Ahmad Sayed-Kassem

Abdallah Elorf

Pascale Gillon

Mahmoud Idir

Brahim Sarh

Virginie Gilard

Numerical modelling to study the effect of DC electric field on a laminar ethylene diffusion flame

Keywords: Laminar diffusion flame, Electric field, Numerical simulation, Ethylene

Highlights

• A numerical model was developed to simulate ethylene diffusion combustion and validated with experimental measurements. • The developed model was extended to cover the effect of a DC electric field on ethylene nonpremixed combustion. • An applied electric field deforms the flame structure, promotes the combustion rate and modifies flame dynamic. • Ionic wind is responsible for the observed modifications acting essentially near the burner exit.

Introduction

With the worsening pollution crisis and fossil fuel reserves decline, combustion control technologies become mandatory in order to reduce emissions and enhance combustion efficiency. Among different ways of control such as plasma assisted combustion [START_REF] Ju | Plasma assisted combustion: Dynamics and chemistry[END_REF], aerodynamic controlled combustion [START_REF] Chen | Gas/particle flow and combustion characteristics and NOx emissions of a new swirl coal burner[END_REF], or even dilution [START_REF] Liu | Review of modern low emissions combustion technologies for aero gas turbine engines[END_REF], electric assisted combustion has proven to be a promising technique. As demonstrated in the literature, applying an electric field to combustion is able to control the stability [START_REF] Kim | Flow instability in laminar jet flames driven by alternating current electric fields[END_REF][START_REF] Kim | Electric fields effect on liftoff and blowoff of nonpremixed laminar jet flames in a coflow[END_REF][START_REF] Won | Effect of electric fields on reattachment and propagation speed of tribrachial flames in laminar coflow jets[END_REF], to reduce soot emissions [START_REF] Saito | Control of soot emitted from acetylene diffusion flames by applying an electric field[END_REF][START_REF] Xiong | AC electric field induced vortex in laminar coflow diffusion flames[END_REF][START_REF] Park | Soot Reduction Under DC Electric Fields in Counterflow Non-Premixed Laminar Ethylene Flames[END_REF] and pollutants [START_REF] Chien | Ion current and carbon monoxide release from an impinging methane/air coflow flame in an electric field[END_REF][START_REF] Vega | NO emission of oxygen-enriched CH4/O2/N2 premixed flames under electric field[END_REF] and to change the flame structure [START_REF] Kuhl | On the effect of ionic wind on structure and temperature of laminar premixed flames influenced by electric fields[END_REF][START_REF] Lacoste | Transfer functions of laminar premixed flames subjected to forcing by acoustic waves, AC electric fields, and non-thermal plasma discharges[END_REF][START_REF] Tinajero | Non-premixed axisymmetric flames driven by ion currents[END_REF].

Investigations of diffusion flame behavior when submitted to electric fields have been extensively studied. Although methane has been investigated as the most referenced fuel, ethylene has also been studied [START_REF] Park | Soot Reduction Under DC Electric Fields in Counterflow Non-Premixed Laminar Ethylene Flames[END_REF] in the objective to control soot particle formation. Various configurations of the diffusion flame have been tested: upward and downward [START_REF] Yamashita | Numerical prediction of ion current from a small methane jet flame[END_REF] vertical injection, without and with co-injection of air and counter-flow flames [START_REF] Park | Soot Reduction Under DC Electric Fields in Counterflow Non-Premixed Laminar Ethylene Flames[END_REF] coupled with a variety of electric field direction. The ion driven motion due to the drift velocity of the produced ions in flames appears to be the most important factor that profoundly modifies the flow field. Despite the fact that the majority of the studies reported in literature are experimental, some numerical studies were developed to investigate the nature of the effects. Yamashita et al. [START_REF] Yamashita | Numerical prediction of ion current from a small methane jet flame[END_REF] developed a numerical model in order to explain the voltage-current characteristic behavior of a capillary-fed methane diffusion flame. Considering only the positive ions, they reported that the ion driven motion is responsible for the increase of ion production rate which resembles the flame current observed experimentally. Belhi et al. [START_REF] Belhi | Direct numerical simulation of the effect of an electric field on flame stability[END_REF] performed a direct numerical simulation (DNS) focusing on the mechanisms responsible for the improvement of the stability of a methane diffusion flame when submitted to DC electric fields. The authors showed that an electric field induces a decrease in the lift height and favors the flame anchorage to the burner. In another study, Belhi et al. [START_REF] Belhi | Modelling of the effect of DC and AC electric fields on the stability of a lifted diffusion methane/air flame[END_REF] studied the effect of an AC electric field on the same flame. They tested three different values of ions' mobility and the effect of negative species on the ionic wind. They conclude that negative ions play a crucial role especially when a negative or alternating potential is applied to the electrode. Di Renzo et al. [START_REF] Renzo | An efficient flamelet progress-variable method for modeling non-premixed flames in weak electric fields[END_REF] developed a new model for non-premixed flames under a weak electric field based on flame let progress-variable method. The model showed an agreement with the results of 1D and 2D numerical test cases and the ability to reduce the computational time by a factor of 40. The authors advanced another numerical study applied to Park et al. experiment [START_REF] Park | Bidirectional ionic wind in nonpremixed counterflow flames with DC electric fields[END_REF]. The calculation was performed on a counter-flow diffusion flame of methane subjected to a DC electric field. Taking into account the role of anions, the model was able to reproduce qualitatively the effects of bi-directional ionic wind on the flow field and the combustion process. The authors also discussed the plausible sources of discrepancies with the experiment. The same model was employed to check the self-similarity of an electrified counter-flow diffusion flames [START_REF] Renzo | The breakdown of self-similarity in electrified counterflow diffusion flames[END_REF]. The authors showed that the self-similarity breaks under electrified conditions. More recently, Di Renzo et al. [START_REF] Renzo | A mixture fraction space model for counterflow diffusion flames with incident electric field[END_REF] developed a mixture fraction space model for counter-flow diffusion flames with sub-breakdown electric fields.

