Heat and Mass Transfer
Editorial Board D. Mewes F. Mayinger
“This page left intentionally blank.”
John Shrimpton
Charge Injection Systems Physical Principles, Experimental and Theoretical Work
ABC
Dr. John Shrimpton University of Southampton School of Engineering Sciences Highfield Southampton United Kingdom SO17 1BJ E-mail:
[email protected]
ISBN 978-3-642-00293-9
e-ISBN 978-3-642-00294-6
DOI 10.1007/978-3-642-00294-6 Heat and Mass Transfer
ISSN 1860-4846
Library of Congress Control Number: 2009921600 c 2009 Springer-Verlag Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typeset & Coverdesign: Scientific Publishing Services Pvt. Ltd., Chennai, India. Printed in acid-free paper 987654321 springer.com
Nomenclature
Preface
Cp D Dd D E E F G I J K kB Li Lo P Q Q rp T T U V W X
Specific heat at constant pressure Displacement field Diffusion coefficient Orifice diameter Electric field Electron charge Force Acceleration due to gravity Current Current flux Conductivity Boltzmann constant Atomizer geometry: length from electrode tip to orifice plane Atomizer geometry : length of orifice channel Polarization Flow rate/Heat flux Charge Atomizer geometry : electrode tip radius Time Temperature Velocity Voltage Energy Distance
Nomenclature (Greek)
β ε
ε
κ
Thermal expansion coefficient Permittivity Permutation operator
ijk
Ion mobility
VI
λD μ ρ σT σ τ ω
Nomenclature
Debye length Dynamic viscosity Mass density Surface tension Electrical conductivity Timescale Vorticity
Nomenclature (Subscripts)
ϕo ϕijk ϕv ϕs ϕl
ϕc ϕinj
Reference state Cartesian tensor notation
ϕ per unit volume) Surface density ( ϕ per unit area) Linear density ( ϕ per unit length) Volume density (
‘critical’ state Bulk mean injection
Nomenclature (Superscripts)
ϕ
Time or ensemble averaged
Contents Contents
1
Introduction…………………………………………………………. 1.1 Introduction and Scope………………………………………….. 1.2 Organization……………………………………………………..
1 1 3
2
Electrostatics, Electrohydrodynamic Flow, Coupling and Instability………………………………………………………... 2.1 Electrostatics…………………………………………………….. 2.1.1 The Coulomb Force……………………………………… 2.1.2 Permittivity……………………………………………… 2.1.3 Conductors, Insulators, Dielectrics and Polarization…….. 2.1.4 Gauss’s Law……………………………………………... 2.2 Mobility and Charge Transport…………………………………. 2.2.1 Introduction……………………………………………… 2.2.2 Convective Transport by Fluid Motion………………….. 2.2.3 Mobility and the Drift Term……………………………... 2.2.4 Diffusion and the Debye Length………………………… 2.2.5 Charge Conservation…………………………………….. 2.3 Momentum and Energy………………………………………… 2.3.1 Introduction……………………………………………… 2.3.2 Electrical Forces…………………………………………. 2.3.3 Momentum Conservation………………………………… 2.3.4 Energy Conservation…………………………………….. 2.4 Electrical Timescales…………………………………………… 2.4.1 Introduction……………………………………………… 2.4.2 Ohmic-Charge Relaxation……………………………….. 2.4.3 Space-Charge Relaxation………………………………... 2.4.4 Ionic Diffusion Timescale……………………………….. 2.4.5 Ionic Transit Timescale………………………………….. 2.4.6 Electro-viscous Timescale………………………………..
5 5 5 6 6 8 10 10 10 10 11 12 13 13 13 14 14 15 15 15 15 16 16 17
VIII
3
Contents
2.4.7 Electro-inertial Timescale……………………………….. 2.5 Non-dimensional Transport Equations…………………………. 2.5.1 Introduction……………………………………………… 2.5.2 Momentum Conservation: Free Flow…………………… 2.5.3 Momentum Conservation: Forced Flow………………… 2.5.4 Non-dimensional Parameters……………………………. 2.6 Electrohydrodynamics…………………………………….. 2.6.1 Introduction……………………………………………… 2.6.2 Fundamentals……………………………………………. 2.6.3 Instability…………………………………………………. 2.6.4 Plumes…………………………………………………… 2.6.5 Transition to Turbulence………………………………… 2.7 Electrohydrodynamic Turbulence in a Propagating Flow Front… 2.7.1 EHD Vorticity……………………………………………. 2.7.2 EHD RANS in a Propagating Flow Front……………….. 2.7.3 Transient Turbulence……………………………………. 2.7.4 AC Turbulence……………………………………............ 2.7.5 Current and Voltage……………………………………..... 2.8 Chapter Summary………………………………………………..
17 17 17 18 19 19 21 21 22 22 26 29 30 30 31 32 33 33 35
Charge Injection into a Quiescent Dielectric Liquid……………… 3.1 Charge and Field Distribution…………………………………... 3.1.1 Field Emission and Ionization…………………………… 3.1.2 Electrochemical………………………………….............. 3.1.3 Ohmic Conduction…………………………………......... 3.1.4 Space-Charge…………………………………................. 3.1.5 Point Sharpness………………………………….............. 3.1.6 Hyperbolic Field Expression…………………………….. 3.2 IV Characteristics of Point-Plane Systems……………………. 3.2.1 Steady-State Behavior………………………………….... 3.2.2 Current Instabilities…………………………………........ 3.3 Vapor Bubble Creation and Pressure Dependence in Liquids….. 3.3.1 Vapor Bubble Formation………………………………… 3.3.2 Vapor Bubble Growth: Pulsed Voltage Operation………. 3.3.3 Vapor Bubble Growth: Constant Voltage……………….. 3.3.4 Vapor Bubble Evolution………………………………… 3.4 Chapter Summary………………………………….....................
37 37 37 38 39 39 39 40 40 40 44 49 49 51 52 58 60
Contents
4
Single Charged Drop Stability, Evaporation and Combustion…... 4.1 Maximum Spherical Drop Charge…………………………….... 4.2 Maximum Spheroidal Drop Charge…………………………….. 4.3 Spheroidal Deformation of Non-stationary Charged Drops……. 4.4 Models for Products of Charged Drop Disruption…………….... 4.5 Combustion of Single Drops………………………………......... 4.6 Summary……………………………….......................................
5
Charge Injection Atomizers: Design and Electrical Performance………………………………………………………… 5.1 Overview: Electrostatic Atomization for Electrically Semi-conducting Liquids………………………………………… 5.2 Overview: Electrostatic Atomization for Electrically Insulating Liquids…………………………………………………………... 5.3 Atomizer Construction………………………………………….. 5.4 Nozzle Design…………………………………………………... 5.5 Rig Design………………………………………………………. 5.6 Liquids Used……………………………………………………. 5.7 Breakdown Limits and Typical Current-Voltage Response……. 5.7.1 Sub-critical Breakdown…………………………………. 5.7.2 Super-critical Breakdown……………………………….. 5.7.3 Overview of the Breakdown Regimes…………………… 5.8 Total Current Versus Voltage: Observations…………………… 5.9 Total Current Versus Voltage: Comparison to Quiescent Fluid Data……………………………………………………….. 5.10 Effect of Flow-Rate/Injection Velocity………………………... 5.11 Specific Charge Regimes……………………………………… 5.12 Specific Charge: Summary…………………………………… 5.13 Variation of Electrode Gap Ratio (Li/d), L0/d=2, d=500μm, Version 1 Design………………………………………………. 5.14 Variation of d: Version 1 Design: Constant Q, Li, L0/d………. 5.15 Variation of Electrode Gap Ratio (Li/d): Version 2 Design, d=500μm………………………………………………………. 5.16 Variation of Electrode Gap Ratio (Li/d): Version 2 Design, d=250μm………………………………………………………. 5.17 Performance Evaluation: Version 1 and Version 2……………… 5.18 Point-Plane Atomizer Design Modifications………………….. 5.19 Beyond the Point-Plane Atomizer Concept…………………… 5.19.1 Single Hole Electrostatically Enhanced Pressure Swirl Atomizers………………………………………………. 5.19.2 Multi-hole Charge Injection Atomizers………………..
IX
61 61 69 70 72 76 77
79 79 81 82 84 85 86 87 87 91 94 94 96 100 101 106 107 110 112 114 116 117 121 121 122
X
Contents
5.19.3 Pulsed Spray Charge Injection Atomizers……………… 5.19.4 Other Developments within Charge Injection Atomization…………………………………………….. 5.20 Chapter Summary………………………………………………
122 123 123
6
Jet Instability and Primary Atomization………………………….. 6.1 Measured Characteristics…………………….…………………. 6.2 Orifice Channel Space Charge Distribution Model…………….. 6.3 Chapter Summary……………..……………..………………….
125 125 132 137
7
Spray Characterization and Combustion…………………………. 7.1 Spray Visualization and Prediction of Expansion Rate………… 7.2 Quantitative Spray Characteristics……………………………… 7.3 Estimation of the Radial Profile of Spray Specific Charge…….. 7.4 Models for Drop Diameter and Charge Distributions………….. 7.4.1 Energy Minimization Methods………………………….. 7.4.2 Spray Theory of Kelly…………………………………… 7.4.2.1 Correlations and Simplifications……………….. 7.2.4.2 Analysis of the Lagrangian Multipliers………… 7.4.2.3 Energy Considerations…………………………. 7.4.2.4 Performance of Kelly’s Model…………………. 7.5 Spray Combustion………………………………………………. 7.6 Summary…………………………………………………………
139 139 146 154 160 160 163 167 169 172 172 173 178
8
Conclusions and Future Outlook…………………………………... 8.1 Conclusions……………………………………………………... 8.2 Future Outlook…………………………………………………...
181 181 183
References……………………………...…………………………….. Index…………………………………...……………………………..
185 195
Chapter 1
Introduction 1 Intro ductio n
Abstract. This monograph covers the literature and patents relevant to a specific type of liquid and a specific method of atomization. The liquids are dielectrics; poor electrical conductors, typically vegetable oils such as corn, soy and rapeseed, or petroleum products, such as petrol/gas, aviation fuel and Diesel oils. The liquids need not be ‘doped’ to enhance their electrical conductivity. The ‘charge injection’ atomizer injects electric charge into the poorly conducting liquid and the liquid atomization, spray dispersion and combustion are significantly influenced by the presence of the injected electric charge. The monograph initially covers electrohydrodynamic basics, fundamental studies of charge injection into quiescent liquid, and the design and operation of the atomizer itself. The review then continues by surveying studies of the primary atomization process, spray characterization, and effect on combustion before finally discussing measurements of the radial distribution of spray charge and modeling of the drop diameter and charge probability distribution. The review concludes that whilst some fundamental understanding still requires more research, sufficient knowledge exists to design and operate practical devices.
1.1 Introduction and Scope The creation and atomization of electrically charged insulating liquid jets and the dynamics of the charged sprays so produced are not subjects that have been widely reported in the literature. Such a technique, if widely available, could well be valuable in a number of applications, for instance in molten plastic production and a range of fuel spray combustion systems. The benefits of electrically charged fluid mechanics are well known and are used successfully in a range of industrial applications, such as production of nano-particles [1], to spray coating applications and flue gas treatments [2] and include; 1) 2) 3) 4)
Low drop concentration within the plume. Lack of drop agglomeration. Controllable and constrained particle size distribution. Controllable spray plume shape and deposition.
J. Shrimpton: Charge Inject. Sys.: Phy. Principles, Experi. & Theore. Work, HMT, pp. 1–4. springerlink.com © Springer-Verlag Berlin Heidelberg 2009
2
1 Introduction
These benefits, unique to electrically charged sprays, are also attractive for combustion applications, where a few large drops and regions of high drop concentration induce non-uniform combustion kinetics within a burner. This in turn leads to non-optimum combustion and production of unwanted products. Historically it has been problematic to extract charge from the charging electrode, a metallic conductor, into an electrically insulating liquid using an electrostatic atomization method. The traditional way to circumvent this problem has been to add small amounts of an additive that would reduce the electrical resistivity, from a high value of ~1010 Ωm, typical of an insulating liquid, to a more useful range, typically ~107 Ωm. The doped insulating liquid, from an electrical viewpoint now ‘semi-conducting’, permits the liquid to be atomized using standard contact charging techniques. For further information on this, and the induction charging of drops, the reader is referred to the work of Law [3]. For an explanation of the role of electrical conductivity in electrostatic atomization see Ganan-Calvo et al. [4]. A range of hydrocarbon oils have been sprayed by doping the fuel [5-7] and although the flow rates were very small, limited by the atomization method, combustion was readily achieved. Research has also suggested that the presence of electric charge on the drops that form the spray plume enhances the overall evaporation rate [8] and also may reduce the amount of soot produced [9] due to the increase of inter-particle separation. The use of corona is known to reduce NOx emissions of natural gas flames [10], and extraction of soot by applying an external field across a flame has been demonstrated by Lawton & Weinburg [11]. Work on charged spray combustion of hydrocarbon oils, using methods other than charge injection, have been confined to unrealistic, low flow rate studies of doped liquids, due to the inability of producing sprays of insulating liquids at practical flow rates. The 'charge injection' method, at the stage of development described here, allows hydrocarbon fuels such as kerosenes and gas oils to be atomized without the use of additives to alter the electrical resistivity, or additional atomization methods such as centrifugal force [12]. Commercial grade fuel oils, used ‘off the shelf’ with various species of dissolved polar contaminants, seem to be well suited to this form of atomization. A method which makes use of electric charge for the purpose of liquid atomization and spray dispersion and encourages preferential modification of combustion kinetics and also enhanced flue gas treatment, could thus have application in combustion systems. At the heart of the charge injection technique is a balance between hydrodynamic and electrical convection, both of which may generate flow instabilities, leading to turbulence. This requires co-design of the atomizer internal geometry from both hydrodynamic and electrical perspectives, the optimization of which leads to a maximum in the generated spray charge per unit volume contained by the spray. This monograph seeks to link several research areas together to provide an integrated summary of the knowledge relevant to electrostatically assisted atomization of electrically insulating liquids, from fundamentals to applications. The emphasis of the review leans towards explanation of physics and description of experimental work, and model developments are only included where it is felt they add to the broader understanding of the defined scope.
1.2 Organization
3
Necessarily, some subjects are not discussed in sufficient detail, and undeniably, this effort in information collation has revealed more areas for scientific exploration. Although a researchers work is never complete, it is hoped that the information contained within this manuscript will provide a useful starting place to explore the field, and the references provided will enable the readership to delve further into areas of specific interest.
1.2 Organization The section above has briefly introduced how sprays that contain drops that are electrically charged are proven to be useful in both a research and a commercial context. The remainder of the monograph is organized as follows : Chapter 2 covers the background to electrical forces in a fluid continuum and provides an overview of how electric charge and the electric fields generated by a non-homogenous electric charge distribution interact with single phase fluid motion. The concepts of theoretically pure electrical conductors and insulators, polarization in dielectrics, and the range of electrical timescales present in an electrical fluid continuum are all introduced. The electrical equations relevant to the quasi-electrostatic approximation typical of electrical fluids employed for charge injection systems are then discussed. It is then shown how electrical forces influence the fluid continuum and how these forces scale using an analogy with thermal buoyancy. An outline is given of how interactions between the space charge gradient and the electric field produce vorticity which in turn can generate instability throughout the bulk of the continuum – provided the vorticity generation is strong enough. Finally, a limited example of the Reynolds averaged forms of the governing equations are discussed, highlighting how the variance of space charge has an important role, and that this variance cannot be treated as a passive scalar. Whilst chapter 2 covers the interaction of electrical variables and a fluid continuum in a volume, chapter 3 reviews the literature for knowledge of the charge injection process itself into a fluid without an imposed bulk flow. The chapter covers generic charge injection concepts relevant to dielectric liquids and current-voltage relations before finishing with a discussion of vapor bubble formation when the liquid is below its critical pressure. Chapter 2 and 3 cover the coupling between electrical variables, such as space charge, voltage gradient/electric field, permittivity and fluid variables, such as (mass) density, fluid velocity and pressure gradient, with the bulk (chapter 2) and charge injection at a metal-liquid interface (chapter 3). Chapter 4 covers the basic processes particular to electrically charged single drops. It introduces the classical “Rayleigh Limit” defining the maximum charge a drop may hold. Various extensions relaxing the assumptions of sphericity, an external electric field, and that drop liquid may be a dielectric are discussed. The chapter concludes by reviewing the literature on charged drop fission, evaporation and combustion. In chapter 5, the theme of the review evolves from the fundamental to the applied. Chapters 1 to 4 are to some extent introductory, and bring together a broad
4
1 Introduction
range of information. Chapter 5 covers the primary interest for the review, the design and electrical operation of charge injection atomizers for ‘electrostatically assisted atomization of electrically insulating liquids’. The emphasis of the monograph is to understand the basic principles of how a geometrically simple atomizer operates, and where possible to relate the phenomena of charge injection into liquid without imposed bulk motion discussed previously in chapter 3. Once the basic principles are outlined, a review of the technology advancements published in the scientific literature and also via patents is provided. The remainder of the monograph centers on the behavior of the electrically charged liquid downstream of the atomizer. Chapter 6 focuses on the unique manner in which primary atomization occurs, and the theoretical basis for optimizing the atomizer design further. Chapter 7 summarises the spray characteristics generated by charge injection atomizers. Both qualitative and qualitative information is provided. Finally measurements and a model of the drop diameter-charge distribution are presented and the suitability of the sprays for combustion applications is assessed. The monograph concludes with a brief comment on the future outlook for this subject.
Chapter 2
Electrostatics, Electrohydrodynamic Flow, Coupling and Instability
Abstract. Electrostatics is a term used to describe the physics of charge in motion and at rest in the absence of significant magnetic field effects and can be used to refer to any type of phase where this is the case [13]. Electrohydrodynamics (EHD) is a more specialized term that is generally used to refer to the role of electrostatics in liquid media. A good introduction to electrostatics can be found in texts such as Crowley [13] and Chang et al. [10] with a more EHD orientated approach taken in Melcher [14] and specific to dielectrics, Castellanos [15]. Much of what follows in this chapter can be found in these references, but for the benefit of those new to the subject is repeated here in a more concise form to enable a better understanding of the literature discussed in chapters 3 onwards. A discussion of instability due to EHD interactions is included, since the internal flow within charge injection atomizers is generally within this regime.
2.1 Electrostatics 2.1.1 The Coulomb Force The basic principles governing charge interaction are covered concisely by Crowley [16] where more detail can be found on equations discussed below. How electrical forces arise is an obvious starting point in the discussion of electrostatics and electrohydrodynamics. Electrical forces can exist only if charge is present, the two most basic types of charged particles being monopoles and dipoles. The former contains only a single charge and the latter usually has two equal but opposite charges [16]. The force experienced between two charged monopoles, q1 and q2, can be expressed as,
f i ,1 = where
q1 q 2 4πε xi ,1 − xi , 2
2
ii , 21
(2.1)
xi ,1 represents the position of charge q1. Positive forces indicate repulsion,
and negative forces attraction. If ii,21 is taken as the unit vector from q2 to q1 then this can be interpreted as the force experienced by q1 resulting from the presence J. Shrimpton: Charge Inject. Sys.: Phy. Principles, Experi. & Theore. Work, HMT, pp. 5–35. springerlink.com © Springer-Verlag Berlin Heidelberg 2009
6
Electrostatics, Electrohydrodynamic Flow, Coupling and Instability
of q2. Generalizing eqn. (2.1) to give the Coulomb force acting on an arbitrary charged point, q, as a result of all the surrounding charges produces,
f i ,1 = qEi
(2.2)
where the electric field, Ei, created by the surrounding charges is defined as,
E =∑ i
m
q
m
4πε x − x i ,1
2
i
i , m1
(2.3)
i ,m
2.1.2 Permittivity The permittivity, ε, which arises in eqn. (2.1) and eqn. (2.3) is a function of the electrical characteristics of the medium that the point charges are dispersed in. For free space, i.e. a vacuum, it is assumed that all charges are free to act directly on one another without any external interference. However, for real materials the permittivity is a function of the molecular structure, the number density and the degree of freedom of the charges contained on the molecules in the medium. These factors act to reduce the force that the charges experience relative to that which would be present in a vacuum, defining the relative permittivity as,
ε = r
F F
i , medium
i , vacuum
=
ε ε
(2.4)
0
Typical values for hydrocarbon fluids lie around εr ≈ 2.2, and for air εr ≈ 1.
2.1.3 Conductors, Insulators, Dielectrics and Polarization A perfect insulator would be a medium with infinite resistivity through which no conduction current would flow [17]. Conversely, a perfect conductor is a medium with no resistivity through which, potentially, an infinite conduction current could flow. A more practical definition is provided by Crowley [18] who defines a conductor to be a material that exhibits a conductivity greater than 10-12/Ωm. Those with an electrical conductivity equal to or less than this quantity are, by convention, termed insulators. In conduction the charge carriers in the material are set into motion by an applied field, and this motion continues as long as the field is applied. In many materials, however, the charge carriers cannot continue to move indefinitely and may not be able to move at all in some circumstances. These restrictions on the charge motion give rise to the phenomena which are collectively referred to as polarization [13]. Materials in which equal, but opposite, charged monopoles or dipoles are separated by neutral entities are called dielectrics [13]. Polarization can occur over a range of scales, from a bulk volume down to the atomic level, however, its
2.1 Electrostatics
7
E +ve y
x
fx
Fig. 2.1 Dipole in a non-uniform electric field
effects are easiest to understand at the molecular level. For example, take the case where an external uniform electric field acts upon a dielectric molecule where the centre of the negatively charged electron cloud is not coincident with the positively charged centre of mass. In this case, electric forces act to push and pull the negatively-charged electrons and positively-charged centre-of-mass in opposite directions. The molecule will not move but it will orientate itself with respect to the field and so enters a polarized state. Dipoles are, in general, neutrally charged and so in a uniform electric field, the net volumetric force is zero. If however the situation in Fig. 2.1 arises, where the field is now non-uniform, the polarization of the dipole becomes relevant, producing a dielectrophoretic force in the x-direction [16].
Fx = qd
dE x dx
(2.5)
The product qd is known as the dipole moment and so eqn. (2.5) can be rewritten as,
Fx = p x
dE x dx
(2.6)
8
Electrostatics, Electrohydrodynamic Flow, Coupling and Instability
Generalizing this yields,
Fi = p j
∂E i ∂x j
(2.7)
A continuum consists of many dipole particles and so eqn. (2.7) may be modified to reflect this by multiplying through by the volume number density, n,
f i = Pj
∂Ei ∂x j
(2.8)
Further detail on the derivation of the dielectrophoretic force can be found in Crowley [13]. If the permittivity is isotropic then it may be assumed that polarization is a linear function of the applied electric field and the following expression holds.
Pi = εEi − ε 0 Ei = ε 0 (ε r − 1)Ei
(2.9)
2.1.4 Gauss’s Law Gauss’s Law is an expression that relates the instantaneous electric field to the space-charge at any given point in a continuum. A single point charge, q, in space will create an electric field with strength E at distance r,
E=
q 4πεr
(2.10) 2
D
q
Fig. 2.2 D-field around a charged sphere
2.1 Electrostatics
9
If a hypothetical spherical shell (see Fig. 2.2) encloses this electric field then the product of the field strength over the surface area is constant,
q
EA =
4πr = 2
4πεr
2
q
ε
(2.11)
This can be generalized [16] to,
∫∫ (εE ) ⋅ dS = q
v
(2.12)
The bracketed quantity in eqn. (2.12) is the displacement field vector and essentially represents the charge per unit area normal to the surface it acts upon (C/m2).
D = εE i
(2.13)
i
By comparing eqn. (2.9) with eqn. (2.13) it is possible to decompose the displacement field vector into two components; those independent of polarization effects and those due to it, (2.14)
D =ε E +P i
0
i
i
It should be noted that the surface integral in eqn. (2.12) is also equivalent to,
∂Di = qv ∂xi
(2.15)
where qv is the charge density or ‘space charge’ (C/m3). This is the most common form of Gauss’s Law. Substituting eqn. (2.13) into eqn. (2.15), and assuming that the permittivity is spatially independent, Gauss’s Law for charge conservation results in,
∂Ei q = v ε ∂ xi
(2.16)
The electric field is simply the negative derivative of the electric potential,
Ei = −
∂V ∂x i
(2.17)
Substitution of eqn. (2.17) into eqn. (2.16) gives an alternative form of the Poisson equation in terms of electric potential, or ‘voltage’.
q ∂ 2V =− v 2 ε ∂ xi
(2.18)
10
Electrostatics, Electrohydrodynamic Flow, Coupling and Instability
If no space-charge is present then eqn. (2.18) reverts to the Laplace equation. As eqn. (2.16) shows, the Poisson equation is important as it provides a means of coupling the space-charge present in the medium to the instantaneous electric field.
2.2 Mobility and Charge Transport 2.2.1 Introduction The previous section dealt with the interaction between the stationary charge and the electric field via the Poisson equation, however, this is of no use when considering how dynamic charge is conserved within an elemental control volume. There are several factors influencing the transport of charge and each are discussed in turn below. Since the scope of this review is to cover liquid systems, in particular dielectrics, then this medium is assumed.
2.2.2 Convective Transport by Fluid Motion Naturally, if charge is present in a convected fluid then the convective flux, i.e. current flux, for a specific species is simply that carried by the bulk motion of the fluid
ji = q v u i
(2.19)
2.2.3 Mobility and the Drift Term When a field is applied, the charge at first accelerates but eventually reaches a terminal velocity which in turn depends on the nature of the surrounding material [19]. After this initial transient, and as long as no other flux processes contribute, the electrical convection is related to the applied field by,
u = κE i
i
Table 2.1 Typical ion mobilities [10]
Medium Polymer General liquid Water Ultra-pure hydrocarbons Liquefied rare gases Gases at STP Ordered materials
Mobility (m2/Vs) 10-12 10-8 2 x 10-7 10-7~10-6 10-2~1 10-4 1
(2.20)
2.2 Mobility and Charge Transport
11
The mobility of a fluid is usually taken to be isotropic and constant and varies depending on the physical properties of the charge carrier and medium being traversed. Typical values for electron mobilities are summarized in table 2.1. Melcher [14] has a useful guide for calculating ionic mobilities in highly insulating liquids based on Walden’s Rule. The expressions in eqn. (2.21) give the relations for negative and positive ions respectively,
κ=
3 ×10 −11 , 1.5 × 10−11 κ= μ μ
(2.21)
If it is assumed that the charged fluid element is traveling at its terminal velocity, then it is said to have reached the “mobility limit” [13] and the flux of current can be expressed as,
ji = qvκEi
(2.22)
Normally, the number density, charge density and mobility are all a function of the physical system. However, if there is no bulk or electrical convection of charge density, then it is possible to define a conductivity, σ [18], hence,
ji = σEi
(2.23)
It should be stressed that this relation is only valid for linear, Ohmic conduction. For dielectric liquids the literature will show in section 3.2 that Ohmic behavior only occurs for very low values of electric fields before convective effects start to dominate the current flux.
2.2.4 Diffusion and the Debye Length A discussion of the charged double layer that forms around the injecting electrode can be found in Melcher [14] and also section 3.1.3. This thickness is characterized by the Debye length and is defined as the distance over which the potential developed by separating a charge density from the background charge of the opposite polarity is equal to the thermal voltage kBT/e.
The Einstein relation Dd = κkBT/e can be used to estimate the ionic diffusion co-efficient and therefore the ionic diffusion timescale. Melcher states that the Debye length is important when the ratio of the ionic diffusion timescale to the charge relaxation timescale is large. As a result, diffusion is only of significance in EHD processes occurring close to the charge injecting electrode. The current flux due to diffusion can be written as,
j i = − Dd
∂qv ∂xi
(2.24)
12
Electrostatics, Electrohydrodynamic Flow, Coupling and Instability
As discussed previously, the diffusion term can usually be eliminated if only bulk effects are of interest. This can be proved by performing an order of magnitude analysis on the drift and diffusion terms [20, 21]. It is assumed in the analysis that the Debye length, λD, represents the diffusion length scale, and the drift length scale, d, by a bulk dimension such as the electrode spacing. Equation. (2.25) uses the Einstein relation for ideal gases and liquids [14] and linear approximations, to express the ratio of the diffusion term to the drift term as,
Dd ∇qv Dd ⎛ d ⎜ ~ qvκE κV0 ⎜⎝ λ D
⎞ 0.025 d ⎟⎟ ~ V0 λ D ⎠
(2.25)
The constant 0.025 arises from the ratio of the ionic diffusion to mobility of the fluid and this is the value for a single negatively-charged ion at approximately 20°C [14]. Thus, the diffusion term is of interest only if the ratio of diffusion and drift currents is of order unity and so in most cases can be disregarded.
2.2.5 Charge Conservation The total steady-state current flux can be defined by combining the drift eqn. (2.22), diffusive eqn. (2.24), and convective eqn. (2.19) components,
j i = q v κE i − Dd
∂q v + qv ui ∂x i
(2.26)
Space-charge is a scalar and so the scalar transport equation can be invoked to incorporate eqn. (2.26),
∂q v ∂ji =0 + ∂t ∂x i
(2.27)
This can be expanded to give [20],
∂qv ∂u i ∂q v ∂E i ∂q v ∂ 2 qv E i − Dd + qv + ui + qvκ +κ =0 ∂t ∂xi ∂xi ∂xi ∂x i ∂xi ∂xi (2.28) For an incompressible fluid the second term disappears and the Poisson eqn. (2.28) can be used to replace the 4th term,
∂qv ∂q ∂q ∂ 2 qv q κ + u i v + v + κ v E i − Dd =0 ∂t ∂x i ε ∂xi ∂xi ∂xi 2
(2.29)
The fifth term in eqn. (2.29) represents the ionic diffusion and is usually small enough to be neglected. The first and second terms combine into the material
2.3 Momentum and Energy
13
derivative for the space-charge leaving a modified form of eqn. (2.27) that links the space-charge, electric field and fluid velocity,
∂qv q ∂q + κ v + κ v Ei = 0 ∂t ε ∂xi 2
(2.30)
Clearly the space charge equation is non-linear due to its effect in the second term of eqn. (2.30).
2.3 Momentum and Energy 2.3.1 Introduction The Poisson eqn. (2.16) and charge transport eqn. (2.27) equations have been presented, the former linking the space-charge to the electric field. The latter also contains this coupling, but more importantly introduces a link to the bulk flow, an important effect when considering the transport of charge within an electrostatic atomizer. The purpose of this section is to revise the classical thermofluid equations and introduce the effect of electrical forces acting on the medium. For simplicity it will be assumed that the liquid is incompressible. Electrical forces have no effect on the mass continuity equation and so this remains as,
∂u =0 ∂x i
(2.31)
i
In addition to incompressibility, this analysis will be limited to single, that is, liquid phase physics and, bar the charge carrier species in this phase, consider only single component media. Research has been carried out for EHD flows in twophase systems and for further details references [22-25] provide a good starting point.
2.3.2 Electrical Forces The volume electrical force for a linear medium can be expressed as,
f i = qv Ei −
1 2 ∂ 1 ∂ ⎡ ⎛ ∂ε ⎞ 2 ⎤ E ε+ ⎢ρ ⎜ ⎟ E ⎥ 2 2 ∂xi ⎢⎣ ⎜⎝ ∂ρ ⎟⎠T ⎦⎥ ∂xi
(2.32)
Castellanos [15] provides a full derivation and explanation of eqn. (2.32). The left-most term is the Coulomb force which has already been discussed in section 2.1.1. This term usually dominates the other two. The central term is known as the dielectric force and is only of significance if an alternating electric field is applied with a period much shorter than the charge relaxation time and/or the ionic transit time. Finally, the right-hand term is the electrostrictive pressure, so
14
Electrostatics, Electrohydrodynamic Flow, Coupling and Instability
described because it is a scalar derivative. Because of its form Castellanos remarks that it is not unusual to group this with the standard pressure derivative when solving the equation set.
2.3.3 Momentum Conservation The Navier-Stokes equations provide a final link between the transport of momentum in the dielectric medium and the net force caused by the presence of electrical charge, hence,
∂ui ∂ui ∂p ∂ 2ui + ρu j =− +μ + ρg i + f i ρ ∂t ∂x j ∂xi ∂x j ∂x j
(2.33)
As the liquid is being treated as incompressible it is usual to replace the gravity body force in eqn. (2.33) by a buoyancy term, with reference quantities denoted by a zero subscript,
ρg i = ρ 0 [1 − β (T − T0 )]g i
(2.34)
2.3.4 Energy Conservation Chang [22] provides a comprehensive derivation of the thermal transport equation relevant to EHD flows, which is repeated below,
∂T ∂T k ∂ 2T +uj = +Q ∂t ∂x j ρC p ∂x j ∂x j
(2.35)
Here Q represents the energy change due to the presence of an electric field,
Q = ( ji − q v u i )E i −
⎡ d ⎛D ∂ ε ijk ε lmk E j u l Dm + ⎢ Ei ⎜⎜ i ∂xi ⎣ dt ⎝ ρ
[
]
⎞⎤ ⎟⎟⎥ ρ ⎠⎦
(2.36)
For a non-conducting liquid the author states that eqn. (2.36) can be simplified to,
⎡ d ⎛ D ⎞⎤ Q = ⎢ E i ⎜⎜ i ⎟⎟ ⎥ q v ⎣⎢ dt ⎝ q v ⎠ ⎦⎥
(2.37)
Generally speaking, the energy equation can usually be ignored for incompressible liquid systems if no external heating mechanisms are present. Note that this also eliminates the gravity term from eqn. (2.33). A revised treatment of the energy equation including entropy considerations can be found in Castellanos [26].
2.4 Electrical Timescales
15
2.4 Electrical Timescales 2.4.1 Introduction There are many timescales associated with EHD flow in addition to the existing hydrodynamic definitions. The Institution of Electrical and Electronic Engineers [27] recently drew up a draft standard for EHD numbers and associated definitions, however, from this document it is clear that there are many interpretations and versions depending on the application. The following attempts to define the fundamental timescales relevant to incompressible liquid flows.
2.4.2 Ohmic-Charge Relaxation Crowley [18] provides a concise definition of the Ohmic-charge relaxation timescale. Ultimately, this quantity represents the time taken for charge within the liquid to be neutralized by opposing and neutral polarity charge carriers. Naturally, this timescale is of greatest relevance in conducting liquids, which the author defines as those with a conductivity greater than 10-12 /Ωm and is defined as,
τ
OC
=
ε σ
(2.38)
2.4.3 Space-Charge Relaxation Dielectric fluids are, generally, highly insulating and therefore the Ohmic-charge relaxation timescale is of little relevance. In such cases the charge does not decay by neutralization, but by spreading out in response to its self-repulsion [13]. The space charge relaxation timescale is defined as,
τ SC =
ε q v 0κ
(2.39)
As Crowley [13], and other authors such as Castellanos [21] explain, the spacecharge relaxation timescale in effect determines the rate at which charge decays from a given origin; a large initial space-charge will decay much more quickly than one with only a small value. Such a concept is presented in examples by Castellanos [21], Atten [28] and Crowley [13]. Firstly, the diffusion term is omitted from eqn. (2.26) for the reasons described in section 2.2.4. Vector algebra can then be used to manipulate eqn. (2.26) and eqn. (2.15), producing the following identity, an alternative form to eqn. (2.30),
∂qv ∂q κ 2 + (κEi + u i ) v = − qv ∂t ∂xi ε
(2.40)
16
Electrostatics, Electrohydrodynamic Flow, Coupling and Instability
Since the characteristic line is defined as,
dxi = κEi + ui dt
(2.41)
then the spatial derivative can be eliminated from eqn. (2.40). The solution of this gives the decay of space-charge along eqn. (2.41),
qv =
q v,0 1+ t
τ sc
(2.42)
Taking the limit of eqn. (2.42) as t >> τsc, implies that the space-charge, qv, tends to ε/κt, independent of the initial quantity. As Castellanos [21] states, using this model, a point in the fluid will only contain space-charge if a characteristic line can be traced back to a charge injection surface. The obvious limitation of this analysis is that it does not couple charge and velocity, that is, the latter is not dependent on the former and so is not an accurate reflection of EHD flow. However, it illustrates the meaning of the space-charge relaxation timescale.
2.4.4 Ionic Diffusion Timescale Near to the electrodes, a thin layer exists where electro-chemical reactions predominately occur, enabling charge to be injected into the fluid. The Debye length scale is of interest here because of the molecular processes occurring, hence the ionic diffusion timescale is,
τd =
Dd
λD 2
(2.43)
2.4.5 Ionic Transit Timescale The ionic diffusion timescale is usually not of interest when analyzing the bulk charged flow. What is of more interest is the ionic transit timescale, that is, the timescale associated with drift of the ions with respect to the characteristic length of the system, l0,
τκ =
102 κV0
(2.44)
Sometimes this is expressed as a ratio of the characteristic length to the drift velocity κE; it is termed the migration time [14] but still represents the same process of ionic diffusion.
2.5 Non-dimensional Transport Equations
17
2.4.6 Electro-viscous Timescale When viscous forces are comparable to the electric field forces then the electroviscous timescale is of importance,
τ = eυ
μ εE
2
(2.45)
This is significant when a voltage is first applied to an electrode arrangement as it controls how quickly the liquid accelerates to a constant velocity [29].
2.4.7 Electro-inertial Timescale In a similar manner to the electro-viscous timescale, when the inertial forces are of the same magnitude as the electrostatic forces then the electro-inertial timescale [14] is influential,
τ ei = l
ρ εE 2
(2.46)
2.5 Non-dimensional Transport Equations 2.5.1 Introduction The momentum conservation equation incorporating electrical body force terms was introduced in section 2.3. When no body forces are present then only four reference quantities are needed to non-dimensionalize this equation two of these being a length (l0) and velocity (u0). Dielectric fluids are assumed to be incompressible and temperature effects usually ignored so the density (ρ) and dynamic viscosity (μ) are defined as fixed reference variables. If electrical body forces are also present then this list must be expanded to include space charge (qv0), electrical potential (V0), permittivity (ε0) and ionic mobility (κ). Merging eqn. (2.32), eqn. (2.33) and rearranging for density produces the following nondimensionalization [15],
⎛q V τ ⎞ ∂u i τ ∂u p τ ∂p τ ∂ 2 u i gτ + + + 0 g i + ⎜⎜ v 0 0 ⎟⎟qv Ei uj i = − 0 ∂t τ m ∂x j ρu 0 l 0 ∂xi τ ν ∂x j ∂x j u 0 ⎝ ρu 0 l 0 ⎠ ⎛ ε V 2τ ⎞ 1 ⎛ ε V 2τ ⎞ 1 ∂ ⎡ ⎛ ∂ε ⎞ 2 ⎤ ∂ ε + ⎜⎜ 0 0 3 ⎟⎟ − ⎜⎜ 0 0 3 ⎟⎟ E 2 ⎢ρ ⎜ ⎟ E ⎥ ∂xi ⎝ ρu 0 l 0 ⎠ 2 ⎝ ρu 0 l 0 ⎠ 2 ∂xi ⎣ ⎝ ∂ρ ⎠ T ⎦ (2.47)
18
Electrostatics, Electrohydrodynamic Flow, Coupling and Instability
In eqn. (2.47) τ, τν and τm represent arbitrary, viscous (l02/ν) and mechanical /u (l0 0) timescales respectively. An arbitrary timescale has been introduced to demonstrate the effect that this has on the non-dimensional numbers formed.
