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 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, 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 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
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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
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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
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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