The proposed model shows a good agreement with the 2D calculation's results [START_REF] Renzo | The effects of incident electric fields on counterflow diffusion flames[END_REF] with an important reduction in calculation cost. The same case was simulated by Belhi et al. [START_REF] Belhi | A computational study of the effects of DC electric fields on non-premixed counterflow methane-air flames[END_REF]. A skeletal mechanism for methane combustion was employed with a specific approximation to predict ions production rate and a different approach to compute transport properties. Several parameters of their model (CH prediction, chemi-ionization reaction rate, recombination reaction and transport coefficient) were tested in a 1D-numerical calculation of premixed flames [START_REF] Belhi | Analysis of the current-voltage curves and saturation currents in burner-stabilised premixed flames with detailed ion chemistry and transport models[END_REF]. The authors found that these parameters play an important role in predicting ions formation in a flame. More recently, the same model was integrated into an unsteady 3D simulation for a premixed methane flame under a transverse electric field in the saturation regime [START_REF] Belhi | Three-dimensional simulation of ionic wind in a laminar premixed Bunsen flame subjected to a transverse DC electric field[END_REF]. A quantitative agreement with experiments was obtained. Rao and Honnery [START_REF] Rao | A simplified mechanism for the prediction of the ion current during methane oxidation in enginelike conditions[END_REF] developed a simplified model to predict the ion current characteristics during premixed methane oxidation in engine-like conditions. The model was able to predict the ion current under different conditions of pressure and equivalence ratio. A qualitatively agreement with experimental tendency was obtained but with a notable difference in ion current magnitude. Xu et al. [START_REF] Xu | Effects of electric field on micro-scale flame properties of biobutanol fuel[END_REF] investigated numerically and experimentally the effect of a DC electric field on a micro scale biobutanol diffusion flame. The authors assumed a constant electric field in the domain and described the flame behavior at different fuel flow rates. The electric field was proved to decrease the quenching flow rate and modify the flame highest temperature and the flame length. The model developed by Luo et al. [START_REF] Luo | Effects of electric field intensity and frequency of AC electric field on the smallscale ethanol diffusion flame behaviors[END_REF][START_REF] Luo | Investigation of the effect of DC electric field on a small ethanol diffusion flame[END_REF] for a small ethanol diffusion flame includes the electric body forces generated by applied DC and AC electric fields. An enhancement in fuel/oxidizer mixing was observed coupled with an increase in chemical reaction rate.

In our latest work [START_REF] Sayed-Kassem | On the Effect of a DC Electric Field on Soot Particles' Emission of a Laminar Diffusion Flame[END_REF], a laminar ethylene diffusion flame was experimentally investigated under the influence of a DC electric field. Here, a co-flow burner was used as a cathode. A grounded electrode was mounted at 140 mm above the burner exit in order to generate an upward electric field. Flame imaging was employed to monitor flame structure and Extinction/ Rayleigh Scattering optical diagnostic was applied to measure soot characteristics. Relevantly, when we applied an electric field, flame shortness with an enlargement was detected. Additionally, a reduction in soot volume fraction and soot number density was noted. These observations were mainly attributed to the ionic wind acting on the flow field and the soot formation process.

The main objective of this study is to investigate numerically the effects of the electric body forces on a flame which was studied previously experimentally [START_REF] Sayed-Kassem | On the Effect of a DC Electric Field on Soot Particles' Emission of a Laminar Diffusion Flame[END_REF]. In this work, a numerical model was developed for a laminar diffusion ethylene flame using the CFD software FLUENT. The electric field equation with the source terms for the governing equations were integrated to the model by implementing User Defined Functions (UDF) codes in the solver and resolving User Defined Scalar (UDS) equation for the potential. Here, the impact of electric field on gas dynamic during combustion is studied. The effect of electric field on the flame shape, combustion rate and flame temperature are also elucidated by comparing results obtained under different values of potential.