2.5.2 Momentum Conservation: Free Flow For the free-flow case the reference velocity must be defined first. There are a number of alternative definitions, but the most common is to assume that the characteristic velocity is given by the ionic drift contribution, κE. The arbitrary timescale in eqn. (2.47) becomes the mechanical timescale, but for the free-flow case this is now defined as,
l0 l0 l02 τm = = = u 0 κE0 κV0
(2.48)
Substituting eqn. (2.48) into eqn. (2.47) results in the following,
⎛ q l2 ⎞ p ∂p g l3 ∂ui ∂u ν ∂ 2u i + u j i = − 02 + + 20 02 g i + ⎜⎜ v 02 0 ⎟⎟qv E ρu0 ∂xi κV0 ∂x j ∂x j κ V0 ∂t ∂x j ⎝ ρκ V0 ⎠ ⎛ ε ⎞1 ⎛ ε ⎞ 1 ∂ ⎡ ⎛ ∂ε ⎞ 2 ⎤ ∂ − ⎜⎜ 0 2 ⎟⎟ E 2 ε + ⎜⎜ 0 2 ⎟⎟ ⎢ ρ ⎜⎜ ⎟⎟ E ⎥ ∂x i ⎝ ρκ ⎠ 2 ⎝ ρκ ⎠ 2 ∂xi ⎣ ⎝ ∂ρ ⎠ T ⎦ (2.49) Introducing non-dimensional parameters and assuming that the pressure and kinetic energy terms are approximately equal yields,
1 ∂ 2 ui 1 ∂ui ∂ui ∂p +uj =− + + g i + CM 2 qv E ∂t ∂x j ∂xi Re E ∂x j ∂x j FrE ∂ 1 1 ∂ ⎡ ⎛ ∂ε ⎞ 2 ⎤ − M 2 E2 ε +M2 ⎢ρ⎜ ⎟ E ⎥ ∂x i 2 2 ∂x i ⎣ ⎜⎝ ∂ρ ⎟⎠ T ⎦
(2.50)
The electrical Reynolds number in eqn. (2.50) is defined as,
Re E =
κV0 ⎛ μκ ⎞ ⎛ ε 0 1 ⎞ T ⎟⎟ ⋅ ⎜⎜ ⎟= 2 = ⎜⎜ 2 ⎟ ν ⎝ ε 0V0 ⎠ ⎝ ρ κ ⎠ M
(2.51)
The new T, C and M parameters introduced in eqn. (2.50) and eqn. (2.51) are discussed further in section 2.5.4. Under certain circumstances it is difficult to estimate a reference charge density and so in this case it is assumed to be equal to εV0/l02. The implication of this is that the C parameter disappears from eqn. (2.50). Other forms of eqn. (2.50) exist and one example can be found in Atten [28].
2.5 Non-dimensional Transport Equations
19
2.5.3 Momentum Conservation: Forced Flow If, in eqn. (2.47), the arbitrary timescale is chosen to match the mechanical timescale defined by the characteristic length and velocity of the system then eqn. (2.47) becomes,
⎛q V ⎞ ∂u i ∂u i p 0 ∂p g l ν ∂ 2ui +uj =− 2 + + 02 0 g i + ⎜⎜ v 0 20 ⎟⎟q v E ∂t ∂x j ρu 0 ∂x i u 0 l 0 ∂x j ∂x j u0 ⎝ ρu 0 ⎠ (2.52) 2 2 ⎛ ε 0V0 ⎞ 1 2 ∂ ⎛ ε 0V0 ⎞ 1 ∂ ⎡ ⎛ ∂ε ⎞ 2 ⎤ ε + ⎜⎜ 2 2 ⎟⎟ − ⎜⎜ 2 2 ⎟⎟ E ⎢ρ ⎜ ⎟ E ⎥ ∂xi ⎝ ρu 0 l 0 ⎠ 2 ⎝ ρu 0 l 0 ⎠ 2 ∂x i ⎣ ⎝ ∂ρ ⎠ T ⎦ As the liquid is assumed to be incompressible then it is reasonable to assume that the reference kinetic energy is of the same magnitude as the reference pressure. For a forced-flow regime with an imposed characteristic velocity this results in,
∂ui ∂u ∂p 1 ∂ 2ui 1 Gr +uj i = − + + g i + E2 qv Ei ∂t ∂x j ∂xi Re ∂x j ∂x j Fr Re
T2 1 2 ∂ T 2 1 ∂ ⎡ ⎛ ∂ε ⎞ 2 ⎤ − 2 2 E ⎟ E ⎥ ε+ 2 2 ⎢ρ ⎜ Re M 2 Re M 2 ∂xi ⎣ ⎜⎝ ∂ρ ⎟⎠T ⎦ ∂xi
(2.53)
2.5.4 Non-dimensional Parameters Several non-dimensional parameters are immediately apparent in the preceding equations. Alongside the definition for the classical Reynolds and Froude numbers are their electrical equivalents and four new electrical quantities, the electrical Grashof (GrE), as well as the T, C and M parameters. Each of these will be dealt with in turn. Note that although the permittivity of vacuum, ε0, arises from the nondimensionalization process it is common practice to use the material permittivity, ε, as this has more meaning. Only a brief explanation of the T, C and M parameters will be given with their main relevance left for discussion in section 2.6. The classical Reynolds and Froude numbers need no explanation. The electrical Reynolds number simply represents the ratio of the ionic drift timescale to the viscosity timescale. Similarly, the electrical Froude number is the squared ratio of the ionic drift timescale to the gravity timescale. The electrical Grashof number is analogous to its thermal variant and is defined as the ratio of the Coulombic body force to the viscous body force,
GrE =
q v 0V0 l 02
μν
⎛ q V ⎞ ⎛ l3 ⎞ = ⎜⎜ vo 0 ⎟⎟ ⋅ ⎜⎜ 0 ⎟⎟ ⎝ l 0 ⎠ ⎝ μν ⎠
(2.54)
20
Electrostatics, Electrohydrodynamic Flow, Coupling and Instability
This non-dimensional quantity is also discussed by Zhakin [30] however he does not go so far as to call it the electrical Grashof number. The T parameter is a measure of the stability of the system, is effectively an electrical Rayleigh number and is defined as follows,
T=
εV0 μκ
(2.55)
This parameter will be discussed in further detail in section 2.6. Zhakin [30] also defines an electrical Prandtl number,
PrE =
τc εν = τ ν κq v 0 l02
(2.56)
For free-flow electroconvection the product PrEGrE is equal to T, therefore a direct comparison can be made between this and the thermal case. If qvo=ε0V0/l02 is substituted into eqn. (2.56) another Prandtl number variant, derived by Schneider & Watson [31], is defined. The M parameter is characterized in many papers and a concise overview of non-dimensionalization, including a discussion of this, can be found in Atten [28]. It represents the ratio of the hydrodynamic mobility to the ionic mobility,
M =
(ε / ρ )1 / 2 = κ H κ
κ
(2.57)
A limiting value for the hydrodynamic mobility can be obtained if it is assumed that all of the electrical energy (1/2εE2) is converted to kinetic energy (1/2ρu2). As explained by Atten, this parameter is a measure of the extent of turbulence in the system. A slightly later paper by Atten et al. [32] suggests that the M parameter can be used to differentiate between liquid and gas EHD systems. Typical values for gases usually obey M << 1 and for liquids M > 3. The implication of this is that the charge and mass transport equations are coupled in liquid systems hence the treatment given in this section. In gaseous systems this link is less important and the two variables can be solved independently. Electrostatics is a term usually reserved for problems such as these. Felici [20] discusses the M parameter in depth and the associated hydrodynamic mobility, but essentially eqn. (2.57) is most relevant for strong injection when the system has developed to a fully turbulent state. The injection strength parameter, C, also arises from the electrical body force Navier-Stokes equations when non-dimensionalized,
C=
q l τ = τ εV κ
SC
2
v0 0
0
(2.58)
2.6 Electrohydrodynamics
21
This clearly shows that it is a timescale ratio of the ionic drift to the Coulombic relaxation of charge so the C parameter represents the strength of charge injection. or C << 1 the charge injection regime is termed weak injection and the electric field within the fluid is determined by the external power supply. Conversely for C >> 1, a strong injection regime is present and the space-charge within and around the fluid determines the electric field over this volume [15]. Tobazéon [17] provides a slightly more precise definition for the different regimes of C and this is summarized below, - strong injection: - medium injection: - weak injection:
5
2.6 Electrohydrodynamics 2.6.1 Introduction At the beginning of the section a distinction was made between electrostatics and electrohydrodynamics. As discussed, the latter term is generally employed when the system of interest exhibits charged liquid motion. Electroconvection is an alternative term used in some literature. A fundamental study into charge-driven convection was performed by Schneider & Watson [31] and Watson et al. [33] at the beginning of the 1970s which demonstrates electroconvection in action. Figure 2.3 is taken from their work and shows a top-down cellular convection pattern in a thin film of silicone fluid. A comparison between this and the equivalent thermal situation with Rayleigh-Bernard cells is attempted, where the fluid is heated from below. As the authors remark, in the thermal case, the hotter fluid rises as the cooler fluid near the top falls, creating the observed convection cells. They suggest that an analogous situation exists for the electrically charged system; highly charged liquid is repelled from the injecting electrode displacing less heavily charged liquid at the other, so forming convective cells akin to the
Fig. 2.3 Convection cells in a 1mm film of Dow Corning 703 charged by positive corona. The diameter of the glass dish is 0.5in [31]. Reproduced with permission from the American Institute of Physics
22
Electrostatics, Electrohydrodynamic Flow, Coupling and Instability
thermal case. The purpose of what follows will be to discuss the flow regimes that occur in a charged flow, such as that described above and inform the reader of the work that has been carried out in this area.
2.6.2 Fundamentals As Atten [28] explains, the simplest electrohydrodynamic system is that of a dielectric liquid contained between two parallel plane electrodes. If it is also assumed that the liquid between the plates is stationary then taking the curl of the resulting Navier-Stokes equations eqn. (2.33) produces the identity for an electrohydrodynamic system at rest,
ε ijk qv
∂E k ∂q = ε ijk E j v = 0 ∂x j ∂xk
(2.59)
Felici [20] provides a vigorous derivation and discussion of eqn. (2.59), but essentially this expression means that space charge must be present if an electric field is active and vice-versa. As can be seen from the transport equations given in section 2.5, the presence of space-charge and an electric field results in a force upon the liquid, and as will be covered in the next sub-section, under certain conditions, electroconvection. The type of electroconvection present depends on a number of factors, the most important of these being the medium present and injection method. For liquids, the Coulomb force induces velocities very much greater than those created by ionic drift, whereas the opposite relation is found in gases [32]. The relevance of this effect on solving the transport equations was discussed in section 2.5.4.
2.6.3 Instability Fundamental studies into the onset of electroconvection in insulating liquid have been carried out by Watson et al. [33]. The apparatus used can be seen in Fig. 2.4, they used an electron beam to charge the free surface of a silicone fluid film. This effectively creates a cathode on the free surface with the grounded conducting plate under the film acting as an anode. Charge from the free-surface migrates to the anode via ionic drift, entraining liquid in the process and setting up an electroconvective, or more appropriately termed, electrohydrodynamic system. A second electron beam was passed over the free-surface cathode to ascertain the potential of this surface. As with the thermal analogy, a threshold parameter must be exceeded for the system to become unstable and electroconvection to ensue. Schneider & Watson [31] derive the equations of motion in non-dimensional form and a number of features arise from this. The most important is the stability parameter, T, introduced in section 2.5. Atten [28] shows that this can be derived by balancing the viscous and Coulombic forces and assuming that the reference space-charge is given by εV0/l02,
2.6 Electrohydrodynamics
23
Electron Gun No. 1 Recorder Voltage Amp.
Writing Beam
Differential Electrometer
Read-out Beam Electron Gun No. 2
Fluorescent Screen Split Anode
A Optical Path
a)
Grid
Electron Beam Ground Plates
Liquid
Glass
Conductive Layer
A
(b) Fig. 2.4 (a) Schematic diagram of the dual electron beam system (b) Detailed view of the sample holder [33]
T~
εV0 E / l02 εVE εV0 qv E ~ ~ = μ∇ 2 u μu / l02 μκE κμ
(2.60)
where the characteristic voltage (at the free surface) is given by, 1/ 2
⎛ 8 Jl03 ⎞ ⎟⎟ V0 = ⎜⎜ 9 κε ⎝ ⎠
(2.61)
24
Electrostatics, Electrohydrodynamic Flow, Coupling and Instability
This voltage is the space-charge limited voltage (see section 3.1.4) arising from applying boundary conditions of the experimental setup to the Poisson eqn. (2.17) and charge transport equation, eqn. (2.27). In the derivation it has been assumed that charge transport is steady and dominated by ionic drift, consequently the convection and diffusion terms have been ignored. It should be noted that the space charge limited density in eqn. (2.61), discussed more fully in section 3.1.4 is not instantaneously reached upon application of the voltage and in fact peaks before settling down to the steady state value [17]. A dimensionless wavenumber also appears in the analysis carried out by Schneider & Watson [31] and they show in Fig. 2.5 that a neutral stability curve can be plotted of this versus the electrical Rayleigh number given in eqn. (2.60). The important feature to arise from Fig. 2.5 is the minimum stability criterion, that is T ≥ 99 for the system to be unstable and for electroconvection to initiate. Schneider & Watson [31] show that the electrical Rayleigh number represents an energy balance by multiplying numerator and denominator by the ionic transit timescale eqn. (2.44) yielding,
10 8 α
6 4 2 0 0
200 400 Electrical Rayleigh No.
600
Fig. 2.5 Neutral stability curve of electric Rayleigh number as a function of dimensionless wavenumber α [31]
2.6 Electrohydrodynamics
T=
25
εV0 τ κ κV0 16 ⎛ 3εV02 ⎞⎛ τ κ ⎞ ⎟⎜ ⎟ = ⎜⎜ κμ L2 3 ⎝ 16 L ⎟⎠⎜⎝ μL ⎟⎠
(2.62)
As the authors explain, the first parenthesized factor is the electrostatic stored energy per unit area in a space-charge layer of thickness L and the second is the reciprocal of the viscous energy dissipated per unit area in the same thickness. In effect, the number represents the ratio of the stored electrostatic energy to the viscous dissipation energy for a convecting system. The dependence of surface potential on current density is shown in Fig. 2.6 from Watson et al. [31]. As they comment, eqn. (2.61) holds for low current densities. They account for the deviation at high current density values due to the film being distorted by electrostatic forces and a correlation of this can be seen in the same plot. In the original analysis they discounted the effect of bulk liquid convection on the transport of charge. However, discussing this with Fig. 2.7 in mind, as can be seen the relation eqn. (2.61) does not take into account bulk liquid convection. Figure 2.7 shows that the conduction (JL3/2) begins to increase for current densities above 6 x 10-10 A/cm2 suggesting that the effects of bulk liquid convection take hold at this point. Atten [28] states that for parallel electrodes and weak injection the electroconvective system becomes unstable for a theoretical value of TcC2 = 220.7. For strong injection, instability is only dependent on T, and occurs for a
1
1000 JαV 2
0.1
100 Thickness at Peak Steady State
10 -11 10
-10
-9
0.01
10 10 Current Density (Amps/cm3)
Fig. 2.6 Relationships among sample potential, thickness and current density [33]
Thickness (cm)
Surface Potential
Peak Voltage Steady Voltage
JL3 / V2 (Amp cm/Volt2) × 1018
26
Electrostatics, Electrohydrodynamic Flow, Coupling and Instability
Peak Values Steady Values
2.0
1.5
1.0 10
-11
-10
-9
10 10 Current Density (Amps/cm3)
Fig. 2.7 Space charge limited conductance JL3/V2, as a function of current density [33]
value of 161. This second criterion contrasts with the value defined by Watson et al. Note that in both of these cases the parameters were derived under the assumption of a linear stability analysis. Tobazéon [17] found that for step voltages there exists a time delay between application of the voltage and a response in the current flux across the EHD system. He suggests that this is characterized by,
τ R ≈ 2Tc
μ εE 2
(2.63)
Where Tc is the stability parameter at which electroconvection initiates. From eqn. (63) it can be seen that the time delay is proportional to the electro viscous timescale described in section 2.4.6. A more detailed discussion of the transient regime can be found in Tobazéon’s paper.
2.6.4 Plumes When the critical stability parameter for the system is attained electroconvective forces exceed the viscous and inertial damping forces and flow to the anode occurs when the cathode is the injecting electrode. Vazquez et al. [34] state that this takes the form of a bi-dimensional charged plume when the injector is a blade or wire and an axisymmetric shape when the electrode is a point. They highlight
2.6 Electrohydrodynamics
27
Fig. 2.8 Charged plume in blade-plane geometry in silicone oil of kinematic viscosity ν = 10cSt (d= 13mm, V=20kV). a) Perspective representation of the blade b) trough c)-f) snapshots taken at different instants of the flow (Schlieren technique): one can see that the plume is not steady and that its structure is not strictly bidimensional: on the view d) appear 3 individual plumes; on e) and f) we can more distinctly see rolls of return flow close to the plane [35]. Reproduced with permission from the American Institute of Physics
some features of these EHD plumes, one of these being that the bulk velocity dominates the ionic drift velocity, as well as a double boundary layer structure that forms around the cathode and continues to the anode. For laminar plumes charge
28
Electrostatics, Electrohydrodynamic Flow, Coupling and Instability
is assumed to only exist in a narrow core. Motion produced by the electrical field acting on this charged core entrains surrounding uncharged fluid creating the second, much thicker layer. Atten [28] points out that in the case of the blade and point geometry the injection surface is non-planar leading to a highly non-uniform distribution of space-charge on the injector and in the liquid bulk. Atten et al. [28] observed thin, unsteady plumes with a diameter of approximately 1mm. In addition to the existence of the plume they also noted a recirculating toroidal structure around the plume, regardless of whether the plume impacted on a plate or passed through a grid. The authors debate whether the presence of the grid had an effect or ultimately caused the observed recirculating eddy, and remark that if this was not the case then a different plume structure would be expected. Specifically they state that they would expect a laminar portion to develop before unsteadiness developed. In the observed behavior, the plume was seen to be unsteady along its whole length. One explanation they put forward is that the plume was completely turbulent, and this, it is suggested, is why their estimates for its theoretical velocity (2m/s) seem to be smaller than its observed velocity (0.3m/s). Schlieren images of blade-plane plumes have been obtained by Malraison et al. [35]. These are shown in Fig. 2.8 and illustrate the time dependent nature of the plume. Malraison et al. conclude that these plumes operate in the laminar regime however they were clearly unsteady and comment that that they were not entirely
a
z
z W0
b
x
W0
x
Fig. 2.9 Schematic representation of laminar (a) and turbulent (b) plumes with profiles of charge distribution indicated [35]
2.6 Electrohydrodynamics
29
bi-dimensional. Within this paper the authors illustrate their understanding of the laminar and turbulent plume structure and for completeness these are reproduced in Fig. 2.9. So far no conclusive experimental evidence of the velocity profiles produced by such plumes is known to exist, due to the difficulty of obtaining accurate data from existing velocimetry techniques. Despite this, attempts have been made to derive analytical expressions for the centerline charge and velocity magnitude as well as the estimated plume width for blade-plane and point-plane systems. Vazquez et al. [36] provide a useful comparison between thermal and electrical laminar plume characteristics for blade-plane bi-dimensional and pointplane axisymmetric profiles. They use the similarity method which differs from the separate scaling law arguments presented by Malraison et al. [35] and Atten et al. [32].
2.6.5 Transition to Turbulence Atten [28] notes that experimental results show that finite amplitude convection is always observed to be unsteady and the presence of this time dependence is not influenced by the test cell aspect ratio. He also remarks that in the case of a unit ratio between the electrode gap distance and tank radius, typically only one electroconvective cell is seen to form. This initially only has one periodic vacillation (discrete power spectra of current fluctuations) but further increases in voltage cause extra vacillations until a continuous spectra is formed, indicating a chaotic flow. The author suggests that when the test cell aspect ratio is increased beyond unity then any convection present displays fully chaotic characteristics regardless of the voltage applied. A concise review of observed turbulent behavior in EHD systems can be found in Castellanos [21], however only the main points will be discussed here. He reaches similar conclusions to Atten stating that current fluctuations are present when the liquid is in motion, and when the motion is dominated by viscous forces the power spectra of these obeys an exponential decay law. These power spectra are continuous and have a peak frequency, f1, which has been found to be a direct function of the mean velocity to gap distance ratio. In the inertial EHD regime the power spectra takes on a power law relation proportional to f-α, where α is a constant. As the injection is increased it is found that, in the case of plane electrodes, that the hexagonal convection cells become gradually distorted before eventual transition to the turbulent regime. For strong injection, i.e. C >> 1, the electric Nusselt number relation in eqn. (2.64) is obeyed until the space-charge limit is attained [28] as can be seen in Fig. 2.10,
⎡T ⎤ NuE ≈ ⎢ ⎥ ⎣ Tc ⎦
1/ 2
(2.64)
30
Electrostatics, Electrohydrodynamic Flow, Coupling and Instability
(x)
10
Ne
(ix) (viii)
3
(iv)
(v)
(vii) (vi)
(iii) (ii)
1 1
2
10
10
(i) 3
10
T / Tc Fig. 2.10 SCL injection in various liquids. Electric Nusselt number Ne vs. the stability parameter T/(Tc)exp for various couples of liquids and injected ion species: (i) methanol/H+; (ii) cholorobenzene/Cl-; (iii) ethanol/H+; (iv) nitrobenzene/Cl-; (v) ethanol/Cl-; (vi) propylene carbonate/Cl-, 35°C; (ix) Pyralene 1500/Cl-, 20°C; (x) Pyralene 1499/Cl-. [28]
2.7 Electrohydrodynamic Turbulence in a Propagating Flow Front 2.7.1 EHD Vorticity In the preceding section the general concepts of electroconvection and the transition to turbulent EHD were introduced. In this section fully turbulent EHD systems will be discussed in the context of a liquid medium and the main features identified by researchers in this area. Vorticity is the obvious starting point when looking at turbulent behavior. Taking the curl of the extended Navier-Stokes eqn. (2.33) and accounting for fluid motion in this instance results in the following [20],
(
)
∂ωi ∂ui ∂ 2ωi ∂q ∂ E 2 / 2 ∂ε − ρω j −μ = ε ijk Ek − ε ijk ρ ∂t ∂x j ∂x j ∂x j ∂x j ∂x j ∂xk
(2.65)
Felici notes that the temperature dependent term is usually very small and so can be neglected. The main point arising from eqn. (2.65) is that the resultant system is highly rotational with vorticity being continually generated within the flow wherever space-charge and the electric field exist. Felici infers that the vorticity and first term on the right of eqn. (2.65) form a positive feedback loop. When the critical voltage is exceeded, the feedback loop is self-sustaining.
2.7 Electrohydrodynamic Turbulence in a Propagating Flow Front
31
However, when it is not then feedback does not occur and he comments that it is then the geometry and conductive behavior of the fluid that controls the first term on the right of eqn. (2.65).
2.7.2 EHD RANS in a Propagating Flow Front There are very few publications that discuss the application of the RANS equations to EHD flow. However, Hopfinger & Gosse [29] provide a reasonably detailed analysis of a flow front propagating between two plane electrodes upon application of a potential difference between the two. They assume that the variables in eqn. (2.33) can be decomposed into mean and fluctuating parts,
u i = ui + ui ' p = p + p'
(2.66)
qv = q v + qv ' Ei = Ei + Ei '
where turbulent mean square fluctuations of velocity and field strength are given by,
u i ' 2 = u1 ' 2 + u 2 ' 2 + u 3 ' 2 E i ' = E1 ' + E 2 ' + E 3 ' 2
2
2
(2.67) 2
The turbulence between two electrode plates is assumed homogeneous in any plane parallel to these and so the turbulent kinetic energy equation for this arrangement becomes,
(
)
∂u ' ∂u i ' ⎛1 ∂ u '2 / 2 ∂ p' ⎞ + u1 ' ⎜⎜ u ' 2 + ⎟⎟ + ν i ρ⎠ ∂t ∂x1 ⎝ 2 ∂x j ∂x j
(
1 ∂ 2 u '2 1 − ν − E u1 ' qv ' + E ' u i 'q v + E ' ui ' qv ' 2 ∂x1 2 ρ In eqn. (2.68) it is reasoned that the correlation
)
u1 ' qv ' is positive and so
production of turbulent kinetic energy is given by the Conversely, it is argued that the remaining terms,
(2.68)
E u1 ' qv ' term.
E ' ui 'q v + E ' ui ' qv ' are
negative and it is suggested that energy contained in these terms is converted to electrical turbulent energy. It can be shown by using scaling laws that E’ can be neglected, the implication being that the turbulent kinetic energy is transferred directly from the mean electric field energy. Because of this assumption it is
32
Electrostatics, Electrohydrodynamic Flow, Coupling and Instability
reasonable to ignore the turbulent electrical energy equation and concentrate on its kinetic counterpart. The points discussed reduce eqn. (2.68) to,
(
)
⎛1 ∂ u '2 / 2 ∂ ∂u ' ∂ui ' p' ⎞ 1 + u1 ' ⎜⎜ u '2 + ⎟⎟ − E u1 ' q ' + ν i =0 ∂t ∂x1 ⎝ 2 ρ⎠ ρ ∂x j ∂x j
(2.69)
Hopfinger & Gosse characterize the nature of the turbulence generated by stating that, The propagation of the turbulent front during the first transient stage can be described as a Lagrangian diffusion process when using certain assumed statistical properties of the turbulence in the fully turbulent region.
As a by-product of their analysis they also define equations for the mean square charge fluctuations and charge flux,
(
)
∂q ∂ q v ' 2 / 2 1 ∂ u1 ' q v ' 2 + + u1 ' q v ' v 2 ∂ x1 ∂t ∂ x1 ⎛1 ∂ q '2 q '3 ⎞ 2 + κ ⎜ E1 v + qv '2 q + v ⎟ = 0 ⎜2 ∂x1 Ei '2 E i ' 2 ⎟⎠ ⎝
∂u1 ' qv ∂u1 '2 qv ' ∂q ∂p ' qv '2 E1 1 + + u1 '2 v + qv ' − − ν qv ' ∇ 2u1 ' ∂t ∂x1 ∂x1 ρ ∂x1 ρ ⎛ 2 u 'q ' ∂q ' u1 ' qv 'qv + 1 v + κ ⎜ E1 u1 ' v + 2 ⎜ ∂x1 Ei ' Ei '2 ⎝
2
(2.70)
(2.71)
⎞ ⎟ ⎟ ⎠
2.7.3 Transient Turbulence Hopfinger and Gosse [29] also analyze the transient period that turbulence occupies from the instant that a voltage is applied to the electrodes. They divide this period up into two stages (i) the first transient, representing the phenomena from the time after the liquid has been set in motion until the instant when the turbulent front reaches the receptive electrode; (ii) a second transient where the turbulence adjusts to steady-state conditions. The first transient stage is shown in a series of Schlieren images in Fig. 2.11. It is assumed that the smallest turbulent eddies are comparable to the product of the steady state bulk velocity and the electro-viscous timescale. A paper by Felici [20] notes, in reference to Hopfinger and Gosse [29] that the turbulent front travels ahead of the charge front in the transitionary regime.
2.7 Electrohydrodynamic Turbulence in a Propagating Flow Front
33
Fig. 2.11 Schlieren photographs in nitrobenzene, Va = 18kV, L = 0.56cm; (a) 0.95ms, (b) 1.9ms, (c) 2.9ms, (d) steady state [29]
2.7.4 AC Turbulence Qualitative and quantitative data on the behavior of turbulence when an alternating voltage is applied to any electrode arrangement is in limited supply. Only Felici [20] seems to make any observations, stating that, perhaps obviously, for a plane-plane gap it is found that turbulence occurs at a field strength considerably lower than the liquid breakdown strength.
2.7.5 Current and Voltage The electric Nusselt number was introduced by eqn. (2.64) but can also be defined as,
Nu E =
I I0
(2.72)
34
Electrostatics, Electrohydrodynamic Flow, Coupling and Instability
I0 is usually taken as the maximum conduction current prior to the occurrence of electroconvection; as Felici [37] comments, the point at which electroconvection starts is identified by the sudden jump in observed current as the voltage is increased. The electric Nusselt number tends to grow with increasing EHD motion until reaching a saturated value for fully turbulent behavior in the system as discussed previously. Felici [37] argues that the quantity M, introduced in section 2.5.4, measures the “efficiency” of electroconvection. For unipolar convection Felici quotes that for steady state and strong turbulence the electric Nusselt number is √(M/3). The average local velocity of the turbulent flow is estimated to be 1/3 E√(ε/ρ) and exceeds the equivalent ionic drift velocities by a factor of √(M/3). This can be seen in Fig. 2.10. The electric Reynolds number, eqn. (2.51), that signifies the transition from a partially turbulent to fully turbulent system is [28, 38],
Re E ,t ≈
I (μA)
1st Transient
Tc ≈ 10 9
(2.73)
2nd Transient
2
90 80 70 60 50 40 30 20 10
1
0
2
4
6
8 10 t (ms)
12
14
Fig. 2.12 Variation of the current with time. (1) Observed current, (2) Calculated, --difference between (1) and (2) [29]
2.8 Chapter Summary
35
Once fully turbulent conditions have been attained then a mean charge density exists in the convected liquid bulk. Experimental data on the injection current as a function of time is illustrated in Fig. 2.12. The second line plots a theoretical relation derived by Hopfinger & Gosse [29] and more details can be found in their paper. As can be seen, the fit is debatable however this does provide an approximate magnitude for the steady-state current. In their analysis of the turbulent structure Hopfinger & Gosse state that they do not expect the turbulence to be isotropic as the transient timescales are much faster than those needed to achieve the isotropic turbulence state. As a result they anticipate that
u1 ' 2 , the electrode normal mean squared velocity fluctuation, is
much greater than
u 2 ' 2 and u 3 ' 2 , the in-plane quantities.
2.8 Chapter Summary The fundamental equations governing electrostatics and the physics associated with this subject for a pure incompressible dielectric liquid were introduced first, establishing that charge must be present in the liquid for electrical forces to have any effect. This was followed by an explanation of how the charge is transported and a description of the various terms that contribute to the associated transport equation. The extended Navier-Stokes equations with incorporated electrostatic terms have been defined to demonstrate how the charge, momentum and electric field variables are highly coupled. A discussion of the energy equation is also made here to show how charge also affects this. Relevant timescales and nondimensional numbers associated with the EHD system have been defined, in particular a brief introduction is provided concerning the T, C and M parameters widely quoted in the available literature. The final sub-sections discuss the underlying physics associated with EHD flow including the initiation of the charged convective plumes and transitionary regime to turbulence before discussing the fully random chaotic events observed after this. The overall aim of this section is to address the physics and flow types occurring in the charged liquid upstream of the atomizer and highlight the role that charge has on this when present.
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Chapter 3
Charge Injection into a Quiescent Dielectric Liquid 3 Charge I njection into a Q uiescent Dielectric Liquid
3 Charge Injection into a Quiescent Dielectric Liquid Abstract. Upstream of the atomizer orifice the dielectric liquid remains in a continuous phase. The physics acting on this region were discussed in the preceding sections however the analysis given automatically assumes that an electrical charge is present. The aim of this chapter is to describe how the electrical charge actually enters the dielectric liquid, the effect on the measurable electrical quantities and, depending on the operating conditions of the atomizer, additional physical phenomena that may arise.
3.1 Charge and Field Distribution 3.1.1 Field Emission and Ionization How charge enters the liquid is an important factor as this affects the whole operation of the atomizer. Field emission and field ionization processes in gases are well understood and explanations for them can be found in numerous text books. Kuffel et al. [39] is a good source for general concepts whereas Schmidt [40] explains these in the context of dielectric liquids. Field emission occurs when high electric fields are present on the metal surface. The field reduces the potential barrier of the metal and so allows the electrons to escape via a quantum tunneling process. Generally field strengths of 109-1010V/m are required for emission to take place and these can be attained in applications involving gaseous mediums such as in electrostatic precipitators. Field emission was believed to be responsible for the injection of space-charge into liquid dielectrics at one time but this was found not to be the case for the majority of dielectric liquid applications as the field strength required for this process is much greater than those observed. Evidence for this can be found in experimental references such as Atten [32]. The Fowler-Nordheim equation is used to relate the field emission current density, J, to the field at the surface, E, and the work function of the emitting electrode metal, Ø.
J=
2
AE e 2 φt ( y )
⎡ φ 3/ 2 ⎤ ⎢ − B E v ( y )⎥ ⎣ ⎦
(3.1)
All other variables in the above expression are non-empirical parameters [41]. Field ionization is in effect the opposite process to field emission. Electrons from liquid particles near the anode are emitted into the metal surface therefore generating positive ions in the process [40]. Again, it is not believed that this is a standard mechanism by which space-charge enters dielectric liquids.
J. Shrimpton: Charge Inject. Sys.: Phy. Principles, Experi. & Theore. Work, HMT, pp. 37–60. springerlink.com © Springer-Verlag Berlin Heidelberg 2009
38
3 Charge Injection into a Quiescent Dielectric Liquid
3.1.2 Electrochemical Alj et al. [42] take a combined experimental and theoretical approach to provide evidence that charge carriers are created either in the bulk of the liquid or at the metal-liquid interface when the field strength is not high enough for field emission to occur. The charge creation mechanism in both cases is similar but there are subtle differences. In the first method an ionic-dipole can form within the liquid, which is then separated by the applied field, creating the charge carriers. In the case of the metal-liquid interface, the charge carriers form and are extracted out of the image-force region. Strictly speaking only the steady-state charge-injection mechanism can be termed an electrochemical process. Denat et al. [43] suggest that the transient current build-up identified in Fig. 3.1 is a slightly different process. They reason that because the steady-state rise time is so short that the current cannot be caused by electrochemical reactions. Instead, it is implied that dissolved ions are already present at the electrodes and so the field only extracts these rather than creating them when operating in the transient current region.
300
V=4.9kV
V=4.0kV
I (nA)
200 V=3.2kV
100
0 0
V=2.0kV
10 20 30 40 50 60 70 80 90 t (ms)
Fig. 3.1 Time dependence of conduction current in cyclohexane with 10-2 Ml-1 tetramethylphenylendiamine and with 2.10-4 Ml-1 triisoamylammonium picrate, between stainless steel electrodes (spacing d = 1.5mm) [43]
3.1 Charge and Field Distribution
39
3.1.3 Ohmic Conduction Denat et al. [43] have carried out a comprehensive study into charge injection under low strength electrical fields. Within this region the current follows Ohmic behavior and the authors note that only the supply current influences how much charge is present in the liquid. Time dependence is also identified, an example of which is shown in Fig. 3.1. Atten and Seyed-Yagoobi [44] take a slightly different approach to this subject in their paper and examine the instance of exclusive conduction through slightly conducting liquids. They state that heterocharge layers of finite thickness appear in the vicinity of the electrodes and that the heterocharge mechanism is only of significance when the electrode surfaces have comparatively large radii.
3.1.4 Space-Charge The fundamentals of electrohydrodynamics including the space-charge concept are covered well in a number of references already mentioned in chapters 2 and 3 as well as others such as Felici [37]. As explained above, ions are created via an electrochemical process either at the injecting electrode or, depending on the electric field strength, within the liquid. As Felici remarks, the presence of these ions does not necessarily mean that the liquid is “electrified” and it is only when there is an imbalance of positive and negative ions within the dielectric that the term space-charge is appropriate. However, both processes are contingent on the electric field, and hence the voltage being present. If the voltage is removed from a space-charged system then the charge imbalance will gradually reduce until none is present. The presence of space-charge reduces the local electric field strength, therefore for a constant voltage a steady condition will be reached. This is known as the space-charge limited condition (SCL) where the space-charge generated is in equilibrium with the electric field strength. An increase in the former will reduce the latter, preventing further space-charge injection.
3.1.5 Point Sharpness The point sharpness is an important factor in determining what charge injection process is occurring and the characteristics of this. Denat et al. [45] conducted a series of experiments varying the pressure and tip radius of a point-plane electrode arrangement. They found a distinct change in behavior at a tip radius of 0.5μm. For tip radii smaller than 0.5μm the current-voltage characteristics suggest that the Fowler-Nordheim equation is obeyed for negative polarity points (see section 3.1.1) and so field emission is thought to be occurring. In the anodic case field ionization occurs instead. Above 0.5μm, and for negative polarity points, the current-voltage relationship can no longer be described by the Fowler-Nordheim eqn. (3.1) suggesting that field emission is no longer occurring and that electrochemical processes are dominating. In addition to the features highlighted here, above a specific threshold voltage, high frequency current pulses are also experienced. As explained further on in this section, these coincide with the
40
3 Charge Injection into a Quiescent Dielectric Liquid
formation of bubbles at the tip and the resulting dynamics will be covered in section 3.3. The tip radius also has an effect on the current-voltage relationships as will be discussed in section 3.2.
3.1.6 Hyperbolic Field Expression For a point-plane arrangement Coelho & Debeau [46] derive an expression for the electric field strength along the axis between the two. This assumes a hyperbolic tip profile which is usually the case in experimental work and is approximately,
E (x ) =
2aV0 ⎛ 4a ⎞ ln⎜ ⎟[x(2a − x ) + (a − x )r ] ⎝ r ⎠
(3.2)
The potential, V0, has its usual meaning, a is the distance between the tip radius centre and the plane surface, r is the tip radius and x is the distance from the tip to the point of interest. In most cases r << a and so a can be simply approximated as the axial distance between the tip and plane. Note that eqn. (3.2) is the expression for an electric field with no space-charge present and is sometimes used as an approximation carried out in EHD analytical work. An attempt to produce an analytical expression for the case where space-charge is present is also attempted in the same paper by Colheo & Debeau, however this is more involved than that given in eqn. (3.2) and so will not be repeated here.
3.2 IV Characteristics of Point-Plane Systems 3.2.1 Steady-State Behavior Atten et al. [32] have carried out a comprehensive study utilizing a point-plane arrangement. A typical current-voltage plot can be seen in Fig. 3.2 consisting of, in order, the initial Ohmic, transitional quasi-Ohmic and, finally, space-charge limited regions. The departure from Ohmic current is found by the authors to occur at a threshold voltage, Vth = 7 +/- 1kV. Substituting values of 3μm for the tip radius and setting the gap at 12mm gives a non-space-charge limited field strength of 4.8 +/- 0.7MV/cm using eqn. (3.2). Voltages past this threshold were observed to create non-regular current pulses and cause a weak blue light near the cathode tip which is suggested by other workers in the field to be a liquid corona effect. Work carried out by Atten et al. focuses on characteristics beyond the threshold voltage identified, specifically the dependence of point-plane gap distance with the injection current. A plot characterizing this dependence can be seen in Fig. 3.3. They suggest that the injection current follows the equation,
3.2 IV Characteristics of Point-Plane Systems
41
-6
10
+2 -7
Current (A)
10
-8
10
-9
10
+1 -10
10
1
10 Voltage (kV)
100
Fig. 3.2 Typical current-voltage curve showing the Ohmic part, a steep increase, and a V2 variation, at d = 42mm [32]
8
5mm 8mm
I1/2 × 104 (A1/2)
7
30mm
6
60mm
5 4 3 2 1 0 0
5
10
15 20 25 Voltage (kV)
30
35
40
Fig. 3.3 Variation with applied voltage V of the square root of the injected current Ip1/2 [32]
42
3 Charge Injection into a Quiescent Dielectric Liquid
-15
I2 (A2)
10
10
-16
10 d (mm) Fig. 3.4 Variation with distance of the ratio A2=Ip/(V-Vth)2. The continuous line indicates a d-1 variation law, the dashed one a variation like d-0.7 [32]
I = [ A(d )] (V − V 2
p
)
2
th
(3.3)
The injection current, Ip, is that after the threshold voltage has been subtracted having a square power law dependence on the applied voltage difference. The constant, A, is assumed to depend only on the separation distance, d: in this case, however it should be pointed out that this study was only for a tip radius of 3μm. Figure 3.4 shows the variation of the constant in eqn. (3.3) with point-plane gap. Atten et al. identify two distinctive regimes; one up to 20mm where A2 is proportional to d-1, and one beyond 20mm where A2 is proportional to d-0.7. Atten et al. suggest that the variation in this law could be due to a change in charge distribution or to a laminar-turbulent transition of the plume. Atten [28] states that for parallel-plane electrodes,
Ip ∝V 2 /d3
(3.4)
therefore the arrangement has a voltage relation similar to that in eqn. (3.3) although a considerably different gap dependency, compared to the point-plane arrangement.
