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Multiphysics simulation of corona discharge induced ionic wind Davide Cagnoni, Francesco Agostini, Thomas Christen, Nicola Parolini, Ivica Stevanovi, and Carlo de Falco Citation: Journal of Applied Physics 114, 233301 (2013); doi: 10.1063/1.4843823 View online: http://dx.doi.org/10.1063/1.4843823 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/114/23?ver=pdfcov Published by the AIP Publishing

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JOURNAL OF APPLIED PHYSICS 114, 233301 (2013)

Multiphysics simulation of corona discharge induced ionic wind Davide Cagnoni,1,2,a) Francesco Agostini,1 Thomas Christen,1 Nicola Parolini,2 Ivica Stevanovic´,1,3 and Carlo de Falco2,4 1

ABB Switzerland Ltd., Corporate Research, CH-5405 Baden-D€ attwil, Switzerland MOX - Dipartimento di Matematica “F. Brioschi,” Politecnico di Milano, 20133 Milano, Italy 3 Laboratory of Electromagnetics and Acoustics, Ecole Polytechnique F ed erale de Lausanne, CH-1015 Lausanne, Switzerland 4 CEN - Centro Europeo di Nanomedicina, 20133 Milano, Italy 2

(Received 9 October 2013; accepted 25 November 2013; published online 17 December 2013) Ionic wind devices or electrostatic fluid accelerators are becoming of increasing interest as tools for thermal management, in particular for semiconductor devices. In this work, we present a numerical model for predicting the performance of such devices; its main benefit is the ability to accurately predict the amount of charge injected from the corona electrode. Our multiphysics numerical model consists of a highly nonlinear, strongly coupled set of partial differential equations including the Navier-Stokes equations for fluid flow, Poisson’s equation for electrostatic potential, charge continuity, and heat transfer equations. To solve this system we employ a staggered solution algorithm that generalizes Gummel’s algorithm for charge transport in semiconductors. Predictions of our simulations are verified and validated by comparison with experimental measurements of C 2013 AIP Publishing LLC. integral physical quantities, which are shown to closely match. V [http://dx.doi.org/10.1063/1.4843823] I. INTRODUCTION AND MOTIVATION

Cooling of electric and electronic devices is a continuous challenge for researchers and engineers. Power electronics trends indicate a continuous increase of power densities and a shrinking of component dimensions. These conditions make thermal management essential to guarantee a safe, reliable, and affordable operation of electronic components where suitable cooling schemes must be applied. Forced convection air cooling is probably the oldest and still one of the most used approaches for electronic systems cooling. Usually, forced convection is driven by a fan, but, for some applications as, for example, the cooling of hot spots or enclosure-contained devices, alternative methods based on Electro-Hydrodynamic (EHD) forces have been recently studied and exploited.1 A representative example of such methods is based on ionic wind induced by a so called corona discharge. Figure 1 schematically illustrates the phenomenon of corona discharge occurring between two electrodes in air. The gas ions formed in the discharge are accelerated by the electric field and exchange momentum with neutral fluid molecules, initiating a drag of the bulk fluid which is referred to as ionic wind. The choice of a positive corona is favored in industrial applications as it leads to significantly reduced ozone production and increased durability of the metal electrodes in comparison to negative corona devices.2 Therefore, in this study, we focus on the case of DC positive corona wind, where the applied voltage at the electrodes is stationary, gas ionization occurs at the anode, and charge carriers are mainly Oþ 2 ions, as described in Fig. 1(a). Both experimental and numerical studies of EHD phenomena have been presented in recent literature. For example, a)

Author to whom correspondence should be addressed. Electronic mail: [email protected]

0021-8979/2013/114(23)/233301/10/$30.00

Adamiak and others3–6 studied the DC and pulsed corona discharge between a needle and a plate collector, using different numerical methods for the approximation of each equation in the partial differential equation (PDE) system; Ahmedou and Havet7,8 used a commercial software to investigate the effect of EHD on turbulent flows; Moreau and Touchard,9 Huang and others,10 and Kim and others11 experimentally studied different EHD devices designed for cooling or air pumping purpose; Chang, Tsubone, and others12,13 made extensive experimental study of the forced airflow and the corona discharge in a converging duct; Jewell-Larsen and others14–16 conducted both experimental and numerical studies aimed at designing and applying ionic wind cooling devices to thermal management of electronic devices. In this paper, we use a numerical approach based on a multiphysics mathematical model that accounts for all relevant electrostatic, fluid, and thermal aspects of the phenomena being considered. Particular attention is devoted to correctly modeling the relation between the electric field at the anode and the amount of charge injected from the anode corona into the neutral gas region. The accuracy of such relation is crucial for increasing the predictive capability of numerical simulations. Here, we present a novel approach for modeling charge injection, which is based on enforcing Kaptsov’s hypothesis19 and is shown to provide good simulation accuracy using few free model parameters. Our approach to the charge injection modeling is compared to others existing in literature on a set of benchmark device geometries for which experimental data are available. II. GOVERNING EQUATIONS