Model description

The experimental apparatus is constituted of two main parts. The first segment is the combustion part including the burner, the gas bottles and the flow controllers, while the second is the electric part composed of the DC power supply, the downstream electrode and the burner as a second electrode.

Geometrical and numerical model

The numerical geometry, based on the experimental system, is a 2D axisymmetric domain comprising the burner and the upstream electrode. Owing to the symmetrical geometry of the burner, only one half of the geometric domain was modeled to minimize the calculation cost (CPU). Since the flow is steady and the flame is stationary and laminar, the steady state was considered when resolving the governing equations. The governing equations are conservation of mass (1), momentum (2-3), energy [START_REF] Saito | Control of soot emitted from acetylene diffusion flames by applying an electric field[END_REF] and species [START_REF] Park | Soot Reduction Under DC Electric Fields in Counterflow Non-Premixed Laminar Ethylene Flames[END_REF].

Continuity: ( ) + 1 ( ) = 0 ( 1
)
Where ρ is the density of gas, u is the x-direction of flow velocity and v is the r-direction of flow velocity.

x-direction momentum:

( ) + 1 ( ) = - + 2 + 1 + 1 - + (2)
r-direction momentum:

( ) + 1 ( ) = - + + + 2 - 2 + ( 3
)
Where µ is the dynamic viscosity, p is the pressure, g is the gravitational acceleration and and are the electric body force components.

The source terms in momentum equations can be given by Lorentz force as:

= ( -) = ! ( 4
)
Where Ei is the axial or the radial electric field intensity, e is the elementary charge, n+ is the positive charge density, n-is the negative charge density, and nc is the net charge density.

The electric field strength is calculated by the mean of Poisson's equation:

∂ ∂ # + 1 # = - ! $ % (5)
Here, V is the electric potential and ԑ0 represents the permittivity of the free space.

The electric field is related to the electric potential by the simple differential equation:

& = -∇# (6)
Energy:

( ℎ) + 1 ( ℎ) = ) * + , ℎ -+ 1 ) * + , ℎ --. /ℎ 0 1 0 2 0 -. 1 /ℎ 0 1 0 2 0 + (3 ) + 1 (3 ) + + 4 (7)
Where h corresponds to the total sensible enthalpy of the gas, k is the thermal conductivity, cp is the heat capacity, Jj and hj are the diffusion flux and the sensible enthalpy of species j respectively, τ is the viscous dissipation stress, corresponds to the heat release due to combustion and f is the electric contribution in total energy transport equation.

Since the energy from the applied electric field f is weak compared to the burning thermal energy of ethylene, the electric field contribution in the total energy variation has been neglected.

Species:

(

5 ) + 1 ( 5 ) = - /1 0 2 - 1 ( 1 0 ) + 6 ( 8
)
Where Yi is the mass fraction of species i and ωi is the net rate of production of species i by chemical reaction. The net rate of production, ωi, is obtained by applying the laminar finite rate model using Arrhenius expression.

For neutral species, the diffusion flux of species is calculated using the dilute approximation (Fick's law) to model mass diffusion due to concentration gradients. For charged species, the electric diffusion flux was introduced to the molecular diffusion by the following relation:

7 8 = -9 ∇5 + : ; 5 & ( 9
)
Where Di is the molecular diffusion coefficient, qi is the species charge (negative if the species is negatively charged) and κi is the electric mobility.

In this study, a simplified model was employed to compute the molecular diffusion and the electric mobility [START_REF] Belhi | Direct numerical simulation of the effect of an electric field on flame stability[END_REF]. This model is debatable but, in the same time, can be easily implemented and computed by the solver UDF DEFINE_DIFFUSIVITY. For all the species except electrons, Di was calculated by assuming a constant Schmidt number Sc=0.7. For the electrons, the diffusion coefficient D e was computed by the mean of Einstein's relation as presented by equation ( 11):

9 = ; * < = ( 10
)
Where kb is the Boltzmann constant (1.38×10 -23 J.K -1 ).

As supposed by Fialkov [START_REF] Fialkov | Investigations on ions in flames[END_REF], a constant value of 1 cm 2 .s -1 .V -1 for the electric mobility was admitted for the ions. For the electrons, the relation of Delcroix [START_REF] Delcroix | Physique des plasmas[END_REF] was employed to compute the electronic mobility:

; = ) > % > - %.@ ; (11)
Where m 0 is the mass of gas (molar mass of gas mixture divided by Avogadro's number) and m e is the electronic mass.

