Ahmad Sayed-Kassem
email: ahmad.sayed-kassem@cnrs-orleans.fr

Pascale Gillon
email: pascale.gillon@cnrs-orleans.fr

Mahmoud Idir

Abdallah Elorf

Virginie Gilard

Numerical simulation of the effect of a 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 | Modelling of the effect of DC and AC electric fields on the stability of a lifted diffusion methane/air flame[END_REF][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. Rao and Honnery [START_REF] Rao | A simplified mechanism for the prediction of the ion current during methane oxidation in engine-like 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 height. The model developed by Luo et al. [20,[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. 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] performed a 2D numerical simulation of counter-flow non-premixed flames under a DC electric field. 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, chemiionization reaction rate, recombination reaction and transport coefficient) were tested in a 1Dnumerical 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.

In the 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 in 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 different results 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 E i 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 n c is the net charge density.

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

∂ ∂x 1 (5)
Here, V is the electric potential and ԑ 0 represents the permittivity of the free space.

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

Energy:

1 1 1 1 (7)
Where h corresponds to the total sensible enthalpy of the gas, k is the thermal conductivity, J j and h j 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:

1 1 (8)
Where Y i is the mass fraction of species i and ω i is the net rate of production of species i by chemical reaction. The net rate of production, ω i , 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:

(9)
Where D i is the molecular diffusion coefficient, q i is the specie charge (negative if the specie is negatively charged) and κ i is the electric mobility.

For all the species except electrons, D i 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):

(10)
Where k b 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 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.

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

To simulate ethylene combustion, four reaction schemes were tested: a two-steps 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:

4 , ( 12
)
Where σ is the Stefan-Boltzmann constant (5.669x10 -8 W.m -2 .K -4 ), T is the local temperature, T b is the atmospheric temperature (fixed at 298 K), p i is the local partial pressure of species i, a pi is the Plank mean absorption coefficient. In the first three cases, only the radiation energy lost by H 2 O, CO 2 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. 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]; k s,Plank was calculated by applying the following expression [START_REF] Demarco | Modelling thermal radiation and soot formation in buoyant diffision flames[END_REF]:

, 1464 (13)
Where f s 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] Boedeker | Soot distribution and cars temperature measuremnts in axisymmetric laminar diffusion flames with several fuels[END_REF][START_REF] Smooke | Computational and experimental study of soot formation in a coflow, laminar diffusion flame[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).

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 + , H 3 O + (the most abundant cation) and electron. These reactions include two reactions for H 3 O + formation and four reactions for charge recombination as described in the Table 2. In this study, only the positive ions are taken into consideration. If negative ions have been shown to play an important role in the effect of electric field on flames, they were not taken into account here for two reasons: their concentration is systematically lower than the positive ones and they are rapidly dissipated due to the proximity of the flame to the positive electrode.

Boundary conditions were adapted to the experimental conditions [START_REF] Sayed-Kassem | On the Effect of a DC Electric Field on Soot Particles' Emission of a Laminar Diffusion Flame[END_REF] as presented in the Figure 5.

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.

Figure 5. Computational geometry reproducing the experimental setup of [25] (distance in mm)

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.

Table 3. Boundary conditions

Boundary Conditions

Fuel 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 ethylene-air 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.

Electric effects on ethylene flame

Figure 8. Simulated OH mole fraction field under different values of applied potential

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 8.

The same observation was remarked during the experiments at 5 and 10 kV as shown in Figure 9. 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 10. 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 10 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. Two main mechanisms have been proposed [START_REF] Sayed-Kassem | On the Effect of a DC Electric Field on Soot Particles' Emission of a Laminar Diffusion Flame[END_REF][START_REF] Gillon | Electric field influence on the stability and the soot particles emission of a laminar diffusion flame[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 C 2 H 4 consumption rate is presented as function of the applied voltage in Figure 11. 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 8. The influence of the electric field on the flame propagation velocity was also reported by several experimental studies [START_REF] Li | Effect of DC electric field on laminar premixed spherical propagation flame at elevated pressures up to 0.5 MPa[END_REF][START_REF] Li | Experimental study of positive and negative DC electric fields in lean premixed spherically expanding 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 12-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 positive ions motion is directed downwards in a positive voltage. 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 from the injection. 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 12-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 height 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 13). 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

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 Figure 4 .

 44 Figure 4. Accordingly, the developed numerical model coupled with the skeletal mechanism can be

Figure 3 .

 3 Figure 3. Comparison between (a) computed temperature and (b) temperature field measured by nTLAFtechnique[START_REF] Sun | The effect of oxygen concentration in the co-flow of laminar ethylene diffusion flames[END_REF]

Figure 4 .

 4 Figure 4. (a-d) Radial distribution of the computed and measured temperature (by CARS [38] and nTLAF [39] techniques) at HAB of 10, 20, 40 and 50 mm. (e) Axial evolution of the computed and measured temperature along the flame axis.

Figure 6

 6 Figure 6 shows the comparison between a visible flame image captured during an experiment and the

Figure 6 .Figure 7 .

 67 Figure 6. Simulated temperature field and visible flame image without an electric field

Figure 7 .

 7 Figure 7. Radial distribution of luminosity and computed temperature at HABs 15.5, 19.5, 29 and 39 mm.

Figure 8 .