3.2 IV Characteristics of Point-Plane Systems
43
-6
10
d=4mm Transformer Oil (20°C)
A
A: r = 10μm B: r = 25μm
-7
10
C
Current (A)
LN 2 (77.3 K) C: r = 10μm
B
-8
10
-9
10
-10
10
-11
10
0
2
4
6 8 10 Voltage (kV)
12
14
Fig. 3.5 Plots of conduction current I vs. applied voltage V in the negative needle to plane configuration. d: gap length, rt: tip radius [47]
Takshima et al. [47] have provided the most recent comprehensive study into the current-voltage relationship for point-plane and needle-plane systems. Figure 3.5 shows the current-voltage dependence for their two test liquids transformer oil and liquid nitrogen. An alternative plot of the data is shown in Fig. 3.6 and illustrates the V-V0 = kI1/2 relation accepted by researchers in the field for a needle-plane arrangement and compares well with eqn. (3.3). It can be seen in Fig. 3.5 that a larger tip radius reduces the current injected into the dielectric for the same applied voltage. Similar plots for a blade-plane arrangement can be seen in Figs. 3.7-3.8. A current-voltage law for the space-charge limited conduction regime is deduced to have the form V-V0 = k’I1/3, where k’ is a constant.
44
3 Charge Injection into a Quiescent Dielectric Liquid
16 14
B
C
A
V (kV)
12
V0 V0 V0
10 8 6 4 2 0 0
1
2 1/2 ×
I
-4
10 (A
3 1/2
4
)
Fig. 3.6 Plots of V vs I1/2 from figure 17. [47]
3.2.2 Current Instabilities Studies by Haidara & Denat [48] found that above a threshold voltage (different to the threshold voltage for the onset of electroconvection) a regular current pulse regime occurs, as confirmed by other workers in this field. In addition to other findings they noted a dependence of this threshold voltage on the medium density as well. Qureshi & Chadband [49] provide a brief overview of current instabilities seen in all insulators subject to high electrical stresses. Chiefly, they identify pulses and trains of pulses, which are seen to be a precursor to eventual breakdown of the dielectric medium via the formation of vapor bubbles at the charge-injecting electrode tip. Through their investigations they found that for negative electrode points the number and magnitude of the pulses increase with applied DC voltage. This was not observed however for positive points. For this experiment it was found that positive point pulses formed more readily at lower applied voltage magnitudes as compared to negative points. Also of note in this investigation by Qureshi & Chadband was the effect of the electrode tip radius. As explained by the authors, when the point radius is less than 10μm and the voltage is set around the point required to cause pulse behavior, individual pulses are formed with fast
3.2 IV Characteristics of Point-Plane Systems
45
-5
10
A B
-6
10
-7
Current (A)
10
-8
Transformer Oil (20°C)
10
A: d = 4mm B: d = 6mm -9
10
-10
10
-11
10
0
4
8
12 16 20 Voltage (kV)
24
28
Fig. 3.7 Plots of conduction current I vs. applied voltage V in the razor blade to plane configuration. d: gap length [47]
rise times. Increasing the voltage results in increased frequency of these pulses eventually merging all of the individual pulses into a larger single pulse with, unsurprisingly, a slower overall rise time. Larger point radii result in higher inception voltages and slower rise times. Oliveri et al. [50] and Kattan et al. [51] studied positive and negative polarity charge injection however concentrated on cathodic processes. As with Watson et al. [52] they found that regular primary pulses behavior had a lower limit, not existing for points less than 0.5μm. Kattan et al. [51] highlight the role of a threshold field in their work, the field above which current pulses and bubble
46
3 Charge Injection into a Quiescent Dielectric Liquid
25 B A
V (kV)
20
15
10 V0
5
0 0
5 1/3
I
10 (× 10-3) A1/3
15
20
Fig. 3.8 Plots of V vs I1/2 [47]
formation start to appear in negative polarity systems. They note that the primary current pulse and bubble formation shows a marked dependence on the thermal electronic mobility of the dielectrics under test. For low electronic mobilities the duration of primary pulses was found to be < 4ns, the primary pulse train frequency in the kHz. It was found that the frequency of the primary pulses increased with voltage and, as confirmed by other researchers in this field, was also directly proportional to the mean current. The injected charge per pulse is independent of tip radius, voltage and ambient pressure under sub-critical conditions. This contrasts with high electronic mobility fluids where, firstly, the onset of current pulses was found to occur at a lower threshold field and, secondly, the occurrence of the primary pulses was highly irregular. The frequency of the primary pulses was found to be in the 0.1-10 kHz region and independent of the mean current. The primary current pulse accompanies the formation of vapor bubbles and it is accepted that the energy contained within this first pulse leads to formation of the bubbles. Through experiments it has been observed that internal discharges also occur within the bubble over time, manifesting themselves in secondary rebound pulses accompanying each primary pulse. Watson et al. [52] provide a comprehensive description of how they believe the two events are linked:
3.2 IV Characteristics of Point-Plane Systems
47
When a cavity discharge occurs the discharge deposits ions and electrons on the cavity wall, the internal cavity field collapses to near-zero, and the cavity surface potential approaches the cathode potential. After the cavity discharge the charge density decreases due to the expansion of the cavity and the drift of the space-charge in the field. As a result the cavity field builds up until the next discharge occurs. The [secondary] pulse frequency must therefore be governed by the charge relaxation time, and that in turn must be related to the dynamics of the cavity and the flow of charge from the cavity surface.
Fig. 3.9 Typical current bursts (a) 0.65cS 40nS/div (b) 10cS 200ns/div (c) 100cS 400ns/div (d) 1000cS 4μs/div [52]. Reproduced with permission from the IEEE.
48
3 Charge Injection into a Quiescent Dielectric Liquid
1000cS
4
Time of Occurrence (ns)
10
100cS
10cS
3
10
0.65cS 2
10
1
3 Pulse Number
10
Fig. 3.10 Time of occurrence of pulses in current bursts in 0.65, 10, 100 and 1000cS silicone fluids. The first pulse in a train defines zero time for a sequence. Applied voltage, negative dc on point electrode, 15 to 22kV, 0.6mm gap [54]
Work carried out by Qureshi et al. [53] found that the initial series of secondary pulses is regularly spaced as can be seen in Fig. 3.9. However, as these go on they become more irregularly spaced in time and the authors postulate that this coincides with instability in the cavity wall as discussed in section 3.3.4.
3.3 Vapor Bubble Creation and Pressure Dependence in Liquids
49
Denat et al. [45] performed some detailed studies into the characteristics of secondary current pulses. They conclude that the initiating primary pulse is independent of voltage, temperature and hydrostatic pressure. What is more interesting is that the number of secondary pulses following the initial one is a function of voltage, temperature and pressure; for pressures of 5MPa in cyclohexane these are eliminated altogether. The secondary pulse burst frequency was found to be proportional to the mean current, but inversely proportional to the tip radius. Typical pulse frequency data is shown in Fig. 3.10. These discharges are relevant because they allow estimates of the initial vapor bubble diameter to be made. Watson et al. [52] use the pulse train data to arrive at the conclusion that the vapor bubbles have an initial diameter of 0.5 to 1μm and pressure of 50MPa. They found that the estimated pressure value falls within the theoretical range calculated by Aitken et al. [55].
3.3 Vapor Bubble Creation and Pressure Dependence in Liquids 3.3.1 Vapor Bubble Formation Watson et al. [52] have published a comprehensive paper linking vapor bubble formation and the discharge pulses (identified in section 3.2.2) in silicone fluids of differing viscosities. They found that typically a nanosecond duration current pulse precedes the formation of a vapor cavity on the electrode surface and that the energy contained in this current pulse creates the vapor bubble. Aitken et al. [55] state that 80% of the available energy goes into heating of the liquid with the remainder forming a pressure transient. This whole process is estimated to take approximately one picosecond with the resultant plasma zone having the same dimensions as the radius of the injecting point. These authors, as well as Watson et al. [56] state that the energy injection takes the form of an electron avalanche and so reason that the resultant cavity is composed of ionized plasma. Note that this is significantly different to the standard electrochemical charge injection method highlighted in section 3.1.2. Immediately after creation of the vapor bubble Aitken et al. assume that the temperature and pressure follow a Gaussian distribution within the plasma zone. It is suggested that the pressure transient then moves outwards with a falling pressure filling the volume behind it due to dissipation effects as shown in Fig. 3.11. During this time it is assumed that the pressure acts on a much faster timescale compared to the temperature and so the latter remains relatively unchanged. Whether the pressure front can be classed as shock wave is of little relevance in the scope of this review but the interested reader should consult Aitken et al. for a detailed discussion. A falling pressure behind the pressure transient allows the liquid in the cavity to vaporize, which in turn causes further expansion of the bubble and hence further vaporization. As Aitken et al. state, the process terminates when the internal pressure falls below the ambient hydrostatic pressure and the bubble begins to collapse assuming that no more energy is injected into the system.
50
3 Charge Injection into a Quiescent Dielectric Liquid
a
b
p mo - p ∞
p mo - p ∞ VC
C p2 2
2 p io - p ∞ VC
C p1 1
1 r0
0
r
r
r0
0
Fig. 3.11 Simplified evolution of the pressure distribution in the plasma leading to shock formation: (a) initial Gaussian distribution with sound velocity denoted as Cp, (b) formed pressure front with velocity Vc; the pressure behind the front drops as the wave travels outwards. The areas under each curve should be equal [55]
80
(%)
Win
40 WV WSW
0 0
2 P∞ (MPa)
Wb
4
Fig. 3.12 The balance between the different components (mechanical (Wb + Wsw) and thermal (Win + Wν)) of the total injected energy Wi (= Wb + Wsw + Win + Wν) as a function of the applied hydrostatic pressure P∞ in cyclohexane [55]
The thermal energy injected into the vapor bubble can be further decomposed into two components; the internal energy of the vapor and the latent heat of vaporization [55]. Figure 3.12 shows a graphical breakdown of the energy types involved for bubble formation in cyclohexane. The distribution of all the different energy types within cyclohexane is given as a function of ambient pressure. Here
3.3 Vapor Bubble Creation and Pressure Dependence in Liquids
51
Win represents the internal thermal energy of the bubble, Wν the latent heat of vaporization, Wsw the energy in the pressure transient and Wb the remaining mechanical energy contained within the bubble. Aitken et al. compute an initial temperature rise of 1000K upon bubble formation but this still remains to be validated experimentally. Haidara & Denat [48] have carried out a parametric study on bubble formation characteristics varying phase and pressure and using injection points of radii 0.520μm. Experiments were carried out on anodic and cathodic point-plane systems but for brevity only the latter are reported here. They found that bubble formation is found to be pressure dependent in liquids and state that for pressures ≤1MPa the bubbles are always observed after a current pulse is detected. When the pressure exceeds the critical pressure for the liquid (≥ 12MPa) they remark that bubbles are no longer observed however the current pulses still occur.
3.3.2 Vapor Bubble Growth: Pulsed Voltage Operation If the electric field is removed immediately after vapor bubble formation then the dynamics of the system follow those defined by Rayleigh [57] and no significant electrical influence is observed [52]. In experiments carried out by Watson et al. they found that this was the case for low viscosity silicone fluids equal to and below 10cS. Figure 3.13 illustrates this mechanism for an applied short duration pulse of 632ns [53]. Similarly, if electrostatic forces cause the charged vapor cavity to detach from the surface of the electrode then the same bubble collapse sequence will occur. Oliveri et al. [50] and Kattan et al. [51] have carried out a parametric study of vapor bubble formation varying injection tip radius and ambient pressures. Both papers share common authors and so a lot of the points made overlap between the two. Through experiment they found that for sharp points of 0.1μm and less that the injection current was continuous and that no vapor bubbles were formed, reasoning that a field emission or ionization process was occurring (as outlined in section 3.1.1). They suggested that during the growth, collapse and rebound phases of the bubble, Fig. 3.14, shock waves radiate from it which correlates with Aitken et al. [55] assessment. However, above the critical fluid pressure no bubbles are seen with only the shock wave and current pulse being detected. From their investigations Oliveri et al. [50] and Kattan et al. [51] also concluded that erosion seen on the injection tip is only a problem when the ambient pressure falls below its critical value and attribute it to cavitation damage. An example of this is shown in Fig. 3.15. Oliveri et al. [50] also found that as the hydrostatic pressure is increased that the volume, hence the radius, and the lifespan of the bubbles are reduced. However, they also discover that the linear relationship between the first and second rebound pressure peaks remains constant regardless of the initiating primary pulse energy and the ambient pressure.
52
3 Charge Injection into a Quiescent Dielectric Liquid
80
Discharge Radius (μm)
60
40
20
0 0
10
Time (μs)
20
30
Fig. 3.13 Showing cavity radius vs. time in 0.65cS fluid. 632ns long single pulses of 15kV amplitude. Illustrates growth and collapse of a cavity plus several rebounds [53]
3.3.3 Vapor Bubble Growth: Constant Voltage Watson et al. [56] identify the major components affecting cavity expansion as the kinetic energy of the fluid surrounding the cavity, viscous losses in the fluid, the volume work done against ambient pressure and the work done against surface tension. Carrying out an order of magnitude analysis the authors demonstrate that the kinetic energy in the initial formation stage is approximately five times larger than the PV work for ambient pressures less than 1atm and several orders of magnitude greater than the surface tension. From this they justify only considering inertial and viscous forces in the energy balances carried out on the cavity expansion in systems under standard atmospheric conditions. In terms of the force driving the expansion a reasoned argument is presented by Watson et al. [52] over whether this is by internal pressure or by the applied
3.3 Vapor Bubble Creation and Pressure Dependence in Liquids
53
4.34 R 'm
R (μm)
Rm
t1
0
336 t (ns)
Fig. 3.14 Bubble radius vs. time in isooctane for a series of 30 events; P0 = 519kPa and W = 6.88 nJ [50]
electric field. They arrive at the conclusion that the expansion of the bubble following the initial growth stage is driven by the electric field for the case of a constant applied voltage. Some contribution is made by the internal vapor pressure but the authors conclude that expansion is only a weak function of the fluid volatility. Two possible forces are seen to act against this electrostatic expansion; inertia and viscosity. For low viscosity liquids, less than 10cS, inertial forces dominate and for high viscosities the opposite is found. For a continuously applied electric field Watson et al. [52] split the growth into three stages. Stage one is, as explained previously, the energy injected into the vapor bubble via the primary current pulse. Following this, field driven expansion occurs, and finally EHD instability at the vapor bubble wall manifests itself leading to streamer formation and ultimately breakdown across the electrically stressed gap. As discussed before, for low viscosity liquids inertia at the bubble wall is the main force acting against the electrostatic expansion of the cavity. Watson et al. [52] arrive at an expression for the cavity radius as a function of time by assuming that space-charge effects can be ignored and that the field in the cavity is at or near zero due to internal discharges. This allows the hyperboloid tip field approximation eqn. (3.2) to be used and after some further manipulation of the energy balances yields,
54
3 Charge Injection into a Quiescent Dielectric Liquid
Fig. 3.15 (a) Profile and (b) photographs of the needle before and (c) after an experiment [51]. Reproduced with permission from the IEEE
⎡ 2εV 2 t 2 ⎤ R(t ) ≈ ⎢ 1 / 2 ⎥ ⎣ a ρ ⎦
2/7
(3.5)
where a is the tip radius. Figures 3.16-3.17 are taken from Watson et al. [52] and show the relationship of cavity diameter with elapsed time. As can be seen, the (V2t2)2/7 relationship has a tendency to over-predict the cavity size at any time point and this is attributed to the fact that at the lower voltages the assumption about the space-charge is not an adequate approximation. The authors suggest that at the limit of breakdown the effect of the space-charge on reducing the localized electric field is small in comparison to the magnitude of the imposed electric field hence the better fit at higher voltages. For high viscosity fluids (i.e. ~1000cS) the electrostatic work done is balanced against the viscous energy losses instead of the inertial losses to give,
⎡ 3εV 2 t ⎤ R(t ) ≈ ⎢ 1/ 2 ⎥ ⎣ 32μa ⎦
2/3
(3.6)
3.3 Vapor Bubble Creation and Pressure Dependence in Liquids
55
18kV
180 160
15kV
Cavity Diameter (μm)
140 120
13kV
100 80 60 40 20 0 0
1
2 3 μ Time ( s)
4
5
Fig. 3.16 Cavity diameter vs. time for 0.65cS silicone fluid. Point-plane gap, length 0.6mm. Pulse voltages 13, 15 and 18kV [52]
Cavity Diameter (μm)
56
3 Charge Injection into a Quiescent Dielectric Liquid
160 120
13kV 15kV 18kV
80 40 0 0
1
2
3
4/7
(100 Vt)
Fig 3.17 Cavity diameter vs. (V2t2)2/7. Data from Figure 28. The solid line, based on (3.5) indicates the growth of an inertial cavity at the space-charge limit [52]
Figures 3.18-3.19 show the correlation for high viscosity fluids. Here it can be seen that the model prediction improves with increasing electric potential in a similar manner for the inertially limited growth model. Naturally, a transition regime exists for cavity formation. Watson et al. [56] infer that this point is sharply defined and that the properties that define whether expansion is inertia or viscosity controlled are defined by meeting the following criterion,
ρRU 16 = μ 3
(3.7)
where density, viscosity and cavity radius are easily defined. The characteristic velocity, U, is defined as R/τ, where τ is the timescale over which viscous losses occur i.e. the time elapsed since bubble formation. A more detailed analysis of the cavity expansion models in Watson et al. [56] identifies other limitations of the scaling laws used in the preceding analysis. Another deviation they observe is for low viscosity liquids such as n-hexane and 0.65cS silicone fluid. They find that the scaling law actually under-predicts the rate of cavity growth and suggest that in order to improve the accuracy of the analysis the effect of the vapor pressure should be considered for volatile liquids such as these.
3.3 Vapor Bubble Creation and Pressure Dependence in Liquids
57
25kV 20kV
Cavity Diameter (μm)
60
18kV
40
15kV
20 13kV
0 0
2
4
6 Time (μs)
8
10
12
Fig 3.18 Cavity diameter vs. time for 1000cS silicone fluid. Point-plane gap, length 0.6mm. Pulse voltages 13, 15, 18, 20 and 25kV [52]
58
3 Charge Injection into a Quiescent Dielectric Liquid
Cavity Diameter (μm)
200
150
25kV 20kV 18kV
100
50
0 0
50
100 (V2I)2/3
150
200
Fig 3.19 Cavity diameter vs. (V2t)2/3. Data from Figure 30. Two theoretical lines are shown, based on (3.6). The upper curve is calculated for shear viscosity; the lower curve is based on elongational viscosity (3μ) [52]
3.3.4 Vapor Bubble Evolution Pre-breakdown and breakdown phenomena in dielectric liquids are varied and only a brief overview will be presented here. The formation of the vapor bubble is only the first step in the formation of a breakdown path in the electrically stressed gap and Watson et al. [56] discuss these issues. Essentially, over a period of time wave instabilities manifest themselves on the vapor bubble surface. These disturbances then grow into finger-like branches, ‘streamers’. If composed of ionized vapor these structures will concentrate the electric field at their tips and propagate into the fluid, in this case usually being referred to as streamers. These protrusions will then branch out eventually forming an extremely low conduction path between the two electrodes and hence breakdown of the gap. Figure 3.20 illustrates this sequence well whereas Fig. 3.21 shows the decay of these into clusters of bubbles.
3.3 Vapor Bubble Creation and Pressure Dependence in Liquids
59
3μsec a
0
0.3mm
6μsec b
7 to 8μsec
c
Fig. 3.20 Streamer development in 2cS DC-200. These shadowgraph images show (a) the developing interfacial EHD instability at 3μs, leading to (b) streamers at 6μs, and (c) streamers that have almost bridged the gap by 8μs. Applied voltage, 11.5kV. Gap length 0.6mm [56]
Watson et al. [56] remark that the streamers are commonly seen to have a uniform radial spacing and hypothesize that this is caused by them growing from the regularly-spaced wave instabilities on the vapor bubble surface. They also mention that different bush-like structures can also form and propagate but are different to the regularly spaced streamers already discussed. This behavior is in contrast to that observed by Qureshi et al. [49] where the negative streamers are seen to propagate at right angles to the axis between the two electrodes. This is
60
3 Charge Injection into a Quiescent Dielectric Liquid
Fig. 3.21 Shadowgraph sequence following the occurrence of a negative discharge/ Illustrating the effect of adding a time delay between the occurrence of the trigger pulse and the taking of the shadow picture (a) 2.0, (b) 10, (c) 24, (d) 100, (e) 500, and (f) 1000μs [49]. Reproduced with permission from the IEEE
illustrated in Fig. 3.21 and the authors argue that the presence of space-charge on the axis reduces the electric field in the vicinity. The result is that the electric field is higher at right-angles to the axis and so streamer growth is favored in this direction. Kattan et al. [51] suggest that the successive discharges that take place in expanding and collapsing bubbles cause slow streamer growth. In this mechanism they hypothesize that the discharge energy promotes the formation of a string of bubbles which extend from the cathode. These in turn undergo internal discharges as they expand and collapse forming a path of ionized vapor from the cathode. Increasing the ambient pressure is found to retard the growth of slow negative streamers.
3.4 Chapter Summary The aim of this chapter was to provide an elementary understanding of how charge can enter the liquid to be atomized as well as the effect of the point geometry on the process and the resultant electric field. It is then shown how the voltage, point geometry and electrode gap affect the injection current regime. A discussion of the current instabilities that accompany the space-charge limited regime at higher voltages is also present, which leads onto the topic of vapor bubbles. It is shown that vapor bubbles have two major effects that should be considered when designing charge injection atomizers. The first is that they are initiated by current pulses which inject further deposits of charge into the liquid, in addition to the steady current observed, and so may have a further effect on transport of the charged flow. The second is that these vapor bubbles are thought to be precursors to breakdown of the dielectric liquid and so are of importance when considering atomizer design, operation parameters and methods of injecting charge into a dielectric liquid.
Chapter 4
Single Charged Drop Stability, Evaporation and Combustion 4 Single Charged Drop Stability, Evaporation and Combustion
Abstract. Before moving onto more applied and empirical aspects of the monograph, in this chapter the effect electrical charge has upon the stability, and also the evaporation and combustion processes of a single liquid drop is discussed. This may be viewed as a precursor to a discussion of the characteristics of spray plumes, which are generated by electrostatic atomizers and are comprised of electrically charged drops.
4.1 Maximum Spherical Drop Charge For ‘large’ drops, the theoretical charge a drop may hold is defined by Rayleigh Limit [58],
qray = 8 π (ε 0 σ T ) r 3 / 2 1/ 2
(4.1)
and assumes 1. 2. 3.
the liquid is a pure conductor the drop is in a vacuum the drop is not in the presence of an externally applied electric field.
A simplistic interpretation of the Rayleigh Limit is that it is the charge at which the effective surface tension becomes zero, and the drop disrupts. When this occurs the charge and mass of the parent drop must distribute amongst the products of the break up in some way. On the other hand, the charge present on the drop surface generates an electric field, normal to the surface, and this field must remain below the breakdown strength of the fluid comprising the drop and the continuum [59],
q field = 4 π ε 0 E s r 2
(4.2)
In this case, it is possible that charge can be lost from the drop, via an electrical discharge but the drop volume could remain intact. It is noted that there must exist a critical crossover radius where the Rayleigh limit eqn. (4.1) and the breakdown limit eqn. (4.2) for a drop must be equal. This critical radius is defined as, J. Shrimpton: Charge Inject. Sys.: Phy. Principles, Experi. & Theore. Work, HMT, pp. 61–77. springerlink.com © Springer-Verlag Berlin Heidelberg 2009
62
4 Single Charged Drop Stability, Evaporation and Combustion
rx =
4σ T 2 ε 0 Es
(4.3)
Early in the 20th century Zeleny [60] claimed to confirm the Rayleigh limit by studying the reaction of a meniscus at the tip of a glass capillary. The meniscus oscillated at lower voltages and eventually steadily sprayed charged particles at a critical potential. His proposal that this occurred when the instability grew when the internal pressure of the droplet was the same as the external pressure was erroneous. This was proved later by Taylor [61] in his theoretical prediction of spheroidal instability where he proved that the pressure difference must exist. Hendricks [62] measured the charge and mass of hundreds of octoil drops by using a hollow steel hypodermic needle held at +12 kV. It was conclusively proved that the Rayleigh limit was obeyed in practice (only 0.3% were slightly above). His graphical result, Fig. 4.1, has been extensively reproduced since. Doyle et al. [63] investigated the Rayleigh Limit by using a Millikan type force balance and correlated the potential gradient required to keep an evaporating charged drop between the plates with loss of charge at fission and measured the drop size at the end of the experiment using filter paper. They concluded that a charged drop lost approximately 30% of its charge after each Rayleigh Limit disruption. A similar approach was used by Abbas & Latham [64] with the additional information on drop diameter variation being recorded by observation, and the Rayleigh Limit was confirmed to an accuracy of 15%. More accurate work was presented by Schweizer & Hanson [65] who found good agreement with the Rayleigh Limit, with a scatter of about ±4%, and that the disruption process resulted in a 33% charge loss accompanied by a 5% mass loss. More recently doubt has been expressed regarding the accuracy of the calibration of the early equipment and it has been suggested by Taflin et al. [66] that the inherent errors in the size could be as much as 10%. Taflin et al. [66] reported an experiment that for the first time allowed determination of the mass and charge loss associated with the explosion in a more reliable and accurate way. The experimental apparatus is similar to that given in Doyle et al. [63], but in Taflin et al. [66] optical resonance spectroscopy was used to continuously measure droplet size, to an accuracy of better than 1 part in 104. The disruption of droplets in the range 20μm to 66μm diameter resulted in a mass loss of approximately 1% to 2.3%, and a charge loss in the range 9.5% to 18%. All the droplets were observed to burst before reaching the Rayleigh Limit, the actual critical charge being scattered around 80% of the theoretical maximum. The mass loss results are consistent with most of the earlier investigators who all found little mass loss, with the exception of Abbas & Latham [64] who measured between 20% and 30% loss. It is notable that the measured fraction of the Rayleigh Limit at which drop disruption occurred for a particular liquid was similar, even for different sized droplets. That it was different for each liquid may suggest the critical charge depends in some way on the material properties. Measurements by Richardson et al. [67] of charged drop stability were made using an electrodynamic levitator. As in the previous instance by Taflin et al. [66], light scattering techniques were used
4.1 Maximum Spherical Drop Charge
63
Charge-to-Mass Ratio (C/kg)
3
1
0.3
0.1
0.03
0.01 0.1
0.3
1 3 Drop Radius (μm)
10
Fig. 4.1 Experimental confirmation of the Rayleigh Limit (solid line) [62]
to measure droplet size, which gave good accuracy. This set of experiments was carried out in a vacuum, however, which had not previously been attempted. Experiments were carried out with two liquids, firstly with dioctylphthalate and secondly with sulfuric acid. For the first time, a good conductor, sulfuric acid, was tested and the results obtained for this liquid are somewhat different. They are not directly relevant to this study which is concerned with the stability of dielectric liquids, although they do highlight the dependence of material properties on the
64
4 Single Charged Drop Stability, Evaporation and Combustion
break up process. The droplets were held in a quadropole trap, the radius was monitored by an accurate light scattering method, and the charge was determined by periodic weight balancing. The oil droplets were found to lose approximately 15% of their charge and 2.25% of their mass on bursting, independent of their original size. Gomez & Tang [68] carried out experiments with heptane, including an additive to increase electrical conductivity, in ambient air and at atmospheric pressure. Here phase Doppler anemometry was used to measure the size of the droplets, although it is a less accurate technique than that used by Taflin et al. [66] still gives size information to an accuracy of ~1 part in 102. In other respects, however, this experiment was slightly different to the preceding examples in that the droplets were sprayed from a capillary and mean stable drop charge was found to exist between 70% and 80% of the Rayleigh Limit. Significantly an external field was present in the spray and was several orders of magnitude greater than that applied in levitation experiments with single droplets. De Juan & Fernandez de la Mora [69] report that using mean flow and current are poor indicators of mean drop charge due to the formation of satellite drops, via either primary or secondary atomization mechanisms. With this in mind, the mean charge of drops can be thought of as accurate at best, or an overestimate at worst, since the charge on the satellite drops is considered negligible by Gomez & Tang [68]. De Juan & Fernandez de la Mora [69] show the drop charge broadly varies with the cube of diameter, and that the drop charge of the large drops varies from 98% down to 55% percent of the Rayleigh Limit, decreasing with increase in flow rate. Clearly for the larger flow rates, coulomb explosions were occurring, to the extent that the satellites carry up to 30% of the charge. The findings of the above work are summarized, in chronological order, in Table 4.1. Here data is sourced from the experimental papers, or Lide [70]. The recent, accurate experimental results show that break-up occurs before and not at the Rayleigh Limit. When break-up does occur, then mass in the range 1% to 5% is ejected from the parent drop, carrying a charge approximately 15% that of the parent. Most of the experimental evidence is mutually supporting, with a few notable exceptions. Abbas & Latham [64] were alone in reporting a much larger mass loss, and Richardson et al. [67] were alone in testing an ionic liquid, which showed a much larger but more random charge loss. It is noted that the drop will break up when it reaches a critical charge level but that the charge may be ‘real’ or ‘induced’. ‘Real’ charge is free charge present on the drop surface. ‘Induced’ charge, relevant for dielectric drops, is polarization charge created by an electric field passing through the surface. Clearly, for electrically charged drops of dielectric liquids, in external electric fields there will exist both ‘real’ and ‘induced’ charge at the drop surface. Uncharged dielectric (ferroelectric) drops, in the presence of electric (magnetic) fields, as investigated by Sherwood et al. [71] and Stone et al. [72] is first considered. The break up mechanism is a function of the dielectric (conductivity) ratios between the two fluids, e.g. εd/εg (σd/σg) and liquid viscosity. Considering henceforth solely dielectric liquids and electric field interactions, where the permittivity ratio is very
4.1 Maximum Spherical Drop Charge
65
Table 4.1 Summary of Experimental Results
Source
[63]
Procedure
Droplet Immediately Before break-up γ*
Liquid
mN
m
Results
εr *
Aniline
42.12
7.06
Water
71.99
80.1
Single drop levitation and final ink spot calibration
d ( μm )
Q0 QRay
Q1 Q0
M1 M0
n
60 200
1
~0.7
~1
110
60 – 400
1
0.75
0.75
0.95
Water
71.99
80.1
[64]
Single drop levitation and regular aerodynamic balancing
Aniline
42.12
7.06
Toluene
27.93
2.379
[65]
Single drop levitation and ac spring voltage plate
n-octanol
27.10
10.3
15 – 40
1
0.77
Bromododecane
30.5
4.07
44
0.715
0.78
0.84
0.98
[66]
[67]
Single drop electrodynamic balance and optical resonance spectroscopy
Single drop electrodynamic levitator and light scattering
Dibromooctane
34.0
7.43
33
0.857
Dibutyl phthalate
34.09
6.58
20
0.747
Hexadecane
27.44
2.05
42
0.737
0.83
0.98
Heptadecane
27.9
2.06
33
0.795
0.88
0.98
1–8
1.02
0.85
0.975
0.84
0.506
0.999
Dioctyl phthalate
28.3
5.22
Sulphuric Acid
55.4
84
[68]
Electrospray and phase Doppler anemometry
Heptane and 0.3% Stadis 450
18.1
1.93
5130
[69]
Electrospray and DMA
Benzyl alcohol and Dibutyl sebacate
33.9
6
~1
0.6 – 0.8
~15
0.980.55
large, εd/εg→∞, and approximating a perfectly conducting drop in a vacuum, then a conical point forms, as predicted by Taylor [61], and charge and mass may issue from this point. When the conductivities are finite, i.e. there exists an electric field inside the drop, then the drop still deforms. Providing εd/εg≥20 conical points still form, though the apex is different from the ‘Taylor-Cone’ angle of 49.3°. Below these permittivity ratios conical points do not form, and instead the drop extends along the field direction to form a cylindrical shape. Similar behavior may be observed when free charge is present on the drop surface [73] by applying scaling laws for ‘electrosprays’, formed from cone-jets [74] to drops. Here, ‘polar’ and ‘non-polar’ models are required to explain the electrospray behavior. In a similar manner to the shape deformation of uncharged drops of Sherwood [71] and Stone et al. [72], very non-polar liquids cannot form electrosprays. To summarize, analogous behavior is observed between uncharged dielectrics in the presence of electric fields and charged dielectrics, with self-generated electric fields, and suggests that within charged spray plumes both space charge generated electric fields and externally applied electric fields affect the stability of dielectric charged drops. This is beyond the assumptions of the Rayleigh limit, which applies only to a pure conductor, with no external field disturbances.
66
4 Single Charged Drop Stability, Evaporation and Combustion
By noting the similarity in behavior between drops with electric charge in the absence of electric fields, and uncharged drops in the presence of fields, Shrimpton [75] sought to establish the magnitude of charge required to destabilize a given drop in a given electric field. Because of the fluid motion occurring inside dielectric drops [71], driven by the surface fields, a single dynamic drop analysis would require a significant computational effort.
Induced charge
Free charge Fig. 4.2 Drop break up nomenclature
The net charge at the surface was analyzed using a simplistic static approach and shows how limiting solutions match the dynamic analysis. A perfectly conducting fluid droplet has an infinite supply of charge that can move within it, and so there will never be a steady state internal electric field. This is the situation to which the Rayleigh Limit applies, but it is not true for a dielectric, but charged, drop. In the liquids used in the fission experiments listed in Table 4.1, the permittivity was finite and low, so an electric field may have existed inside the drops due to the presence of the external field, originating from either the levitating field in the case of single drop experiments, or from the potential applied to the capillary, for electrosprays. This causes polarization of the drop fluid due to the realignment of the molecules in the liquid with the electric field. As a result, a charge is induced on the boundary of the liquid and this is shown diagrammatically in Fig. 4.2, where on the right hand boundary the positive ends of the molecules are aligned, and similarly the negative ends of the molecules on the left hand boundary. It is important to note that the induced charge increases the total surface charge on one side of the drop and simultaneously reduces it on the opposite side. In terms of drop stability this means that one side of the drop is
4.1 Maximum Spherical Drop Charge
67
always more unstable than the same drop with no electric field present. Overall, the drop is always more unstable, and the location of the point of maximum surface charge density is a function of the sign of the free charge and the direction of the external electric field. It is proposed that this induced polarization surface charge, q p , can contribute, with the free charge on the surface, q f , towards the fission of the droplet. This principle leads to the simplistic expression for the critical free charge locally on the drop surface:
qray = q f + q p
(4.4)
This is stating that the drop stability condition is still the Rayleigh Limit, but that the induced charge, due to the dielectric nature of the liquid and the applied electric field, may contribute. This ensures that for a non-zero induced charge, the amount of free charge that may be carried is less than that predicted by the Rayleigh Limit. It is also shown that this approach, for zero free charge, correlates with the induced charge necessary for the disruption of uncharged drops in the presence of external electric fields. The drop surface the induced charge density is given as,
⎛ ε −εd qs , p = E g ε 0 ⎜⎜ g ⎝ εd where
ε d and ε g
⎞ ⎛ ε −1⎞ ⎟⎟ + qs , f ⎜⎜ d ⎟⎟ ⎠ ⎝ εd ⎠
(4.5)
denote the relative permittivity of the drop and continuum
respectively. It was asserted that the electric field outside the drop, at and normal to the surface, E g comprises elements due to the external field and the free surface charge. The electric field at the outer surface of an uncharged drop, in a uniform field Eo, is given by,
⎛ εd E g = −3 E 0 ⎜ ⎜ ε + 2ε g ⎝ d
⎞ ⎟ cosθ ⎟ ⎠
(4.6)
Now, considering the same drop, holding a charge qf, but with no external field,
Eg =
qs , f
(4.7)
ε gε o
Here it was assumed that the free charge qf is distributed evenly over the surface and thus does not generate an electric field inside the drop. Since eqn. (4.6) and eqn. (4.7) are solutions to the Laplace equation, they may be superimposed, and then substituting for the total E g in eqn. (4.5) to give,
⎛ ε −εg ⎛ ε g −1 ⎞ ⎟ + 3E0ε 0 ⎜ d qs , p = qs , f ⎜ ⎜ ε + 2ε ⎜ ε ⎟ g ⎝ d ⎝ g ⎠
⎞ ⎟ cosθ ⎟ ⎠
(4.8)
68
4 Single Charged Drop Stability, Evaporation and Combustion
eqn. (4.8) gives the surface charge density around the drop surface in terms of θ for a given externally applied field E0 and free charge density qs,f. The maximum will depend on the relative sign of qs,f and the complementary sign of cosθ. For qs,f >0, the maximum charge density occurs when cosθ=1, i.e.
⎛ ε g −1⎞ ⎛ ε −εg ⎟ + 3E0ε 0 ⎜ d qs , p = qs , f ⎜ ⎜ ε ⎟ ⎜ ε + 2ε g ⎝ g ⎠ ⎝ d
⎞ ⎟ ⎟ ⎠
(4.9)
Substituting into eqn. (4.4) gives the following result which predicts the maximum free surface charge a dielectric droplet in an electric field can hold:
qf qray
⎛ ε g ⎞⎧⎪ 3 ⎟⎨1 − E0 rε 0 =⎜ ⎜ 2ε − 1 ⎟⎪ 2 ε gσ T ⎝ g ⎠⎩
⎛ εd − εg ⎜ ⎜ ε + 2ε g ⎝ d
⎞⎫⎪ ⎟⎬ ⎟⎪ ⎠⎭
(4.10)
From eqn. (4.10) as E0→0, εg→1 and as εd→∞ then the Rayleigh Stability limit is recovered and as such the new relationship is well bounded. From the experimental work of Inculet and Kromann [76] if an uncharged droplet is placed in a sufficiently strong electric field, the induced charge will be sufficient to cause break-up. Using eqn. (4.10), the electric field strength to cause fission with zero free surface charge can also be obtained:
E0 =
2 ε gσ T ⎛⎜ ε d + 2ε g 3 ε 0 r ⎜⎝ ε d − ε g
⎞ ⎟ ⎟ ⎠
(4.11)
From experimental observations Taylor [61] suggests that for an uncharged drop in a uniform field, at the point of disintegration, the following correlation seemed to hold,
E0
r
σT
= C where C is a constant
(4.12)
By comparison with eqn. (4.11), an expression for this constant is obtained:
C=
2 ε g ⎛⎜ ε d + 2ε g 3 ε 0 ⎜⎝ ε d − ε g
⎞ ⎟ ⎟ ⎠
(4.13)
For a water droplet in oil, Inculet & Kromann [76] presented the experimental
1.91x10 5 VN −1 / 2 . By applying eqn. (4.10), for a water drop in oil, with permittivities estimated as ε d = 50 and oil with value of this dimensional constant as
ε g = 2.2 ,
then
C = 3.74 ⋅ 10 5 VN −1 / 2 . Although these values differ by a
factor of two the agreement is still thought reasonable since from the photographic evidence presented in Inculet & Kromann [76] the drop is highly non-spherical.
4.2 Maximum Spheroidal Drop Charge
69
Equation (4.10) shows that large charged drops are highly unstable, unlikely to ever attain their Rayleigh charges. Inspection of eqn. (4.9) shows the reason to be that drop charge varies with r3/2 whereas the induced charge varies as r2. The location of break up will always be where the surface tangent is normal to the direction of E0, and which side depends on the relative signs of qf and E0. This behavior is observed experimentally in the photographic evidence provided by Gomez & Tang [68] where all drop disruption occurs away from the spray axis. The approximate stability limits as a function of E0 and r also seem to be well predicted and it is asserted that charged drop stability in spray plumes is well represented by the modified Rayleigh Limit expression. In particular the detailed work of de Juan & Fernandez de la Mora [69] and Gomez & Tang [68] are well explained by using this model. In the latter paper, estimates of electric fields inside the spray plume are available.
4.2 Maximum Spheroidal Drop Charge The Rayleigh limit is rarely reached in practice since external forces, either electrical or aerodynamic act to deform the drop away from the perfect electrical and physical symmetry that is assumed. The thorough paper of Sir Geoffrey Taylor [61] examined the break-up of conducting water drops, and finally laid to rest the false surmise proposed by Zeleny [60] that the equalization of internal and external pressure of a droplet was the cause of drop instability. He discovered an 'accidental' solution of the Legendre function derived by, where the pressures happen to be equal and the spherical drop begins to assume a spheroidal shape. It does not occur for other integer values of the polynomial, nor when a spheroid of finite eccentricity becomes unstable in a uniform electric field. Taylor considered two approximations, one where he considered the case where the equilibrium equations are satisfied at the poles and at the equator, and secondly where the equilibrium at the equator was replaced by balancing the internal pressure, surface tension and electric field forces which acted over half the spheroid. For an ellipsoid of major and minor axis a and b and of eccentricity e, I1 represented the ellipticity parameter for approximation I, the more relevant case.
