design	p. 107
5.14 Variation of d: Version 1 Design: Constant Q, L i , L 0 /d	p. 110
5.15 Variation of Electrode Gap Ratio (L i /d): Version 2 Design, d=500¿m	p. 112
5.16 Variation of Electrode Gap Ratio (L i /d): Version 2 Design, d=250¿m	p. 114
5.17 Performance Evaluation: Version 1 and Version 2	p. 116
5.18 Point-Plane Atomizer Design Modifications	p. 117
5.19 Beyond the Point-Plane Atomizer Concept	p. 121
5.19.1 Single Hole Electrostatically Enhanced Pressure Swirl Atomizers	p. 121
5.19.2 Multi-hole Charge Injection Atomizers	p. 122
5.19.3 Pulsed Spray Charge Injection Atomizers	p. 122
5.19.4 Other Developments within Charge Injection Atomization	p. 123
5.20 Chapter Summary	p. 123
6 Jet Instability and Primary Atomization	p. 125
6.1 Measured Characteristics	p. 125
6.2 Orifice Channel Space Charge Distribution Model	p. 132
6.3 Chapter Summary	p. 137
7 Spray Characterization and Combusion	p. 139
7.1 Spray Visualization and Prediction of Expansion Rate	p. 139
7.2 Quantitative Spray Characteristics	p. 146
7.3 Estimation of the Radial Profile of Spray Specific Charge	p. 154
7.4 Models for Drop Diameter and Charge Distributions	p. 160
7.4.1 Energy Minimization Methods	p. 160
7.4.2 Spray Theory of Kelly	p. 163
7.4.2.1 Correlations and Simplifications	p. 167
7.2.4.2 Analysis of the Lagrangian Multipliers	p. 169
7.4.2.3 Energy Considerations	p. 172
7.4.2.4 Performance of Kelly's Model	p. 172
7.5 Spray Combustion	p. 173
7.6 Summary	p. 178
8 Conclusions and Future Outlook	p. 181
8.1 Conclusions	p. 181
8.2 Future Outlook	p. 183
References	p. 185
Index	p. 195

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