Modeling of EHD systems requires accounting for a number of interplaying phenomena of different physical nature. Figure 2 summarizes such phenomena and their interactions:

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FIG. 1. Schematic representation of positive wire-to-plane corona dischargeinduced ionic wind. In ambient air, X represents primarily O2 or N2 molecules, and the dominant ionization reactions are of the type e þ X 2e þ Xþ .20

electric current due to drifting ions generates bulk fluid flow which, in turn, contributes to ion drift; thermal energy is transported by the flowing fluid while, at the same time, temperature gradients give rise to buoyancy forces; finally electric conduction properties of the gas are influenced by temperature while electric currents act as heat sources via Joule effect. The system of partial differential equations governing the behavior of each subsystem is introduced below together with most of the constitutive relations for the system coefficients. The differential problem is set in an open bounded domain X whose typical geometry is shown in Fig. 3; the domain X represents the region occupied by bulk neutral fluid and drifting positive ions. In our model, the thickness of the ionization layer around the anode is considered to be negligible with respect to the length scale of the overall system. Such region is therefore represented as a portion of the boundary, denoted as CA in Fig. 3, and the process of ion generation is modeled by enforcing a suitable set of boundary conditions on CA. Existing and new models for such boundary conditions (BCs) are discussed in Sec. III. Unipolar (positive) electrical discharge in a fluid is described by Poisson’s equation

~ the electric field, / the where e is the electric permittivity, E electric potential, q the elementary (proton) charge, Np the number density of (positive) ions. The current density ~ j is given by the sum of three contributions: drift due to electric field, advection, and diffusion ~ þ~ ~ v qDrNp ; (2) j ¼ qNp lE

~ ¼ r ðer/Þ ¼ qNp ; r ðeEÞ

where is the kinematic viscosity, p~is the modified (nong ~ x , and q is the gas mass hydrostatic) pressure pq1 ~ density, and ~ g the gravity acceleration. The volume force term on the right hand side of the first equation of (1c) consists of the sum of electrohydrodynamic force ~ f EHD and buoyancy force ~ f B . As we consider single-phase flows with limited temperature gradients, ~ f EHD can be expressed as22

(1a)

coupled with current continuity equation @qNp þ r ~ j ¼ 0; @t

(1b)

l being the ion mobility in the fluid and ~ v the fluid velocity field. The diffusivity D is related to mobility and temperature T through Einstein’s relation D ¼ lkB Tq1 ;

where kB is Boltzmann’s constant. The flow of incompressible Newtonian fluids is described by the Navier-Stokes equations, stating the conservation of momentum and mass density 8 ~ > v þ~ fB f < @~ ; þ ð~ v rÞ~ v ¼ D~ v r~ p þ EHD (1c) q @t > : r ~ v ¼ 0;

~ ~ f EHD ¼ qNp E:

FIG. 2. Relations between the variables in the EHD system, with arrows pointing to an influenced subsystems from the influencing one. Thicker arrows indicate stronger interactions while thinner ones indicate minor influence (adapted from Ref. 21).

(3)

(4)

FIG. 3. Example domain where all the five possible kinds of boundaries are depicted.


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For the buoyancy force, due to the limited temperature variations, the Boussinesq approximation can be adopted ~ g ½qðTÞ q ¼ ~ g qbexp ðTref ÞðT Tref Þ ; fB ¼~

(5)

where bexp is the thermal expansion coefficient, and the dependence of the gas density q(T) on temperature T is linearized around a certain reference temperature Tref, at which the reference density q is taken. Finally, the temperature equation, which describes heat transfer, reads [resume] @T k Q_ þ~ v rT ¼ DT þ ; @t qCV qCV

(1d)

where k is the heat conductivity and CV the specific heat per unit mass. The thermal power production Q_ on the right hand side of (1d) can be expressed as a balance of terms accounting for the Joule heating caused by the current density ~ j and the mechanical power provided by the EHD force ~ f EHD ~ ~ ~ qDrNp Þ E: ~ Q_ ¼ ~ jE v ~ f EHD ¼ ðqNp lE

(6)