I1 =
1 ⎛ (1 + e) ⎞ ⎟ ln⎜ 2e ⎜⎝ (1 − e) ⎟⎠
(4.14)
Equilibrium equations for the normal stress at x=0 and x=a were formed and the pressure difference was then eliminated, and then a and b where expressed in terms of r0, and eˆ (=1-e2), the normalized eccentricity. Ultimately, for the charged spheroid, approximation I gave 1/2
1⎛2−e ˆ1 / 2 − eˆ3 / 2 ⎞ V (π r0 σ T )−1/ 2 = 2 2 I 1 eˆ 3 ⎜ ⎟ 1 − eˆ ⎝ ⎠
(4.15)
There is no stationary value for V at a/b≠1, consequently the only stable equilibrium condition was when the drop was spherical (V(πr0σT)-1/2=4), which is
70
4 Single Charged Drop Stability, Evaporation and Combustion
the Rayleigh limit eqn. (4.1). Taylor attributed the misconception made by Zeleny of stable spheroidal drops to the fact that they were not isolated. By calculation he found the angle of the apex (49.3°) that a charged fluid cone must possess when in equilibrium with an applied electric field. This was demonstrated earlier, unknown to Taylor, by Vonnegut & Neubauer [77] when they observed that the droplets "develop a tiny bump, and at this point it can be seen that small droplets in a mist are being ejected." Further work on conducting drop break-up was undertaken by Inculet & Kromann [76] and Inculet et al. [78]. In the first paper a large (r = 5 to 8mm) water/alcohol droplet was immersed in a neutral density oil bath and an increasing potential was applied across its horizontal axis. The drop elongated and at a critical tip field strength the surface deformed identically at both ends. Jet filaments, very similar to those observed by Zeleny [79] and Taylor [61] were produced and broke up into very fine droplets. In another experiment the drop was displaced nearer to one electrode and jet formation occurred at one end only. At the other, part of the droplet breaks off to form a small satellite drop. This was caused by charging of the (neutral) drop due to jet extraction from one end. In the symmetrical case, emission was from both ends and the drop was maintained at near earth potential. The second paper continued the work in a microgravity environment and models the deformations up to the formation of a Taylor cone and close agreement was found. Law [80] looked at the loss of charge from an evaporating drop of water. His model predicted that for conducting fluids the droplet charge density has no effect on the rate of evaporation, and concluded that the vapor was not a charge carrier.
4.3 Spheroidal Deformation of Non-stationary Charged Drops Cerkanowicz [81] extended the Rayleigh limit to non-stationary drops by linking the work of Taylor to established drop shattering equations based on the Weber and Ohnesorge number correlations published by Hinze [82-83]. It was demonstrated that charge, its effect magnified by spheroidal eccentricity at the poles of a prolate ellipsoid, may be incorporated into an effective (reduced) surface tension term, σT,eff. 2
qS 2 (8π ) ε 0 σ T,eff r 3
2
⎛N ⎞ = ⎜ I1 ⎟ ⎝ 4 ⎠
(4.16)
Using approximation I from Taylor [61], where
2 eˆ2/3 (2 − eˆ1/2 − eˆ3/2 ) 2 N = 1 − eˆ
(4.17)
Hinze showed that for a non-charged drop the maximum distortion (Δr) was found to be dependent on the Weber (We) and Ohnesorge (Oh) numbers. Cerkanowicz, using σT,eff asserted that for a charged drop
4.3 Spheroidal Deformation of Non-stationary Charged Drops
71
2 ⎛ ⎞ ⎛ NI 1 ⎞2 qS ⎟ ⎟ σ T,eff = σ T ⎜⎜ 2 3 ⎟⎜ ⎝ (8π ) ε 0 σ T r ⎠ ⎝ 4 ⎠
(4.18)
For a prolate spheroid, (Δr/rd)max was related to the major and minor axis, a and b such that 2 ⎛ ⎞ ⎛ NI 1 ⎞2 qS ⎟ ⎟ σ T,eff = σ T ⎜⎜ 2 3 ⎟⎜ ⎝ (8π ) ε 0 σ T r ⎠ ⎝ 4 ⎠
(4.19)
The constant K relates to the Ohnesorge number, such that,
2 ρ g u 2rel r ⎛ Δr ⎞ ⎜ ⎟ = K We = K σ T,eff ⎝ r d ⎠max K = 0.0850
Oh ≤ 1, K = 0.0475
(4.20)
Oh ≥ 1
⎛ Δr ⎞ ⎛a⎞ ⎜ ⎟ =1− ⎜ ⎟ ⎝ r ⎠max ⎝b⎠
1/3
(4.21)
By combining the above equations the behavior of charged nonstationary drops could be assessed. 2 ⎛ ⎛ a ⎞1/3 ⎞⎛ ⎛ 2 ρ g u 2rel r ⎞ ⎛ NI 1 ⎞2 ⎞ qS ⎟ ⎜ 1 − ⎜ ⎟ ⎟⎜ 1 − ⎜ ⎟ = K (4.22) ⎜ ⎝ b ⎠ ⎟⎜ ⎜ (8π )2 ε 0 σ T r 3 ⎟ ⎜⎝ 4 ⎟⎠ ⎟ σT ⎠ ⎝ ⎠⎝ ⎝ ⎠
Rayleigh Limit Fraction
This locus is plotted in Fig. 4.3 showing the relative reduction in the Rayleigh limit as a function of drop Weber number.
1.0 0.9 0.8 0.7 0.6 0.5 0
0.05
0.10 0.15 2 0.085 ρgUrel D/σr
Fig. 4.3 Rayleigh Limit reduction [81]
0.20
0.25
72
4 Single Charged Drop Stability, Evaporation and Combustion
The transient deformation of uncharged drops in the presence of an electric field is covered by Haywood et al. [84] for a wide range of dispersed and continuous phase relative permittivities, densities, viscosities and electric field strengths. For the case of conducting liquids the numerical model and an approximate analytical model agree very well with experimental data. For the case of most interest here, dielectric drops in air, the model predicts that for permittivity ratios less than 20 (εd/εg≤20) no critical field strength exists, at which point disruption would occur. Rather the drop will elongate to accommodate any field strength, and then fluid inertia will cause the drop to oscillate.
4.4 Models for Products of Charged Drop Disruption By assuming that the parent drop broke up into n identical siblings Roth & Kelly [85] developed a drop break up model by defining the change in distribution of energy, momentum, mass and charge in the system. It was assumed that the total system energy was constant. The energy of a drop was assumed to be defined by the surface tension and electrostatic forces. Equating the initial and final states gave 2 ⎡ q s,0 ⎤ 2 4 + π ⎥ ⎢ σ T r0 8π ε 0 r 0 ⎦⎥ ⎣⎢
= t=0
2 ⎡ ⎤ q s,1 2 4 + + π ⎢ σ T r1 ⎥ 8π ε 0 r1 ⎢ ⎥ 2 ⎢ ⎛ ⎞⎥ ⎢ n⎜ 4π σ T r 2 + q s,2 + W ⎟⎥ 2 ⎟⎥ ⎢ ⎜ 8π ε 0 r 2 ⎠⎦ ⎣ ⎝
(4.23)
t=∞
The subscripts 0, 1 and 2 refer to the parent before and after disintegration and the siblings respectively. The charge conservation, since all siblings are identical was given by,
q ⎞ 1⎛ = ⎜ 1 − s ,1 ⎟ q s,0 n ⎜⎝ q s,0 ⎟⎠ q s,2
(4.24)
Similarly assuming no evaporation, mass conservation gives, 3 3 ⎛ r 2 ⎞ 1 ⎛⎜ ⎛ r1 ⎞ ⎞⎟ ⎜ ⎟ = 1−⎜ ⎟ ⎝ r 0 ⎠ n ⎜⎝ ⎝ r 0 ⎠ ⎟⎠
(4.25)
By specifying further that the average surface charge density was constant over all drops then (equation corrected from Roth & Kelly [85]),
4.4 Models for Products of Charged Drop Disruption
q s ,1 qs,0
=
73
1
(r ) (1 − r
)
(4.26)
nqs,1 q s,2 1 M N q2s,2 + ∑∑ 4π ε 0 s02 8π ε 0 m=1 n≠ m smn
(4.27)
1/3
1+ n
2
0
3 1
/ r 03
1/3
The total sibling Kinetic Energy was defined as
W=
where the first term considers the parent-sibling repulsion and the second the sibling-sibling case. Here s02 was the distance separating the centers of parent and sibling and smn the centre to centre distances between the mth and nth siblings. The break-up radius (s02) was calculated by assuming that each sibling was joined to the parent drop by a catenary surface of cross-section of the form acosh(x/a). The parameter a was chosen to maximize s02 at zero surface thickness and was equal to r2. The assumption yields,
⎛ x1 ⎞ ⎤ ⎛ x1 ⎞ ⎡ s02 = a sinh ⎜ ⎟ ⎢cosh⎜ ⎟ − 1⎥ + x1 ⎝a⎠ ⎦ ⎝ a ⎠⎣ ⎛x ⎞ ⎤ ⎛ x ⎞⎡ + a sinh ⎜ 2 ⎟ ⎢cosh⎜ 2 ⎟ − 1⎥ + x2 ⎝a⎠ ⎦ ⎝ a ⎠⎣
(4.28)
where 1/2 ⎡ ⎤ −1 1 + (1 + 4 r k /a ) = a cosh ⎢ xk ⎥ 2 ⎣ ⎦
k =1,2
(4.29)
Only solutions for n = 2 to n = 7 were possible and the results were compared to the experimental correlations of Schweizer & Hanson [65]. They claim agreement, even though Octanol is not an insulating liquid. For insulators, charge mobility effects would be significant and a fewer number of drops would be expected to be produced, though no experimental evidence was offered. The presumption that less conductive liquids would produce smaller numbers of siblings cannot be incorporated into the model, the only liquid parameters defined are the drop charge and surface tension. Further to the work of Roth & Kelly, Elgahazaly & Castle presented a set of three papers dealing with the theory of single and multi-sibling drop break-up, and an experimental analysis to support their results. In the model of Roth & Kelly several of the assumptions limited its applicability, in particular, 1) The original charge was assumed to be distributed among the siblings and residual drop such that the final average surface charge density was constant. 2) The single sibling case was not possible.
74
4 Single Charged Drop Stability, Evaporation and Combustion
3) The model considers that the total energy before the disintegration equals that after the disintegration, thus neglecting all losses. 4) All siblings were assumed to be emitted simultaneously in geometrically regular patterns. The first paper [86] examined single sibling break up. The assumptions of Roth & Kelly are modified in several respects. The fluid was conductive so that the droplets after disintegration were assumed to possess the same electrical potential. Its effect on the charge distribution and on the energy equation was taken into account. The drop, the surface charge density (qS,0) was at the Rayleigh limit, and the initial energy being specified as
W0 = 4π σ T r0 +
q s ,0
2
2
(4.30)
8π ε 0 r 0
The initial drop disintegrates into residual (new parent) and sibling drops of radius r1 and r2 respectively. Mass and charge loss from evaporation was neglected, hence
r0 = r1 + n r 2
3
3
3
(4.31)
q0 = q1 + n q 2
(4.32)
Elgahazaly & Castle defined s02 the same way as Roth & Kelly. The difference between the models was the way that charge was apportioned to the discrete surfaces. Elgahazaly & Castle assumed the fluid was a conductor, and a constant electrical potential at breakup, hence an instantaneous charge distribution.
V=
q1
4π ε 0 r 1
=
q2
(4.33)
4π ε 0 r 2
Since the potential, V was kept constant, iteration occurs on q1 and q2. The effective total charge (q1,eff , q2,eff) can then be calculated by modifying the capacitance of each drop to account for the interaction effect of one drop charge on the other. After one iteration the estimated charges are
qT1 = ( C 11 + C 21 ) V
(4.34)
qT2 = ( C 12 + C 22 ) V
(4.35)
CXX is the self capacitance, i.e. the charge/potential ratio on the Xth conductor when the others are present but earthed. CXY is the mutual capacitance, and may be defined as the ratio of induced charge on the Yth conductor to the potential of the Xth, when all except the Xth are earthed. The final energy was then defined :-
1 2 2 2 W f = 4π σ T ( r 1 + r 2 ) + V 2
(
C 11 + 2 C12 + C 22
)
(4.36)
4.4 Models for Products of Charged Drop Disruption
75
An iterative scheme was used, stepping along all possible radii of the sibling to find the minimum final energy for the system using the conditions :1) Final energy ≤ Initial energy. 2) Final drop charge 〈 drop Rayleigh limit. The model predicts that for sibling mass ratios ≥11.1% single sibling solutions exist. For smaller ratios the sibling charge is above its Rayleigh limit and further disintegration is necessary. Varying the initial drop size (up to 200μm) or the type of conducting fluid had little effect on the results, which agreed with the results of Abbas & Latham [64] and their own. Multi-sibling break up was considered in the second paper [87]. They classify drop break-up into three states, initial, semi-stable and final. In the initial state they account for drop break-up occurring at charge density levels less than the Rayleigh limit by using a modified surface tension term to account for the effects of external forces. In the semi-stable state the parent drop disintegrated to residual and siblings. It was found that that for sibling mass ratios 〉 11.1% the sibling final charge (q2) was higher than the Rayleigh limit. The effective charge at break up (q2,eff) however, was less than the limit for all mass ratios. As the sibling and parent drop move away from each other capacitance effects reduce and q2,eff→q2. This causes further disintegrations until all siblings have charge densities less than the Rayleigh limit. These disintegrations, due to the small size of the droplet, neglect the external force effect and only have to satisfy the single sibling conditions at minimum energy. The final state was defined such that there exists a residual parent drop and a cloud of siblings divided into groups, and each of these was described as a secondary residual or a secondary sibling according to the last break up. Interaction effects are assumed to occur only within a group. The residual drop carries a charge q1, while siblings will carry effective charges less than their final ones. Elgahazaly & Castle demonstrated good agreement with Abbas & Latham [64] for initial drop sizes of 100μm and σT,eff = 0.07 N/m, although it was reported that the surface tension parameter was not critical due to the method of normalization used. The discontinuities at 11.1% in Fig. 4.4 pinpointed the multi/single sibling break up transition which exists due to the smaller interaction effects in the multisibling case which increased s02 and hence the effective charge. The more drops that are emitted the smaller the mass ratio, up to a maximum of 20. This was as reported by Ryce & Patriache [88] and Pfeifer & Hendricks [89]. For such systems where forces such as externally applied electric fields are present more siblings were expected. For drops of D≥200μm, high relative humidity or low pressure the assumption of negligible charge loss may not be valid. An experimental study has also been published by the same authors, supporting the theories published in the previous two papers . Elgahazaly & Castle [90] used a 150μm capillary to produce water drops, and correlated mass and charge information. The critical mass ratio of 11.1% was confirmed but the agreement of theory and experiment decreased sharply with the number of siblings, down to 40% error for n=6.
76
4 Single Charged Drop Stability, Evaporation and Combustion
Sibling Charge/Initial Charge
1.0
0.8
0.6
0.4
Experimental Data
Model
0.2
0 0
0.1
0.2 0.3 Sibling Mass Ratio
0.4
0.5
Fig. 4.4 Comparison of sibling to parent charge ratio (curve) with other workers [87]
4.5 Combustion of Single Drops There are only two published papers describing the effects of DC electric fields on the combustion of single uncharged fuel droplets [91-92] and one concerning such effects on the combustion of droplet pairs [93]. In 1959 Nakamura dropped burning alcohol droplets between plane-parallel electrodes and observed that the flame was extinguished if the electrodes were oppositely charged so as to produce an electric field. Recently, Ueda et al., [92] reported microgravity observations on the combustion of single droplets of ethanol, n-octane, and toluene in electric fields. The authors explained the observed increases in the burning rates of ethanol and n-octane on the basis of the convective enhancement by the electric wind. The magnitude of this velocity, which results from the motion of the positive ions, was between 10 and 20 mm/s. For toluene, the most highly sooting fuel, the
4.6 Summary
77
burning-rate enhancement is much larger than for the other two fuels. The experiments showed strong streaming of soot toward both electrodes for this fuel. The authors attribute this burning-rate enhancement to radiant energy transfer. This enhancement, however, could also be caused by the electric wind generated by positively and negatively charged soot particles creating an axisymmetric flow field with radially inward motion on the side that draws the hot flame closer to the droplet surface. Anderson et al. [94] report some preliminary findings for burning single charged drops in the presence of an electric field and also suggest that the burning rate is correlated with the strength of the electric field, though in this case the drop charge was very small.
4.6 Summary A drop can hold a maximum charge, defined by the Rayleigh Limit, or a limit defined by an electrical breakdown strength. The Rayleigh limit assumptions may be relaxed to provide a more general form. Past and present day work on charged drop disruption suggests that drop break up can be viewed as two linked subjects, depending on the conductivity of the liquid and therefore the way in which charge is held on the drop surface. For both cases repeatability of the charge and mass ratio of original to residual drop for a range of diameters is preserved. For conducting liquids the work of Taylor, Cerkanowicz, Haywood and Elgahazaly & Castle give a good description of the processes controlling drop break up of conducting liquids, with broad agreement found with a range of published work. For an uncharged dielectric drop Haywood showed that in the presence of an electric field, providing εd/εg≤20, no disruption will occur. The drop will instead elongate until an equilibrium is reached and oscillate about that. Since the charge injection method is specific to electrically insulating liquids, which have low dielectric constants, this disruption mechanism is directly relevant to drops within sprays generated by this atomization method. For stationary isolated charged drops the theory of Roth & Kelly agrees well with the results of Taflin. However the theory does have inconsistencies, in particular why the single sibling case is 'impossible' . As far as the authors are aware no experimental work has been published on the break up of moving charged dielectric drops. The reported literature concerning drop combustion, either charged or uncharged, in the presence of an electric field is very sparse. That which exists suggests that the burning rate is increased.
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Chapter 5
Charge Injection Atomizers: Design and Electrical Performance 5 Charge I njection Ato mizers: Design and Electrical Performance
Charge Injection Atomizers: Desig n a nd Electrical Performa nce
Abstract. The ease and efficiency by which fluids and solids can be atomized and manipulated by electrostatic fields has long been recognized and there now exists a plethora of applications and systems for the spraying of liquids that are at least electrically semi-conducting. A brief review is presented here, which is by no means exhaustive, and serves to demonstrate the dearth of systems able to electrostatically charge and hence spray electrically insulating fluids. The remainder of the chapter then summarises the empirical knowledge available concerning simple charge injection atomizer designs, required to electrostatically atomize electrically insulating liquids.
5.1 Overview: Electrostatic Atomization for Electrically Semi-conducting Liquids For pure conductors, simple electrostatic methods of liquid charging may be utilized to produce mono-size aerosols such as applying a potential to a metal nozzle. For semi-conducting liquids Meesters et al. [95] made use of the Taylor cone phenomena to produce ethylene glycol sprays of conductivity ~10-4(Ωm)-1 [96]. The method has advantages over mechanical micron size droplet generators since it can produce high flow rates and drop frequencies with simple operation and no clogging. The drop size distribution was usually narrow enough to produce higher order Tyndall spectra. The mono-disperse nature of many electrostatic sprays has made them ideal for ink jet printing [97] in which the ink possessed an electrical conductivity of ~10-2 (Ωm)-1 and also liquid-liquid mass transfer operations [98]. Law [3] gives a thorough engineering basis for the design and operation of liquid electrostatic atomizers for electrically conducting and semi-conducting liquids. In general, as shown by Fig. 5.1, a jet of liquid (J) issues at velocity (u) from the nozzle (N) and moves toward a sharp discharge electrode (P). By the application of an electric field the liquid is disrupted in the length of (L1). The cylindrical electrode C may also influence the shape of the liquid surface. By connecting L1, L2, and L3 (the nozzle, cylindrical and discharge electrodes) in defined combinations, three induction charging modes can be obtained. As such these combinations represent the method by which the majority of electrostatic spraying systems atomize pure and semi-conducting liquids. J. Shrimpton: Charge Inject. Sys.: Phy. Principles, Experi. & Theore. Work, HMT, pp. 79–124. springerlink.com © Springer-Verlag Berlin Heidelberg 2009
80
Charge Injection Atomizers: Design and Electrical Performance
lJ lC C N
J rJ
rc
u
Z P
L1
L2
L3
Fig. 5.1 Generalized electrode geometry [3]
a) Ionized Field Charging :- L1 and L2 earthed, high potential to L3. b) Contact Charging :- L2 and L3 earthed, high potential to L1. c) Electrostatic Induction Charging :- high potential to L2 and L1 of opposite polarity, L3 absent. Research has also been carried out using centrifugal force with the electrostatically charged fluid influencing the break-up of the ligaments coming off the disc. Bailey & Balachandran [99] described the modes of break up and the influence of an applied field. A model for estimating the drop size considered centrifugal and surface tension forces. The inclusion of an electrostatic force term was used in the calculation of the growth rate of the ligament instability. The disruption was then modeled on the varicose perturbations of the liquid column (i.e. Rayleigh break up). For tests on distilled water (conductivity of 10-4 (Ωm)-1), a 70μm size reduction was observed by the application of a 30kV potential over the non-charged case. Bailey & Balachandran [100] also analyzed the atomization efficiency of electrostatic centrifugal devices as a function of liquid. The liquid was not charged, an electric field was imposed on the disc edge by means of a ring electrode. An applied voltage of between 15 and 30 kV DC and liquids of electrical conductivities in the range of 10-4 to 10-6 (Ωm)-1 were tested. They concluded that the most efficient conductivity for atomization was 10-7 (Ωm)-1. Here ligament spacing was at a minimum. For more conductive fluids the charge relaxed and intense electric fields arose at liquid cusps, inducing corona and disrupting ligament formation. Conversely for low conductivity fluids (10-9 (Ωm)1) insufficient charge relaxation time was available for the liquid at the disc edge to accumulate enough charge to be fully affected by the electric field produced by the ring.
5.2 Overview: Electrostatic Atomization for Electrically Insulating Liquids
81
Sato [101] has made use of centrifugal force in the design of an atomizer which produces a highly mono-disperse size distribution. The atomizer consists of a hollow disc with 100 identical nozzles placed equally around its outer circumference. To obtain a narrow distribution he maintained the flow rate such that the jet issuing from each nozzle was laminar. A DC biased AC field was applied to the disc and the frequency was optimized such that the drop production rate and hence diameter was constant. The system gave drop distributions that centre on D = d = 200μm, and not d=1.89D as expected from pure Rayleigh jet break up. The multi-nozzle system in contrast gave a slightly wider distribution due to some nozzle imperfections but at a high flow rate of 230 mL/min.
5.2 Overview: Electrostatic Atomization for Electrically Insulating Liquids The charge injection method has evolved from the pioneering work of Kim & Turnbull [102] who sprayed Freon 113 using a needle with a chemically etched tip radius less than 1μm, which was placed in a glass capillary. However only extremely small flow rates and currents of 10-3 ml/s and 10-9 A respectively, could be attained. In addition the atomization performance quickly degraded because the needle tip became blunt due to the high current flux emitted. This technique should not be confused with the addition of inserts for spraying semi-conducting liquids, as proposed by Cloupeau [103] where the intention was to assist the formation of a small liquid cone and not to supply electrical charge. For charge injection devices a central electrode is maintained at a negative potential, typically 10kV magnitude, and is the source of electric charge in the liquid. The charge is ‘injected’ due to the strong and highly non-linear electric field present at the emitter tip but the mechanism is as yet not fully understood. The low flow rate limitation [102] was improved by Kelly [104-106] who made the atomizer orifice contraction an additional earthed electrode and denoted his patented design a 'Spray Triode'. Instead of using a single charge injection site [102] a eutectic material [107] was used, and it was claimed that this acted as a vast array of injection points, with the typical tip radius less than 1μm. Using this approach the atomization performance and more importantly the flow rate of the ‘Spray Triode’ were not limited by the charge supply rate because the current flux injected, per tip, was very small, and therefore the electrode did not degrade. Kelly [106] proposed that the 'Spray Triode' design operated essentially as a massively multiplexed version of the single point concept pioneered by Kim and Turnbull and therefore that an electron/ion mechanism was responsible for the charge injection. The remainder of chapter 5 is taken primarily from the publications of the author, specifically Shrimpton [108], Rigit [109], Shrimpton & Yule [110-111], Rigit & Shrimpton [112]. This specific set of references is used to give a concise overview of the basic principles of operation, and are used for the simple reason that complete data is available to the authors. Romat [113-115] has developed a similar, higher pressure design. Also of relevance is a set of patents filed by Kelly
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Charge Injection Atomizers: Design and Electrical Performance
whilst employed at Exxon [104-105,116-119] and two journal publications [106,120] relating to his ‘Spray Triode’ device. To conclude the chapter more specific technology advancements published in the literature and in patent disclosures are reviewed.
5.3 Atomizer Construction A generic atomizer, Fig. 5.2, was fabricated in three sections and the conducting metal parts were fabricated from brass for ease of machining and to eliminate
Fig. 5.2 Photograph of the atomizer
5.3 Atomizer Construction
83
corrosion problems. The nozzle sections, containing the version 1 and 2 profiles, Fig. 5.3, were threaded to allow removal and replacement onto the body of the atomizer, and these internal parts are shown schematically in Fig. 5.4. The liquid flow inlet was electrically isolated from the atomizer part on the left of the schematic that contained the high voltage electrode mount via a PTFE spacer and nylon bolts. A hole was drilled in the centre of the PTFE spacer of diameter slightly smaller than that of the steel needle electrode such that the needle was movable coaxially in both directions by a micrometer. The fit was such that the liquid seal was maintained up to the maximum liquid supply pressure of 0.6MPa. The needle was mounted on the micrometer via an insulating section of PTFE so that the electrode gap between the needle tip and the inlet to the orifice channel, Li was adjustable. This was done in order to quantify the response of the spray and leakage currents, IS and IL (defined below) to the varying electrode geometry. The charge emitter potential magnitude was varied in the range 0-25kV, after which the likelihood of breakdown along the PTFE surfaces insulating the electrode from the earthed atomizer body increased. A distinct advantage of the charge injection atomizer design is that the high voltage electrode is immersed in the electrically insulating liquid and that it is also completely encased within the atomizer body, which is earthed. This, for combustion applications has the advantage that the possibility of a corona discharge from the high voltage electrode to the ionized combustion environment is precluded. For non-combustion applications, the fact that the high voltage electrode is deeply imbedded within the atomizer body has obvious safety advantages.
EHT
MΩ L0
Li d a
b
c
Fig. 5.3 Variation of the generic charge injection design for (a) Version 1, (b) Version 2 and (c) Version 3
84
Charge Injection Atomizers: Design and Electrical Performance
PTFE Spacer Electrically Insulating Bolts
EHT Cable
Liquid
Atomizer Body ( I L Sink) Removable Nozzle
Atomizer Housing
HV Electrode
Fig. 5.4 Atomizer schematic
5.4 Nozzle Design Referring to Fig. 5.3, the geometrical variables are d, the orifice diameter, L0, the length of the orifice channel and Li the distance between the back face of the atomizer orifice channel and the tip of the emitting electrode. This contrasts with the much more compact ‘Spray Triode’ of Kelly [121] shown in Fig. 9 of that paper. The atomizers use a very simple charge injection electrode, a stainless steel needle, and this was mounted on a micrometer to allow variation in the electrode gap Li. The electrode tip varied from a radius of 60μm ± 5μm down to 20μm ± 5μm, depending on the needle used. In the charge injection process, a negative DC voltage V is applied between a sharp charge-emitter (the needle) at a point (the tip) of known radius rp and a charge-receiver plate, a plane translated a distance Li from the point, resulting in an injection current IT, as discussed previously in chapter 3. Figure 5.3 shows the key differences between the various internal geometry of the nozzle versions. The version 1 nozzle design was based on the work of Jido [122-123] and was reasonably effective providing the orifice diameter was large (~500μm). Smaller orifice diameters presented problems, as discussed below, and a ‘version 2’ geometry similar to Kelly [106] was employed. This proved more successful and the ‘version 3’ design provided improved co-axial alignment between electrode tip and orifice, and permitted successful atomizer operation down to ~100μm. Classical point-plane atomizers of d<100μm are extremely difficult to operate successfully, though other designs are capable of this, and will be discussed later.
5.5 Rig Design
85
5.5 Rig Design A typical experimental layout and electrical connections for producing a continuous charged spray of a dielectric liquid are shown in Fig. 5.5. A pressurized vessel was used as a reservoir for the liquid and for inducing liquid flow in the system. The liquid was filtered using sintered metal filters of nominally 7 μm porosity and a standard needle valve was used to maintain the required steady flow, monitored with a rotameter of range 0.10 ≤Q≤ 0.35 ml/s, at worst ±6.0% and 0.15 ≤Q≤ 2.0 ml/s, at worst ±9.0%. Nylon pipe work and push-in fittings connections ensured effective isolation of the nozzle from ground. Two high voltage supplies were used during different phases of the research. A switchable polarity Brandenburg high voltage DC supply (V=±0-30kV) and a Spellman Model SL300PN high voltage power supplied negative voltage up to – 24 kV ±1.0%. A grounded sink captured the charged spray and this measured the “spray” current IS, via a Keithley Model 6514 electrometer with an accuracy of ±0.1%. The error arising from not capturing all the spray current was assumed to be negligible and the fluctuations were inherently damped by applying a 10 point moving average to the incoming measurement for the displayed current. The leakage current IL is defined as the current that passes through the liquid, inside the nozzle, to the grounded inner surfaces of the nozzle body and measured with an accuracy of ±7.4% via a Wavetek Model 23XT digital multimeter. The upstream conduction of electric charge was assumed to be negligible.
HV Electrode
Flow Meter
HV Supply
R
μA
Leakage Current ( I L ) Filter
Spray Current ( I S ) R
μA
Liquid Supply
Fig. 5.5 A typical experimental layout and electrical connections
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Charge Injection Atomizers: Design and Electrical Performance
Regarding leakage current measurement there was a necessary compromise between accuracy and robustness of the instrument since when ‘probing’ the limits of operation of the atomization device large current transients were occasionally observed, leading to tripping of the power supply. This compromise was possible due to the fact that the leakage current is of less importance than the spray current in determining atomizer characteristics.
5.6 Liquids Used Several test liquids, physical properties of which are listed in table 5.1, were used during the experimental program [108,109] with a range of viscosities and purities. A common UK domestic heating kerosene fuel marketed by Shell as 'Homeglow 28' was used for the majority of the research. This has chemical, physical and electrical properties similar to saturated hydrocarbons with a molecular weight around 140 (~C10). Standard UK diesel oil (~C12) was also used to provide information on the effects of physical property variation. Tests were also carried out using commercially available ‘white spirit’ which is a low grade UK cleaning solvent with a mixture of aliphatic and aromatic hydrocarbons, the mass fraction of the latter being around 18%. It was noted during preliminary tests that if a liquid was recycled several times it became easier to charge, which gave an indication that molecular degradation occurred and that the charge injection mechanism is electrochemical in nature. This enforced the experimental restriction that all liquid should be sprayed only once. The resistivities of the kerosene and diesel oil were found, by measuring currents between planar electrodes, to be relatively invariant to the value of the DC potential up to 20kV, as shown in Fig. 5.6. There was a non-linear response at the lower applied potentials for white spirit and this is thought to be due to the diversity of molecular structures that comprise this liquid, and differences in their respective ion mobilities. This was largely irrelevant since the regime of atomizer operation of most interest generally used potentials of ⏐V⏐≥6kV where the resistivity profile is relatively flat. The relatively lower resistivity of the diesel oil is thought to be due to the presence of traces of detergents and other additives, however all three liquids could be reasonably classified as ‘insulators’ based on these values of resistivity. Table 5.1 Liquid physical properties Property
Unit
Reference
-
ρ
kg/m3
780
800
μ
Ns/m2
0.0009
εr
-
2.2
*estimates
White spirit
Kerosene
Diesel No.1
Diesel No.2
Mineral oil
Marcol-87
Measured
[168]
[120]
840
815
850*
850
0.0011
0.0024
0.0026
0.0250
0.0300
2.2
2.2
2.2
2.0*
2.0
[108]
5.7 Breakdown Limits and Typical Current-Voltage Response
87
11
10
10
σ (Ωm)
10
9
10
8
10
0
5
10 V (kV)
15
20
Fig. 5.6 Resistivities (σ) of the white spirit ( ), kerosene (Δ) diesel oil (◊) versus DC applied potential. [110]
5.7 Breakdown Limits and Typical Current-Voltage Response The voltage-current response for charge injection atomizers was investigated by first defining a given atomizer geometry and flow through the device. The high voltage was progressively increased and the spray and the leakage current response was recorded, and the total current is assumed to be the sum of these, as stated in section 5.5. In many ways this voltage-current analysis is a study similar to chapter 3 of this review, the difference here being that the internal geometry is more complex, and an imposed bulk flow exists. It was found that two mechanisms for limiting the maximum current, defined by different types of electrical breakdown were present. These, and a typical voltage-current relationship for each, are now discussed.
5.7.1 Sub-critical Breakdown Figure 5.7 shows the total emitted current, IT, plotted against the potential applied to the charge injection electrode, V, for kerosene, white spirit and Diesel oil for selected flow rates in a ‘sub-critical’ regime, for a ‘version 1’ nozzle design Fig. 5.3. The terminations of the curves are highlighted by circled data points and the loci of these curve terminations for each liquid over the voltage range are represented by dashed lines. Each of the curve terminations is the result of a high
88
Charge Injection Atomizers: Design and Electrical Performance
4
IT (μA)
3 White Spirit Kerosene
2
Diesel Oil
1 0
1
2
3
4 5 V (kV)
6
7
8
9
Fig. 5.7 Total current versus applied potential for kerosene (◊=0.42ml/s, ♦=0.63ml/s, ×=0.83ml/s), diesel oil (∆=0.42ml/s,▲=0.63ml/s, +=0.83ml/s) and white spirit ( =0.42ml/s, ■=0.63ml/s =0.83ml/s) in the sub-critical regime : version 1 nozzle, Li/d=6.6, L0/d=2, d=500μm [110]
voltage breakdown event. Two issues are discussed in this section (1) the voltage current response as a function of liquid properties and (2) the form of the ‘subcritical’ breakdown event. It is observed that the total emitted current is a function of applied potential (similar to the work discussed in chapter 3), but not of the bulk velocity. This suggests that at the charge injection point the charge convection by the bulk flow is unimportant and the charge injection is similar to that discussed in section 3, in particular the voltage-current response may be similar to eqn. (3.3). It is also observed that the current-voltage response is a function of the liquid properties. While the charge injection process nominally is only a function of Li and the electrode tip radius (e.g. eqn. (3.3)) the charge injected into the fluid in the small volume near the high voltage electrode tip, it must also be extracted from this region at the same rate. Here the ion mobility has an important role to play, and as such controls the overall rate of charge injected. This is reproduced in Fig. 5.7, with higher viscosity (lower ion mobility, eqn. (2.9)) liquids carrying smaller total currents for a given voltage, independent of applied bulk flow rate. The effect noted above, where a lower viscosity liquid (higher ion mobility) liquid supports a larger total injected current also has an impact on the electrical
5.7 Breakdown Limits and Typical Current-Voltage Response
89
efficiency of the device, i.e. the spray to total current ratio. An ion trajectory may be decomposed into two ‘convective’ components thus :
v i = κE i + u i
(5.1)
Here diffusive contributions are neglected and the instantaneous ion velocity vector vi is the sum of two components, the electrical mobility velocity, κEi , and the velocity vector of the bulk flow, ui . The electric field Ei may be thought of as the sum of two contributions from the potential gradient between the emitting electrode and the earthed nozzle wall and from the space charge field generated by the molecular ions themselves. Since lower viscosity liquids consist of smaller molecules their ions are more sensitive to the imposed electric field. Therefore a higher flushing bulk flow, with a predominately axial velocity component, is required to prevent molecular ions being deflected by the local total electric field, which due to the cylindrical geometry of the nozzle, has a predominantly radial component. It can be shown that this is particularly applicable near the symmetry axis, where the charged particles are created at the cathode surface. Taking a typical ion mobility of liquid hydrocarbons to be ~10-8 m2/Vs [10] and a maximum field strength of 108 V/m, which is near the breakdown strength of most hydrocarbon oils, a maximum electrical velocity component of the order 1m/s is estimated.
0.5
IS (μA)
0.4 0.3 0.2 0.1 0 0
1
2
3
4
5 6 V (kV)
7
8
9
10
Fig. 5.8 Spray current versus applied potential for kerosene (◊=0.42ml/s, ♦=0.63ml/s, ×=0.83ml/s), diesel oil (∆=0.42ml/s,▲=0.63ml/s, +=0.83ml/s) and white spirit ( =0.42ml/s, ■=0.63ml/s =0.83ml/s) in the sub-critical regime : version 1 nozzle, Li/d=6.6, L0/d=2, d=500μm [110]
90
Charge Injection Atomizers: Design and Electrical Performance
Evidence of the effect that differing ion mobilities have can be obtained by examining the spray current plotted against applied voltage as given in Fig. 5.8, for the data-set of Fig. 5.7 for white spirit and Diesel oil. The kerosene case has been omitted on the grounds of clarity and again the dashed circles and lines represent the maximum spray currents and their loci respectively as in Fig. 5.7. It is shown that the spray to total current ratios are approximately 0.1, and this is the same order as the ratio κEr u x , and as such the interpretation of eqn. (5.1) has some support. In contrast to the results for the total current response, which was a function of applied voltage only, the spray current is now a function of both voltage and flow rate. Spray currents are in general higher for less viscous liquids, however the spray to total current ratios are larger for more viscous liquids. This trend may be understood by stating that the spray current is primarily a function of axial drag forces from the bulk motion and the leakage current is due to electrical forces acting predominantly in the radial direction. Thus an increase in viscosity reduces the radial ion velocity and increases the residence time of the ion, thus moving the ion further downstream and making it more likely to contribute to the spray, rather than the leakage current. This implies that electrostatic atomization is inherently more efficient, in terms of the fraction of injected current that leaves with the spray, for more viscous fuel oils, and that the technique could have application in the atomization of the heavier fuel oil fractions. Conversely this, together with the fact that the total injected current is larger for less viscous fuel oils implies that there should be a fundamental limit regarding how low a viscosity can be and be successfully sprayed by the charge injection technique. As shown in Fig. 5.7 the quantity of charge injected in the sub-critical regime is a function of liquid physical properties for kerosene and diesel oil, and change of flow rate seems to have an effect on the lowest viscosity fuel only, white spirit. This suggests that viscosity has a key role in the charge injection process in controlling the current where the flow rate or more accurately the flow velocity at the charge injection point is low. The thickness of the high space charge concentration layer around the charge injection point (the needle tip) is thought to be influenced by the velocity profile of the boundary layer. The thickness of the space charge layer may be reduced by a thinning of the boundary layer on the electrode, which may be achieved by either a reduction in the liquid viscosity or an increase in the bulk flow velocity. The thinning of the boundary layer, for a given applied potential, would then allow more current to be injected because the charge injection process is space charge (diffusion) limited. This correlates with the observed flow rate and injected current trends and viscosity variations. This gives an explanation of the observed flow rate dependence on injected current for the relatively low viscosity white spirit, which should have a thinner laminar electrode boundary layer than the other two hydrocarbon oils, where the total injected current is invariant with flow rate. This may also explain the enhanced dependence of injected current on flow rate for the super-critical regime as discussed in the next section. Turning now to the breakdown mechanism, Fig. 5.7 shows the maximum current at breakdown is affected by the flow field with higher bulk flow velocities permitting greater current injection prior to electrical breakdown. This charge
5.7 Breakdown Limits and Typical Current-Voltage Response
91
limiting mechanism is a form of electrical breakdown and is typical of a high voltage insulation breakdown event and occurs when the electric field at a point equals the electrical breakdown strength of the hydrocarbon liquid. Since the electric field passing through the dielectric fluid is dependent on the emitting electrode, the internal geometry and the space charge distribution, which is strongly coupled to the flow field, the location of the breakdown location is difficult to pinpoint in both space and time. It was observed however that the electrical breakdown event may be approached by either maintaining a given voltage and reducing the liquid flow rate, or by maintaining a liquid flow rate and increasing the voltage applied to the cathode. Both of these operations have exactly the same effect, that is to increase the space charge concentration and this must lead to an increase in the electrical field magnitude inside the nozzle. At the critical point, the space charge concentration is such that the field exceeds the breakdown strength of the insulating fluid and precipitates an electrical short circuit inside the nozzle. This produces a surge in leakage current, the tripping of the power supply, and the effective cessation of the atomization process. The breakdown event for the sub-critical flow rate charge injection regime is a ‘catastrophic breakdown’, and is due to complete electrical failure of the insulating liquid. For liquids at rest electrical breakdown is known to occur at smaller electric field magnitudes, and therefore the voltage applied, for less dense hydrocarbons [124]. As shown in Fig. 5.7 an increase in flow rate delays the onset of catastrophic breakdown to higher applied potential, and this trend is universal for all test liquids. This is despite an increase in the total current at higher voltages which must increase the space charge averaged over the volume of liquid in the atomizer. The precise cause for the ‘catastrophic breakdown’ for the sub-critical atomizer operation is unknown. Currently it is thought that since the liquids are pressurized to only few bar gauge pressure, and significantly below their critical pressure, vapor cavities form (as proven in quiescent point-plane systems, e.g. section 3.3) and at elevated voltages streamers form and reach the earthed nozzle body. It is suggested, but not demonstrated, that larger bulk flow rates inhibit the formation of streamers and delay the onset of ‘catastrophic breakdown’. Investigation of the sub-critical regime is instructive since it has suggested the existence of a maximum ionic mobility and hence a lower limit for liquid viscosity. However it is not of useful engineering significance in terms of atomization performance due to the unstable nature of the EHD physics and therefore atomizer operation, the low maximum injected spray currents possible and the relatively poor atomization of the liquid. The remainder of this review concentrates on the super-critical regime, which is not subject to these problems.