Chemical kinetic model and validation case

The numerical model was validated against the experimental case of Santoro et al. [START_REF] Santoro | The Transport and Growth of Soot Particles in Laminar Diffusion Flames[END_REF]. Santoro's experiment was performed using a co-flow burner with inner diameters of 10.5 mm and 97.7 mm for the central tube and the outer tube respectively. This burner was described in detail at the International Sooting Flame (ISF) Workshop [START_REF]Laminar Flames | International Sooting Flame (ISF) Workshop[END_REF]. The ethylene flow rate was fixed at 3.45 cm 3 /sec corresponding to a mean velocity of 3.98 cm/sec with an air co-flow rate of 713.3 cm 3 /sec. An 88 mm ethylene diffusion flame was developed and radial temperature and velocity measurements were taken at several HABs (Height Above the Burner) using thermocouple rapid insertion and laser velocimetry techniques. The computational domain for Santoro's case is about 14×6 cm as shown by the Figure 1 and a non-uniform quadrilateral grid was created with a resolution of 180×100 control volumes. To simulate ethylene combustion, four reaction schemes were tested: a two-step reaction scheme, a ten-step reaction scheme and a skeletal scheme Z66 with (designed by 2) and without (designed by 1) soot radiation loss. The tested schemes are presented in Table 1. The transport and thermodynamics data are taken from GRI-mech 3.0 [START_REF] Smith | GRI-Mech 3.0 n.d[END_REF].

For an important part of the produced energy is lost by radiation, the thermal radiation loss was computed under the assumption of optically thin heat transfer between the flame and the cold surroundings. The thermal energy loss due to radiation under this approach is given by the following expression:

ABCC = 4E/= F -= < F 2(./ G , 2 + * C,IAJKL ) (12)
Where σ is the Stefan-Boltzmann constant (5.669x10 -8 W.m -2 .K -4 ), T is the local temperature, Tb is the atmospheric temperature (fixed at 298 K), pi is the local partial pressure of species i, api is the Plank mean absorption coefficient. In the first three cases, only the radiation energy lost by H2O, CO2 and CO (if it exists in the scheme) was considered, whereas in the fourth case, the radiation loss by soot is added to the gaseous species radiation loss. In this case, soot characteristics were calculated by resolving a simplified Moss-Brookes model. The Plank mean absorption coefficient for gaseous species was fitted using n-th degree Gaussian functions based on [START_REF] Chmielewski | Planck Mean Absorption Coefficients of H2O, CO2, CO and NO for radiation numericalmodeling in combusting flows[END_REF]; ks,Plank was calculated by applying the following expression [START_REF] Demarco | Modelling thermal radiation and soot formation in buoyant diffision flames[END_REF]:

* C,IAJKL = 1464 N 4 C = ( 13
)
Where fs is the soot volume fraction and T is the local temperature.

The choice of the chemical schemes and the validation of the numerical model were performed according to the measured axial velocity and temperature profiles of Santoro's experimental data [START_REF] Santoro | The Transport and Growth of Soot Particles in Laminar Diffusion Flames[END_REF].

In the Figure 2, the computed results are presented for several reaction schemes against Santoro et al.

measurements [START_REF] Santoro | The Transport and Growth of Soot Particles in Laminar Diffusion Flames[END_REF]. Obviously, the more the chemical reaction model is detailed, the more it is accurate. As expected, the skeletal mechanism gives the most precise results and well predicts the temperature and the axial velocity throughout the flame.

At the region near the burner tip (HAB= 3 and 5 mm), the calculated values of the axial velocity peaks are overestimated. In this region, velocity and temperature are particularly sensitive to the injection conditions since several aerodynamic phenomena (burner preheating and friction loss at the burner wall) could interact and influence the experimental measurements. Indeed, the effect of burner preheating is observed in the difference of temperature values between the experiment and the model at the flame axis. This effect was also observed by other numerical studies [START_REF] Singh | Quasiglobal reaction model for ethylene combustion[END_REF][START_REF] Zettervall | Small Skeletal Kinetic Reaction Mechanism for Ethylene-Air Combustion[END_REF]. A good agreement with the experiment is obtained especially in temperature profiles at HAB 20, 50 and 70 mm. The difference between the experimental and numerical curves could be attributed to the soot model which plays a key role in estimating the radiation loss and therefore the temperature calculation. This aspect is highlighted by the significant difference between curves with and without considering soot. In order to check the role of burner preheating and the ability of the model to reproduce the experimental results, the temperature of the fuel at the burner exit was increased to 600 K as proposed by Boedeker et al. [START_REF] Boedeker | Soot distribution and cars temperature measuremnts in axisymmetric laminar diffusion flames with several fuels[END_REF]. Then, the computed temperature with fuel preheating are compared to the measured values of Snelling et al. [START_REF] Hadef | The concept of 2D gated imaging for particle sizing in a laminar diffusion flame[END_REF] and Sun et al. [START_REF] Sun | The effect of oxygen concentration in the co-flow of laminar ethylene diffusion flames[END_REF]. Figure 3 presents the computed temperature distribution compared to the temperature field measured by Sun et al. [START_REF] Sun | The effect of oxygen concentration in the co-flow of laminar ethylene diffusion flames[END_REF] using non-linear Two-Line Atomic Fluorescence (nTLAF). As a result, the skeletal kinetic reaction mechanism (2) was selected to model ethylene-air combustion in this study. In this mechanism, the two radicals CH and O which are the parents of the first ion are included.