In addition to the bulk thermal energy generation given _ thermal energy may also be exchanged with an exterby Q, nal body; it is worth pointing out that the contribution of the injected energy through the system boundary usually out_ for the small elecweighs the bulk power production rate Q, tric currents flowing in EHD systems. The coupled system of PDEs (1a)–(1d) presented in this section needs to be completed by a suitable set of initial and boundary conditions. Referring to the example geometry of the domain X depicted in Fig. 3, we partition the boundary @X into five subregions @X ¼ Cin [ Cout [ CI [ CC [ CA , on which different conditions are enforced. Initial conditions, which are to be set for ion density, velocity, and temperature, are chosen as uniform fields with values based on the expected “device off” state. The fluid inlet is represented by the boundary region Cin, where Dirichlet conditions are enforced for the velocity ~ v and the temperature T, and a homogeneous Neumann condition is enforced for p~. Since the inlet is supposed to be far from the electrodes, and thus from the region where major electrical phenomena are localized, the electrical variables are also considered to have vanishing gradients along the outward normal direction ~ n on the boundary @X. At the fluid outlet Cout, we require the normal component of the fluid stress tensor and of the temperature, charge density, and electric potential gradients to vanish, considering again a region which is far from the major phenomena in the system. The boundary region denoted as CI represents an electrically insulating wall and both drift and diffusion current den~ þ~ v) and qDrNp , respectively, are supposed sities (qNp ðlE to independently vanish. Since CI is also a solid wall, the no-penetration condition ~ v ~ n ¼ 0, which we impose on the ~ ~ fluid flow, allows for the drift current to vanish if E n ¼ 0. Diffusion currents are instead damped by the homogeneous n ¼ 0. Additionally, Neumann condition for ion density rNp ~

fluid flow is subject to a no-slip condition k~ v ð~ v ~ n Þ~ nk ¼ r~ p ~ n ¼ 0. Temperature can either attain an imposed value or satisfy a condition of imposed thermal energy flux ein through the wall surface, depending on the situation at hand. Finally, the regions CC and CA represent the cathode and anode contacts, respectively. At both electrodes, we enforce Dirichlet condition for the electrostatic potential and no-slip, no-penetration conditions for the fluid flow. The cathode CC often coincides with the surface to be cooled, in which case we may impose either fixed heat flux through the surface, or fixed temperature, as we do on CI. With regard to the ion density, a homogeneous Neumann condition is enforced on CC. Physically, this means that the only current allowed through the cathode is due to ion drift: since mass is not allowed to cross the boundary, though, this results in imposing each one of the positive ions hitting the cathode to recombine with an extracted electron. Boundary conditions for ion density on the anode are instead more complicated, and Sec. III is entirely devoted to the derivation and comparison of different models for such boundary conditions. A final note about the physical model presented in this section is in order, concerning the multiscale nature of the time evolution of the EHD system. The characteristic time scales at which the electrical and fluid subsystems evolve fall into widely separated ranges. For the former we can define a characteristic time sE ¼ ðlEon Þ1 r, r being a characteristic length of the ionization region, whereas for the latter vj1 L, with L a characteristic time can be expressed as sF ¼ j~ the length of the entire device. In cases of interest to us, a typical value of sE is 107 s whereas sF falls in the range of 102–1 s. Since we are mainly interested in simulating devices operated in DC conditions, and in a regime where the electric variables have reached steady-state, we could, in principle, decide to consider the time derivative in Eq. (1b) to be negligible and use the steady-state model. Yet, we have found the time-dependent formulation to enjoy better numerical stability properties than its stationary counterpart; therefore, we have used the full form of Eq. (1b) in our simulations. III. MODELING OF CHARGE INJECTION

To trigger the corona discharge, the voltage drop between anode and cathode must exceed a threshold (or onset) value, which we denote by Von, while the corresponding magnitude of the electric field at the anode is denoted by Eon. The generally accepted Kaptsov’s hypothesis19 states that free charge, emitted by the steady-state corona for voltages higher than Von, causes a shielding of the anode that results in “clamping” of the anode electric field at the onset value Eon. While Von depends strongly on the overall device geometry, experimental evidence indicates that Eon is strictly correlated with the curvature radius of the anode contact.23 At the microscale, corona discharge is generated by the impact ionization of gas molecules and avalanche multiplication of electrons. According to the avalanche model first developed by Townsend,24 cations are generated in an area characterized as the locus of points ~ x 2 X such that


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!

ð cT exp

aT ð~ r Þ d~ r 1;

(7)

Lð~ xÞ

where cT and aT are parameters depending on the applied electric field, the pressure, and the chemical composition of the gas and the electrodes whereas Lð~ x Þ is the trajectory of a negatively charged particle, which leaves from the cathode and drifts to ~ x due to the force exerted on it by the electric field. Although not of much practical interest when space charge is not negligible, relation (7) provides a rough estimate for the thickness of the ionization region, where the gas can effectively be considered to be in plasma state. Such thickness depends on the geometry of the anode as well as on the gas pressure and on the electric field; in the corona discharge regime, it is so small in comparison to the length-scale of the neutral fluid region that it makes sense to adopt a lumped model for the ionization region and to represent it as a portion of the anode surface. Under such approximation, the only charge carriers within the bulk fluid region are cations.25 In this section, we discuss several possible options for modeling the rate at which such cations are injected into the bulk fluid region. In order to ease the comparison of the different models, we express them in the common form of a Robin-type boundary condition for Eq. (1b) a Np jCA þ b @~n Np jCA ¼ j;