5.7.2 Super-critical Breakdown As noted above, in the sub-critical regime, breakdown occurred before achieving a maximum spray current on a continuous IS-V curve. Experiments were conducted at higher flow rates using the same version 1 atomizer geometry, Li/d=6.6, L0/d=2,
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Charge Injection Atomizers: Design and Electrical Performance
2.0 1.6 IS (μA)
White Spirit Kerosene
1.2 0.8 0.4
Diesel Oil
0 0
4
8
12
16
20
12
16
20
V (kV) 10
IL (μA)
8 6 4 2 0 0
4
8 V (kV)
Fig. 5.9 (a) Spray current in the super-critical flow rate regime using a version 1 nozzle, Li/d=6.6, L0/d=2, d=500μm, Q=2.00ml/s for kerosene, white spirit and diesel oil, (b) Leakage current in the super-critical flow rate regime using a version 1 nozzle, Li/d=6.6, L0/d=2, d=500μm, Q=2.00ml/s for kerosene, white spirit and diesel oil [110]
5.7 Breakdown Limits and Typical Current-Voltage Response
93
d=500μm, as used above, and a new breakdown mechanism was encountered. Here the catastrophic type of breakdown described in the previous section does not occur and a maximum (optimum) spray current is always achievable for any flow rate and liquid. Figure 5.9a shows the spray current response where in this case kerosene, white spirit and diesel oil were sprayed at a flow rate of Q=2ml/s for the three liquids. The least viscous oil, white spirit, exhibits the lowest maximum spray currents and for this liquid the maximum spray current changes more gradually with applied voltage, in contrast to the more extreme responses of kerosene and diesel oil. This is attributed to the diversity of aromatic and aliphatic molecular structures, and commensurately wide range of ion mobilities, present in the white spirit. Figure 5.9b shows the leakage current responses for the same cases Fig. 5.9a and in contrast to the sub-critical regime these are relatively unaffected by changes in liquid viscosity. In fact this observation is true for the version 1 atomizer design, because the electrode gap Li is relatively large, and is not applicable to the version 2 design, where Li is much smaller. This is discussed in more detail in section 5.13. Before reaching the optimum operating condition for the atomizer (the maximum spray current) both the spray and leakage current rise steadily. After this condition the spray current is significantly reduced and the majority of the current flows to the nozzle body and to earth, although the spray current stays at a low, non-zero and near-constant value. It has been experimentally determined that for operation in this regime a stable and repeatable voltage-current-flow rate response can be obtained for all three liquids, i.e. the voltage may be increased above, then reduced below the breakdown value repeatedly and the spray responds with no observable hysteresis effect. Unlike for the sub-critical regime, there is no change in the total current, IT, drawn from the power supply after the spray current maximum occurs. This means that there is no disturbance to the charge injection mechanism near the needle cathode when the ‘super critical’ breakdown occurs. This absence of feedback between breakdown and the charge injection itself, has led the authors to refer to this regime as super-critical, as opposed to sub-critical, operation at lower flow rates. It can be inferred that the electrical and flow conditions local to the charge injection point are unchanged and the breakdown event must occur far from the charge injection site. Since within the confines of the atomizer the flow and electrical parameters are highly coupled, this suggests that the breakdown event does not occur inside the atomizer. The character of the breakdown event is completely different to the ‘catastrophic breakdown’ of the Fig. 5.10 Spark shadowgraph of the partial breakdown event using kerosene : version 1 nozzle, Li/d=6.6, L0/d=2, d=500μm, Q=2.00ml/s. Reproduced with permission from the Begell House
94
Charge Injection Atomizers: Design and Electrical Performance
sub-critical regime and it is termed ‘partial breakdown’ since from Fig. 5.9a there is some residual charge in the spray. This means that in contrast to sub-critical electrohydrodynamics, operation in the super-critical regime is inherently robust since the charge injection mechanism is completely decoupled during breakdown mechanism. Corona discharge is present in air around the liquid jet during the partial breakdown event and this was detected as static recorded in the radio frequencies. In addition a faint purple-blue glow was observed under reduced lighting conditions around the liquid jet as it emerged from the orifice. Figure 5.10a is a shadow photograph using the type 54 Polaroid film of the liquid jet under partial breakdown conditions and Fig. 5.10b shows a self-illuminated image of the corona discharge, using the type 57 Polaroid film, under reduced lighting and an exposure time of 15 minutes. It is observed that the highest light intensity occurs where the jet emerges from the orifice and this reduces along the jet and along the nozzle wall. By comparing the two images it is clear that the air at the orifice surrounding the liquid jet breaks down, however whether this is precipitated by electrical breakdown of the space charge laden liquid jet is unclear. The corona provides a path for charge to transfer from the liquid jet to the earthed nozzle body via the air, thus reducing the spray current and increasing the leakage current. The fraction of the useful spray current lost to earth via the surface corona discharge may be influenced both by the mean velocity, as discussed below, and the orifice diameter.
5.7.3 Overview of the Breakdown Regimes To summarize section 5.7, in the sub-critical regime, a complete electrical breakdown occurs due to electrical failure of the liquid inside the nozzle. In the super-critical regime, a corona discharge occurs outside the nozzle. The magnitude of voltage V that can be applied before a complete electrical breakdown of the liquid itself inside the atomizer, or a partial breakdown by corona discharge of the surface charge on the jet to the earthed atomizer body is defined as the ‘critical voltage’, VC. Figure 5.11 shows how the critical voltage VC varies as a function of flow rate Q and orifice diameter d, and also highlights the domains where ‘subcritical’ and ‘super-critical’ atomizer operation occurs. Generally, VC increased with Q and reduction in d for the version 2 and 3 nozzle designs. The value of VC for d = 140 μm is similar with d = 116 μm for the version 3 design, implying that the electric field at the needle tip E, and hence VC, is dependent on the tip radius rp [45]. The regime transition was found to approximately coincide with a mean velocity in the orifice channel of around uinj ≈ 6 m/s.
5.8 Total Current Versus Voltage: Observations The variation of the total (injected) current, IT, as a function of the applied voltage, V, for different Li/d ratios for d = 116, 140 and 254 μm at uinj ≈ 10 m/s, for a ‘version 3’ nozzle design. Fig. 5.3, using Diesel oil are shown in Fig. 5.12a-c respectively. Results in Fig. 5.12d using the version 2 nozzle design and kerosene
5.8 Total Current Versus Voltage: Observations
95
Sub-critical Regime Super-critical Regime
24 Version 2 Version 3
V (kV)
18
12
6
0 0
30
60 90 Flow Rate (mL/min)
120
Fig. 5.11 The critical applied voltage, VC before a partial (super critical regime) or complete (sub critical regime) breakdown occurred versus applied flow rate Q at Li/d = 1.0 using kerosene and the version 2 nozzle design with rp ≈ 60 μm for d = 300 (▲), 250 (□) and 150 (♦) μm and using diesel and the version 3 nozzle design with rp ≈ 5 – 20 μm for d = 254 (■), 140 (0) and 116 (+) μm. [112]
as a test liquid for d = 250 μm and uinj ≈ 10 m/s are included to quantify the effect of liquid physical properties. The termination of the curves represents the final measurement point where a partial breakdown event occurs within 1 kV of the critical voltage VC, or at the maximum applied voltage of Vmax = 24 kV if the partial breakdown did not occur, as in the case for d = 254 μm at Li/d > 3.1. The shape of the IT-V response is similar for Fig. 5.12a-d and a conclusion is that the electrode gap has much more influence on the charge injection law than the orifice diameter.
96
Charge Injection Atomizers: Design and Electrical Performance
The shape of the IT-V response is also similar to other point-plane data of previous work, and is approximately parabolic e.g. as defined in section 3, and this correspondence is now discussed more fully. 7.5
7.5
5.0
L i /d = 4.0 L i /d = 1.2 L i /d = 1.0 L i /d = 0.8 L i /d = 0.4
IT (μA)
IT (μA)
5.0
L i /d = 4.0 L i /d = 1.2 L i /d = 1.0 L i /d = 0.8 L i /d = 0.4
2.5
2.5
0 0
6
12 V (kV)
18
0 0
24
7.5
18
24
12 V (kV)
18
24
7.5 L i /d = 4.0 L i /d = 2.4 L i /d = 1.2 L i /d = 0.4
L i /d = 4.9 L i /d = 3.1 L i /d = 2.4 L i /d = 1.6 L i /d = 1.0 L i /d = 0.6 L i /d = 0.2
5.0
2.5
0 0
12 V (kV)
IT (μA)
IT (μA)
5.0
6
2.5
6
12 V (kV)
18
24
0 0
6
Fig. 5.12 Total current, IT versus the applied voltage, V at a constant injection velocity of uinj ≈ 10 m/s using diesel and the version 3 nozzle design with rp ≈ 5 – 20 μm for (a) d = 116 μm, (b) d = 140 μm, (c) d = 254 μm, (d) d = 250 μm [112]
5.9 Total Current Versus Voltage: Comparison to Quiescent Fluid Data For quiescent liquids two regimes of charge injection processes were observed [45] and later confirmed by Bonifaci et al. [125] for point electrodes held at a negative polarity, and the same is observed here.
5.9 Total Current Versus Voltage: Comparison to Quiescent Fluid Data
97
A relationship of the variation of IT with V may be developed of the form
I T = A(V − V0 )
n
(5.2)
where A, n and V0 are constants. As discussed in section 3 and above, n~2, A is a function of the geometry (and the fluid), and V0 is a ‘threshold voltage’ (c.f. eqn. (3.4)) that delineates the two regimes of charge injection processes (not the ‘subcritical’ and ‘super-critical’ atomizer operating regimes). The threshold voltage V0 appears to occur at V ≈ 1.9, 2.9 and 4.1kV for d = 254, 116 and 140 μm respectively using the version 3 nozzle design. The slight increases in the value of V0 are due to the order in which the experimental work occurred, started with d = 254 μm followed by d = 116 and 140 μm. As the experiment proceeded the needle tip gradually melted and as a result increased the size of the tip radius, rp from approximately 5 to 20 μm. For the version 2 nozzle design using kerosene as the test liquid with rp ≈ 60 μm, V0 appears to occur at V ≈ 4.0kV for d = 250 μm. From Fig. 5.12c, where d = 254 μm, there seems to be two regions of dependence where the parabolic profile of IT-V is different in each case. Where Li/d is relatively large, i.e. Li/d ≥ 3.1, there is a parabolic profile of IT-V over the range 0 ≤ V ≤ 24 where n ≈ 2. This behavior has also been observed in earlier work using kerosene for similar atomizer geometry [108] as shown in Fig. 5.12d for Li/d ≥ 3.2. The relationship in eqn. (5.2) has also been observed in charge injection work of other workers for quiescent liquids [47], where A in eqn. (5.2) is described as a function of Li with an essentially quadratic variation of IT-V (i.e. n = 2). Here the medium is in motion, and for larger Li/d a I T = AV relationship is apparent. For instance, taking the Li/d = 2.4 data, denoted by the (◊) data points in Fig. 5.12c, the relationship between IT and V is approximately parabolic, where n ≈ 2 for V<4.8kV. However, for smaller Li/d, n > 2, which is similar to the results of Higuera [126] for applied voltage V well above the threshold value V0 . A regime transition [45] was found to occur at a threshold voltage V0 for emitter tip radius of rp > 0.4 μm, which together with the liquid physical properties, inter-electrode distance Li and the size of rp determined the magnitude of electric field Ep at the emitter surface [45] as 2
Ep =
2V0 r p ln 4 Li r p
[
]
(5.3)
The IT ∝ V2 relationship well above the threshold value V0 with larger tip radius of rp > 0.4 μm was reproducible [45] and was also found in an earlier work by [47] and later by Atten et al. [32] for needle-to-plane configurations. The constant A in a
I T = A(V − V0 ) relationship was found to be a function of inter
electrode distance Atten et al. [32] with a A2 = Li-m variation law, where m ≈ 0.7 to 1.0 for insulating transformer oil of viscosity μ = 0.024 Ns/m2. The value of VC was found to be a function of liquid viscosity [127], electrode geometry [52] and polarity [128], and inter electrode distance [129]. The breakdown processes lead
98
Charge Injection Atomizers: Design and Electrical Performance
to the melting of the needle material [130], and avoiding the complete electrical breakdown meant that the value of the applied voltage in the present investigation was limited to the critical value VC. The needle tip radius was rp ≈ 5 μm ± 1 μm for the work discussed here [112] compared to rp < 1 μm for the specialized UO2 emitter material in the “Spray Triode” [106] and rp ≈ 60 μm in the case of Shrimpton & Yule [110]. Figure 5.13 shows the electron micrograph of a used needle tip, obtained using a JEOL Model JSM 5300 scanning microscope with a 350× magnification at the end of the experimental work. The electrode surface (i.e. in the ring) seems to reveal melting processes at the needle tip, which increased rp from approximately 5 μm before the start of the experimental work to approximately 20 μm towards the end. Breakdown processes with high injected currents IT are believe to lead to the melting of the needle material [131]. However, the size of rp is known only to effect the IT-V characteristic [45] and the value of the threshold voltage V0 as previously. We assume it does not have an effect
Fig. 5.13 Electron micrograph of needle tip with a 350× magnification, and the measured tip radius, rp ≈ 20 ± 1 μm. [112]
on the critical (i.e. optimum) atomizer operation point as the applied voltage is well above the threshold voltage V0. The critical point here is defined as the point where VC is well above V0 but below a certain value at a breakdown or discharge event. The optimum value VC is defined as 1 kV before a complete electrical breakdown occurs (i.e. in the sub-critical regime), or before the charges in the liquid jet induce corona in the surrounding gas (i.e. in the super-critical flow regime).
5.9 Total Current Versus Voltage: Comparison to Quiescent Fluid Data
99
The derivation of a A2 = Li-m variation law was compared with the result of Atten et al. [32] who used insulating transformer oil of viscosity approximately 10× higher than diesel oil and similar tip radius size of rp ≈ 3 μm. Our rate of IT increase with V is predicted to be higher than that from Atten et al. [32] as the applied bulk convection in the present study helps to strip ions from the charged layer in the vicinity of the electrode tip especially at small Li/d ratios. Small Li/d and small d conditions are more efficient because the needle tip, the charge injection site, has been translated into a region of faster moving liquid. Thus, the comparison was made for data obtained at a low applied bulk flow and at Li/d ratios bigger than 1.0. Figure 5.14a shows an I-V relationship for the version 3 nozzle design using d = 254 μm with 1.0 ≤ Li/d ≤ 4.9 μm at a low and steady liquid injection velocity of uinj ≈ 6 m/s, up to the maximum IT. Large relative errors at low voltages, V < 4 kV, i.e. below the threshold voltage, also make it unlikely for the data points to be included in this analysis. When subtracting the low Ohmic current from IT, the variation of IT = A2(V-V0)2 is shown by the straight lines in Fig. 5.14b. The I T = A (V − V0 ) relationship was used by Atten et al. [32] for analyzing the variations of IT above the threshold voltage V0. By plotting A2 as a function of Li, Fig. 5.15, a continuous line with a L-0.6 variation for Li < 800 μm and a dashed line with a L-1.2 variation for Li > 800 μm
6
L i /d = 4.9 L i /d = 3.9 L i /d = 3.1 L i /d = 2.4 L i /d = 1.6 L i /d = 1.0
I1/2 (A)1/2 T
IT (μA)
4
2.4 L i /d = 4.9 L i /d = 3.9 L i /d = 3.1 L i /d = 2.4 L i /d = 1.6 L i /d = 1.0
2
0 0
8
16 V (kV)
24
0 0
8
16
24
V (kV)
Fig. 5.14 (a) IT and (b) IT1/2 versus applied voltage V at a constant injection velocity of uinj ≈ 6 m/s using diesel and the version 3 nozzle design with rp ≈ 5 μm for d = 254 μm [112]
were found. For comparison, Atten et al. [32] reported L-1 and L-0.7 variations for L < 20 mm and L > 20 mm respectively for 5 ≤ Li ≤ 60 mm in a quiescent liquid of higher viscosity. The differences in the correlation may be due to a laminarturbulent transition of the EHD plumes as suggested by Atten et al. [32]. A refined experimental investigation using more viscous liquids with the charge injection atomizer is required in order to confirm the variation laws.
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Charge Injection Atomizers: Design and Electrical Performance
0.04
A2
0.03
0.02
0.01
0 0
250
500 750 L (μm)
1000
1250
Fig. 5.15 Variation with point to plane gap Li of the ratio A2=I/(V-Vo)2 where the continuous line indicates a L-0.6 variation law for Li < 800 μm and the dashed one a L-1.2 variation for Li ≥ 800 μm at a low and steady liquid injection velocity of uinj ≈ 6 m/s for d = 254 μm using diesel and the version 3 nozzle design with rp ≈ 5 μm. [112]
5.10 Effect of Flow-Rate/Injection Velocity The variation of injection current IT with V for different volume flow rate Q is briefly investigated here to further elucidate the empirical charge injection law for V > V0, particularly for small Li/d ratios, where the atomizer is most efficient, as a function of applied bulk flow. Figure 5.16a-c show IT versus V for different Q at Li/d = 1.0 using the version 3 nozzle design with diesel oil as the test liquid for d = 116, 140 and 254 μm respectively. Results from Shrimpton [108] in Fig. 5.16d using the version 2 nozzle design and kerosene as a test liquid for d = 150 μm and for different Q at Li/d = 1.0 is also shown here for studying the effect of liquid physical properties on the charge injection process. The general trend of the graphs shows that the rate of IT rise with V increases and is almost independent of applied flow rate Q and weakly dependent on liquid
5.11 Specific Charge Regimes
101
physical properties. Higher magnitudes of maximum IT are obtainable at higher Q, which is simply understood due to higher magnitudes of VC and increases the ‘flushing’ velocity. 8
6
IT (μA)
IT (μA)
6
8 4.2 mL/min 6.3 mL/min 9.3 mL/min 15.6 mL/min
4
2
0 0
5
10 V (kV)
15
0 0
20
6
IT (μA)
IT (μA)
5
10 V (kV)
15
20
10 V (kV)
15
20
8 14 mL/min 19 mL/min 45 mL/min 75 mL/min
4
2
0 0
4
2
8
6
6 mL/min 9.6 mL/min 13.8 mL/min 22.2 mL/min
11 mL/min 20 mL/min 30 mL/min 40 mL/min
4
2
5
10 V (kV)
15
20
0 0
5
Fig. 5.16 Total current IT versus the applied voltage, V/Vmax for different volume flow rate at Li/d = 1.0 using diesel and the version 3 nozzle design with rp ≈ 5 – 20 μm for (a) d = 116 μm, (b) d = 140 μm, and (c) d = 254 μm and (d) using kerosene and the version 2 nozzle design with rp ≈ 60 μm for d = 150 μm. [112]
5.11 Specific Charge Regimes Figure 5.17 shows the spray volumetric specific charge qv=IS/Q, plotted against voltage. There is a tendency to exhibit self similarity for the two groups of flow rates. Three curves are plotted showing the sub-critical and two ‘families’ of
102
Charge Injection Atomizers: Design and Electrical Performance
0.8 Super-critical Re < 4600
QV (C/m3)
0.6
Super-critical Re > 4600
0.4
0.2 Sub-critical
0 0
5
10 15 V (kV)
20
25
Fig 5.17 Spray specific charge versus applied potential using kerosene : version 1 nozzle, Li/d=6.6, L0/d=2, d=500μm [110]
super-critical regimes. As noted above, in the super-critical regime there is a maximum followed by a sharp reduction, and the sub-critical regime curves are truncated where catastrophic breakdown occurs inside the nozzle. The profiles in the super-critical regime, for Q≥1.25ml/s after partial breakdown has occurred exhibit similar shapes and all show that the charge remaining within the spray is now independent of both flow rate and applied voltage. There is a trend for the optimum operating points, defined as when the spray contains its maximum charge before breakdown to occur at higher applied potentials for higher flow rates. For the super-critical regime two sub-groups of data are apparent, one, with flow rates Q ≥2.27ml/s exhibit reduced spray specific charges relative to other sub-group at lower voltages. For Q =2.27ml/s, Re≈4000 (based on the orifice channel) and turbulent flow will become increasingly prevalent between the entrance of the orifice channel and the atomizer exit. Due to the existence of an additional turbulent transport mechanism for charge for the higher flow rate ranges it is thought that the radial transport of charged particles away from the centerline will be enhanced. Thus a greater charge concentration will occur near the surface of the jet as it emerges from the orifice, and this will make the surface partial breakdown condition more likely and reduce the (volumetric) spray specific charge in the jet at the breakdown condition.
5.11 Specific Charge Regimes
4
d = 150μm d = 250μm
103
d = 300μm d = 500μm
Max QV (Cm-3)
3
2
1
0 0
2000
4000 Re
6000
8000
Fig. 5.18 Maximum spray specific charge versus atomizer Reynolds number using kerosene: version 2 nozzle, Li/d=1.0, L0/d=2 for orifice diameters of 150μm, 250μm, 300μm and 500μm [110]
The use of the version 1 atomizer geometry was confined to larger orifice diameters (d≥500μm) as noted in the introduction to chapter 5 and the version 2 atomizer was used with kerosene for a complete data set of flow rates and diameters using the fixed geometry ratios Li/d=1.0 and L0/d=2. Changing to the version 2 atomizer while keeping d=500μm raised the spray to leakage current ratio for any flow rate and increased the spray current by approximately 15% relative to the version 1 nozzle. The trends of variations in maximum spray charge with orifice diameter and flow rate described below are typical of both version 1 and 2 designs. In Fig. 5.18 the maximum spray specific charge, for each flow rate and diameter of the version 2 atomizer, is plotted against Reynolds number for kerosene. As shown, significant improvements in the maximum spray specific charge can be made by using smaller orifice diameters. These improvements can be further magnified by using larger flow rates to a much greater extent than for the larger orifice diameters. This trend has been found to be common to the other test liquids with the added factor that more viscous, denser hydrocarbons were able to contain larger maximum specific charges. As noted above, this is assumed
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Charge Injection Atomizers: Design and Electrical Performance
to be caused by the reduction in ion mobility at increased viscosity which reduces the electric component of radial charge transport. The greater efficiency of the charge injection method for more viscous insulating liquids makes the charge injection atomization technique eminently suitable for such liquids. The reason for the reduction in the spray specific charge shown in Fig. 5.18 for the highest flow rate and the d=150μm diameter nozzle is hydraulic flip, the mean injection velocity being 37m/s through an orifice with no inlet chamfer. This is supported by experimental observation that the spray angle was markedly narrower for this operating condition. The inter-relationship between spray specific charge and orifice diameter may be clarified by deriving an approximate expression for the radial electric field with the liquid jet where it emerges from the nozzle orifice. The Poisson equation represents the relationship between space charge and the radial electric field in the orifice channel,
q 1 1 § ∂V · ¨r ¸ = − v r ∂r © ∂ r ¹ ε
(5.4)
If the radial profile of the specific charge in the orifice channel is assumed to be uniform, then the following solution exists for boundary conditions Vr=0 is finite and Vr=d/2=0, representing the earthed nozzle wall just prior to the emergence of the liquid jet.
Er =
qv r 4ε
(5.5)
Dr =
qv r 4
(5.6)
Equations (5.5) and (5.6) give the radial component of the electric (E) and displacement (D) fields due to a uniform radial space charge. The calculated radial electric field strength at r=d/2 is plotted against Reynolds number in Fig. 5.19 and as shown the data collapses quite satisfactorily. The numerator of the right hand side of eqns. (5.5) and (5.6) is roughly constant and has units C/m2, thus relates to displacement field, D, or surface charge density. It is this quantity that limits the maximum spray specific charge in the super-critical regime. From the dependence of spray specific charge on the liquid injection velocity it is deduced that the increased flow velocity delays the build up of charge near the jet surface and the partial breakdown event. The data distributions in Fig. 5.19 have been given linear fits and these indicate that two relationships apply and that the boundary between these occurs when Re≈4000. This agrees with arguments made above concerning Fig. 5.17 and it may be concluded that there is indeed a laminar/ turbulent transition at Re≈4000 and that the turbulent transport mechanism detrimentally affects the maximum spray charge that the spray can contain. These linear curve-fits give the following empirical relationships between spray specific charge and Reynolds number for the version 2 design where Li/d=1,
105
qV =
qV =
ε
(3.5 × 10 r
ε
(8 × 10 r
6
6
+ 2200 Re
+ 4200 Re
)
)
: Re ≥ 4000
(5.7)
: Re ≤ 4000
The results obtained in Fig. 5.19 apply to the radial component of a cylindrical electric field in an insulator of permittivity 2.2 at r=d/2 where the radial electric field is non-linear at the jet surface due to the cylindrical geometry. A commonly quoted electrical breakdown strength for air [132] is 3×106V/m for atmospheric pressure and planar geometry and this compares closely with the intercept at small Re on Fig. 5.19. The similarity of these values are probably co-incidental however
6
E×106 (V/m)
5
4 d = 150μm d = 250μm
3
d = 300μm d = 500μm
2 0
2000
4000 Re
6000
8000
Fig. 5.19 Surface electric field strength versus atomizer Reynolds number using kerosene : version 2 nozzle, Li/d=1.0, L0/d=2 for orifice diameters of 150μm, 250μm, 300μm and 500μm [110]
since correlations are available for the surface electric field required for corona onset of a cylindrical conductor of radius r, of which the best known is that proposed by Peek [132] and for air at 760mmHg and 20°C,
106
Charge Injection Atomizers: Design and Electrical Performance
0 . 308 ⎞ ⎛ E r = 31 ⎜ 1 + ⎟ r 0 .5 ⎠ ⎝
( kV / cm )
(5.8)
Clearly the satisfactorily collapse of the data shown in Fig. 5.19 implies that eqn. (5.8) is not applicable in the present case since it predicts a limiting surface electric field proportional to r –1/2, where it is shown from Fig. 5.19 the limiting field is proportional to r.
5.12 Specific Charge: Summary The properties of Diesel oil used in Bankston [120] are assumed to be similar to that of the fluid used in the present work (i.e. Diesel no.2). The properties of Kerosene oil used in reference [133] are assumed to be similar to that used in Shrimpton [108]. The ranges for viscosity, density and relative permittivity are 0.0009 ≤ μ ≤ 0.030 Ns/m2, 780 ≤ ρ ≤ 850 kg/m3 and 2.0 ≤ εr ≤ 2.2 respectively. The critical spray specific charge qv versus Re for various data obtained using charge injection atomizers for various fuels and sources as summarized in table 5.2. Table 5.2 Atomizer operating conditions
Description Orifice diameter (d, μm)
250
250
150
Flow rate ( QL, mL/s)
0.5
1.67
0.5
Mean injection velocity (m/s)
10
34
28
Specific charge (ρV, C/m3) Reynolds Number Weber Number
-1.20 1500 1.2
-1.80 5100 13.9
-3.00 2520 5.6
The critical specific charge qv is enhanced for smaller orifice diameter d as shown in Fig. 5.20. The results also show that the magnitude of qv for the version 3 nozzle design using Diesel oil as a test liquid is higher than that of the version 2 design using kerosene for similar d in the super-critical regime of operation. This confirms that the spray, generated by a charge injection atomizer, operating in this regime holds more electric charge if the liquid viscosity is larger. Overall the dominance of orifice diameter as the primary parameter is shown.
5.13 Variation of Electrode Gap Ratio (Li/d), L0/d=2, d=500μm, Version 1 Design
107
qV (C/m3)
4.5
0 0
3000
6000
9000
Re Fig. 5.20 Critical spray specific charge qV versus Re for version 3 nozzle design using diesel no.2 with rp ≈ 5 – 20 μm for d = 254 (●), 140 (■) and 116 (▲) μm; from reference [108] using kerosene and version 2 for d = 500 μm with rp = 60 ( ) and 1 (◆) μm; with rp = 60 μm for 300 (★), 250 (+) and 150 (✕ ) μm; using diesel no.1 for d = 250 μm (⊗); and for version 1 for d = 500 μm using diesel no.1 with rp = 60 (▦) and 4 (◊) μm; with rp = 60 μm using kerosene (Δ) and white spirit ( ) for d = 500 μm; and for d = 1000 μm using kerosene (✠); and for version 1 using kerosene for d = 1500 (◍), 1000 ( ) and 630 (✻) μm; from Spray Triode of reference [168] using Marcol-87 for d = 300 μm (✩), and reference [120] using mineral oil for d = 422 μm (✴) and diesel no.2 for d = 173 μm (O) [112]
5.13 Variation of Electrode Gap Ratio (Li/d), L0/d=2, d=500μm, Version 1 Design The trends introduced in section 5.7 concerning the link between the spray and leakage currents and the electrode gap (IS, IL and Li) are now explored in more detail. The distance Li between the tip of the emitting electrode and the back face
108
Charge Injection Atomizers: Design and Electrical Performance
of the orifice plate was varied whilst the orifice length to diameter ratio, L0/d, and diameter, d, remained constant at 2 and 500μm respectively. Compared with the version 2 design, Li is larger, and the flow and the electric field strength near to the emitter tip remain essentially constant as Li changes. This is not true for the electric field, permeating the liquid, which acts as electrical insulation between the earthed nozzle body and the emitting electrode. One would therefore expect changes in this internal electric field to give rise to changes in the trajectories of the charge carriers and this should be observable via variations in the measured spray and leakage currents. The total injected current, as discussed in section 5.7 should remain relatively constant. White spirit was used as the test liquid since it had the lowest viscosity of the three test liquids and, following Walden’s Rule eqn. (2.21), the highest ionic mobility. Therefore variations in charge carrier trajectories in this liquid should be most sensitive to spatial variations of the electric field permeating the liquid, produced by modifying Li. A flow rate of Q=2ml/s was used which was near the lower limit of operation in the super-critical regime, thus providing the longest liquid and charge carrier residence times within the nozzle. These two factors contribute to make charge carrier trajectories as sensitive as possible to changes in atomizer operating conditions and thus allowing identification of trends in dependent variables most apparent. It is recalled [110] that the spray current (IS) is convected charge that exits the atomizer in the liquid jet and the leakage current (IL) is charge radially transported through the liquid and reaches the nozzle wall. Therefore an increase in the radial component of the electric field inside the nozzle or a reduction in the axial bulk velocity would be expected to reduce the spray to leakage current ratio. As shown in Fig. 5.21a the total current IT, defined as IT=IS+IL was only slightly dependent on electrode gap with larger gap geometries producing the smaller currents as expected. This implies that for the version 1 geometry the charge injection process, which is highly localized to the needle tip, the emitting electrode, is not a function of the global nozzle electric field over the Li/d range used. As discussed elsewhere [110], the total current is dependent on liquid properties and the flow rate, thus the slight increase in total current may be attributed to an increase in the bulk flow velocity as Li reduces due to flow crosssectional area reduction at the axial position of the electrode tip (c.f. Fig. 5.3). For the same conditions as shown in Fig. 5.21a, Fig. 5.21b shows how the spray current responds to changes in applied potential and electrode gap, where the dashed line represents the loci of maximum spray currents possible in the supercritical flow regime. The spray current is more sensitive to the electrode gap than total current, however the trends are similar in that the largest electrode gaps generate the smallest spray currents for a particular applied potential. At the largest electrode gap ratio, Li/d=7.1, catastrophic electrical breakdown occurs [110], a characteristic of atomizer operation in the sub-critical regime. The next largest gap, Li/d=6.9 was intermittently prone to this behavior which does not cease completely until Li/d≤6.6. For electrode gap ratios Li/d≤6.6, the spray current always reached a maximum at this flow rate (2ml/s) and then a partial
5.13 Variation of Electrode Gap Ratio (Li/d), L0/d=2, d=500μm, Version 1 Design
109
5
IT (μA)
4 3 2 1 0 0
4
8
12
16
20
V (kV)
1.5 L i /d = 5.8
IS (μA)
1.2 0.9
L i /d = 6.9
0.6 0.3 L i /d = 7.1
0 0
5
10 V (kV)
15
20
Fig. 5.21 (a) Total and (b) Spray current versus applied potential for varying Li/d, version 1 nozzle, L0/d=2, d=500μm, Q=2ml/s for white spirit where Li/d=7.1-5.8. [111]
110
Charge Injection Atomizers: Design and Electrical Performance
breakdown occurs, which is the behavior characteristic of the super-critical regime. Figure 5.21a-b reveal that the maximum spray charge decreases and the total injected current increases slightly for the smaller Li/d values tested. The increase in leakage current at small electrode gaps is thought to be due to the intensification of the potential gradient between the needle tip and the nozzle body which induces larger radial components for the charge carrier trajectories. By examination of the trends in total and spray current response to varying Li/d for the version 1 design shown in Fig. 5.21a-b, it can be concluded that further reduction in Li/d below Li/d=5.8 has only detrimental effects. The explanation lies in the change of direction of voltage gradient vector, as Li/d reduces below 5.8 the direction acquires an increasingly significant radial component. Thus the radial motion of the charge carriers is enhanced, producing higher leakage currents. This highlights the probable error in the approach of Jido [122,123] and the author in pursuing the ‘version 1’ nozzle, which was designed to optimize the internal flow field. In terms of atomization performance of which the primary variable is the spray specific charge [134], flow field optimization implies a hydrodynamically streamlined geometry into which a charge emitting high voltage electrode may be placed. Unfortunately, this also enforces the restriction that the charge injection point, the needle tip, is several orifice diameters upstream of the orifice channel itself. This large Li/d implies long charge carrier residence times and significant radial deflection, caused by expansion of the space charge plume emanating from the needle tip. The accumulation of these radial deflections of the charge carriers trajectories gives low IS/IL ratios. As shown in section 5.15, an ‘electrically optimized’ nozzle, where Li/d is much reduced, permits more efficient atomizer operation, however the effects of other parameters on the ‘version 1’ nozzle performance are first discussed.
5.14 Variation of d: Version 1 Design: Constant Q, Li, L0/d By changing the orifice diameter, but keeping Li and flow rate constant, the effect on the spray current of changes in the near-orifice internal flow may be investigated. Orifice diameters of 1000μm, 500μm and 250μm were tested at a flow rate of 1.25ml/s, using kerosene as the test liquid. Note that this flow rate is in the lower range for the super-critical regime [110], thus effects of changes in atomizer geometry should be easily quantified via the measured spray and leakage currents. The electrode gap was fixed at Li =3.3mm and the orifice length to diameter ratio Lo/d was fixed at a value of 2. All other features of the nozzle geometry remained invariant. The spray current is plotted as a function of applied potential and orifice diameter in Fig. 5.22, which shows that although the larger orifice diameters achieved a spray charge maximum in the super-critical regime the d=250μm orifice diameter could not. The largest orifice diameter has no practical relevance but serve to show the trend of the data. For this smallest diameter however there was no electrical breakdown in neither the sub-critical nor the super-critical
5.14 Variation of d: Version 1 Design: Constant Q, Li, L0/d
111
atomizer regimes for V≤25kV. The rounded shape of the spray charge curve for the d=1000μm case occurred because the jet had very low momentum and tended to attach to the nozzle exterior after it had emerged from the atomizer. This caused the gradual reduction in the spray current over a range of applied voltage, rather than the sudden transition normally experienced as shown in Fig. 5.22 for the d=500μm orifice diameter results. The maximum spray current for the d=500μm and d=1000μm orifice cases agreed well with the values predicted assuming constant surface charge density at breakdown [110]. This is not the case for the 250μm orifice diameter, here the experimental spray current fails to reach the magnitude required to precipitate partial breakdown in the super-critical flow regime. Since the electrode position and the flow rate are unchanged for all the results given in Fig. 5.22, and the electrode tip is relatively far from the back face of the orifice contraction, it may be assumed that the flow field in the region local to the electrode tip is unchanged. Therefore the charge injection process and the charge carrier trajectories near the electrode tip are not a function of orifice diameter for constant flow rate. For large (d≥500μm) diameter nozzles, a certain fraction of charge exits the nozzle and is sufficient to precipitate partial breakdown in the super critical regime. For the smaller orifice diameters a large fraction of injected current paths always terminate on the nozzle wall and contribute to the leakage current.
0.8 0.7
IS (μA)
0.6 0.5
500μm
0.4 0.3
250μm
0.2 1000μm
0.1 0 0
5
10 15 V (kV)
20
25
Fig. 5.22 Spray current versus applied potential for orifice diameters of 250μm, 500μm and 1000μm for a version 1 nozzle, L0/d=2, Li/d=6.62, Q=1.25ml/s, spraying kerosene
112
Charge Injection Atomizers: Design and Electrical Performance
In summary, within the voltage range used, the breakdown mechanism of the sub-critical regime never occurs at this flow rate for version 1 nozzles of any orifice diameter. This occurs regardless of orifice diameter, and the current paths simply terminate on the interior walls of the nozzle, and contribute to the leakage current. To obtain more highly charged and better atomized sprays we require a smaller orifice diameter on our atomizer (c.f. Fig. 5.18), and this is not possible with the ‘version 1’ nozzle design. This was the reason the ‘version 2, 3’ design (Fig. 5.3) was developed.