Kinetics of charged species and boundary conditions

Since the concentration of charged species is always lower than the concentration of any neutral species present in the mechanism, six reactions were added including the three charged species: HCO + , H3O + (the most abundant cation) and electron. These reactions include two reactions for H3O + formation and four reactions for charge recombination as described in the Table 2. In this study, only the positive ions are taken into consideration. The negative anions were demonstrated to have a role in the ionic wind. But considering their small concentration compared to the cations, we assume that only positive ions act on the flow field and are the major responsible of the flame response.

In order to check the skeletal scheme accuracy in predicting CH radical, a 1D calculation for the counterflow diffusion flame of ethylene (SFO flame of Park et al. [START_REF] Park | Soot Reduction Under DC Electric Fields in Counterflow Non-Premixed Laminar Ethylene Flames[END_REF]) was performed by the mean of Cantera code [START_REF]Cantera n[END_REF] for two chemical schemes: the employed one and GRI-mech 3.0 [START_REF] Smith | GRI-Mech 3.0 n.d[END_REF] scheme. The CH mass fraction of the two schemes are in close agreement as presented by the Figure 5. Boundary conditions were adapted to the experimental conditions [START_REF] Belhi | Three-dimensional simulation of ionic wind in a laminar premixed Bunsen flame subjected to a transverse DC electric field[END_REF] as presented in the Figure 6.

The injection conditions were those used in the experiment (ethylene flow rate of 3.1 cm 3 /sec and air co-flow rate of 72 cm 3 /sec). These conditions are detailed in the Table 3. In the experiment, the upstream electrode, a horizontal grid placed at 140 mm above the burner, is set at 0 potential value (grounded electrode). Here, in the numerical model, this condition was satisfied by fixing the outlet potential value to 0 kV. A non-uniform quadrilateral mesh of 30000 nodes (250×120 of resolution) was generated. It was refined at the symmetry axis and near the nozzle tip (flame region). A grid convergence study was performed by simulating the reacting flow with one-step reaction scheme. A medium grid was chosen as the best compromise between results accuracy and calculation cost.

The governing equations were solved using a pressure based implicit solver. For the spatial derivative, the 'Least Squares Cell-Based Gradient Evaluation' method was adopted. The ethylene was ignited above the burner exit by assuming a temperature of 2000 K. Calculations were performed until convergence.

Results and discussion

Flame shape and luminosity

Figure 7 shows the comparison between a visible flame image captured during an experiment and the calculated flame temperature contour where no electric field is applied. It reveals that the computed flame shape is intimately similar to the flame contour given by the image. The high sooting zones in the radial section of the flame correspond to the high luminosity regions where a high concentration of particles is responsible for a large part of the luminous emission of the flame. The luminosity decreases with the decrease of the volume fraction of soot that stops forming by approaching higher temperature zones. A similar trend has been observed by comparing the transmittance curves measured by Santoro et al. [START_REF] Santoro | Soot particle measurements in diffusion flames[END_REF] and the measured temperature at the equivalent HABs [START_REF] Santoro | The Transport and Growth of Soot Particles in Laminar Diffusion Flames[END_REF].

Electric effects on ethylene flame

Figure 9. Simulated OH mole fraction field under different values of applied potential presents the computed OH mole fraction contour of the flame at different voltages applied to the burner: 0, 5, 10, 15 and 20 kV. The solid black line corresponds to a value of 5×10 -3 OH mole fraction; this value corresponds to the stoichiometric front and is attained by calculating the equilibrium state of ethyleneair mixture near the stoichiometry using the detailed scheme (Aramco 2.0 [START_REF]AramcoMech 2.0 -NUI Galway n[END_REF]). This line was added to monitor the flame shape change with the electric field intensity. Interestingly, modifications in the flame form are remarked when we apply an electric field. The line tip appears flat at both 5 and 10 kV instead of its pointed form in normal case as shown in Figure 9.

The same observation was remarked during the experiments at 5 and 10 kV as shown in Figure 10. In this figure, the white line represents a value of 30 pixel light intensity. The areas of the flame tip at 0, 5 and 10 kV are presented in Figure 11. These areas correspond to the surfaces formed by the iso-line (OH contour in numerical case or pixel intensity contour in the experimental case) between the flame tip and 3 mm below this point. These areas are then divided by the flame tip area at 0 kV. Figure 11 represents the same trend of the numerical results compared to the experiment: the flame tip area becomes larger as the applied potential at the burner is increased. Furthermore, a slight decrease in flame length is observed for 5 and 10 kV voltages. This decrease was also noted experimentally but with a more significant contrast [START_REF] Belhi | Three-dimensional simulation of ionic wind in a laminar premixed Bunsen flame subjected to a transverse DC electric field[END_REF][START_REF] Gardiner | Combustion Chemistry[END_REF]. The flame shape at voltage 15 kV shows a local perturbation near the flame tip, while it is globally deformed at 20 kV. The flame at 15 kV marks a transitional state to a new regime as shown in the last two plots in Figure 9. This behavior corresponds to what has been experimentally detected when instabilities start to appear under high voltages. Hence, the numerical model qualitatively reproduces the experimental observations under the effect of an electric body force.