(8)

where @~n Np ¼ r Np ~ n is the component of the ion density gradient normal to CA. Condition (8) is in general nonlinear as we allow the coefficients a, b, and j to depend locally on ~ C and n Ej the normal component of the electric field En ¼ ~ A on the density of ions Np jCA . The most common approach used in numerical studies of positive corona discharge available in the literature16,26 consists in imposing the current at the anode to be equal to the experimentally measured value im. This leads to the following choice of parameters in Eq. (8): ( a1 Np jCA þ b1 @~n Np jCA ¼ j1 (9) a1 ¼ qlEn ; b1 ¼ qD; j1 ¼ im =s; s being the anode surface area. Notice that Eq. (9) is based on the additional assumption that the component of the ion current density jn ¼ ~ n ~ jjCA normal to the contact be uniformly distributed along CA (hence, we will hereafter refer to this model as uniform). This assumption, together with the fact that knowledge of a measured value of the current corresponding to each value of the applied bias is required, strongly limits the ability of simulations based on Eq. (9) to provide useful information about the impact of the anode contact geometry on device performance. One possible approach to overcome the drawbacks of Eq. (9) is to enforce a pointwise relation between jn and the normal component of the electric field on CA. Such relation, as proposed in Ref. 27, accounts for a balance between different contributions that make up the ionic current at the microscale jn ¼ wNp js HðEn Eon Þ;

(10)

where H(x) denotes the Heaviside’s step function. The parameters appearing in Eq. (10) are the maximum allowed current density js, the threshold field Eon, and the proportionality constant w (which has dimensions of a velocity times an electric charge) between the backscattering current and the amount of ions accumulated in the space charge region at the anode. This model is hereafter denoted by Space Charge Controlled Current (SCCC). Using Eq. (10) to determine the coefficients of the general expression (8) leads to (

a2 Np jCA þ b2 @~n Np jCA ¼ j2 a2 ¼ w qlEn ;

b2 ¼ qD;

j2 ¼ js HðEn Eon Þ: (11)

While this model does not require prior knowledge of the current density, thus apparently solving the main issue of model (9), the accuracy of quantitative predictions depends critically on the correct choice of its parameters js and w, and the model can lead to numerical issues in the regime of interest. An alternative approach consists in selecting the coefficients of Eq. (8) in such a way as to enforce, pointwise on CA, the negative feedback relation between normal electric field and space charge that is at the basis of Kaptsov’s hypothesis. This can be done, for example, by defining the following model: ( a3 Np jCA þ b3 @~n Np jCA ¼ j3 (12) a3 ¼ qlEon ; b3 ¼ 0; j3 ¼ qlEn Np jCA : The model (12) has only one parameter, the onset field Eon, whose typical magnitude can be, at least roughly, estimated by means of correlations available in literature.23 On the other hand, Eq. (12) presents a further nonlinearity in comparison to Eqs. (9) and (11) as j3 depends on Np; thus, its implementation requires a suitable linearization approach. Since in this study we are mainly interested in the stationary regime device performance, we adopt the simplest approach and evaluate j3 in Eq. (12) using the latest computed value of Np. This approach will be shown in numerical examples of Sec. V to be very effective in terms of accuracy of the simulation but also needs under-relaxation in order to avoid numerical oscillations, thus resulting in larger computational times. This model was named ideal diode since it allows arbitrary currents over the threshold, and no current under the threshold. The alternative method of solving the nonlinearity adopted, e.g., Refs. 3 and 14, does not seem to reduce such numerical problems. We are therefore led to consider yet one more type of boundary condition at the anode, where part of the predictive accuracy of Eq. (12) is traded off to achieve better numerical efficiency. This latter model is expressed by the following choice of the boundary condition coefficients: 8 > < a4 Np þ b4 @~n Np ¼ j4 En Eon > a ¼ qlE ; b ¼ 0; j ¼ qlE N exp ; 4 on 4 on ref : 4 Eref (13)


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where Eref is a reference electric field and Nref is a reference cation density. It can be easily verified that the set of points in the Np-En plane that satisfy Eq. (13) reduces to the set satisfying Eq. (12) as Eref ! 0; in such sense, an interpretation of this model as a smoother version of the ideal diode model is possible; to highlight the analogy with Eq. (12), thus, this model was named exponential diode. A summary of the different types of boundary conditions considered in this paper is presented in Table I where, for each condition, the corresponding expressions for the coefficients a, b, and j are reported. IV. DECOUPLED ITERATIVE SOLUTION ALGORITHM