5.15 Variation of Electrode Gap Ratio (Li/d): Version 2 Design, d=500μm The version 2 atomizer Fig. 5.3 was developed to permit the operation in the super-critical regime where d≤250μm. This involved electrically optimizing the design by removing much of the internal geometry, allowing Li/d to be drastically reduced and giving a point-plane electrode geometry. The needle tip for the version 2 design is much closer to the orifice channel and lies in a region of higher bulk velocity. Thus the residence time of the charge carriers is relatively short and they are more likely to be swept out of the nozzle and contribute to the useful spray current, in contrast to the version 1 geometry where the charge carriers have time to be pushed toward the nozzle wall by the radial electric field. One would therefore expect the spray to total current ratio to be significantly higher for the version 2 design compared with the version 1 atomizer operating under identical conditions. Therefore experiments were repeated with the version 2 design, using the same orifice length to orifice diameter ratio, L0/d=2, d=500μm and flow rate of Q=2ml/s as discussed in section 5.13. The electrode gap ratios (Li /d) of interest in the version 2 design were found to be in the range 0.0≤Li/d≤1.4, much smaller than those possible for the version 1 design discussed above. Figure 5.23a shows that the total current drawn from the charge emitter is now a stronger function of electrode gap ratio than was shown in Fig. 5.21a for the version 1 design. This supports the conjecture made above that smaller electrode gaps extract more charge from the emitter by allowing more efficient extraction of the space charge due to the higher bulk velocity near the needle tip. This adds weight to the argument that the charge injection process is to some extent space charge (i.e. diffusion) limited and supports the hypothesis of the authors, described elsewhere [110], that an increase in flow rate or decrease in viscosity thins the boundary layer on the electrode, reducing the thickness of the high space charge region and increasing current flow through it. A comparison is now made between Fig. 5.23b, the spray current versus applied potential for the version 2 nozzle and the equivalent conditions for the version 1 design, given in Fig. 5.21b. There are a number of differences in the spray current response between the two versions of nozzle design. For the version
5.15 Variation of Electrode Gap Ratio (Li/d): Version 2 Design, d=500(m
113
4
3
IT (μA)
L i /d = 0
2
L i /d = 1.4
1
0 0
5
10
15
V (kV) 1.5
1.0
IT (μA)
L i /d = 0
0.5 L i /d = 1.4
0 0
5
10
15
V (kV) Fig. 5.23 (a) Total and (b) Spray current versus applied potential for varying Li/d using a version 2 nozzle, L0/d=2, d=500μm, Q=2ml/s, spraying kerosene where Li/d=0-1.4 [111]
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2 design, at small applied potentials (typically V<5kV), the spray current rises in a non-linear fashion with the applied voltage. For V>5kV the relationship is more linear, but for small electrode gaps the distinction becomes small. For Li/d≥1.4 it was found that insulation failure of the liquid, the breakdown mechanism of the sub-critical regime, always occurred. This compares with occurrence at Li/d≥6.9 for the version 1 design. The maximum spray current occurred at Li/d=1.2, however more stable operation was achieved by setting Li/d=1.0 for which only partial breakdown occurred, the breakdown mechanism of the super-critical regime. Note that comparing the loci of the maximum spray currents, the dashed lines of Fig. 5.21b and Fig. 5.23b, super-critical operation occurs over a wider operating window in terms of Li/d ratio and typically at a lower applied voltage for the version 2 design. We may conclude that the sub-critical and super-critical flow regimes and the dependence on geometry, liquid physical properties and operating conditions are generic characteristics of this class of atomizer.
5.16 Variation of Electrode Gap Ratio (Li/d): Version 2 Design, d=250μm The spray and leakage current response for a d=250μm orifice, with L0/d=2 and varying Li /d in the range 0≤Li /d≤1.4 are presented in Fig. 5.24a and Fig. 5.24b for a constant flow rate of Q=0.5mL/s. For Li/d≥1.2 the breakdown event was catastrophic, and this occurred inside the nozzle, indicating that the local electric field strength exceeded the electrical breakdown strength of the kerosene. More stable operation was obtained for Li/d≤1.2. However at this Li/d ratio catastrophic electrical breakdown of the liquid would readily occur as the system was sensitive to small perturbations of the operating conditions. This can occur at cathode potentials below the potential required to achieve partial breakdown. Note that for constant Li/d, halving the orifice diameter halves the inter-electrode gap and doubles the radial voltage gradient and thus the Coulomb force on the charged particles. However the image force, due to the interaction between the charged particle and the grounded nozzle wall, increases by a factor of four. The optimum ratio of orifice gap to electrode diameter ratio (Li/d) where repeatable operation, as described above, was achieved occurred for Li /d=1.2. This is in agreement with similar work with a d=500μm case [108]. At small applied potentials (typically V<5kV) the total current drawn varied approximately as the square of the voltage. For V>5kV the relationship was linear, but for small electrode gaps the distinction became small. The transition point between the IT∝V2 and IT∝V regimes over the range of 0.0≤Li /d≤1.4 was constant at V≈5kV. The maximum current for the 0.5mL/s flow rate occurred at Li /d=1.2, however more stable operation, where only partial breakdown occurred could was achieved by setting Li /d=1.0. Hence at Li /d=1.0, by alternately increasing and decreasing the applied potential around the voltage required for partial breakdown the spray would repeatedly charge, breakdown and re-charge to the maximum specific charge before breakdown.
5.16 Variation of Electrode Gap Ratio (Li/d): Version 2 Design, d=250μm
115
0.9 0.8 L i /d = 0
0.7
IS (μA)
0.6 0.5 0.4 0.3 0.2 L i /d = 1.4
0.1 0 0
4
8 V (kV)
12
16
8 7 6
IL (μA)
5 L i /d = 0
4 3 2 1
L i /d = 1.4
0 0
4
8 V (kV)
12
16
Fig. 5.24 (a) Spray and (b) Leakage current versus applied potential for varying Li/d using a version 2 nozzle, L0/d=2, d=250μm, Q=2ml/s, spraying kerosene where Li/d=0-1.4 [108]
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5.17 Performance Evaluation: Version 1 and Version 2 There are three important atomizer performance advantages of the version 2 over the version 1 design [109-112,135]. The first is that there is a significant improvement in spray charging efficiency, defined as the useful spray current as a fraction of the total injected current, IS/IT. This is assessed at the optimum operating point, defined as that giving the maximum spray current just prior to partial breakdown. Using the results given in Figs. 5.21a-b and Figs. 5.23a-b, for the version 1 and version 2 designs respectively, operating at the same applied potential and flow rate, it is found that IS/IT is 0.62 for the version 2 design, compared with 0.46 for the version 1 design. The second advantage is that in addition to more efficient operation for the version 2 design, the magnitude of the spray current is increased for the version 2 design for a given flow rate. A comparison between the spray current magnitudes at the operating point for the version 1 and 2 designs given in Fig. 5.21b and Fig. 5.23b, show that the spray current is 17% greater in the version 2 design. As discussed in an earlier contribution [110], the partial breakdown is thought to occur at a limiting surface charge density on the liquid jet as it emerges from the orifice. Following on from this, any additional charge produced in the version 2 design must be contained in the interior of the jet, and not at the jet surface. This seems plausible bearing in mind that for the version 2 design the electrode tip is much closer to the orifice than for the version 1 design, therefore more charge should lie near the centerline of the liquid jet. There is no other obvious explanation for the increase in performance of the version 2 design over version 1, operating at the same conditions, assuming that the partial breakdown mechanism is the same in both cases. This novel finding has important implications for the atomization process, where it would seem likely that a more uniform radial charge density profile across the emerging liquid jet should produce more uniform atomization and less significant ‘tail’ to the large drop diameter end of the probability density function. The third advantage is the optimum operating point of the atomizer, requires a potential of approximately 9kV is required to inject sufficient charge to precipitate partial breakdown in the super-critical regime for the version 2 atomizer, against a value of 12kV for the version 1 design. The reasons for this are thought to be twofold. Firstly, the electrode tip for the version 2 design is in a region of relatively fast moving fluid. Assuming that the charge injection process is to some extent space charge limited as discussed earlier, then enhanced removal of the outer layers of this region will indeed increase the total injected current for a given applied potential by reducing the thickness of the charge limiting diffusion layer around the needle tip. The second reason for the enhanced injected current for the version 2 design is thought to be the proximity of the back (upstream) face of the orifice contraction. This increases the electric field gradient and also assists in the charge extraction from the high space charge concentration surrounding the needle tip. Additional tests were undertaken for the 250μm and 150μm diameter orifice nozzles and similar sub- and super-critical flow regimes were observed. The optimum of Li/d ~1 was found to apply to these smaller diameters also. In
5.18 Point-Plane Atomizer Design Modifications
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addition, experiments investigated the effect of the variation of Lo/d as a function of liquid flow rate. An orifice diameter of d=300μm orifice was used, with Li /d=1.0 and Lo/d ratios of 5, 2 and 0. Three average nozzle exit velocities of 6.4, 10.6 and 15.3m/s were tested and examination of the spray, leakage and total injected currents revealed no dependence of these quantities on L0/d for the range of flow rates investigated [108].
5.18 Point-Plane Atomizer Design Modifications In this section the effect on the spray and leakage currents caused by variation of the electrical parameters of the atomizer are reported. Six variants are presented at a constant flow rate of Q=2ml/s, kerosene as the test liquid, and using as a basis the version 1 nozzle design, and a ‘standard’ geometry, Li/d=6.62 and Lo/d=2. The variants are described by table 5.3 and a schematic is given in Fig. 5.25. Table 5.3 List of electrical changes to the version 1 atomizer
Variant
Ring electrode (kV)
Standard 1 2 3 4
Shunt resistance (MΩ) 100 200 384 -
5
-
+5
6
-
-3
Shunt EHT
a
b
a/d~2 b/d~2
Ring Fig. 5.25 Schematic of electrical atomizer modifications, version 1 atomizer design
+3
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Charge Injection Atomizers: Design and Electrical Performance
2.0 +3kV
IS (μA)
1.5
+5kV
1.0 300MΩ 200MΩ
0.5
100MΩ
-3kV
0 0
5
10
15
20
25 -3kV
20
+3kV +5kV
15
100MΩ
IL (μA)
100MΩ
10
384MΩ
5
0 0
5
10
15
20
25
Fig. 5.26 (a) Spray and (b) Leakage current versus applied potential for the version 1 nozzle spraying kerosene where L0/d=2, d=500μm, Q=2ml/s. Standard (-----)[111]
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119
Variants 1 to 3 involve the placement of large resistances (100MΩ, 200MΩ and 384MΩ) between the nozzle and earth (the ‘shunt’ of Fig. 5.25) to modify the nozzle internal electric field. These were fabricated out of smaller high voltage resistors, soldered together and potted in an epoxy compound to prevent flashover. The three other variants made use of an external ring electrode held at varying potentials to alter the external electric field near to the discharge site for breakdown in the supercritical regime. The length scales used for the ring electrode, relative to the orifice diameter are given in Fig. 5.25. Figures 5.26a-b show the spray and leakage current responses to changes in applied potential for the six variants. For comparison results from the standard, unmodified version 1 design are included. Discussing the results of the variants 1 to 3 first, by electrically 'floating' the nozzle body the partial breakdown voltage shifts to progressively higher values as the shunt resistance is increased. The maximum spray current does not increase, suggesting that the electric field external to the nozzle is not modified, since the partial breakdown mechanism is unaffected. It can therefore be concluded that the potential at which the atomizer nozzle is held has a negligible effect on the partial breakdown mechanism, and that the partial breakdown phenomenon is completely governed by the electric field generated by the charged jet and its resultant corona discharge into air. The efficiency (i.e. IS/IT) of the spraying process does improve, however, by virtue of a reduction in the leakage current with increasing resistance. Of particular interest is the spray charge trend after partial breakdown which progressively increases with higher shunt resistances. It seems that the higher nozzle potentials discourage the current to flow from the discharged liquid jet, via the nozzle assembly to earth. The trend in the results suggest that it would be theoretically possible that very large resistances could remove partial breakdown altogether. However this ignores the fact that a large potential difference must still be present between the emitting electrode and the nozzle body, in order to extract charge from the space charge limited region near to the electrode tip. In fact, at V≥20kV the possibility of accidental air breakdown becomes increasingly likely and is to be avoided if stable, robust operation is desired. Experiments with an external ring electrode, variants 4 to 6, were more successful. In contrast to the additional resistances tested, the ring electrodes influenced the spray current while the leakage current was largely unaffected. For the +5kV ring potential the onset of partial breakdown is delayed and the maximum value of the spray current is increased by 17%. These simple electrical modifications were a first attempt to quantify the magnitude of their effect on spray and leakage current and represent a demonstration of the concept. It is thought that further improvements could be achieved with refinement of the principles introduced here. The version 2 nozzle, with Li/d=1.0 and L0/d=2.0, was modified by replacing the stainless steel needle with a eutectic tipped electrode as used by Kelly [106] in his 'Spray Triode'. The electrode material was originally developed by Chapman et al. [136] and consists of small tungsten fibers that are present on a UO2 matrix. The tip radii are sub-micron in scale and the regular arrays that are formed have densities in excess of 107 fibers per cm2. The very small radii of curvature for the
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Charge Injection Atomizers: Design and Electrical Performance
4
L i /d = 0.6
IT (μA)
3
2
1 L i /d = 1.6
0 0
2
4
6 8 V (kV)
10
12
14
6 8 V (kV)
10
12
14
1.2 1.0
IS (μA)
0.8 0.6 0.4 0.2 0 0
2
4
Fig 5.27 (a) Spray and (b) Total current for emitting electrode modifications, version 1 nozzle, L0/d=2, d=500μm, Q=2ml/s, kerosene, where Li/d=0.6-1.6 [111]
5.19 Beyond the Point-Plane Atomizer Concept
121
fiber tips and the nominal 7kV emitter potential reported by Kelly [106] gave a field strength in excess of 109V/m for the reported 'Spray Triode' operation. It is this fact that led Kelly to make the assertion that the charge transfer mechanism from electrode to insulating liquid was field emission. Results are presented for the spray current, Fig. 5.27a, and total current, Fig. 5.27b, of the eutectic tipped electrode. Comparing the results with Figs. 5.23a-b show that the total current emitted is reduced for the eutectic electrode for all values of Li/d and in particular the maximum spray current is also less than for the single needle electrode used here. The shapes of the current versus applied voltage curves are almost identical, for the stainless steel and the eutectic electrodes, which implies that the species and transport of the charged carriers and thus the mechanism are the same. In terms of the mechanism of charge carrier creation in insulating liquids, a survey of the literature shows that the theory and experimental data of Denat and coworkers [42,45,137] predict the shape of the current-applied potential curves experimentally obtained by the authors. In addition, the onset field strength for significant current flow of 107-108 V/m at the electrode tip is also confirmed by these results. These results therefore do not confirm the hypothesis of Kelly [106], since current flow has been recorded at field strengths less than that required for field emission.
5.19 Beyond the Point-Plane Atomizer Concept Two important problems arise with the point-plane atomizer concept described above, where the tip of the high voltage electrode (the ‘point’) and the atomizer orifice (the ‘plane’) exist with a separation of Li/d~1. The first is that as d reduces so does Li, and the precise coaxial alignment required between electrode tip and orifice centre becomes increasingly difficult to achieve. The second is that with the point plane concept, multi-orifice atomizers are difficult to construct in order to maintain a uniform hole to hole mass and charge flow. Therefore, there is an unavoidable compromise between large mass flow (requiring a large orifice diameter) and highly charged sprays (requiring a small orifice diameter, Fig. 5.23). Whilst the work based around the point plane concept outlined above is useful from the viewpoint of understanding the physical basis for their operation, within the context of a forced flow extension of the fundamental studies outlined in section 3, there is limited practical application of this particular method. There have been a number of developments to move beyond the point plane concept and improve the applicability of the charge injection method to practical systems. The developments are now discussed.
5.19.1 Single Hole Electrostatically Enhanced Pressure Swirl Atomizers For a reasonably sized orifice diameter atomizer employing the point plane concept a limitation, discussed further in chapter 6, is that a ‘two zone’ spray tends to be generated. Typically a spray plume consists of a poorly atomized and
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charged ‘core’ surrounded by a highly charged ‘sheath’ of smaller more highly charged drops. One possible solution to this problem is to employ a secondary atomization method and the fusion of a pressure swirl atomizer and a charge injection atomizer has been suggested using a number of strategies. Kelly [106], whilst at Exxon, reported the ‘Spray Triode’ modified to operate as a pressure swirl atomizer but gave very few details of the design and performance. It was claimed the addition of an imposed potential of 7 kV changes the distribution, disrupting all drops where D>150μm. The reported drop size distribution for charged spray atomization shows an absence of drops larger than approximately 120μm but the distribution appears truncated. The drop size measurement or sample size is not quoted, nor are any details of the device. Laryea & No [138-139] took a standard pressure swirl nozzle and inserted a high voltage electrode along the axis. They reported a significant improvement for their application of crop spraying, but the coincidence of the high voltage electrode and the air core of the pressure swirl atomizer may have hindered the performance. Anderson et al. [140-142] modified a gasoline direct injector (Mitsubishi p/n MR560552) to inject electrostatic charge into the fuel by externally mounting an exposed high voltage electrode, featuring a sharp conical tip at the orifice. This electrode made from copper, had a sharp cone at the tip of the electrode protruding 0.5 mm at an angle of 135 degrees with the base. The locally intense electric field that is generated from the sharp conical feature partially polarizes the fuel molecules and injects ions into it, thus transferring a net charge to the liquid. The degree of spray charging appears not to be significant enough to produce meaningful changes in the primary atomization nor spray dispersion. In addition, in contrast to charge injection atomizers, the external exposed high voltage electrode is vulnerable to corona discharge in an ionized combustion environment.
5.19.2
Multi-hole Charge Injection Atomizers
Here the aim is to obtain high volume flow rates at low pressure drop with a finely atomized and highly charged sprays. In the context of electrostatic methods based around the charge injection concept, the only possible solution is to multiplex the number of orifice present. The engineering challenge then moves to ensuring each liquid jet emerging from each orifice carries a similar amount of current as its companions. Two technologies have emerged in the commercial sector, one of which remains protected by patent, the other has been released to to public domain. Kelly [143-146] suggests a number of possible design concepts and two SAE papers explore this [147-148]. Similarly Allen [149-150] suggests some concepts and these have also been evaluated [151-152], which are now in the public domain.
5.19.3 Pulsed Spray Charge Injection Atomizers Here the typical application is fuel injection for internal combustion spark or compression ignition engines. There are a number of reasons for the interest. For
5.20 Chapter Summary
123
compression ignition engines, electric charge in the drops will aid dispersion and mixing, and can possibly improve the combustion emissions, especially NOX. For direct injection spark ignition engines there is the possibility of controlling the spray more effectively moving from an early to a late injection strategy, since electric spray charge is in effect a free parameter. There is also interest in using EHD injectors to enable the downsizing of engines for enhanced fuel economy, for instance within hybrid systems. In this scenario atomizer design and operation is particularly challenging since the flow and indeed the atomizer geometry are time dependent, and the high voltage electrode must be activated correctly. In addition various time delays must be accounted for (start/end of fluid motion, voltage rise time, time for charge to be injected, needle lift response). Kelly, in collaboration with Robert Bosch GmbH, patented a EHD single hole CI injector design [153] and Bosch published some performance data [154]. Experiments were carried out and some tests resulted in very high specific charges (up to 5C/m3) and consequently very small drop sizes (<50μm). Experiments were also conducted for pulsed voltages of several milliseconds. Time delays were apparent, allowing for the voltage rise time (~100μs) and also the time taken for the spray current to rise from its initial to its final value (charge rise time), allowing for the voltage rise time. Typical delays of several 100μs correspond to charge mobility velocities of 1 m/s, correct for the hydrocarbon fuel tested. Increasing the electrode gap increases the time taken for the current to start to rise and increasing the flow rate reduces the charge rise time. The onset of corona discharge typically shifts to higher voltages for shorter duration pulses. Charge density measurements were also found to be higher than for stationary conditions. They note that at extreme conditions of high voltage, short pulse operation leads to as yet undefined time delay effects which limit operation. Allen [155] has patented a multi-hole pulsed EHD atomizer though no data is available, and this is now in the public domain.
5.19.4 Other Developments within Charge Injection Atomization There are a number of other developments in the subject, and all are patented by Kelly. Of most interest is an ‘electron gun’ system, known as the ‘Spraytron’, a photo may be found in Kelly [121], Figure 11 of that paper, and the relevant patents are Kelly [156-159]. In this concept the pointed metal high voltage electrode is replaced with an electron gun and an ‘electron transparent window’. Other patents relate to electronic control systems [160-162].
5.20 Chapter Summary This chapter provides a largely empirical understanding of how a generic ‘pointplane’ charge injection atomizer operates, through largely experimental iteration and interpretation of the results. It is noted that more complex atomizer designs including multi-orifice and more complex high voltage electrode configurations are possible, and for these the reader is encouraged to review the patents that are
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referenced and in the public domain. The only design to be commercially marketed is the Spray Triode of Kelly. The Scion Sprays technology is no longer protected by patents and this provides a route to further develop practical applications of this technology. The use of a geometrically simple ‘point-plane’ atomizer design naturally links to the work outlined in chapter 3, in effect a point-plane charge injection atomizer with zero imposed bulk flow. The data presented in this chapter shows the link between the internal geometry of the atomizer, and the hydrodynamic and electrical coupling present, and highlights that only certain configurations permit operation in the super-critical atomizer regime. In this regime, the corona discharge surrounding the liquid jet limits the maximum spray charge and that the charge injection process, occurring inside the atomizer at the tip of the high voltage electrode is stable and unaffected by this external disturbance. In terms of the important geometrical parameters than define successful atomizer design and operation, the most important is the orifice diameter, and smaller values permit larger spray specific charges. Although not discussed in detail here, practical multi-orifice atomizer configurations are possible, permitting high flow rates and fine atomization. The next most important geometrical parameter is the gap between the high voltage electrode and the back face of the orifice channel. For the point-plane systems Li/d~1 is recommended, for multiorifice systems, using a blunter electrode, smaller gaps are typically required, down to Li/d~0.3. Other geometry parameters such is the length of the orifice channel are relatively unimportant. Although high pressure systems, providing much larger injection velocities, and more typical of combustion applications have not yet been considered, no fundamental problems are forseen to take the technology to this application area. Generation of electrically charged pulsed sprays of electrically insulating liquids has been shown to be possible though to obtain robust operation more research on the electrical and hydrodynamic timescales is required to optimize the system stability, particularly for short pulse width spray operation.
Chapter 6
Jet Instability and Primary Atomization 6 Jet Insta bility a nd Primary Atomization
Abstract. Studies of the break-up of dielectric liquid jets are theoretically more complicated than prediction of instability of conducting liquids because the surface charge distribution is not at an equipotential and the surface area evolves. Therefore the jet surface electrical boundary conditions are not well defined. This chapter covers the experimental data available, and provides qualitative understanding and empirical correlations.
6.1 Measured Characteristics The disruption of a liquid surface by electrical forces has been the focus of researchers for over a century now. With the exception of the theoretical contributions of Rayleigh much of the early work was observational and was of an exploratory nature. In the period of 1950 to 1970, numerous scientific papers on various aspects of electrostatic spraying were published resulting in notable advances in the basic mechanisms of liquid surface instabilities induced by electrostatic forces. Several elementary mathematical models of the electrostatic atomization process can be attributed to this era. From 1970 to 1990 significant advances have been made in the theory as well as the application of electrostatic atomization. Laminar charged conducting liquid jet break up in the presence of electric fields is now predictable in terms of the growth of surface instabilities [163]. Figure 6.1 shows shadowgraphs of the spray formation from the 500μm orifice, where the left hand image, for no charge injection, shows an essentially un-atomized kerosene jet at a flow rate Q=3ml/s. The right hand image shows the effect of a specific charge of 0.5C/m3 at the same flow rate. Clearly the atomization of the jet is caused, or at the very least wholly instigated by the disruptive electrostatic forces. The particular operating condition shown in Fig. 6.1 is near the lower end of the possible performance with regard to atomization quality as shown in Fig. 5.20. More highly charged and more finely atomized sprays have been generated with smaller orifice diameter designs and higher injection velocities, the conditions in Fig. 6.1 being used simply because the core J. Shrimpton: Charge Inject. Sys.: Phy. Principles, Experi. & Theore. Work, HMT, pp. 125–137. springerlink.com © Springer-Verlag Berlin Heidelberg 2009
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6 Jet Instability and Primary Atomization
Fig. 6.1 Comparison between (a) uncharged and (b) qv=0.5C/m3, d=500μm, uinj=15m/s [165]. Reproduced with permission from Begell House
drops are relatively large and hence more visible. An alternative view is shown in Fig. 6.2, here a longer exposure is used, and a highly viscous vegetable oil is being sprayed. The jet break-up dynamics were also studied using the high-speed video camera and the monozoom lens. Within the sub-critical regime, the jet broke up into droplets with diameters D, of the same order of magnitude as orifice diameter d as shown in Fig. 6.3a-h, a distorted Rayleigh jet break-up as previously observed in Ganan-Calvo et al. [4] is evident. The figures show sequential images of the charged liquid jet for nozzle orifice diameter d = 116 μm and liquid injection velocity uinj ≈6 m/s, flowing from right to left, with spray specific charge qv ≈ 2.29 C/m3. The time interval between two consecutive frames is ti = 0.123 ms with an exposure time per frame of te = 0.025 ms. The centre of the image is 1.5 mm from the nozzle orifice plane, which is on the right side of the image and is just visible. As the spray develops from Fig. 6.3a-h pronounced perturbation and good drop dispersal were observed, with droplet diameters estimated from the images to be D ≈ 2.35d 273 m. The atomization is highly regular, and has the possibility of producing near monosize spray plumes when the atomizer is operated in this mode.
6.1 Measured Characteristics
127
Fig. 6.2 A time averaged photograph of the primary atomization of soy oil with a d=250μm orifice
Increasing the jet velocity produces a more chaotic jet break-up process. Figure 6.4a-h show a charged liquid jet with uinj ≈ 15 m/s and qv ≈ 3.16 C/m3 for d, ti, te, image position and flow direction as exactly the same as that in Fig. 6.3a-h. Droplet development starts with the collapse of the charged cylindrical jet into a thickrimmed, ribbon-like structure as the jet emerges from the nozzle as have been previously observed using the Spray Triode [106] using a viscous mineral oil at uinj ≈ 2.4 m/s and qv ≈ 1.5 C/m3. The charged cylindrical jet becomes unstable due to the growth of a surface wave instability that is augmented by the space charge repulsion of the like charges contained within, and on, the liquid jet as further shown by Fig. 6.5a-h. These images were the extension of images corresponding to that from Fig. 6.4a-h, with the edge of the right side of the image is 1.5 mm from the nozzle orifice plane. As the ligaments emerge from the liquid jet, they quickly lose their forward momentum to aerodynamic drag and reverse, back towards the earthed nozzle surface. The ligament tips collect liquid and they break-up via Rayleigh type instability to produce rapidly dispersing droplets. From the above discussion of jet break-up dynamics, it is clear that jet break-up and droplets dispersal at atmospheric pressure and room temperature are strongly effected by uinj and, to some extent, by qv. Figures 6.6a-c show qualitatively the effects of d on the break up length and cone angle (lj and α) at constant uinj ≈ 10 m/s and qv ≈ 1.5 C/m3. These images were obtained using the high-speed video
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6 Jet Instability and Primary Atomization
Fig. 6.3 Jet break-up dynamics at a low flow rate regime of operation for d = 116 μm, uinj ≈ 6 m/s, qv ≈ 2.29 C/m3, te = 0.025 ms and ti = 0.123 ms [195]. Reproduced with permission from Begell House
Fig. 6.4 Jet break-up dynamics at a high flow rate regime of operation for d = 116 μm, uinj ≈ 15 m/s, qv ≈ 3.16 C/m3, te = 0.025 ms and ti = 0.123 ms [195]. Reproduced with permission from Begell House
6.1 Measured Characteristics
129
Fig. 6.5 Jet break-up dynamics at a high flow rate regime of operation with the right side of the frame at a distance 1.5 mm downstream from the nozzle plane for spray described in Figure 68(a) – (h) where d = 116 μm, uinj ≈ 15 m/s, qv ≈ 3.16 C/m3, te = 0.025 ms and ti = 0.123 ms [195]. Reproduced with permission from Begell House
camera with te = 0.222 ms for d = 116 – 254 μm, where the atomizer is visible on the top of the image. In the present study, the break-up length lj is defined as the length of continuous liquid from the nozzle to the breaking up point as shown in Fig. 6.6a. The break up point is straightforward to estimate for these sprays by post-processing a time integrated photograph of the interrogation region shown in Fig. 6.6. The cone spray angle α is defined as the approximate cone angle of the spray plume originated from the liquid breaking up point to an axial distance z ≈ 20 mm downstream the nozzle orifice plane as shown in Fig. 6.6c. As d decreased from 254 to 116 μm, the length of lj was observed to be similar but α was strongly increased from 15° in Fig. 6.6a to 45° in Fig. 6.6c. The increment in is due to space charge repulsion as the initial spray axial velocity is kept constant. The introduction of electric charge to a liquid jet will normally destabilize the jet surface by overcoming the balancing force of liquid surface tension. This usually leads to smaller mean droplet diameters and larger spray cone angles due to space charge induced repulsion of like charged droplets resulting in good interphase mixing [164]. The break-up length of a charged jet is strongly dependent on the quantity of charge transferred to the jet surface and may be correlated [165] as:
l j = uinj t j
:
tj =
ε 0ε r κqv
(6.1)
130
6 Jet Instability and Primary Atomization
Fig. 6.6 Effects of d on lj and α at constant uinj ≈ 10 m/s and qv ≈ 1.5 C/m3 for (a) d = 254, (b) 140 and (c) 116 μm, with te = 0.222 ms [195] Table 6.1 Spray and liquid properties Liquid
Ref
d
uinj (m/s)
(μ μm) White spirit
[108]
Kerosene
QV
ρ
μ
σT
(C/m3)
(kg/m3)
(kg/ms)
(Ns/m2)
500
8.5-10.2
0.34-0.57
780
0.000854
0.0225
500, 250
6.4-34
0.24-0.62
800
0.001056
0.0235
Diesel
[120]
173
21.3
1.5
815*
0.002608*
0.0250*
Calibrati on diesel
[112]
116,140,
6-20
1.1-3.7
815
0.002608
0.0250
254
*estimated
where tj is a characteristic time and uinj is the mean liquid jet velocity, based on applied flow rate and orifice area. The ionic mobility κ for the liquid in eqn. (6.1) is related via Walden’s Rule eqn. (2.21), scaling lj proportional to liquid viscosity, κ = Aμ-1, where A depends on liquid type. Using a typical ionic mobility for hydrocarbon oils [166] , κ ≈ 10-8 m2/V-1s-1, the jet break-up length relationship was found to scale correctly with the experimental data. Similar qualitative studies to Fig. 6.6 were repeated over the parameters ranges 1.13 ≤ qv ≤ 3.16 C/m3, 116 ≤ d ≤ 254 μm, and 6 ≤ uinj ≤ 25 m/s for commercial Diesel fuel. These results are now compared and evaluated with information on break-up length lj from previous studies using the same generic atomizer design but with other hydrocarbon fuels such as kerosene, white spirit and Diesel [110,111,120,164]. The break-up length lj, defined as the length of continuous liquid from the orifice, were estimated by averaging over typically 10 images. A summary of uinj, qv and d used, together with liquid density ρ, viscosity μ and surface tension σT for each liquid is presented in table 6.1. The dependence of jet break-up length to liquid electrical (κ) and physical (μ) properties, qv and uinj relationships have been investigated by the simple relationship of eqn. (6.1). By substituting κ = Aμ-1 into eqn. (6.1), and taking relative permittivity εr = 2.2 and permittivity of a vacuum εo = 8.854 10-12 F/m, the
6.1 Measured Characteristics
131
constant A could be calculated. By using values as given in table 6.1, the average value of the constant A was calculated to be approximately 1.5 ± 0.2 × 10-11, 1.6 ± 0.2 × 10-11, 4.5 × 10-11 and 2.6 ± 0.6 × 10-11 for white spirit, kerosene, diesel and diesel calibration liquid respectively. Figure 6.7 shows a plot of experimental break-up length lj,exp versus calculated lj,calc (i.e. from eqn. (6.1)) for various liquids from references quoted in table 6.1. Sprays generated with bigger orifice diameter such as kerosene and white spirit (i.e. d = 500 μm) generally have longer lj, implying the dominant effect of spray axial momentum over radial momentum. Generally, the calculated and the experimental value of the break-up length are similar, and the maximum difference was 60%.
40
lj exp (mm)
30
20
10
0 0
10
20 lj calc (mm)
30
40
Fig. 6.7 The experimental break-up length lj,exp versus the calculated break-up length lj,calc for white spirit ( ), kerosene ( ), diesel (▲) and diesel calibration liquid (♦)[195]
132
6 Jet Instability and Primary Atomization
6.2 Orifice Channel Space Charge Distribution Model It is thought that electric charge, injected from a point into a liquid positioned above an orifice migrates away from the axis of the cylindrical liquid column before and after it emerges from the atomizer orifice. Circumstantial evidence for this is found in the slight Reynolds number dependence of maximum spray charge Fig. 5.20 and also the ‘two zone’ character of sprays generated from charge injection atomizers (chapter 7), where the smaller drops are highly charged and the large drops relatively poorly. Kelly [167] conceived a ‘two zone’ model to approximate the proposed preferential accumulation of charge near the orifice wall. Figure 6.8 shows the nomenclature, in addition the following non-dimensional parameters are defined:Sheath to total flow area ratio;
⎛ rb ⎞ AR = 1 − ⎜⎜ ⎟⎟ ⎝ rw ⎠
2
(6.2)
rW rb
δ Core
EW
u0 uW
Charge Sheath Velocity Fig. 6.8 Orifice radial charge distribution nomenclature [167]
q V,0
q V,W
6.2 Orifice Channel Space Charge Distribution Model
133
Flow velocity ratio;
UR=
uw u0
(6.3)
Charge density ratio;
qR =
q v,0 q v,w
(6.4)
and a charge density parameter;
GR =
qv,av d q R (1 − AR ) + AR U R = 4 ε 0 ε r Ew ( q R (1 − AR ) + AR )((1 − AR ) + AR U R )
(6.5)
The algebraic result;
UR=
AR 1 + AR
(6.6)
is obtained for the parabolic velocity profiles associated with the operation and the charge density parameter may be expressed
q R (1 − AR2 ) + A2R GR = q R (1 − AR ) + AR
(6.7)
Since the profile is parabolic there exists a maxima for each orifice diameter, and also an equation of their loci, namely
G R = 2 AR
(6.8)
Kelly proceeded to calculate GR in terms of the sheath thickness, and compares this to experimental results (shaded entries of table 6.2). His assumption of constant sheath thickness (namely, rsh =13.4μm ) is considered dubious, the paper itself showed that rsh varied with d. As shown the model does give some realistic results when compared to the actual data. The core to sheath current ratio decreased with increasing orifice diameter (from Icore/Isheath =2.73 at d = 100μm to Icore/Isheath =1.11 at d=1000μm. This has implications for the charge density and hence the droplet size ratio (chapter 7) between the two regions. At d=100μm, DW, the sheath diameter and D0, the core are similar at 23.9 and 27.7μm. But at d=1000μm, the drop sizes have diverged, now DW =19μm and D0 =344μm. This is due to the drastic change in the charge density ratio between the two regions. Note that the sheath droplet to orifice diameter ratio remains constant (within 7%) and is numerically equal to 0.351. With the constraint of constant sheath thickness, as the orifice diameter increases the core to sheath flow rate ratio also increased, from
134
6 Jet Instability and Primary Atomization
Qcore / Q sheath =3.65 at d=100μm to Qcore / Q sheath =35.5 at d=1000μm. Due to the virtual constancy of the sheath charge density and the reduction of the core, an increasing disparity is apparent in the relative droplet population with orifice size, ranging from N=2.37 at d=100μm to N=0.062 at d=1000μm. This is observed experimentally (section 7).
Table 6.2 Model Predictions [167]
rw= 50ȝm
75
100
150
500
8.34
4.02
2.39
0.928
0.650
0.500
1.16 0.341
0.132 0.106
9.86
4.60
2.66
1.21
0.122
AR
0.464
0.325
0.250
0.171
0.053
rsh/rw
0.268
0.179
0.134
0.089
0.027
qR
0.749
0.232
0.111
0.043
0.00313
LR
0.317
0.245
0.200
0.146
0.050
Icore/Isheath
2.73
1.96
1.67
1.41
1.11
17.2
19.0
0.74
0.060
20.2
19.2
97.5
344
0.357
0.352
33.2
35.5
0.296
0.062
Variable↓ qv,av
Units↓ C/m3
GR qv,av
C/m3
qv,w
C/m3
12.3
14.7
15.9
qv,0
C/m3
9.19
3.41
1.77
DW
Ȃm
23.9
21.9
21.0
D0
Ȃm
27.7
45.4
63.0
D0/2rb
Q core / Q sheath
0.378 3.65
0.369 8.47
0.364 15.0
N
2.37
0.946
0.555
Kelly1 [168] improved on the ‘two regime’ model by assuming a continuous distribution of radial charge density profile to be of the form;
q v,r = q v,0 + br e cr f
0 ≤ r ≤ rw
(6.9)
where b, c, e and f are shape parameters, qv,0 the centerline volumetric charge density and rw the radius of the orifice wall. For the above equation there exists a peak charge density q v, p at r = rp, also, at the wall, q v,w .
1
Published earlier, and as a conference paper, but an evolution from Kelly [167].
6.2 Orifice Channel Space Charge Distribution Model
135
The non-dimensional radius ~ r = r/rw and position of the peak charge density, ~ = r /r are defined. The generalized charge density profile was written in rp p w ~ terms of r , ~ r p , qv,w, qv,p, e and f. Accordingly :f (f −e) e f ⎡~ r p (f − e) − f ~ rp ~ r +e~ r ⎤ qv,~r = qv, p + ( qv,p − qv,w )⎢ ⎥ ~(f −e) ~f ⎣ f r p − e − r p (f − e) ⎦
(6.10)
Kelly assumed that all charge profiles are limited by the maximum electric field that can be sustained by the liquid. This occurred at the wall and is defined by the electrical breakdown strength of the liquid. qv,w is constrained at the wall, by using Gauss' law,
ε 0 ε r Ew = rw
1~ ~ ∫0 r qv,~r dr
(6.11)
The charge density at r, relative to the peak charge density was defined :
qv,r~ =1− qv, p 2ε E w qv, p ⎛ ⎜⎜ 1 − rw ⎝
⎤ ⎡ ⎥ ⎢ f (f e) e f − ⎞ ~ (f − e) − f ~ rp ~ r +e~ r ⎥ ⎟⎟ ⎢ r p f −e 2f ~ 2e ⎥ rp f ⎠⎢ ~ ⎢ r p (f − e) − (e + 2) + f + 2 ⎥ ⎦ ⎣
(6.12)
And the ratio of a uniform charged orifice to the peak charge density is denoted qR and is defined as
qR =
2 ε o ε r E w qv, p rw
≤1
(6.13)
Kelly ‘simplified’ by supposing that the charge profile is similar to a generalized velocity profile, and that in these, (e - f) « e. He then set e+ = f where e+ was infinitesimally larger than f and that f 〉 1 to allow finite gradient at the centerline. As such a three parameter ( ~ r p , e, qR) charge density profile was defined;
⎡ ⎤ ⎢ ⎥ ~ e e e ~ ~ ~ ~ qv,~r r p − r + e r ln(r / r p ) ⎥ = 1 − (1 − q R ) ⎢ 2e ⎥ r p) qv, p ⎢ ~e − 2(1 + e ln ~ − 2 ⎢r p (e + 2) (e + 2) ⎥⎦ ⎣
(6.14)
136
6 Jet Instability and Primary Atomization
Kelly calculated the algorithm on the basis of qR being independent and iterating on e and ~ r until a minimum value of GR was reached. His results show for GR,min ≥ 0.2, the wall to peak charge density ratio is tightly constrained, indicating that the majority of charge lies near the wall. Using experimental data for d and GR, together with εr=2 and Ew=15 MV/m. Table 6.3 shows the relative invariance of rwp, setting a constant of 5.23 ± 0.28 μm. Table 6.3 Parameters for Lmin [168] d (ȝm)
GR
qR
e
~ rp
rwp
100
0.785
0.730
2.7
0.894
5.30
300
0.326
0.265
15.3
0.967
4.95
1000
0.124
0.095
51.5
0.989
5.50
qv/qv, peak
1.0 d = 300μm
0.5
d = 1000μm d = 100μm
0 0
0.5 r/rwall
1.0
Predicted charge profiles for d=100, 300 and 1000μm are shown in Fig. 6.9a, where the larger orifice diameters have no practical relevance but serve to show the trend of the data. The cores of the larger diameter orifices are virtually uncharged since all the charge is attracted to the wall. In reality this would produce a strongly bimodal size distribution, the larger drops forming from the central region. The outer annulus is highly charged and produces much smaller drops. Good charge distribution across the whole diameter was not predicted until a diameter of around 100μm. For sizes less than this the centerline
6.3 Chapter Summary
137
charge density assumes non-zero values. Limited by his method of calculation in that the gradient of the profile at the centerline must have finite gradient, the smallest orifice diameter Kelly was able to calculate its profile was 84 μm, shown in Fig. 6.9b.
qv (C/m3)
15
10 Wall
5
0 0
10
20 r (μm)
30
40
Fig. 6.9 Orifice normalised radial charge density profile as a function of orifice radius [168]
6.3 Chapter Summary There exists a large body of work that has analyzed the instability modes of electrically conducting charged liquid jets, where the surface of the jet may be assumed to be at an equipotential, and all of the charge resides at the surface. The surface boundary conditions, and indeed the proportion of the surface charge and the volume charge are not well defined for dielectric charged liquid jets. From experiment work a number of instability mechanisms are identified, and an approximate model of the charge dielectric jet structure has been shown to be useful in understanding the physics. Experiments measuring the mean spray charge density for different orifice diameters, and also spray measurements (next chapter) suggest that the charge density profile from centerline to orifice channel wall is not uniform. Approximate models of the charge density profile provide predictions which for the most part provide a reasonable picture of the physics.