Two main mechanisms have been proposed [START_REF] Belhi | Three-dimensional simulation of ionic wind in a laminar premixed Bunsen flame subjected to a transverse DC electric field[END_REF][START_REF] Gardiner | Combustion Chemistry[END_REF] to explain the decrease in flame length: an enhancement in mixing and an elevation in burning rate. In order to investigate this effect, the maximum C2H4 consumption rate is presented as function of the applied voltage in Figure 12. As expected, the fuel consumption rate, presented here by the reaction rate, increases with the electric potential. The higher the consumption rate is, the greater the amount of fuel per unit volume and time is burned. This fact could indicate a better mixing between fuel and air which is generated by the electric field action on the interaction between the charged species and the flow. This mixing enhances the combustion process and could entrain an increase in the maximal OH mole fraction as shown in Figure 9. The influence of the electric field on the flame propagation velocity was also reported by several experimental studies [START_REF] Pedersen | Simulation of electric field effects in premixed methane flames[END_REF][START_REF] Prager | Modeling ion chemistry and charged species diffusion in lean methane-oxygen flames[END_REF]. The electric field tends to slow down the flow in front of the flame front though the velocity is increased on both sides. Figure 13-a shows that the electrically-induced reduction of velocity is located specifically at the reaction zone. Hence, it is related to the production rate of ions by chemiionization. Besides, when the electric voltage rises, the maximum velocity decreases. The drift velocity due to the ions motion is directed by the electric field vector. It is proportional to the net charge density and field intensity as described by equation ( 4). The ionic wind that results from the positive charge displacement and their collisions with neutral leads to the reduction of fluid flow in the reaction region and an acceleration near the flame axis. The increased velocity at the periphery of the flame is causing an increase of the air entrainment along the flame, entrainment already described also by Yamashita et al. [START_REF] Yamashita | Numerical prediction of ion current from a small methane jet flame[END_REF]. Accordingly, better blending leads to an enhanced combustion rate.

In the temperature profiles of Figure 13-b, no obvious changes were detected in the reaction zone. However, an increase of temperature, systematically observed on the outer side of the flame, affirms the extension of the reaction zone. Therefore, this confirms the existence of an enhanced mixing by the entrained air.

Linking experimental and numerical results

In the experiment, a noticeable reduction of soot volume fraction was detected with an important change in flame structure (flame shortness and enlargement). Yet, in the numerical approach, several modifications are detected which are consistent with the experiment. Soot formation and oxidation is highly dependent on velocity, temperature and species distribution. By providing the modified flow field and thermal profiles including the action of the ionic wind; the numerical model could explain several experimental observations: the diminution in flame length may be referred to an elevation in burning rate due to the electric body forces effect especially in a zone near the burner, and the increase in OH mole fraction promotes the oxidation of particles and participates in the reduction of soot [START_REF] Shaddix | Laser-induced incandescence measurements of soot production in steady and flickering methane, propane, and ethylene diffusion flames[END_REF].

The electric field effects are extremely intensive in the low flame region where the charged species are concentrated (Figure 14). As most of particles and soot precursors are positively charged as reported by Chen et al. [START_REF] Chen | Ion chemistry in premixed rich methane flames[END_REF], the proximity of the fuel electrode where they are attracted by the Lorentz force reduces their residence time and the potential for their formation. Nevertheless, the kinetic of these particles should be developed with the transport properties to be able to simulate the interaction between an electric field and particles. Such work remains an open topic of combustion researchers to model the electric aspect of soot formation and the effect of electric field on this process.

Conclusion

A simplified model has been developed to elucidate the effect of electric body forces on an ethylene non-premixed combustion. The kinetic model was first validated against collected experimental data from literature. The comprehensive model including the electric field calculation and the charged particles distribution was able to reproduce the experimental observations. The electric field was proved to affect the flow field at the burner exit by acting on both the flame structure and the burning process. Future works should embrace a more detailed model comprising an electrical diffusion model of charged species, more detailed charged particles chemistry and a soot model with charged particles to have a better understanding of the influence of electric field on diffusion combustion and soot formation.

Figure 1 .

 1 Figure 1. Computational geometry of Santoro burner experiment (distance in mm)

Figure 2 .