The algorithm we developed for the solution of system (1a)–(1d) is constructed by analogy with iterative algorithms used for the solution of similar systems of coupled PDEs that arise in modeling of semiconductor devices by driftdiffusion or hydrodynamic models28–31 or in electrochemical models for ionic transport in biological systems.32 The algorithm is constructed by a sequence of four steps: (1) Time semi-discretization is performed to reduce the initial/boundary value problem (1a)–(1d) to a sequence of boundary value problems. (2) The sub-problems comprising the overall system are decoupled, and a strategy to iterate among them in order to achieve self consistency is chosen (outer iteration). (3) The decoupled sub-problems are still nonlinear and need again to be solved by some fixed point method (inner iteration). (4) Finally, as the initial problem has been reduced to a set of linear problems, a proper spatial discretization scheme is chosen to solve them numerically. We chose the backward Euler scheme for time discretization, and a time step in the range of 105–103 s, so that the fluid-dynamics phenomena are sufficiently well resolved in time. For the sake of convenience, we summarize below the full system (1a)–(1d) as it appears after applying time-discretization and enforcing the boundary conditions discussed above (we denote CS ¼ CA [ CC [ CI ; CP ¼ CI [ Cin [ Cout , and A ). CcA ¼ @XnC Poisson’s equation 8 > r ðer/Þ ¼ qNp > > > /¼0 > > > :@ / ¼ 0 ~ n

on X on CA on CC on CP

Time – discretized, current continuity equation 8 > qðNp Npold Þ > > v ¼ 0 on X þ r ~ j /; Np ;~ < dt : aN þ b@~n Np ¼ j on CA > p > > :@ N ¼ 0 on CcA ~ n p Time – discretized, Navier – Stokes equations 8 > ~ v ~ v old > > r ðr~ vÞ > > dt > > > > ~ > þ~ fB f > > þ ð~ v rÞ~ v r~ p ¼ EHD on X > < q : r ~ v¼0 on X > > > > > ~ v¼0 on CS > > > > > on Cin v ¼~ v in >~ > > : @~n~ v þ p~~ n¼0 on Cout Time – discretized, heat equation 8 > qCV ðT T old Þ > > > > > dt > > > > þ r ðkrT þ~ vqCV TÞ > < ~ p DqrNp Þ E ~ ¼ ðlEqN > > > k@~n T ¼ ein > > > > > T ¼ Tin > > > : k@ T ¼ 0 ~ n

on X : on CS

(1b0 Þ

(1c0 Þ

(1d0 Þ

on Cin on Cout

The outer iteration strategy for decoupling system (1a0 )–(1d0 ) is graphically represented in Fig. 4. The equations are subdivided into three blocks representing the electrical, fluid and thermal subsystems. In Figure 4 each subsystem is identified in terms of its solution map, namely E for the electrical subsystem (1a0 )–(1b0 ), F for the fluid subsystem (1c0 ) and T for the thermal subsystem (1d0 ). Each of such maps operates on a subset of the components of the complete system state vec~ ¼ ½/; Np ;~ v; p~; T, and iteration is performed by applytor w ing the fixed point map M ¼ T F E until the prescribed tolerance is achieved. The main advantage of decoupling the system according to the physics as outlined above is that each subproblem can then be treated following a specifically tailored approach,

(1a0 Þ

:

TABLE I. Summary of the coefficients for the four boundary models presented. Model name

Equation

a

b

j

Uniform SCCC Ideal diode

(9) (11) (12)

qlEn w qlEn qlEon

qD qD 0

Exponential diode

(13)

qlEon

0

iexp =s js HðEn Eon Þ qlNp En En Eon qlEon Nref e Eref

FIG. 4. Block diagram representing the composite fixed point iteration used to solve the system (1a0 )–(1d0 ).


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which is known to be the most appropriate in its respective field. The map F is composed of an incompressibilityenforcing iteration based on the standard PISO (Pressure Implicit with Splitting of Operators) scheme,33 well established for the solution of incompressible Navier-Stokes equations and the map T represents the solution of temperature equation, which is treated as a linear equation, neglecting the gas coefficients variations. ~ðkÞ ~ðkþ1=3Þ of the initial guess w In particular, the image w under E is characterized as the fixed point of an inner itera N p ; 7! ½/; Np ; , which is a variant of tion map G : ½/; the well-known Gummel map, widely used in computational electronics.28,29,31 The map G is implicitly defined by the solution of the following problems: Nonlinear Poisson’s equation 8 /Þq > ð / > > r ðer/Þ ¼ qN p exp > > > kB Tref < / ¼ VA > > > /¼0 > > > :@ / ¼ 0 ~ n

on X on CA :