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Chapter 7
Spray Characterization and Combustion
7 Spray Characterization and Combustion Abstract. For charge injection atomization, of electrically insulating liquids, it is typical that a polydisperse drop diameter distribution is present in the spray. It is also typical that the mean trajectories of the drops are a function of the mean drop diameter, and that the smaller drops tend to be found on the spray periphery, and the larger drops nearer the spray axis. These observations are now discussed in more detail.
7.1 Spray Visualization and Prediction of Expansion Rate Spray visualization is described for the three cases described in Table 5.2. It is noted that an uncharged 'spray' from the atomizer, at the conditions described in Table 5.2, consists of stream of large drops moving along the spray centerline. Figure 7.1 shows the case for a flow rate of 0.5mL/s and a specific charge of qv=1.20C/m3 and it is seen that the plume expands significantly. This figure highlights the dual-zone nature of these particular charged sprays. Recirculation is also observed near the injector, and this is due to small (D≈5μm), highly charged drops, produced from jet break-up, being attracted back towards the earthed atomizer body. For higher flow rates and specific charge, as shown in Fig. 7.2, the amount of scattered light increases significantly which implies higher drop concentrations. The spray, under the influence of space charge forces, expands to fill the volume illuminated. It is suggested that the axial drop velocity components are generated primarily by the jet flow, while the radial deflections shown in the figures are caused by electrical forces acting on the drop trajectories. This proposition is supported by CFD modeling of this type of sprays [169] which revealed that the radial electric field, due the approximate cylindrical symmetry, is at least an order of magnitude greater than the axial electric field generated by the charged drops near the centerline. The drop trajectories are very well ordered, typically emerging normally from the spray core and diverging at larger radial displacements. Based on the method developed by True [170], Shrimpton & Yule [165] developed a simple force balance model that was sufficient to explain the interplay between axial momentum and radial dispersion. This is achieved using the following simplifying assumptions :1) Monosized charged spray, where drop and orifice diameter are equivalent. 2) Axial velocity changes are due to aerodynamic drag forces only. J. Shrimpton: Charge Inject. Sys.: Phy. Principles, Experi. & Theore. Work, HMT, pp. 139–179. springerlink.com © Springer-Verlag Berlin Heidelberg 2009
140
7 Spray Characterization and Combustion
3) Radial velocity changes are due to space charge forces and drag forces. 4) The gas phase is quiescent. 5) There is no drop evaporation or break up. The axisymmetric charged spray plume was modeled as a sequence of cylinders, each containing the same mass of drops, each of charge q, which is uniformly spread throughout the volume of the cylinder. This assumption allows an analytic solution of the radial electric field. Bearing in mind the above assumptions, the equations of motion for the axial (u) and radial (v) velocity components may be written,
du x π 2 = CD D ρ g | ux | ux dt 8
(7.1)
π D2 d ur m =- q Er +CD ρg | ur | ur dt 8
(7.2)
m
Fig. 7.1 Time averaged free spray dynamics, Q=0.5mL/s, qV=-1.20C/m3, d=250μm [108]
7.1 Spray Visualization and Prediction of Expansion Rate
141
Fig. 7.2 Time averaged free spray dynamics, Q=0.5mL/s, qV=-3.00C/m3, d=150μm [108]
where m, q and D are the drop mass, charge and diameter, and ρg is the gas density. The radial component of the space charge field generated by the charge on the drops within the spray plume, Er was obtained by applying Gauss's Law to a cylindrical volume containing an average volumetric charge qv, such that
Er =
ql
(7.3)
2 π ε0 r
where ql is simply the charge per unit length, r is the radius of the volume. The drag coefficient used is a standard empirical form [171],
CD =
24 Re d
⎛ 1 2/3 ⎞ ⎜1 + Re d ⎟ ⎝ 6 ⎠
(7.4)
where Red is the drop Reynolds Number. A model jet break-up length, lj may be defined from defining characteristic velocity and break-up times eqn. (6.1) and initial conditions are typically defined from global nozzle parameters: D=d
; ux,o=uinj ; ur,0=0 ;
q = qv
πD 6
3
(7.5)
142
7 Spray Characterization and Combustion
The trajectory of the modeled spray plume boundary, considering the simple form of the model, is representative of the experimental data. Evaluation of the model [165] showed that the simplistic model is successful in capturing the evolution of spray shape as a function of the initial conditions. This shows that the axial motion is governed by drop momentum and aerodynamic drag and the radial motion by an approximate balance between electrostatic and aerodynamic forces. It has been shown that for highly charged sprays (qv≥1C/m3) drop deflection away from the central core is significant and for efficient combustion of these charged plumes some form of spray confinement may be necessary. Disruption of the spray core/sheath boundary is shown in Fig. 7.3 for the case of a flow rate of Q=0.5mL/s and a spray specific charge of qV=1.20C/m3 for an orifice diameter of d=250μm. The disruption is caused by the placement of an earthed bar of
Fig. 7.3 Time averaged spray dynamics near and earthed corner and an earthed cylinder, Q=0.5mL/s, qV=-1.20C/m3, d=250μm [108]
7.1 Spray Visualization and Prediction of Expansion Rate
143
diameter 12mm placed at x=0.08m, y=0.08mm, in addition to earthed surfaces along x=y=0.15m. The attractive force felt by a charged particle to a flat earthed plate (the 'image' force) is only a quarter of the force of attraction of two oppositely but equally charged particles acting over the same distance. This ensures that space charge forces, generated by the charge on the drops, will be dominant except in the region near the earthed surface. The preliminary test shown in Fig. 7.3 showed the effect of altering the electric field at discrete regions rather than planes and this principle is now investigated for a system of ring electrodes. The ring electrodes were 100mm in diameter and fabricated from 1/4" (6.25mm) steel bar. They were placed symmetrically along the axial centerline of the spray and results are shown for the same nozzle diameter (d=250μm), flow rate of Q=1.6mL/s and spray specific charge qv=1.80C/m3. Figure 7.4 shows the spray for both electrodes earthed. In the region
Fig. 7.4 Time averaged spray dynamics near a pair of earthed co-axial rings, Q=1.6mL/s, qV=-1.80C/m3, d=250μm [108]
144
7 Spray Characterization and Combustion
bounded by the nozzle, the upper electrode and the plume boundary, the small drops follow the field lines emanating normally from the spray plume boundary to either the earthed nozzle electrodes or the earthed upper ring electrode. The illuminated region is distinctly bounded by a distorted quadrant. Similar behavior was observed underneath the upper electrode and in the vicinity of the lower electrode. However for these positions the further away the drop is from the earthed electrode the less is the magnitude of the image force, and drop momentum thus has a greater influence on the drop trajectory. This is not observed for the region above the upper electrode since the drops are very small and quickly obtain a terminal velocity from the interaction of drag and electrical forces. The zero normal gradient that exists along the horizontal line halfway between the two electrodes and away from the spray core is observed. The essentially symmetrical behavior of the drops near the electrodes above and below the symmetry line shows that the image forces are dominant in the near electrode region. A fraction of the drops in the plume recirculate underneath the lower
Fig. 7.5 Time averaged spray dynamics near a pair of co-axial rings, upper +7kV, lower 7kV, Q=0.5mL/s, qV=-1.20C/m3, d=250μm [108]
7.1 Spray Visualization and Prediction of Expansion Rate
145
electrode and ultimately reverse their trajectories. Drops that have recirculated and possess radial displacements greater than the ring electrode radius (50mm) are either attracted or repelled from the electrode and this is understood by considering image and space charge forces. Drops near the rings follow the field lines and impinge onto the electrodes. For drops further away from the rings, where there are additional charged drops between the drops considered and the ring electrodes, the space charge repulsion between the drops dominates over the image charge force and the drops are repelled away from the electrodes. For drop trajectories in the diffuse outer region the tracks show a 'wiggle'. It is unclear whether this is due to changes in drop shape, and hence the centre of mass, or, due to velocity fluctuations caused by the non-uniformity of the space charge field in the diffuse region.
Fig. 7.6 Time averaged spray dynamics near a pair of co-axial rings, upper -7kV, lower +7kV, Q=0.5mL/s, qV=-1.20C/m3, d=250μm [108]
146
7 Spray Characterization and Combustion
The effect of applying positive and negative potentials to the ring electrodes on the spray dynamics is now presented. Figure 7.5 shows the case where the potentials of the upper and lower electrodes are positive and negative and have a magnitude of 7kV. The flow conditions for Fig. 7.5 are a flow rate of Q=30mL/min and a spray specific charge of qv=1.20C/m3. The large recirculation caused first by the additional radial deflection from the interaction of the upper positive electrode, and secondly by the repulsive force from the interaction of the negatively charged lower electrode, is similar for both potentials. Higher drop numbers in the recirculation zone were observed for the higher potential case, and similar behavior was observed for higher flow rates and specific charges. Figure 7.6 show the case for the same flow conditions and the same magnitudes of the potentials as Fig. 7.5 respectively, but with reversed polarity of the rings. Above the upper electrode the small drops populating this region are deflected upwards, away from the like charged ring and towards the earthed nozzle. Once drops are at axial displacements greater than the upper electrode they are attracted towards the lower electrode. However it is noted that in the region between the lower electrode and the injector, the spray is effectively confined. This has importance for the possible application of electrostatic atomization of hydrocarbon oils to spray combustion systems, where the divergent plumes could lead to incomplete combustion of the spray plume and the possibility of tailoring the spray shape may offer advantages and solutions.
7.2 Quantitative Spray Characteristics Predominantly this summary is taken from references [108-109,165,172,173]. Phase Doppler Anemometry (PDA) results are now presented for as cases 1-5 in table 7.2. Figure 7.7 shows the drop diameter frequency distributions at a position 0.12m downstream of the atomizers on the spray axis, where the photographic results showed that the majority of the ligament structures had disappeared for all cases. In terms of a representative position for the spray, the axis represents the Table 7.1 Atomizer operating conditions Property Liquid Orifice diameter Flow rate Mean velocity * Specific charge ** Reynolds number* * **
Unit s -
Case 1
Case 2
Case 3
Case 4
Case 5
Case 6
Case 7
kerosene
Kerosene
Kerosene
Kerosene
kerosene
kerosene
Diesel oil
μm ml/s m/s
500 1.3 6.7
500 2.0 10
500 3.0 15
250 0.5 10
250 1.7 34
150 0.5 28
250 1.0 20
C/m3
0.4
0.5
0.5
1.2
1.8
3.0
1.4
-
2700
4900
5720
1900
6800
3400
1800
Based on flow rate and orifice diameter, to 2sf. These are the maximum specific charge values possible for the particular orifice diameter.
7.2 Quantitative Spray Characteristics
147
“worst case” in terms of atomization quality because a high proportion of the more highly charged smaller drops have migrated away from the central core. The drop diameter frequency distribution, for the 500μm orifice diameter nozzle (cases 1-3 of table 7.1) are very different from the other cases. Figure 7.7a,b-c show two distinct drop populations. The population of very small drops, D<5μm, was found throughout the spray plume. It is thought to be generated during the primary atomization of the jet by a Rayleigh–type break-up of charged ligaments. The small drop end of the spectrum also includes drops in the size range 2040μm, and these drops are thought to be the product of secondary atomization of the larger droplets. At the other extreme of the size range there is a population of large drops with the peak of the distribution at approximately 440μm for the lowest flow rate, case 1 of table 7.1. The position of the peak decreases slowly as the flow rate rises. These large droplets are the product of the primary atomization of the relatively poorly charged spray core. There is a low peak slightly to the right of this second distribution. Although small on a number basis, the volume of the large drops in this subsidiary peak equates to approximately twice the volume of smallest drops. Instability in the jet core is thus producing drops of diameter slightly less than that of the orifice. This contrasts with the drop diameter for uncharged jet break-up predicted by Rayleigh, D=1.89d. In general the low flow rate, low specific charge atomization (case 1) generates comparable numbers of very large and very small drops but few intermediate sized drops. As the flow rate is increased, more small drops of diameter range 20-40μm are generated. Given that the specific charge for cases 2 and 3 is identical, the additional secondary atomization for case 3 (giving more drops between 20μm and 40μm) must be at least partially aerodynamically driven, and caused by the higher nozzle velocity for this case. The nature of the resultant drop diameter distribution in case 3 is unlike the products of uncharged bag/strip secondary atomization modes, where the resultant drops have a much greater range of diameters [174]. Therefore there exists a secondary atomization mechanism which is particular to charged drops, but is also a function of aerodynamic effects. Table 7.2 Atomizer operating conditions Case
(i) (ii) (iii) (iv)
D ( μ m)
254 140 140 116
uinj (m/s)
10 10 15 15
QV (C/m3)
m&
V
(g/s)
(kV)
0.4115 0.1304 0.1875 0.1263
10 9 11 10
1.13 2.25 2.67 3.16
148
7 Spray Characterization and Combustion
0.5
a) Case 1
0
0.5
b) Case 2
0.5
0
FDF
CDF
c) Case 3
0
d) Case 4
0.1
e) Case 5
0.1
0
0 0
100
200
300
400
500
Fig. 7.7 Drop diameter frequency distributions at x=0.12m, r=0.00m for (a) case 1, (b) case 2, (c) case 3, (d) case 4 and (e) case 5, all defined in table 3 [165]
There are similarities between the sprays of cases 1, 2 and 3 of table 7.1, generated via a charge injection mechanism, and the ‘electrosprays’ of doped heptane [7]. Both data-sets possess a generic dual-zone spray character, with a bimodal drop diameter frequency distribution with the ratio of diameters between the large and small drop populations being approximately 10:1 in each case. Gomez & Tang [68] presented photographs of the secondary atomization of the large primary drops, which showed this value for the ratio of drop sizes before and after the process. Thus it is likely that this mechanism is responsible for the 20≥D≥40μm size class shown in Fig. 7.7b and Fig. 7.7c. It seems that despite two very different methods of charging the liquids, and the absence of doping to decrease the liquid resistivity in the present case, the sprays are remarkably similar. This is despite a three order of magnitude difference in resistivity between
7.2 Quantitative Spray Characteristics
149
doped heptane [7] and the kerosene of the present work. A significant difference however is that of the sizes of the largest drops. In the case of the present hydrocarbon sprays, the large drops for cases 1, 2 and 3 of table 7.1 are of the order of the orifice diameter (500μm) while for the charged heptane sprays, the largest diameters were approximately 40μm [7]. This suggests that a change in liquid electrical resistivity is primarily relevant to the liquid charging process, and different ranges of resistivity are required for different electrostatic atomization techniques, as discussed in the introduction. However once the liquid contains charge, added by either the cone-jet or the charge injection method, apart from the scale consideration, the spray structures are similar. There is an absolute limit regarding the performance of liquid atomization via a Taylor-Cone. For a given atomizer geometry, an increase in flow rate will generally produce an increase in both the mean diameter and the breadth of the drop diameter frequency distribution if a Taylor cone is used [4]. In the present work it is found that this is not the case for sprays generated using the charge injection technique, since the maximum spray specific charge attainable generally increases with flow rate. This was found to be most apparent for charge injection atomizers where d≤250μm, for example for cases 4 and 5 for which drop diameter distributions at x=0.12m, r=0m are given in Fig. 7.7d-e respectively. These show that for the higher flow rate, case 5 of table 7.1, the distribution is skewed more towards the smaller drops and the most frequent drop diameter is D≈140μm compared to D≈250μm for case 4 of table 7.1, Fig. 7.7d. The maximum specific charge achieved increases with flow rate, thus this shows that promotion of liquid atomization occurs via the combination of increased aerodynamic and increased electrostatic forces. This is the opposite behavior to standard contact and induction charging techniques and is the reason that charge injection techniques are so well suited to high flow rate operation. Comparing the drop diameter frequency distribution results for the lower (cases 1, 2 and 3 of table 7.1) and Fig. 7.7a-c and higher specific charge sprays (cases 4 and 5 of table 7.1) and Figs. 7.7d-e reveal that in the latter cases the frequency distributions are continuous. This behavior cannot be due to a laminar/turbulent transition occurring within the nozzle leading to enhanced atomization for some cases. This is because the Reynolds Number, based on conditions at the atomizer orifice for case 2 is double that of case 4. Superficially there seems to be no obvious linking trend between the PDA data of cases 4 and 5, shown in Fig. 7.7d-e, and either, (1) the low flow rate PDA data of cases 1, 2 and 3, shown in Fig. 7.7a-c) or, (2) the similarity with electrosprays noted above. However electrosprays with continuous drop diameter frequency distributions have been reported in the literature [175, 176] where ethanol was sprayed at a high flow rate, 0.06ml/s, from a capillary held at a very high applied potential, typically 20kV. In this case the atomization mode has been characterized as ‘rim-emission’ [177]. Thus cases 4 and 5 of table 7.1, the sprays of Grace were obtained by operating at relatively high liquid throughput and at a high spray specific charge for the respective atomization methods. In contrast to the work of Gomez & Tang [7,68], Grace & Dunn [176] observed drop generation from ligaments emanating from the electrified meniscus rather than a single jet as in the case of a Taylor Cone, and this produced a polydisperse spray, with a
150
7 Spray Characterization and Combustion
maximum diameter of approximately 40μm. A similarity of the spray structure and drop diameter frequency distribution at high charge densities (cases 4 and 5 of table 7.1) with the results of Grace is apparent. In turn, a similarity exists for the low charge density results and the work of Gomez & Tang [7] and others. Thus the classification of sprays as either a well-ordered dual zone character or a more chaotic polydisperse form, occurs in both semi-conducting liquids, atomized by applying an electrical potential to metal capillary, and highly insulating liquids, atomized using a charge injection technique. A further insight into the spray character may be obtained by examining the variation of mean drop diameter along the axis of these cylindrically symmetric sprays. Figure 7.8 shows contours of numerical mean diameter D10 obtained by processing the PDA data for cases 2, 4 and 5 of table 7.1. The stratification of drop diameter with radial displacement is clear and this is more marked and different than for other atomization methods. For example for full cone and hollow cone pressure jet sprays, and two-fluid atomized sprays, there is a tendency for smaller droplets to occur at the spray centre. Since all drops are injected with a negligible initial radial velocity component the radial stratification of drop diameter must mainly be due to electrostatic forces. The sprays are steady-state, so that the electric field is approximately constant, the only variations being due to local space charge fluctuations. The main parameter that may affect the trajectory of a charged drop is thus its specific charge. Since a radial stratification of drop diameter occurs, the charge to mass ratio is unlikely to be constant and the trend is explicable by smaller drops having larger charge to mass ratios. The variation in mean diameter along the spray axis is due not only to smaller, more highly charged drops moving away from the axis, but also to larger drops becoming “visible” to the PDA when they attain the required sphericity. For case 2, shown in Fig. 7.8a, the boundary of the inner core of large drops is clearly visible, showing the dual-zone spray characteristic, while for the more highly charged sprays this boundary becomes blurred and gradients are more gradual. Figure 7.7-7.8, and the above discussion, shows that the atomization mechanism for producing drop diameter distributions for these charged sprays is a combination of aerodynamic forces and the disruptive electrical forces caused by the presence of free surface charge. The specific charge attainable increases with injection velocity, for a all orifice diameters. Therefore the combination of larger electrical forces that larger higher aerodynamic shear at larger injection velocities improves atomization performance significantly. As described in a section 7.1, these charged, well atomized sprays permit a degree of manipulation and control by the application of additional applied electric fields. They are also ignitable such that stable combustion can be achieved. It is also worth noting that since the charge to mass ratios of drops within the polydisperse spray are size dependant, the possibility of essentially sorting drops by size classes is available. From the spray visualization and drop diameter results it has been shown that although a range of drop sizes is present, the trajectories vary gradually with respect to the spatial position in the spray. Figure 7.9a-b show, respectively, the mean axial and radial velocity profiles for the cases of Q=0.5cc/s, qv=1.20C/m3, and Q=1.67mL/s, qv=1.80C/m3. The general pattern of spray behavior is similar to
7.2 Quantitative Spray Characteristics
151
that first observed by Shrimpton et al. [108] for the 500μm orifice diameter atomizer. Thus high axial velocity components and large radial accelerations occur in the spray core, even though the drop distribution is continuously varying, and radically different from the lower charged sprays.
r (cm)
6 23
4
54
2
77
107
130
0 0
2
4
6
8 Z (cm)
10
12
14
r (cm)
6 12
4
87
149
2
198
5
23
0 0
2
4
6
8 Z (cm)
10
12
14
r (cm)
6 18
4 2
36 81
54
214
107
0 0
2
4
6
8 Z (cm)
10
12
0
28
14
Fig. 7.8 Spatial distribution of arithmetic mean diameter for (a) case 2, (b) case 4 and (c) case 5 all defined in table 7.1 [165]
152
7 Spray Characterization and Combustion
0 z/d = 0
u/u0
-0.10 -0.20
1.00
z/d = 120
0.75
-0.30
0.50 0.25 0
z/d = 240
1.00 0.75 0.50 0.25
1.00
z/d = 360
0
0.75 0.50 0.25 0
z/d = 480 0.5mL/s
1.00 0.75 0.50
1.67mL/s
0.25
1.00
z/d = 600
0
0.75 0.50 0.25 0
0
100
200
300
r/d Fig. 7.9 (a) Axial and (b) Radial velocity profiles for Q=0.5mL/s, qv=-1.20C/m3 and Q=1.67mL/s, qv=-1.80C/m3 for d=250μm [173]
7.2 Quantitative Spray Characteristics
153
0
z/d = 0
v/u0
-0.01 -0.02
0.40
-0.03 -0.04
z/d = 120
0.30 0.20 0.10
0.30
0
z/d = 240
0.20 0.10 0.20
z/d = 360
0
0.15 0.10 0.05 0
z/d = 480 0.5mL/s
0.20 0.15 0.10
1.67mL/s
0.20
0.05 z/d = 600
0
0.15 0.10 0.05 0
0
100
200 r/d
Fig. 7.9b (continued)
300
154
7 Spray Characterization and Combustion
An overview of the mean axial velocity profiles Fig. 7.8b shows that the axial velocity, in general, decreases with increasing axial displacement from the atomizer. This suggests that the initial momentum, from the liquid injection at the atomizer, is transferred to the gas by drag forces and this process predominates over any axial acceleration due to the electric field. Indeed axial velocity versus drop diameter correlations at particular points along the spray centerline [108] have shown that the drop axial velocity is proportional to the square of drop diameter, suggesting that drag forces dominate. Of interest to charged spray dynamics is that the mean axial velocity, u=20m/s, at x=0.03m for Q=1.67cc/s is lower than the mean axial velocity at the next axial ordinate (x=0.6m) where u=24.6m/s, where, from the drag relation noted above, one would expect it to be lower. This would seem to be due to the larger drops of the diameter distribution not attaining sufficient sphericity for measurement by PDA, and biasing the mean drop velocities at small axial displacements to smaller, and hence slower drops. This behavior is not repeated for the lower flow rate case and it was not observed for sprays produced by the atomizer with an orifice diameter of 500μm. Analysis of the results from a two phase charged spray CFD code [169] shows that the axial electric field caused by the charged spray changes direction between the atomizer and at an axial position 10mm below the atomizer orifice where the spray core has started to spread out. Examination of the directions of the axial electric field and sign of the drop charge showed that charged drops must be axially decelerated near the atomizer due to force between individual drop charges and the spray electric field. This experimental evidence confirms that this phenomenon does have an effect, providing sufficiently high spray specific charges are used and supports further development of the model. Examination of the mean radial velocity profiles Fig. 7.9b shows similar behavior for both flow rates. In general the acceleration of the drops away from the spray centerline is higher for the spray with the larger spray specific charge. Due to the axisymmetric geometry both the mean radial electric field and also the drop radial velocity component at the spray centerline must be zero. However near the centerline, relative to the spray width, the radial electric field is a maximum and decreases rapidly as radial displacement increases. The radial velocity profiles shown in Fig. 7.9b closely resemble in shape the radial electric field profiles generated by a charged spray CFD code [169] which suggests that the radial drop motion is wholly driven by the electric field and the drop charge.
7.3 Estimation of the Radial Profile of Spray Specific Charge Spray charge and mass flow were measured using a purpose-built collecting system made from a set of electrode rings which varied in diameter designed in such a way that it would be possible to measure the drop charge-to-mass ratio as a function of spray radius. The system was made of a set of stainless steel tubes of non-uniform spacing to obtain a nominally constant mass flow rate of liquid for each of the annuli as shown in Fig. 7.10a. In order to measure and differentiate the spray current carried by spray droplets that enter each of the annuli, the surface of the annulus was separated from each other by an electrically insulating silicone
7.3 Estimation of the Radial Profile of Spray Specific Charge
155
Silicone Coating (R~ 10 15Ω/cm) Steel Tube (Inside Surface) Wire wool Nickel Coating
μA
kΩ
Fig. 7.10 (a) Photograph of earthed stainless steel electrode rings with a 150 mm steel rule as a scale and (b) schematic of the coated electrode ring surface [195]. Reproduced with permission from Begell House
156
7 Spray Characterization and Combustion
0.25
25
0.20
20
0.15
15
. . m(r)/m
. . m(r)/m
conformal coating (SCC). The SCC was thermal cured for two hours at 100° C before being coated by an electrically conductive nickel coating. The schematic of the coated electrode ring surface is shown in Fig. 7.10b. The interface resistance of the silicone coating prior to adding the conductive layer was measured at many points using the electrometer and found to exceed 200GΩ. A set of nylon webs was used to hold the outer tubes together and a constant blockage factor of 0.8 is applied on each of the annulus due to the introduction of the webs. Thinly lined wire wool was also placed inside the annulus to ensure a good electrical connection with the electrical charge in the spray. A method to estimate the liquid mass flow through each annuli was devised and tested, and found to be accurate to within 12.5% of the total mass flow rate.
0.10
10
0.05
5
0 0
150
300 r/d
450
600
0 0
150
300 r/d
Fig. 7.11 Variation of normalized (a) annular spray mass flow rate direct measurement and (b) spray mass flux
M D ,r M F
450
600
m D ,r m F from
versus normalized radial
position of annulus centre from spray axis r/d at an axial position z = 100 mm downstream of the nozzle into each annuli for spray case (i) d = 254 μm, uinj ≈ 10 m/s and QV = 1.13 C/m3 ( ), (ii) d = 140 μm, uinj ≈ 10 m/s, qV = 2.25 C/m3 ( ), (iii) d = 140 μm, uinj ≈ 15 m/s, qV = 2.67 C/m3 (▲), and (iv) d = 116 μm, uinj ≈ 15 m/s and qV = 3.16 C/m3 (♦)[195].
Spray charge and mass flow were measured ‘directly’ as a function of spray radius at z = 100 mm using the purpose-built collecting system described earlier. Four spray cases in table 7.2 are further investigated by evaluating spray mass and charge flux as a function of spray radius. The liquid flow into each annulus of the electrode rings set was collected over a period of time t and weighed. Overall, a moderate accuracy with underestimation ranging from 0.1–7.5% was obtained. Figure 7.11a-b show the variation of the normalized annular spray mass flow r from direct measurement and the corresponding spray momentum flux rate m
M r = m r Ar respectively, versus the normalized radial position r/d of the
annulus centre from the spray axis. The different data sets all show the same
7.3 Estimation of the Radial Profile of Spray Specific Charge
157
generic behavior in which the spray axis exhibits a near constant normalized mass r m ≈ 0.05 that must be largely uncharged liquid ligament or flow rate m
r m rises to a maximum of droplets as shown in Fig. 7.11a. As r/d increases, m 0.2 at approximately r/d ≈ 100 – 150, dependent on the spray conditions. The off-axis peak mass flow for all data-sets shows that the majority of the liquid mass is charged since this is deflected to form an expanding cone. The r at r/d > 150 sharply reduces suggesting that although annular mass flow rate m highly charged drops exist outside this spray cone, their size and axial velocity is small, which is sensible considering that the smallest drops will possess the largest charge-to-mass ratios, and therefore be strongly deflected by a given electric field. For charged sprays from a nozzle with an orifice diameter d = 140 μm, the mass flux variation generally shows that the spray is better dispersed at a lower bulk injection velocity (i.e. uinj ∼ 10 m/s as compared with 15 m/s) as shown in Fig. 7.11b. This is despite a lower spray specific charge qv as for the spray case with a lower uinj. This is because the charged droplets reside longer in the spray plume and have radial forces exerted upon them continuously. Sprays with similar bulk injection velocity, uinj were compared with sprays from a nozzle with d = 254 and 140 μm at uinj ≈ 10 m/s, case (i) and (ii) respectively as shown in Fig. 7.11b. For the spray with a lower qv (and larger d) in case (i), the mass flux variation shows that the spray is not so well dispersed as compared with the spray with a higher qv (and smaller d) in case (ii). When the orifice diameter is reduced (i.e. d = 116 μm) but with an increased qv at a similar bulk injection velocity (i.e. uinj ≈ 15 m/s) as in case (iii) and (iv) as shown in Fig. 7.11b, the mass flux variation shows an even better-dispersed spray. However when one considers the magnitude of the variation in the dependent variables, e.g. orifice diameter and spray specific charge, the change in mass flux is not especially dramatic. A core of large uncharged drops, surrounded by smaller more highly charged (and therefore more radially dispersed) companions is a characteristic of these sprays. Having quantified the radial mass flow profiles at z = 100 mm, attention now directed at the radial spray current measurements. The accuracy of the direct annular spray current measurement is first investigated. Overall, the calculated errors show an overestimation in the total current ranging from 16.8 – 19.2%. The variation of annular spray current IS,r versus radial position of the annulus centre from the spray axis for each of the spray cases is shown in Fig. 7.12. The annular spray current IS,r and the radial position r are normalized by the mean spray current IS and nozzle orifice diameter d respectively. The data of Fig. 7.13 shows good self-similarity using the chosen normalization procedure and the same curve shape as shown for annular mass flow rate in Fig. 7.11a. Therefore, the current distribution follows the mass distribution in that there is constant, very small axis electric current, followed by a sharp increase with a peak at r/d ∼ 100200 and thereafter with a gradual decay with increasing radial displacement. Comparing the mass and current data however, there are important differences. Firstly, the axis current is very near zero. Since the normalized mass flow rate data is non-zero on the axis as shown by Fig. 7.11a, it confirms the hypothesis made
158
7 Spray Characterization and Combustion
above that this axis mass is very poorly charged. Secondly, the position of the maximum current and the decay once the peak is passed is at larger r/d than the mass flow rate peak for all data sets plotted. This implies that the charge density (C/m3) increases with normalized radial displacement for all data sets.
0.4
Is(r)/Is
0.3
0.2
0.1
0 0
200
400
600
r/d Fig. 7.12 Variation of normalized annular spray current I r I S versus normalized radial position of annulus centre from spray axis r/d at an axial position z = 100 mm downstream of the nozzle from direct measurements of spray mass flow rate into each annulus for spray case (i) d = 254 μm, uinj ≈ 10 m/s and qV = 1.13 C/m3 ( ), (ii) d = 140 μm, uinj ≈ 10 m/s, qV = 2.25 C/m3 ( ), (iii) d = 140 μm, uinj ≈ 15 m/s, qV = 2.67 C/m3 (▲), and (iv) d = 116 μm, uinj ≈ 15 m/s and qV = 3.16 C/m3 (♦) [195]
This is quantified in Fig. 7.13 by dividing the mass and current rates to give the (r ) I S (r ) . The spray specific charge as a function of radius, i.e. qv (r ) = m annular spray specific charge is normalized by each of the spray mean specific charge as previously summarized in table 7.2. Because Fig. 7.13 is a postprocessed quotient of the data presented previously in Figs. 7.11-12 it is no
7.3 Estimation of the Radial Profile of Spray Specific Charge
159
surprise that the spray specific charge is also self similar. The near axis data is very tightly packed together for all the different test cases, suggesting the droplets formed that are not deflected away from the axis (probably the most massive) are all charged to the same specific charge, regardless of atomizer configuration and operating condition. More scatter is evident as radial displacement increases. It is recalled from Fig. 7.11a that the most significant mass flow occurs for 200 > r/d > 100, therefore it is the variability of the specific charge in this region of the Fig. 7.13 that accounts for the mean specific charge variations of each spray listed in
3
qV(r)/qV
2
1
0 0
200
400
600
r/d Fig. 7.13 Variation of normalized annular spray specific charge
QV , r QV
versus
normalized radial position of annulus centre from spray axis r/d at an axial position z = 100 mm downstream of the nozzle from direct measurements of spray mass flow rate into each annulus for spray case (i) d = 254 μm, uinj ≈ 10 m/s and qV = 1.13 C/m3 ( ), (ii) d = 140 μm, uinj ≈ 10 m/s, qV = 2.25 C/m3 ( ), (iii) d = 140 μm, uinj ≈ 15 m/s, qV = 2.67 C/m3 (▲), and (iv) d = 116 μm, uinj ≈ 15 m/s and qV = 3.16 C/m3 (♦) [195]
160
table 7.2. The fact that
7 Spray Characterization and Combustion
qV (r ) continues to rise is expected since it is the
magnitude of the droplet charge to mass ratio that determines the extent of the radial deflection in a given electric field.
7.4 Models for Drop Diameter and Charge Distributions A non-electrical spray, generated from a non-electrical atomizer initially consists of a population of drops with a distribution of mass (diameter), momentum and energy. The manner in which these integral (spray) measures are distributed amongst the discrete spray components (drops) is a function of the atomizer, manner in which it is operated, and the fluids used. A large body of research work exists concerning the relation of the drop diameter distribution to factors that influence it, and the reader is referred to the textbooks of Lefebvre [178] and Dunkley & Yule [179] for a starting point in this large field. Research work on measuring and predicting the drop charge-diameter distribution of electrical sprays is significantly less widespread in the literature, and that relating to electrically insulating liquids a small component of this. This is required if a user of this technology wishes to predict the trajectory of the spray plume, for instance for spray coating applications. Clearly for polydisperse electrically charged sprays this is a significant challenge, not only does the distribution of drop diameters need to be considered, but also the distribution of drop charge for each and every size class. For electrically semi-conducting liquids the majority of the research effort has centered on predicting the operation of cone-jet mode electrosprays and here a set of scaling laws have been developed that link independent variables such as fluid properties, applied voltage and liquid flow rate to the spray current and an average spray drop diameter [4]. Recently, these empirical scaling laws have been used, with a maximum entropy assumption to predict the complete drop charge – diameter PDF [180] and good agreement with several experimental data sets is shown. The roots of this maximum entropy approach may be found further back in time. It is currently the only available method for predicting independent charge and diameter distributions of electrically charged sprays consisting of drops of an electrically insulating liquid.
7.4.1 Energy Minimization Methods Vonnegut and Neubauer [77] founded the concept of using energy minimization techniques to describe the most likely charged droplet diameter within a spray. They proposed that for a particular system the total energy of each drop within the spray may be defined as :Total Energy (WT) = Surface Energy (WS) + Electric Energy (WE)
(7.6)
And that the minimum WT with respect to rd coincides with the most likely rd for a spray of given mean surface charge density (qs). Numerically WT is defined
7.4 Models for Drop Diameter and Charge Distributions
161
2
1 qs W T = 4π σ T r d + 2 Krd 2
(7.7)
Here the constant K relates to the capacitance between two concentric spheres. The minimum total energy was found by setting the derivative of WT with respect to rd to zero. Then the minimum energy state is defined by a population of drops, with surface charge density qs and radius (corrected from the paper), 1/3
⎛ q s2 ⎞ ⎟ r d = ⎜⎜ ⎟ 16 π K σ T ⎝ ⎠
(7.8)
Another contribution in the late sixties to electrodynamic spray theory was the paper published by Pfeifer and Hendricks in 1967 [89]. It drew together and correlated much of the useful experimental data concerning drop charge to mass ratios of the previous few years. Diverse liquid types such as woods metal, glycerin and Octoil were analyzed using a modified form of minimum energy approach first proposed by Vonnegut and Neubauer which assumed a MaxwellBoltzmann distribution. The energy balance describing the relationship between the parent drop of radius r0, and n identical daughter drops of radius r1 and charge q1 was. 2 ⎞ ⎛ q1 2 ⎟ = (2 n − 2/3 + n 1/3 n ⎜⎜ 4 π σ T r 1 + 8 π ε 0 r 1 ⎟⎠ ⎝
) 4 π σ T r 02
(7.9)
From the energy minimization principles they also conclude that if charge and mass are to be conserved, then the maximum number of daughter drops possible was 20, any more and the free surface energy of the sum of all daughter drops exceeded that of the parent. This was fitted to existing data to calculate charge/mass ratios for the entire spray size distribution. Following the minimum energy approach they derived various maximum, minimum and most probable surface charge density limits, namely :-
6 ( ε 0 σ T )1/2
q s,max = q s, min =
ρ l r 3/2
1.34( ε 0 σ T ) 1/2 ρ l r 3/2
q s, prob =
3( ε 0 σ T )1/2
ρ l r 3/2
(7.10)
162
7 Spray Characterization and Combustion
1.2 1.1
Forbidden Region
1.0 Normalised Energy
0.9 0.8
WT WS
0.7 0.6 0.5 0.4 0.3 0.2
Wq
0.1 0 0
5
10 15 Number of Droplets
20
25
Fig. 7.14 Normalized energy versus number of sibling drops [89]
Figure 7.14 shows the minimum energy theory compared to experimental evidence. Figure 7.15 shows how predicted maxima and minima qs versus r1 agree with theory. The large less highly charged drops may have undergone secondary disruption as noted by Abbas and Latham [64] and others. The data shows very good agreement for octoil and glycerin but only partial agreement for woods metal. This paper has been widely quoted for its experimental work and to the present day still represents a significant contribution to charged droplet work. It was clear from the initial thrust of the work of Kelly (charged metallic fluids) that the poor woods metal correlation was his basis for the extensions in statistical descriptions of charge and size. Bailey [181] commented on several of the above papers and points out the simplicities of the energy minimization approach and that by considering only the initial and final states higher energy intermediary structures are neglected which may inhibit symmetrical formations. He also proposed that the drop itself should be considered and energy sink in the work of Pfeifer and Hendricks and not the capillary.
7.4 Models for Drop Diameter and Charge Distributions
163
100 Wood’s Metal
qm (C/kg)
10 Glycerine
Minimum Energy
1.0
Rayleigh Limit Data Wood’s Metal Glycerine Octoil
0.1 0.01
0.1
Octoil
r (μm)
1.0
10
Fig. 7.15 Agreement of the minimum energy theory with experimental data [89]
Noting that theory of Pfeifer and Hendricks does not give a good description of the behavior of woods metal, Kelly [182] set out to improve on the work and to develop a theory for all electrostatically charged metallic sprays.