 2 Figure 2. Computed axial velocity (a) and temperature (b) of the validation cases compared to Santoro'smeasurements[START_REF] Santoro | The Transport and Growth of Soot Particles in Laminar Diffusion Flames[END_REF]

Figure 4

 4 shows the computed values of radial temperature distribution at HAB of 10, 20, 40 and 50 mm (Figure4(a)-(d)) and the temperature profile along the flame axis compared to Snelling et al. measurements[START_REF] Hadef | The concept of 2D gated imaging for particle sizing in a laminar diffusion flame[END_REF] performed by the mean of Coherent anti-Stokes Raman Spectroscopy technique (CARS) and Sun et al.[START_REF] Sun | The effect of oxygen concentration in the co-flow of laminar ethylene diffusion flames[END_REF] values (Figure4(e)). The computed temperature field is in close agreement with the measured one in terms of values and distribution as revealed by the Figure4. Accordingly, the developed numerical model coupled with the skeletal mechanism can be used to simulate ethylene combustion in a co-flow configuration with computing soot radiation loss.

Figure 3 .

 3 Figure 3. Comparison between (a) computed temperature and (b) temperature field measured by nTLAF technique [44]

Figure 4 .

 4 Figure 4. (a-d) Radial distribution of the computed and measured temperature (by CARS [43] and nTLAF [44] techniques) at HAB of 10, 20, 40 and 50 mm. (e) Axial evolution of the computed and measured temperature along the flame axis.

Figure 5 .

 5 Figure 5. Calculated CH mass fraction in 1D counterflow diffusion flame of ethylene using employed skeletal and GRI-mech 3.0 schemes

Figure 6 .

 6 Figure 6. Computational geometry reproducing the experimental setup of [30] (distance in mm)

  T = 298 K; Potential flux = 0; Axial velocity = 6,5 cm/sec; Radial velocity = 0 cm/sec Air exit VELOCITY_INLET YO2 = 0,23; T = 298 K; flux de potential = 0; Axial velocity = 6 cm/sec; Radial velocity = 0 cm/

Figure 8 .

 8 Figure 8. Radial distribution of luminosity and computed temperature at HABs 15.5, 19.5, 29 and 39 mm. displays the radial distribution of deconvoluted luminosity (by Abel's inversion) taken from flame imaging during an experiment and computed temperature at several HABs without an electric field. The high sooting zones in the radial section of the flame correspond to the high luminosity regions where a high concentration of particles is responsible for a large part of the luminous emission of the flame. The luminosity decreases with the decrease of the volume fraction of soot that stops forming by approaching higher temperature zones. A similar trend has been observed by comparing the transmittance curves measured by Santoro et al.[START_REF] Santoro | Soot particle measurements in diffusion flames[END_REF] and the measured temperature at the equivalent HABs[START_REF] Santoro | The Transport and Growth of Soot Particles in Laminar Diffusion Flames[END_REF].

Figure 7 .Figure 8 .

 78 Figure 7. Simulated temperature field and visible flame image without an electric field

Figure 9 .

 9 Figure 9. Simulated OH mole fraction field under different values of applied potential

Figure 10 .Figure 11 .

 1011 Figure 10. Flame images taken during an experiment at 3 different values of applied potential

Figure 12 .

 12 Figure 12. Maximum computed consumption rate of C2H4 as function of applied potential In order to better understand the effect of electric field on the flow and flame behaviors, Figure 13 presents the computed velocity (a) and temperature distribution (b) at different HABs (presented by line colors) under 5 voltage values (presented by line styles). Noticeably, the electric field affects the

Figure 13 .

 13 Figure 13. (a) Computed velocity and (b) temperature distribution at different HABs and 5 potential values

Figure 14

 14 Figure14represents the distribution of H3O + molar concentration at 0 and 10 kV and the difference of H3O + molar concentration between 0 and 10 kV. As evidenced by the figure, most of H3O + molecules are concentrated near the burner zone. Hence, the electric field could act directly on the flow field in this low region of the flame as shown in Figure13(a). An increase of H3O + concentration is also detected when an electric field is applied. This elevation is concentrated in the low flame region where an air entrainment favors the mixing, enhances the combustion rate and promotes the ion production. Such enhancement leads to a greater action of the electric force on the flow and could generate flame deformations as observed in the Figure9at higher potential values.

Figure 14 .

 14 Figure 14. H3O + molar concentration distribution at (a) 0kV and (b) 10 kV. (c) difference of H3O + molar concentration between 10 kV and 0 kV cases (unit in kmol/cm 3 ).

For

  higher voltage values (15 and 20 kV), a flame deformation appears as presented by the contour plot of the Figure9. This effect is referred to the dynamic action of the ionic wind on the flow field with the buoyancy force. The same observation was reported by Karnani et al.[START_REF] Karnani | Detailed characterization of DC electric field effects on small non-premixed flames[END_REF] and Belhi et al.[START_REF] Belhi | A computational study of the effects of DC electric fields on non-premixed counterflow methane-air flames[END_REF] who explained this behavior by a competition between electric force and buoyancy. The effect of the ionic wind on flow field could be remarked on the Figure13(a) by the perturbation of velocity profiles in the low region of the flame.