(1a00 Þ

on CC on CP

Time – discretized and boundary – relaxed, current continuity equation 8 old > > > qðNp Np Þ þ r ~ v ¼ 0 on X j /; Np ;~ > > > dt > > < 1 Np ¼ h ðjð@~n /; N^ p Þ b@~n N^ p Þ (1b00 Þ a > > > > > on CA þð1 hÞN^ p > > > : @~n Np ¼ 0 on CcA where the under-relaxation parameter h 僆 (0,1) is employed to enhance the numerical stability. Since G is applied for a limited number of iterations in each time step, small values of h sometimes essential to avoid numerical oscillations may lead to non-physical overestimation of the response time of the electrical subsystem: in this case, the time variable in Eqs. (1b), (1b0 ), and (1b00 ) should be considered as a continuation parameter for our nonlinear iteration algorithm. The results presented in Sec. V have been obtained using the finite volume method (FVM) for space discretization. A custom solver has been implemented within the Cþþ library OpenFOAM.34 However, the algorithm presented in this section is very general and could be extended to different discretization methods. V. MODEL VERIFICATION AND VALIDATION

In this section, three different test geometries are presented, and the results obtained in our simulations are compared to experimental and numerical data. The simulations were obtained with the help of the library swak4Foam35 for the implementation of the boundary conditions while the domain meshes were produced with gmsh.36 In Subsection V A, we describe the procedure that has been adopted to identify the adequate mesh resolution for the different test

cases. Subsection V B compares the ability of the introduced models to predict the ion emission of the corona; finally Subsection V C validates the fluid dynamics results as the ability to correctly predict the airflow is essential for obtaining good thermal estimates. A. Shielding cylinder and wire cathode to flat plate anode arrangement

An important issue in validating the novel charge injection models (12) and (13), presented in our paper, is to make sure the mesh density in the anode region is sufficiently high for resolving the non-uniform charge injection profile along the contact surface. For this reason, much care was taken in verifying that the selected mesh resolution was sufficient to obtain reasonably grid-independent results. As repeating all simulations on a sequence of hierarchically refined meshes can be extremely time consuming, we developed a procedure for rapidly selecting a reasonable initial mesh size in order to minimize the number of refinement steps required for verification. Such procedure is based on the empirical observation that the electric field along the anode surface attains the most varying distribution for applied voltages slightly lower than the threshold bias, while above threshold the local shielding effect of the injected charge progressively smoothes out the electric field variations. Our physically motivated assumption is that a grid resolution sufficient to well resolve the electric field at the anode in the near sub-threshold regime will be also sufficient to resolve the current density profile in the fully developed corona regime. Indeed, such behavior has been observed in all the test cases considered below. To provide an example of our mesh selection procedure, we use as a benchmark the geometrical set-up of Refs. 37 and 38, which is shown in Fig. 5. It consists of a grounded plate acting as cathode, with a 0.11 mm radius wire as the corona electrode, placed 40 mm away from it. Another high voltage electrode, a cylinder with 12.5 mm radius, is placed 20 mm further away from the anode and acts as a shield for the corona wire. The coarsest mesh used in our simulations has a discretization step of 0.06 mm in both radial and tangential direction, corresponding to 12 uniform subdivisions of the wire surface. A simulation with a bias of 14 kV, slightly below the onset voltage, is carried out; then the mesh is uniformly refined until the electric field at the contact is meshconverged to an acceptable degree. Figure 6 shows the profiles of En on the initial grid, and on three other grids with 0.04 mm, 0.02 mm, and 0.01 mm step (16, 32, and 64 subdivisions, respectively). The “medium” grid of 0.02 mm gives

FIG. 5. Sketch of the computational domain geometry for the device discussed in Subsection V A.


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FIG. 6. Magnitude of the electric field on the corona anode surface for the device of Subsection V A; cylinder direction 0 , plate direction 180 , anode voltage 14 kV.

satisfactory results since the increased computational cost when moving to the finest grid produces only marginal improvements (3.5% if compared to the 0.86 MV m1 maximum radial variation, as opposed to a 16.7% for the coarse grid). In support of our hypothesis, Table II shows the behavior of both ideal and exponential diode models, with 20 kV applied voltage. Even if the total currents are very similar for all three meshes, the peak current density shows how the coarse mesh introduces local errors, when the ideal diode model is used. This is due to the very sharp current density profile produced by the ideal diode model, as it can be appreciated in Fig. 7. The exponential diode model, conversely, produces smoother profiles, which are well approximated even for the “coarse” mesh. Figure 8 reports the current-voltage curve produced by the ideal and exponential diode models. As the difference between the grids is negligible in both steady-state and transient behavior, only the medium grid results are shown. Currents are very well reproduced by both the exponential and the ideal diode model. The results of the latter are almost identical to the numerical data in Ref. 37; this is expected, as the method used there is also based on enforcing the Kaptsov’s hypothesis. The described analysis has been