7.4.2 Spray Theory of Kelly Kelly [182] at this point only considered positive ion emission, wanting to model maximum charge densities. In later work examining non-metallic fluids he realized this assumption was unrealistic and used the electron emission limit for a metal surface, thus defining ES=6.5×109 V/m. Kelly described the spread of droplet charge and size over the whole spray by linking them to MaxwellBoltzmann and Fermi-Dirac distributions, but independent of each other. In both conditions it was assumed that the process was random and the particle charges are indistinguishable, and cited this as a justification that drop formation dynamics are not considered. The size and charge information derived assumes that the
164
7 Spray Characterization and Combustion
droplets are in an 'end state', far from the injector and sufficiently dispersed to exert a negligible effect on each other. In fact the series of papers this 'theory' represent [182-184] give no insight into droplet position or trajectory, nor is the momentum considered at any point. Rather it seems to fit most experimental results and as such is perceived to be a very useful indicator for the efficiency of spray charging. For a description of the charge distribution Kelly used an analogy of statistical mechanics. For the nth size level there are Nd,n drops of radius rn and carrying a total charge number of Nq,n. It is known that the charge per droplet, Nq,n/Nd,n is limited by the Rayleigh ( rˆ > 1 ), or the emission regime, ( rˆ < 1 ). To obtain a probability distribution, gq,n, the availability of occupation sites for the deposition of charge of the Nd,n drops of the nth size level is defined. The degeneracy of the nth level is, for field emission is,
⎛ 64π σ T 2 ⎞ 2 ⎟ g q,n = ⎜⎜ 3 ⎟ rˆn N d,n ⎝ e ε 0 ES ⎠
rˆn ≤ 1
(7.11)
For the Rayleigh limiting case,
⎛ 64π σ T 2 ⎞ 3/2 ⎟ g q,n = ⎜⎜ N d,n 3 ⎟ rˆn ⎝ e ε 0 ES ⎠
rˆn ≥ 1
(7.12)
At all times the total charge of any level must be equal or less than the 'degeneracy' of that level, i.e.. Nq,n
tq = ∏ rˆn
g q,n ! N q,n ! ( g q,n − N q,n ))!
(7.13)
Here rˆ n ranges from 1 atom (n=1) to the maximum possible, where the mass of the entire spray plume was considered as one droplet. The above equation was simplified using Stirlings' approximation, on the basis that Nq,n and gq,n are large. This was true for liquid metals due to their high surface tension and also true for other non-metallic fluids since ES-〈 ES+. The number of possible distributions can then be written
⎡ ⎞⎤ ⎛g − ⎛ g q,n − N q,n ⎞ ⎟ + N q,n ln⎜ q,n N q,n ⎟⎥ f q = ln( t q ) ≈ ∑ ⎢ g q,n ln⎜ ⎟⎥ ⎜ g ⎟ ⎜ g rˆ n ⎢ q,n q,n ⎠⎦ ⎝ ⎠ ⎝ ⎣
(7.14)
It was then asserted that fq would on average be a maximum, being implicit in the assumption of randomness. The constraints on the distributions are the conservation laws of:-
7.4 Models for Drop Diameter and Charge Distributions
165
Charge N q = ∑ N q,n
(7.15)
rn
4 Mass m = ∑ π r n3 ρ l N d,n rn 3
(7.16)
2 ⎞ ⎛ (( N q,n / N d,n )e ) ⎟ Energy W T = ∑ ⎜ 4π σ T r n 2 N d,n + N d,n + N q,n qV + E q ⎟ ⎜ 8π ε 0 r n rn ⎝ ⎠
(7.17) Where Nq, m and WT are the constants of the system. In order of appearance left to right, contributions to WT are :1) Surface energy used in forming the drops. 2) The electrostatic energy associated with the drop formation. 3) Kinetic energy of the droplets resulting from their acceleration through the actual accelerating potential, V. 4) The electrostatic energy of the droplet cloud, where the drops are treated as point charges. For any size level, it was assumed that the available charge was evenly distributed in accordance with energy minimization criteria. The technique of Lagrangian multipliers was then used to determine the maximum number of charge distributions (fq). Designating the multipliers by α, β, and δ, which relate to charge, energy and mass conservation respectively the condition for the maximized number was given by:
∂ fq ∂ N q,n
+α
∂ Nq ∂ ∂m + β ET + δ =0 ∂ N q,n ∂ N q,n ∂ N q,n
(7.18)
Kelly approached the size distribution in the same manner as the charge, except for a different statistical basis. It was assumed that there were many more size levels than there are droplets to occupy them. In addition were are no restrictions to the number of droplets per size level, with the exception of r ≈ rmax where Nd,n →1. Kelly made further assumptions in that the system was dilute and that the charge and droplet distributions are uncoupled. With the above conditions, and no restriction on the population number, Maxwell-Boltzmann statistics were claimed to apply, therefore the number of possible size distributions, td , is:
td = N d ! ∏
g d,Nnd,n N d,n
(7.19)
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7 Spray Characterization and Combustion
By applying Stirlings' approximation as before, and according to the constraints of the system,
∂ Nq ∂ fd ∂ ∂m +α + β ET + δ =0 ∂ N d,n ∂ N d,n ∂ N d,n ∂ N d,n
(7.20)
In describing both distributions Kelly made use of Zn, the fraction of maximum charge possible on the nth size level., i.e..
Z n = N q,n / g q,n
(7.21)
Two sets of equations for the drop number and charge distributions were derived for the Rayleigh and emission limited regimes and were simplified to the following assuming
dZ →0 dln( N d,n )
(7.22)
For the charge distribution :-
⎛ 4σ T e ⎞ − ln[(1 Z n ) / Z n ] + rˆn2 α ′ + β ′⎜⎜ ⎟⎟ Z n rˆn3 = 0 rˆn ≤ 1 (1 Z n ) / Z n Z n ln Z n (1 Z n ) ⎝ ε 0 ES ⎠
[
]
⎛ 4σ T e ⎞ − ln[(1 Z n ) / Z n ] + rˆn3/2 α ′ + β ′⎜⎜ ⎟⎟ Z n rˆn2 = 0 rˆn ≥ 1 (1 Z n ) / Z n Z n ln Z n (1 Z n ) ⎝ ε 0 ES ⎠
[
]
(7.23)
(7.24)
And for the drop number distribution :-
⎤ ⎡ ⎛ σT e ⎞ ⎟⎟ 1 + 2 Z n2 rˆn +⎥ ⎢α ′ Z n + β ′⎜⎜ ⎝ ε 0 BS ⎠ ⎥ 2: ˆ ≤ 1 ln( N d,n ) = ⎢ ⎥ rˆn r n ⎢ ⎛4 ρ σ e⎞ l T ⎥ ⎢δ ′⎜⎜ ⎟ 2 3 ⎟ rˆn ⎥⎦ ⎢⎣ ⎝ 3 ε 0 E S ⎠
(
)
⎤ ⎡ ⎛ σT e ⎞ ⎟⎟ 1 + 2 Z n2 rˆn1/2 +⎥ ⎢α ′ Z n + β ′⎜⎜ ⎝ ε0 FS ⎠ ⎥ 3/2 : ˆ ≥ 1 ln( N d,n ) = ⎢ rn ⎥ rˆn ⎢ ⎛4 ρ σ e⎞ 3/2 l T ⎥ ⎢δ ′⎜⎜ ⎟ 2 3 ⎟ rˆn ⎥⎦ ⎢⎣ ⎝ 3 ε 0 E S ⎠
(
In eqn. (7.23) to eqn. (7.26),
(7.25)
)
(7.26)
7.4 Models for Drop Diameter and Charge Distributions
α ′ = N q,r α , β ′ = N q,r β , δ ′ = N q,r δ
167
where N q,r =
64π σ T 2
ε 0 ES
3
(7.27) where Nq,r is the maximum charge level a droplet of the crossover radius can attain. 7.4.2.1 Correlations and Simplifications Further simplification can be made by considering the limiting cases where one of the derivatives becomes small. Kelly confined himself to where :
− ln [(1 − Z n )/ Z n ] →0 (1− Z )/ Z Z n ln [ Z n (1 − Z n ) n n ]
(7.28)
Liquids of high electrical conductivity were initially to be modeled so the rate of charging was not impeded. Consideration of eqn. (7.23) to eqn. (7.26) reveals that either α', β' and δ' are of different sign. The δ' multiplier must be negative to satisfy vanishingly small populations for large rˆ . Also α' and β' must be of opposing sign for Zn to be real. Further if 0.3 ≤ Zn ≤ 0.7 the first term in eqn. (7.24) could be set to zero, giving the additional simplification :-
Z n ≈−
α ′ ⎛ ε 0 E S ⎞ 1/2 ⎜ ⎟ rˆn β ′⎝ 4 σ T e ⎠
(7.29)
By dividing by the droplet mass, it was found that the specific charge (Zn/m) was proportional to 1/r2, the dependence reported by Krohn [185] when spraying woods metal. By writing Z as the relative charge density ratio (Actual/Rayleigh) then qv represents the mean charge in the spray and D an average diameter.
⎛ α ′ ⎞ 12 ε 0 2 D = ⎜⎜ − ⎟⎟ ⎝ β ′ ⎠ e qv
(7.30)
So far the theory proposed explained three sets of data, all with a drop size less than 10μm. For drop size distributions of larger average diameter the above simplifications could be better quantified. Kelly [184] describes equations applicable to low charge density electrostatic atomization, for liquids such as hydrocarbon oils. The f(Zn) term in eqn. (7.24) was assessed and using his calculated value for -(α'/β') which was ≈10-17 J and typical hydrocarbon properties (σT ≈ 0.03 N/m, ρl ≈ 1000 kg/m3). It was claimed that the term could be neglected as long as the mean drop diameter was 〉 10 μm. This result was applied for metals (i.e. σT ≈ 1 N/m, ρl ≈ 10000 kg/m3) for droplet sizes one magnitude smaller than the hydrocarbon range. The behavior of the equation was shown using data from woods metal (α' = -0.18, α'/β' = -1.06 × 10-17J, σT = 0.45, ES = 6.5 × 109V/m). It was revealed f(Zn) is an symmetrical function of Zn about Z=0.5 which ranges from +∞ to -∞ as the charging parameter (Zn) varies from 0 (no charge) to 1 (the Rayleigh or emission limit). By contrast, the RHS of eqn. (7.24)
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7 Spray Characterization and Combustion
is a linear function of Zn, has an intercept of − α ′ rˆ3/2 and a slope of
− β ′(4 σ T e/ ε 0 E S ) rˆn2 . Since α' is negative and β' positive for a Rayleigh limited system the line can be shown to have negative slope. For Woods metal a single solution occurs from rˆ =1 to rˆ =12.4 (physically D = 0.12μm), at which point two solutions occur, Zn = 0.11 and 0.75 . For rˆ 〉 12.4, three solutions are possible :a) Zn 〈 0.11, Zn decreases with increasing
rˆn . b) Zn 〈 0.75, Zn decreases with increasing rˆ n c) Zn 〉 0.75, Zn increases with increasing rˆ n . As shown by Fig. 7.16 the solution undergoes a "phase transition". In the case of Woods metal, experimental data showed that the 'b' branch was followed. A minimum energy path was calculated and it was found that this was indeed the path of lowest energy. By calculating Zn for rˆ n = 20, Zn = 0.44, the solutions for a, b and c are Zn = 0.06, 0.46 and 0.96. This confirms that b, the minimum energy route was followed. This is clear when demonstrated graphically [184], where the fastest way to the minimum energy line is along b, as both the other solutions recede from it. Kelly examined the tangential solution node, ZT and equated it to
− α ′ ⎡ ⎛ ε 0 ES ⎞ ⎛ α ′ ⎞ ⎤ ⎟ ⎜⎜ − ⎟⎟ ⎥ ZT = 3 ⎢ ⎜ σ T ⎣ ⎝ 4e ⎠ ⎝ β ′ ⎠ ⎦
3
(7.31)
where only α' and σT are variables. For woods metal, for 0.27 ≤ ZT ≤ 0.57 the number and charge distributions are linked and the theory is not valid. Kelly set the limit for valid charged spray behavior to ZT ≥ 0.57. However ZT ≤ 0.27 is still just
1 c
Z
b a
1/2
z α r- 1
r
rˆ for Z
Fig. 7.16 Z versus energy branch [184]
T
> 0.75 (woods metal) showing phase transition and minimum
7.4 Models for Drop Diameter and Charge Distributions
169
1 d
N 1/2 e f r
Fig. 7.17 Z versus
rˆ for Z
T
< 0.27 (glycerine) showing phase transition [184]
as valid. No value of the physical grouping exists for which the tangency does not occur, therefore all liquids will demonstrate such multiple solution behavior. The narrow parameter range associated with the critical ZT range for woods metal is 3
−α′ ⎡ ⎛ ⎞ ⎛ α′ ⎞ ⎤ 0.361 ≤ 3 ⎢ ⎜ ε 0 E S ⎟ ⎜⎜ − ⎟⎟ ⎥ ≤ 0.388 σ T ⎣ ⎝ 4e ⎠ ⎝ β ′ ⎠ ⎦
(7.32)
and defines the boundary between two different types of spray behavior. Woods metal and octoil behave as shown, where ZT≥0.57. Glycerin however has ZT≤0.27 and calculation of eqn. (7.26) shows that the fluid behaves rather differently. The tangency point occurs very close to the Rayleigh limit (Z=1). The solution for greater than the critical radius (physically D = 1.64mm) indicate two similar low charge density solutions as shown in Fig. 7.17. It was then expected that glycerin spray would give a bimodal size distribution since this would be most energetically favorable. 7.2.4.2 Analysis of the Lagrangian Multipliers In extending his theory to cover dielectric electrostatic processes Kelly [184] elaborated on the physical representation of the lagrangian multipliers α', β', and δ' . He analyzed spray data from previous workers [62, 185, 186] for octoil and taking this result as significant, concluded that an appropriate form for this constant was :-
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7 Spray Characterization and Combustion
ª Es e3 º α′ = −2π « » β′ ¬ ε0 ¼
1/2
(7.33)
When he fitted this with existing data he found that an optimal solution occurred when ES = 6.5 ± 0.5 ×109 V/m, which coincides with the field strength limit relating to electron emission processes. Kelly therefore concluded that any electrostatic spraying processes was limited by the lowest value ion emission possible in the system, namely ES-, and not ES+ as assumed in his 1976 paper [182]. This remains sensible providing that the liquid drop or metal particle is metallic in composition. When other fluids such as glycerin and hydrocarbons are modeled the electron emission limit becomes physically unfounded. If the reader imagines a spray of hydrocarbon droplets in air of particular charge and diameter then the breakdown limit of the fluid should be of relevance. The theory pertains to the 'end state', as such the injector is far from the spray and also by implication the charging mechanism is irrelevant. No metallic surfaces are present and the emission limit of an electron from such a metallic material has no place in defining the charge and size distributions within the non-metallic spray. Each of the Lagrangian multipliers were examined. Kelly supposed that if α' was of the order of 10-1 and that if droplet radii below 10 μm the charging eqn. (7.24) was dominated by the α' and β' terms, thus α' was large. Intermediate behavior, exemplified by octoil occurs when α' was of the order 10-3. Small values of α' (typically ≤10-6) result in the specific charge description being dominated by the first term of the RHS in eqn. (7.24) and physically this was a characteristic of a fluid such as glycerin . In spite of the theory being based on the concept of statistical equilibrium, Kelly proceeded to formulate two correlations for α', the first solely based upon fluid transport properties, 1/2
· σT ¸ 2 2¸ ( / ) μ ρ l ε 0 ES ¹ © l §
α ′ = −0.134 ¨¨
This being corrected to eqn. 6 [184]. The second relates α' to
(7.34)
rˆn such that
0.96
§ ˆ · α ′ = −0.0373 ¨¨ σ2T r n ¸¸ © μl / ρl ¹
(7.35)
It must be noted that these two correlations are based on only three data points, and as such are of questionable accuracy. In similar fashion, Kelly tried to correlate a value for δ' based on fluid properties and system parameters:
⎛ 4.2 ε 03/16 ⎞ ⎛ g E S ⎞3/8 δ ′ = −⎜⎜ 7/16 3/4 ⎟⎟ ⎜ ⎟ ⎝ ρ l e ⎠ ⎝ m ⎠
(7.36)
7.4 Models for Drop Diameter and Charge Distributions
171
100
30
qm (C/kg)
10
3 Wood’s Metal
1
Octoil
0.3 0.1 0.03
0.1
0.3 1 3 Drop Radius (μm)
10
Fig. 7.18 Agreement of the Rayleigh Limit and the theories of Pfeifer and Hendricks, and Kelly’s theory with experiment [183]
Kelly used his dimensionally derived expressions to predict the effect of different liquids on the spray properties. 1) 2) 3)
The higher the viscosity and the denser the fluid the broader the distribution. A larger surface tension will result in a droplet population with more massive drops. Temperature effects will act on viscosity, with the result given in a).
172
7 Spray Characterization and Combustion
4)
The higher the charge density (qv), the smaller the droplet diameter. It effectively reduces the influence of -α' and acts in tandem with viscosity.
If the spray operates in the asymptotic regime, where eqn. (7.29) is valid, there exists a closed solution for δ', obtained from the maximum value of Nd,n where d(Nd,n)/d rˆ n =0. The differentiation shows that the value of δ', in terms of α' and α'/β', is
δ′ =
2α ′ ⎛ 2 σ T ( −α ′/β ′ ) ⎞ ⎜⎜ − ε 0 2 2 ⎟⎟ ρ ⎝ ( −α ′/β ′ )D e D ⎠
rˆn ≥ 1 , rˆn ≤ 1
(7.37)
Where eqn. (7.37) is corrected from Kelly [184]. 7.4.2.3 Energy Considerations By defining the total energy required to develop an electrostatic spray as the energy necessary to form the droplet surface plus the energy needed to charge it, Kelly expressed the total energy as WT = VC where V was the equivalent spray voltage drop and C was the capacitance. By assuming that the Rayleigh criteria limited regime was in effect, then the ratio of spray drop energy to charge can be approximated by delta functions as the number distributions in the asymptotic regime are very narrow, then
⎛ ⎞⎛ 1 + 2 Z 2 ⎞ 1/2 V = ⎜⎜ σ T ⎟⎟⎜ ⎟ rˆ ⎝ ε 0 B S ⎠⎝ Z ⎠
(7.38)
and by substituting in the expression for Z eqn. (7.29) and rx eqn. (4.3),
V=
σ T eD ⎡1 + ( −α ′/β ′ ) 2 ε 0 / σ T D ⎤ ⎢ ⎥ q 2 ε 0 ( −α ′/β ′ ) ⎣ ⎦
(7.39)
The equivalent voltage represents the minimum voltage for spray development. This was the case even if enough spray droplet charge was present, assuming operation in the asymptotic, low charge density regime. A field strength parameter can be equated to the first parameter grouping, such that
ES =
σT e 2 ε 0 (-α ′/β ′ )
(7.40)
which for hydrocarbon properties (σT≈0.03N/m) is very near the breakdown limit for these liquids. 7.4.2.4 Performance of Kelly’s Model Droplet flux, diameter and current measurements have been published [187] and compared to the theories of Rayleigh [58] and Kelly [182-184]. They used ethanol drops, of a diameter range 1 to 50 μm and positively charges sprays of charge densities up to 210 C/m3. They examined the spray at two axial positions (15 and
7.5 Spray Combustion
173
30 mm) and 20 lateral positions for two applied voltages, 25 and 30 kV using PDA instrumentation. A comparison between the two spray charge densities shown reveals a progressive increase with increasing specific charge in the number of smaller droplets occurring at the centerline and at the periphery. Dunn & Snarski have found evidence of the proportionality of droplet charge and diameter, which they claimed was in agreement with some unpublished work of Gomez & Tang. Several droplet charge models were investigated. They tested the models of Pfeifer & Hendricks [89] where q ∝ D-3/2 eqn. (7.10) and the asymptotic model of Kelly eqn. (7.30) was also tested where q ∝ D-2. For Kelly, several representative diameters were tested (Dmode, D10, D20, D30, and D32). The linear average diameter (D10) gave the best agreement, to within 5 to 7% when compared to experimental data. Rosin-Rammler fits most of the data very well, with an error of between 2 and 5%. At the higher charge densities, where a progressively larger number of smaller drops were present the distribution was better described by a lognormal fit. Based on this trend it can be expected that at even higher charge densities the drop size distribution will be defined by a normal distribution [182-184]. More recently Kelly has explored the emission limited regime [188-189] though this is outside the scope of the present review.
7.5 Spray Combustion Weinburg conducted research on many aspects of electrical aspects of combustion [6,11,190,191], and recently this work has been used in the development of electrospray combustion systems for small portable power units [192-194]. However these doped electrospray combustion methods rely on the liquid hydrocarbon fuel being ‘doped’ with an additive to increase the electrical conductivity sufficiently to create sprays with conventional electrostatic techniques. Charge injection techniques permit a range of hydrocarbon fuels to be burnt without doping. Combustion tests were carried out for conditions specified as cases 4 and 5 of table 7.1 and the flame shape was similar to the results shown in Fig. 7.19. This is in spite of the spray specific charge being approximately double for the smaller orifice diameter. This supports the previous suggestion that the smaller drops evaporate and burn by the time the flame front is reached. For the case 6 combustion test, a ceramic disk was fitted onto the face of the atomizer, since cold spray recirculation and plume expansion was excessive near the atomizer and this initially raised safety concerns. On ignition however it was found the excessive spray expansion is negated and stable flame was obtained, that did not require the pilot flame for stabilization as shown in Fig. 7.20a. The base of the flame front was 9 to 10mm above the surface of the atomizer, as shown by an enlarged image of the flame seat Fig. 7.20b. Visual examination of the base of the flame front showed that the inside and outside surfaces of the flame front were surrounded by a blue halo and the region where jet break up was occurring was not burning. It is thought that the combustion and atomization zones are segregated by a fuel vapor-rich evaporation zone. These observations are similar to those of Gomez and Chen [192] made for much smaller liquid
174
7 Spray Characterization and Combustion
throughputs using a contact charging method with doped heptane. There is an added advantage of the charge injection design used here, compared to the contact charging method, in that the high voltage electrode is enclosed within the earthed nozzle body, which improves the robustness of the burning spray flow by completely decoupling the charge injection and combustion processes. Diesel oil was also successfully burnt using a 250μm orifice for operating conditions specified in case 7 of table 7.1, and this is shown in Fig. 7.21. This result highlights the applicability of the method for combustion of heavier hydrocarbons fuels and the improvements with respect earlier results [120] where a high
Fig. 7.19 Combustion of kerosene sprays for a d=500μm atomizer with pilot flame for (a) case 1, (b) case 2 and (c) case 3, all defined in table 7.1 [165]. Reproduced with permission from Begell House
7.5 Spray Combustion
175
Fig. 7.20 Combustion of a kerosene spray for a d=150μm atomizer (case 6 of table 9) without pilot, (a) spray flame, (b) close up of flame seat [165]. Reproduced with permission from Begell House
temperature and pressure version of the 'Spray Triode' was used which had smaller orifice diameter of 175μm and reported only intermittent combustion of diesel oil. A study was made of the effect of pre-heating the liquid fuel on the magnitude of the spray charge and this was negligible over the small range tested, of 295≥T≥311K. This suggests that moderate fuel preheating would be possible for the electrostatic atomization and combustion of heavy fuel oils using a charge injection method. The effect of an additional electric field on the flame was also investigated and an approximately uniform field was applied normal to the nominal spray/flame symmetry axis using vertical metal plates. The effect of the presence of the plates when both were earthed was initially investigated. It was found that the presence of the plates caused the spray flame envelope to expand relative to a similar free spray. This is sensible considering the increase in potential gradient caused by the zero potential planes caused by the plate surfaces. The results presented below may be understood more easily with reference to similar effects previously documented [191] on uncharged homogenous flames in the presence of electric fields. To summarize, for a net uncharged flame, positive ions and free electrons are created in equal amounts. However the electrons have much larger mobilities than the positive chemi-ions and are quickly accelerated and removed by the applied electric field. This leaves the flame with a much higher concentrations of positive ions and net positive charge, which ultimately results in positively
176
7 Spray Characterization and Combustion
Fig. 7.21 Diesel oil combustion. Flow conditions as case 7 of table 7.1 [165]. Reproduced with permission from Begell House
charged soot particles. These positive ions will induce body forces and motion, the ‘ionic wind’, and therefore homogenous flames tend to be attracted to sources of negative potential. Stronger electric fields will increase the rate of extraction of negative charge until it equals the rate of production, at which point the current flux and body force magnitude will no longer increase. The electrode on the left of the spray was used with different positive and negative potentials as shown in Fig. 7.22 and Fig. 7.23 respectively. The electrode on the right of each flame was earthed, and for these results the spray defined by case 4 in table 7.1 was used. For the positive applied electric field, Fig. 7.23, spray and flame deflection is small but flame luminosity is reduced at higher applied fields. The reduction in luminosity is thought to be due to the extraction of free electrons but that the deflection was so small was unexpected. Without combustion the spray plume would be deflected to the left progressively as the applied field is increased. For this case, the presence of the ionized environment is decoupling the link between the applied field and the droplets. The application of
7.5 Spray Combustion
177
a negative electric field Fig. 7.23, shows that charge remains on the drops as it can be seen that the drops are deflected by the applied field. Faint drop tracks and streaklines illuminated by combustion of the drops can be seen on the right of the flame. The flame is pulled to the left towards the negative potential sink by, it is thought, the ionic wind induced from the positively charged chemi-ions, the carbon particles. This premise was confirmed by the observation that during this experiment soot deposits were found on this electrode.
Fig. 7.22 Effect of a positive external electric field on spray flame stability. Flow conditions as case 4 of table 7.1. (a) Vr-=+2kV, Vr+=0kV, (b) Vr-=+4kV, Vr+=0kV, (c)Vr=+6kV, Vr+=0kV, (d) Vr-=+8kV, Vr+=0kV, (e) Vr-=+10kV, Vr+=0kV [165]. Reproduced with permission from Begell House
Fig. 7.23 Effect of a negative external electric field on spray flame stability. Flow conditions as case 4 of table 7.1. (a) Vr-=-2kV, Vr+=0kV, (b) Vr-=-4kV, Vr+=0kV, (c)Vr-=6kV, Vr+=0kV, (d) Vr-=-8kV, Vr+=0kV, (e) Vr-=-10kV, Vr+=0kV, (f) Vr-=-15kV, Vr+=0kV [165]. Reproduced with permission from Begell House
178
7 Spray Characterization and Combustion
In general electrostatic effects work best on small particles with high charge to mass ratios. Ionized molecular products probably represent the upper limit in terms for particle charge to mass ratio which implies that the spray shaping, mixing, flame stabilization and optimization should be easily realizable. To have the spray essentially pre-charged as a subsidiary effect of the atomization system encourages the use of electrical flow and combustion modulation. This also raises the possibility that the residual charge in the combustion products could then be used to increase the performance of emissions control systems using robust lower power designs with no moving parts of small unit size. The use of electromagnetic forces in plasma control is very advanced and it is surprising that electrical control of combustion systems has not seen the same success. In particular the use of AC electric fields to synchronize spray placement with combustion oscillations is suggested as a way of reducing soot and NOx simultaneously.
7.6 Summary The spray characterization of liquids electrostatically atomized and charged by charge injection methods is a rich area of fundamental research, because the sprays are polydisperse, and the charge distribution amongst the drop population is complex. The spraying method has direct application to combustion systems, and the possibility of spray shaping permits a unique control method to optimize the combustion process. Providing the sprays are not too highly charged, simple spray plume dispersion models provide reasonable accuracy. Quantitative measurements of sprays, using PDA reveals a radial stratification of mean drop diameter, with the smaller drops having a higher probability of being found on the spray periphery. Sectional measurements of mass and current flow show an approximate self similarity in these profiles, and confirm the radial profile of charge density increases non-linearly away from the spray axis. A model to predict the drop charge and size distributions based upon maximum entropy methods produces good results over a range of conditions. However to be completely general, the use of assumed PDFs and the assumption that the charge and the diameter PDFs are independent should be addressed. The entropy considerations of Kelly are the only feasible way to proceed when trying to predict the global parameters affecting the properties of a spray. A similar approach has been adopted by RW Sellens in a number of papers to describe uncharged sprays with some degree of success. When a spray exists where the contribution of available energy from the electrostatic component is large then the theory of Kelly may adequately describe the spray. Within the limit of the assumptions made by Kelly the derived charge and size distribution predictions for highly charged conducting liquids agree well with experimental data. For sprays of charged insulating liquids there is far less charge present and hence amount of electrostatic energy available for spray description is reduced. In consequence the assumption of negligible mechanical energy contributing to the spray development is not valid and the spray is not well described. The theory of Kelly, nor any other cannot be adequately proved or disproved until instrumentation is available that can measure
7.6 Summary
179
the size and charge and velocity of a significant fraction of the drops within a charged spray plume, or through a posterior testing using charge spray models. The theory of Kelly, based on the charge density level in the liquid was confirmed in the operation of his Triode. Kelly states that the charge density level defines the spray, and not the electrical conductivity of the liquid, and cites this as the major reason why no advances were made while using it as a dependant variable. The theory also predicts that under the condition of radially uniform charge injection, the resultant charged jet should disrupt, giving a uniform droplet distribution having a diameter variation ± 1μm about a mean established by the overall mean chsssssarge density. Electrostatic spray size distributions from charge injection atomizers are narrower than those obtained from conventional designs, but substantially broader than his theory predicts. The charge spray plumes have been shown to burn stably and under very mild operating conditions, and the presence of external electric fields can influence the combustion process.
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Chapter 8
Conclusions and Future Outlook
8 Conclusions and F uture Outloo k
8.1 Conclusions The manuscript brings together several disparate fundamental research areas and presents the information in the context of the development of understanding of a practical technology application; charge injection atomization. The technology permits standard liquid combustion fuels, and any other electrically insulating liquid, to be electrically atomized and dispersed efficiently. Physically, an understanding of the relationship between the various electrical and hydrodynamic timescales pertaining to dielectric liquids that possess a volumetric charge density is required. Chapter 2 introduces the extended NavierStokes equations with incorporated electrostatic terms and defines how the charge, momentum and electric field variables are highly coupled. A discussion of the energy equation is also made here to show how charge also affects this. Relevant timescales and non-dimensional numbers associated with the EHD system have been defined, in particular a brief introduction is provided concerning the T, C and M parameters widely quoted in the available literature. Chapter 3 discussed the mechanism by which charge is injected into an electrically insulating liquid, from which the physics discussed in chapter 2 become relevant. A discussion of the current instabilities that accompany the space-charge limited regime at higher voltages is also present, which leads onto the topic of vapor bubbles. It is shown that vapor bubbles have two major effects that should be considered when designing charge injection atomizers. The first is that they are initiated by current pulses which inject further deposits of charge into the liquid, in addition to the steady current observed, and so may have a further effect on transport of the charged flow. The second is that these vapor bubbles are thought to be precursors to breakdown of the dielectric liquid and so are of importance when considering atomizer design, operation parameters and methods of injecting charge into a dielectric liquid. Whilst chapters 2 and 3 discuss a continuum electrohydrodymics, chapter 4 examines the stability of charged liquid drops, defined by the Rayleigh Limit. Past and present day work on charged drop disruption suggests that drop break up can be viewed as two linked subjects, depending on the conductivity of the liquid and therefore the way in which charge is held on the drop surface. For both cases repeatability of the charge and mass ratio of original to residual drop for a range of diameters is preserved. J. Shrimpton: Charge Inject. Sys.: Phy. Principles, Experi. & Theore. Work, HMT 1, pp. 181–184. springerlink.com © Springer-Verlag Berlin Heidelberg 2009
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Chapter is the main thrust of the manuscript and provides a largely empirical understanding of how a generic ‘point-plane’ charge injection atomizer operates, through largely experimental iteration and interpretation of the results. It is noted that more complex atomizer designs including multi-orifice and more complex high voltage electrode configurations are possible. The use of a geometrically simple ‘point-plane’ atomizer design naturally links to the work outlined in chapter 3, in effect a point-plane charge injection atomizer with zero imposed bulk flow. The data presented in this chapter shows the link between the internal geometry of the atomizer, and the hydrodynamic and electrical coupling present, and highlights that only certain configurations permit operation in the supercritical atomizer regime. In this regime, the corona discharge surrounding the liquid jet limits the maximum spray charge and that the charge injection process, occurring inside the atomizer at the tip of the high voltage electrode is stable and unaffected by this external disturbance. In terms of the important geometrical parameters than define successful atomizer design and operation, the most important is the orifice diameter, and smaller values permit larger spray specific charges. Although not discussed in detail here, practical multi-orifice atomizer configurations are possible, permitting high flow rates and fine atomization. Generation of electrically charged pulsed sprays of electrically insulating liquids has been shown to be possible though to obtain robust operation more research on the electrical and hydrodynamic timescales is required to optimize the system stability, particularly for short pulse width spray operation. Chapter 6 considers the primary atomization of electrically charged dielectric liquid jets, generated by charge injection atomizers. There exists a large body of work that has analyzed the instability modes of electrically conducting charged liquid jets, where the surface of the jet may be assumed to be at an equipotential, and all of the charge resides at the surface. The surface boundary conditions, and indeed the proportion of the surface charge and the volume charge are not well defined for dielectric charged liquid jets. From experiment work a number of instability mechanisms are identified, and an approximate model of the charge dielectric jet structure has been shown to be useful in understanding the physics. Chapter 7 collates spray characterization data of liquids electrostatically atomized and charged by charge injection methods. The sprays are polydisperse, and the charge distribution amongst the drop population is complex. The spraying method has direct application to combustion systems, and the possibility of spray shaping permits a unique control method to optimize the combustion process. Providing the sprays are not too highly charged, simple spray plume dispersion models provide reasonable accuracy. Quantitative measurements of sprays, using PDA reveals a radial stratification of mean drop diameter, with the smaller drops having a higher probability of being found on the spray periphery. Sectional measurements of mass and current flow show an approximate self similarity in these profiles, and confirm the radial profile of charge density increases nonlinearly away from the spray axis. A model to predict the drop charge and size distributions based upon maximum entropy methods produces good results over a range of conditions. However to be completely general, the use of assumed PDFs and the assumption that the charge and the diameter PDFs are independent should be addressed. The entropy considerations of Kelly are the only feasible way to
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proceed when trying to predict the global parameters affecting the properties of a spray. The charged spray plumes have been shown to burn stably and under very mild operating conditions, and the presence of external electric fields can influence the combustion process.
8.2 Future Outlook Electrostatic atomization methods and electrical sprays have been widely used in industry for decades, most notably for painting applications. In science and several high technology industry sectors, electrical sprays are well established and highly regarded, most notably the Nobel prize for chemistry in 2002, given to John Fenn 2002 for his profound impact on the mass spectrometry industry. All of these examples however relate to sprays of electrically semi-conducting liquids, produced typically by flowing the liquid through a metal capillary held at a potential of a few kilovolts. The benefits of electrically charged sprays are well known : excellent dispersion and inter-phase mixing, absence of agglomeration, controllable spray targeting, good coating uniformity, low power consumption, good primary atomization performance. Electrostatic atomization for electrically insulating liquids, the charge injection technique as this manuscript highlights, has been developed over the last 2 or 3 decades, so why does it not have the popularity and acceptance within the same sort of science and industry sectors that is evident for semi-conducting liquids ? This section gives the interpretation of the author. The following points are made : It cannot be denied that a charge injection atomizer is a more complicated device to manufacture and operate than a standard atomizer to do the same job. Much of the research into charge injection atomizers has been directed towards steady state combustion systems. These are systems which generally have large amounts of spare energy to for instance pressurize a liquid to produce a finely atomized spray, without the complexity of dealing with a high voltage electrode embedded deeply within the atomizer. Comparing electrostatic atomizers for semi-conducting and insulating liquids, the latter class of atomizer are more complex. Therefore there has to be a compelling reason for the user to accept this level of complexity. Charge injection atomizers rely on the coupled effects of electrical and hydrodynamic forces acting together to operate effectively, and whilst this manuscript brings together a great deal of the basic physical timescales, nondimensional numbers and empirical data on atomizer operation, these have not been fully understood. Whilst a good empirical understanding of charge injection atomizers is now available the science that underpins and explains the observations is not yet complete. In short, these are complex systems, that are not completely described. A distinct problem has been that the development of much of the early technology was undertaken in the commercial sector, and is effectively lost to the wider community for further development. Some of the theories that have been proposed to explain the spray drop-charge character have rather shaky physical foundations and have obscured the basic electrohydrodynamic coupling that is
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key to understanding the internal flow patterns within the charge injection atomizer and primary atomization characteristics outside it. This monograph is the authors attempt to move away from this purely commercial perspective. The technology, originally patented by Scion Sprays, now in the public domain is another by Jeff Allen. Whilst laboratory work has shown the atomization process to stable and robust on the timescale of hours, to be useful in the commercial sector, the charge injection systems in use need to be able to operate for thousands of hours between service outages, and this level of robustness is completely unproven as of present. Despite the above I am not especially gloomy regarding the prospects of charge injection method becoming more popular, and again, a number of points, my personal opinion again, are made : The charge injection method is the only method able to electrostatically atomize and disperse electrically insulating liquids. Therefore if an application requires excellent spray coating uniformity using an oil based solvent, this is the only option. The charge injection method is the only way to confer all the established advantages of electrically charged sprays onto electrically insulating liquids. My view is that somewhere, sometime soon, someone will need this technology. Several drivers are already present, and here I give a couple of examples : The food industry is one such example, for instance a bread production line. Here the tins need coating with oil before the dough is added to go on to produce bread. The spray needs to coat the tin uniformly with as little overspray and excess deposition as possible. If an electrical spray of a vegetable oil could be produced, this would provide the precise control required. Another example is small (<50cc) internal combustion engines. These currently use carburetors, and have astonishingly poor emissions characteristics. By around 2011 in the EU and the US, these engines will need to meet strict emissions that approach the level required by larger engines. Large engines have met the emissions limits by employing high pressure injection systems, providing fine atomization, good fuelair mixing and cleaner combustion. However one cannot deploy these injection systems to small engines, since they consume too much power. Therefore one needs a low power fuel injection system to provide fine atomization and excellent fuel atomization and mixture preparation. Pulsed EHD systems are one of the very few that can provide this. To conclude, charge injection atomizers are complex EHD systems, which are not fully understood, though this understanding continues to grow. Because of their complexity they will never be adopted where a simpler non-EHD system can perform the same task. Electrical sprays do however have distinct advantages with a high degree of control possible, and charge injection systems are the only way to produce sprays of electrically insulating liquids. Performance of spray systems, in terms of economic or environmental factors (reduction of overspray) quality control (coating uniformity) or efficiency (drop size per watt consumed by the atomizer) will produce niche technology applications where the performance advantage of charge injection systems outweighs the complexity overhead. The adoption will not be driven by choice, but it will arise from the pressures of environmental legislation and economic forces.
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Index Contents
atomizer performance, 117 bubble formation, 46, 49 cavity expansion, 52 streamers, 60 bulk flow, 13 C parameter, 19 centrifugal force, 2, 80, 81 charge density. See space charge charged spray plume, 140 conductivity, 11 conductor, 6 corona discharge, 94 crossover radius, 61 current pulse, 46
electrode gap, 83, 93, 113 electrohydrodynamics, 5 electro-inertial timescale, 17 electro-viscous timescale, 17 energy equation, 15 equation, 9 Gauss’s Law, 9 gravity body force, 14 hexagonal convection cells, 30 injection current, 40 ionic diffusion timescale, 16 ionic transit timescale, 17 jet break-up, 127
Debye length, 11 dielectrophoretic force, 8 Diesel, 85, 107 dipole, 7 displacement field, 9 doping, 2 drop break up, 72 multi sibling, 75 single sibling, 74 drop charge-diameter distribution, 160 drop combustion, 77 electric field, 9, 22 electric Nusselt number, 35 electrical convection, 10 electrical force, 5, 13 electrical Froude number, 20 electrical Grashof number, 20 electrical Prandtl number, 20 electrical Rayleigh number, 24 electrical Reynolds number, 19, 20
kerosene, 85, 107 Laplace equation, 10 leakage current, 85, 90 M parameter, 20 mechanical timescale, 18 mobility, 11, 18, 89, 105 Navier-Stokes equations, 14, 36 Ohmic-charge relaxation timescale, 15 orifice diameter, 105, 111 orifice space charge, 132 partial breakdown. See corona discharge permittivity, 6, 18, 66 point-plane, 118, 123 polarization, 6, 9, 40
196 quantum tunneling, 37 radial mass flow profiles, 157 Rayleigh Limit, 61, 64 soot, 2 space charge, 18, 22 space charge equation, 13 space charge relaxation timescale, 15 space-charge, 10, 13 spray combustion, 173 spray current, 85, 113 maximum, 93 spray formation, 125
Index Spray Triode, 81, 99 spray visualization, 139, 150 strong injection, 21 sub critical breakdown, 87 super critical breakdown, 92 super critical regime, 112 super-critical regime, 103 T parameter, 20 the mass continuity equation, 13 threshold voltage, 98 total current, 87 Walden’s Rule, 11 weak injection, 21