Figure 15

 15 Figure 15 represents the radial distribution of strain rate reported at HAB = 5 mm for different potential values. A noticeable modification can be detected at 15 and 20 kV and may induce the flame deformation observed at these values. The same effect was also remarked by Di Renzo et al. [22] who attributed the flame instability to the electric field influence on the local strain rate of the flow.

Figure 15 .

 15 Figure 15. Computed strain rate at HAB = 5 mm for different values of potential applied to the burner

Table 1 .

 1 Tested reaction schemes

	Reaction scheme	# of species	Radiation model	# of reactions Ref
	Two steps	6	1	2	[35]
	Ten steps	9	1	10	[36]
	Skeletal Z66(1)	23	1	66	[37]
	Skeletal Z66(2)	23	2	66	[37]

Table 2 .

 2 Integrated ions schemes in the model

	Reaction	A	n	E	Reference
	CH+OH ⇔ CO + +e -	2.51x10 11	0	7.12	[45]
	HCO + +H2O ⇔ H3O + +CO	1.51x10 15	0	0	[46]
	H3O + +e -⇔ H2O +H	2.29x10 18	-0.5	0	[47]
	H3O + +e -⇔ OH + H +H	7.95x10 21	-1.4	0	[47]
	H3O + +e -⇔ H2 +OH	1.25x10 19	-0.5	0	[47]
	H3O + + e -⇔ O +H2 +H	6x10 17	-0.3	0	[47]
	Units are cm, mol, s, kJ and K

Table 3 .

 3 Boundary conditions

	Boundary	Conditions

Acknowledgments

The research reported in this publication was supported by labex Caprysses, (Convention ANR-11-LABX 0006-01), CNRS and Orleans University.


Poisoning the surface with different agents varies the growth rates on low index planes such as {111}, {110} and {100}.

Perspectives

▪ The understanding of the growth mechanism of the nanoparticles will be extended by studying the other key parameters mentioned in the "synthesis" section, and will be supported by a theoretical study on the solubility of the synthesis reagents. ▪ The study of control parameters will be explored to produce nanoparticles with controlled size, surface chemistry and morphology with low size dispersion to serve as nanofillers for a polymer matrix.

3. Key parameter effects are exploited to control the characteristics of the nanoparticles to be incorporated. Envisaged characteristics: see the scheme ▪ The incorporation of nanoparticles into a polymer will reveal the evolution of the mechanical and thermal properties of the nanocomposite according to the characteristics of the inserted nanoparticles.

Precursor impurities sakina.meftah@utc.fr + Ligand

  Investigation of the growth of nanoparticles with controlled size and controlled surface chemistry, which will be used as nanofillers in polymers Sakina Mefath, Fahmi Bedoui, Isabelle Lisiecki, Djimédo Kondo, Anh-Tu Ngo To cite this version: Sakina Mefath, Fahmi Bedoui, Isabelle Lisiecki, Djimédo Kondo, Anh-Tu Ngo. Investigation of the growth of nanoparticles with controlled size and controlled surface chemistry, which will be used as nanofillers in polymers. International Symposium on Polymer Nanocomposites 2022: From Elaboration to Applications (ISPn), Sep 2022, Lorient, France. hal-04021776 22h,65°C 72h,65°Cvestigation of the growth of nanoparticles with controlled size and controlled surface chemistry, which will be used as nanofillers in effects of size, surface chemistry and morphology of iron oxide nanofillers on the mechanical and thermal properties of nanoreinforced polymers: J. Phys. Chem. C 2014, 118, 3795-3810 International Symposium on Polymer Nanocomposites 2022: From Elaboration to Applications High-boiling solvent Δ All scale bars are at 100nm

✔

  Series of experiments are conducted to understand the growth mechanism of iron oxide nanoparticles synthesised by thermal decomposition of an organometallic precursor. ✔ The effects of trace of water, temperature & reflux time, impurities in the precursor were observed. Reflux temperature (defined by the solvent boiling point) varies the size of the nanoparticles The trend direction may depend on the nature of the solvent (polar, alkene...etc)... Study in progress MEFTAH Sakina 1,2 -BEDOUI Fahmi 1 -KONDO Djimédo 3 -NGO Anh-Tu 2 -LISIECKI Isabelle 2 Laboratoire Roberval UTC Compiègne 1 -Laboratoire Monaris Sorbonne Université 2 -Institut d'Alembert Sorbonne Université 3 2. Growth mechanisms are revealed by studying the effect of each key parameter (mentioned in the next section) on the nanoparticle characteristics (size, surface chemistry, morphology). 1. Ligand-coated maghemite (γ-Fe 2 O 3 ) nanoparticles are synthesized by the thermal decomposition of an organometallic precursor.Thermal decomposition process (under reflux)