FIG. 8. Anode current vs applied voltage in the device of Subsection V A, obtained on the medium sized mesh for two different BC models. Data from Ref. 37.

carried out for all the models and geometries presented, and all provided analogous results. B. Open wire to wall-embedded collecting electrode arrangement

In this section, we apply our numerical model to the wire-to-plate geometry studied in Refs. 17 and 18. Figure 9 depicts the experimental setup: a flat, insulating plate 125 mm long and 50.8 mm wide, is placed in a laminar, 0.28 m/s air flow, parallel to the plate. In the plate is embedded a 6.35 mm long metal strip, its leading edge 55.25 mm away from the leading edge of the plate, acting as cathode contact. The 0.05 mm diameter wire acting as anode contact is placed 3.15 mm away from the plate, and 4 mm upstream of the cathode strip leading edge. The main conductive channel in the device is highlighted in Fig. 10: electric current flows mainly from the anode to the upstream part of the cathode.41 We attribute the discrepancies from the distribution presented in Ref. 17 (which features two main channels) to nonphysical injection peaks due to a less accurate boundary model.

TABLE II. Peak current densities and total currents for the device of Subsection V A, discretized with different meshes and biased with 20 kV voltage. Peak current density [A m2]

Total current [lA]

Mesh

Ideal

Exp.

Ideal

Exp.

Coarse Medium Fine

1.331 5.256 5.390

0.3637 0.3772 0.3844

22.85 22.79 22.38

13.36 13.26 13.21

FIG. 7. Profiles of current densities jn on the corona anode surface of the device of Subsection V A, with different boundary conditions and imposed voltage of 20 kV.

FIG. 9. Scheme of the computational domain geometry for the device discussed in Subsection V B.

FIG. 10. Ion number density distribution [m3] in a device region near the electrodes, for the arrangement of Subsection V B at a 3.6 kV applied voltage. The ticks on the right show the length scale, each tick is 1 mm. The results shown here were obtained with the exponential diode condition.


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FIG. 11. Anode current vs applied voltage in the device of Subsection V B, computed applying the boundary condition models presented in Sec. III. Data from Ref. 18.

FIG. 13. Profile of the heat transfer coefficient ratio at 3.3 kV applied voltage, and ideal diode BC. Dashed lines indicate the position of the cathode. Data from Ref. 17.

Figure 11 compares the measured data with the results obtained with different models for the boundary condition. Only three models have been successfully used since the uniform model proved especially inappropriate in this very asymmetrical geometry: most of the charge injected from the anode side opposite to the cathode would stagnate, generating nonphysical solution as well as numerical misbehaving, the latter due to the exponential in Eq. (1a00 ). The SCCC model does not suffer from those issues since no charge is injected from the low electric field side of the anode; nonetheless, it fails in this particular regime to reproduce the correct, convex shape of the current-voltage curve, exhibiting an excessive shielding effect. Both the ideal and exponential diode model provide better predictions, both qualitatively, with a convex IV curve, and quantitatively, with the maximum prediction error bounded under 33% of the measured current. Additional accuracy could be obtained with a deeper research for the optimum parameters for both models, but this is beyond the scope of this work. Figure 12 shows the anode quantities En and Np on the anode surface, for the two applied boundary conditions. The ideal diode BC maintains only a small portion of the anode active, providing a very localized ionization near the electric field peak whereas the exponential diode BC yields a Np profile which is more diffuse, with its logarithm proportional to En as expected. The heat transfer enhancement achieved by the corona discharge can be quantified by means of the convective heat transfer coefficient h, which varies upon the position and depends on the local wall and bulk temperatures. Figure 13 depicts the ratio of the heat transfer coefficients with and without corona: this value represents the heat transfer enhancement due to the corona discharge. The computed values agree well with the numerical test in Ref. 17. A direct

comparison with the experimental data indicates that the numerical model captures the trends although it underestimates the relative magnitude of the peak near the corona wire. According to Ref. 17, these discrepancies can be imputable to the nature of the experiment, the measurement technique and the idealization to a 2D case. Downstream the electrodes, the added kinetic energy due to the discharge gradually decreases, because of its continuous dissipation; this manifests in a progressive decrease of the heat transfer coefficient. The heat transfer enhancement is attributed to the flow field distribution resulting from the acceleration of the fluid between the electrodes, affecting the velocity field and the boundary layer in proximity of the wall.

FIG. 12. Comparison of the anode quantities obtained at 3.3 kV applied voltage with the ideal and exponential diode BCs, for the device discussed in Subsection V B.

C. Convergent duct with wire-to-plate electrode arrangement

In this section, we apply our numerical model to the device experimentally studied in Ref. 12. The experimental setup is schematically represented in Fig. 14: a duct enclosed between two insulating non-parallel plates, 33 mm deep and 117 mm long, with the two openings 24 and 12 mm wide, respectively. The wire acting as anode is placed 60 mm away from the smaller opening and has a diameter of 0.24 mm. Two stripes of conductive material, acting as cathodes, are embedded on the non-parallel plates, ranging from 6 mm away from the wider opening to 36 mm away from the smaller one. Figure 15 shows a comparison of the numerical simulation predictions for the anode current to applied voltage characteristics of the device. Simulations were performed with different injection models and compared to measurements from Ref. 12. The uniform model has been useful in this case, thanks to the symmetry of the domain, and matches by construction the experimental current values. The currents predicted by the ideal diode model appear to be in very good agreement with measurements both qualitatively and

FIG. 14. Schematic picture of the computational domain geometry for the device discussed in Subsection V C.


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FIG. 15. Anode current vs applied voltage in the device of Subsection V C, computed applying three of the boundary condition models presented in Sec. III. Data from Ref. 12.

quantitatively, the relative error being consistently bounded under 17% over a wide range of applied voltages. The exponential diode injection model also correctly captures the qualitative behavior of the IV curve, which is approximately parabolic in accordance to approximate analytic solution for totally axisymmetric geometries. The quantitative error with respect to the measurements is, as expected, higher. Such a loss in accuracy, though, is balanced by the better numerical performance. Figure 16 compares the convergence history of the iterative method when the ideal diode or the exponential diode injection model is applied. The number of time steps required for the electric variables to reach a stationary regime is much higher in the case of the ideal diode condition since a smaller under-relaxation coefficient is needed to stabilize the method in this case, and a maximum number of applications of G is allowed per time step. It is interesting to observe how the convergence of the current to its stationary value is non-monotonic over time. Indeed, a possibly high overshoot in the current is usually observed, if the initial value of the cation density is low. In such

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FIG. 17. Steady state outlet velocity for the device with convergent duct with wire-to-plate electrode arrangement discussed in Subsection V C, computed applying three different boundary models from Sec. III and the same applied voltages as in Fig. 15. Data from Ref. 12.

situation, the anode contact electric field is initially much higher than at steady state, and thus more intense charge injection occurs. Also, variations in the space charge distribution due to the evolution of fluid flow, may lead to abrupt current variations.41 Finally, Fig. 17 shows a comparison of the experimental and predicted average velocities on the outlet section, plotted versus the total provided power at the electrical steady state W ¼ iV. The uniform model only provides an approximation of scale of the total flow rate; on the other hand, it underestimates both the high increase in efficiency for smaller applied power and the drop in efficiency at higher power. The ideal diode model, on the contrary, provides a very good approximation for the efficiency of the device, due to the more realistic space distribution of the volume EHD force, even without a-priori knowledge of the expected current. As already stated, this additional accuracy comes at the price of higher computational cost. The exponential diode approach, in the end, provides a flow rate curve quite similar to the one from the uniform model, even if the points are biased towards the low-power region due to the underestimation of the currents. Moreover, the approach is not dependent on empirical data since its parameters depend mainly on the electrode radius and could be estimated from similar cases. This result is in our opinion a fair trade-off between the need of specific empirical data on currents of the uniform model, and the excessive computational effort required by the ideal diode model.

VI. CONCLUSIONS AND ONGOING RESEARCH

FIG. 16. Performance of the iterative algorithm in simulating the device of Subsection V C for an applied voltage of 8 kV. The plots show anode and cathode currents and the total number of iterations of map G at each time step.

In this work, we studied the numerical approximation of the effects of electric discharge on ambient air flow. First, we proposed an algorithm to deal with the multiphysics mathematical model describing the system, by the coupling of the different and particular approaches already used in the fields of electronic device simulation and computational fluid dynamics. Furthermore, we analyzed the particular phenomenon of corona discharge and proposed a phenomenological approach, which allows for the removal of the plasma subdomain and the electron density conservation equation from the computation. Four different models following this approach have been considered, discussed, and compared, providing validation with respect to integral quantities (current, airflow) against literature data. The conclusion is that both the ideal and exponential diode models, proposed in


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this work, are able to reproduce the correct behavior of the corona discharge EHD system without need of measured data for the electric current in the actual device at hand. The purpose of the study was to provide a tool with the flexibility and performance required for the simulation of a relevant industrial device, in particular to the modeling of EHD cooling of a condensation radiator, similar to the ones presented in Refs. 39 and 40. Some preliminary results, referred to a full 3D simulation of a geometrically simplified version of the actual condenser, are reported here as supplemental material.41 In addition, a more thorough study and relevant validation of our methods will take place when data from an upcoming experimental campaign on the physical device are collected. 1

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