 8 Figure 8. Simulated OH mole fraction field under different values of applied potential presents the computed

Figure 9 .Figure 10 .

 910 Figure 9. Flame images taken during an experiment at 3 different values of applied potential

Figure 11 .

 11 Figure 11. Maximum computed consumption rate of C 2 H 4 as function of applied potentialIn order to better understand the effect of electric field on the flow and flame behaviors, Figure12

Figure 12 .

 12 Figure 12. (a) Computed velocity and (b) temperature distribution at different HABs and 5 potential values

Figure 13

 13 Figure 13 represents the distribution of H 3 O + molar concentration at 0 and 10 kV and the difference of H 3 O + molar concentration between 0 and 10 kV. As evidenced by the figure, most of H 3 O + 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 Figure12(a). An increase of H 3 O + concentration is also detected when an electric field is applied. This elevation is concentrated in the low flame region where

Figure 13 .

 13 Figure 13. H 3 O + molar concentration distribution at (a) 0kV and (b) 10 kV. (c) difference of H 3 O + molar concentration between 10 kV and 0 kV cases (unit in kmol/cm 3 ).

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.

Table 1 .

 1 Tested reaction schemes

	Reaction scheme	# of species	Radiation model	# of reactions	Ref
	Two steps	6	1	2	[30]
	Ten steps	9	1	10	[31]
	Skeletal Z66(1)	23	1	66	[32]
	Skeletal Z66(2)	23	2	66	[32]

Table 2 .

 2 Integrated ions schemes in the model

	Reaction	A	n	E	Reference
	CH+OH  CO + +e -	2.51x10 11 0	7.12 [40]
	HCO + +H 2 O  H 3 O + +CO 1.51x10 15 0	0	[41]
	H 3 O + +e - H 2 O +H	2.29x10 18 -0.5 0	[42]
	H 3 O + +e - OH + H +H	7.95x10 21 -1.4 0	[42]
	H 3 O + +e - H 2 +OH	1.25x10 19 -0.5 0	[42]
	H 3 O + + e - O +H 2 +H	6x10 17	-0.3 0	[42]
	Units are cm, mol, s, kJ and K

Acknowledgments

The research reported in this publication was supported by labex Caprysses, (Convention ANR-11-LABX 0006-01), CNRS and Orleans University. The authors acknowledge the participation of B. Sarh in the rereading of the manuscript.


Tomasz G Zieliński

Kamil C Opiela

Piotr Pawłowski

Nicolas Dauchez

Thomas Boutin

John Kennedy

Daniel Trimble

Henry Rice

Bart Van Damme

Gwenael Hannema

Rafał Wróbel

Seok Kim

Shahrzad Ghaffari Mosanenzadeh

Nicholas X Fang

Jieun Yang

Baltazar Briere De La Hosseraye

C J Maarten

Hornikx

Edouard Salze

Marie-Annick Galland

René Boonen

Augusto Carvalho De Sousa

Elke Deckers

Mathieu Gaborit

Jean-Philippe Groby

Reproducibility of sound-absorbing periodic porous materials using additive manufacturing technologies: Round robin study

Keywords: Porous materials, Designed periodicity, Additive manufacturing, Sound absorption

The purpose of this work is to compare the acoustic properties (in particular, sound absorption) for samples manufactured using various Additive Manufacturing (AM) technologies. The work is a round robin test (inter-laboratory test), in which the production of samples (with the same agreed periodic microstructure) and their acoustic measurements are carried out independently by different laboratories.

Most important, however, is that different additive manufacturing technologies (and different types of equipment) are used to make samples. It was found that this is the main but not the only reason for the discrepancy between the results of acoustic measurements obtained for different porous samples based on the same design of periodic micro-geometry.

Introduction

Although standard porous materials, like open cell foams or fibrous material, are still very competitive for acoustic treatments, their efficiency is limited in the low frequency range by their thickness [START_REF] Allard | Propagation of Sound in Porous Media: Modeling Sound Absorbing Materials[END_REF].

If their microstructure can be optimized to meet specific requirements [START_REF] Boulvert | Optimally graded porous material for broadband perfect absorption of sound[END_REF][START_REF] Attenborough | Microstructures for lowering the quarter wavelength resonance frequency of a hard-backed rigid-porous layer[END_REF][START_REF] Attenborough | Macro-and micro-structure designs for porous sound absorbers[END_REF], their manufacturing is now achievable using 3D printing techniques for experimental validation at the scale of a laboratory. Indeed, the development of Additive Manufacturing (AM) technologies [START_REF] Gibson | Additive manufacturing technologies: 3D printing, rapid prototyping, and direct digital manufacturing[END_REF][START_REF] Kumar | A comparison of additive manufacturing technologies[END_REF][START_REF] Ngo | Additive manufacturing (3D printing): A review of materials, methods, applications and challenges[END_REF][START_REF] Wang | 3D printing of polymer matrix composites: A review and prospective[END_REF][START_REF] Mitchell | Additive manufacturing -A review of 4D printing and future applications[END_REF][START_REF] Diegel | A practical guide to design for additive manufacturing, 1st Edition, Springer series in advanced manufacturing[END_REF][START_REF] Jiménez | Additive manufacturing technologies: An overview about 3D printing methods and future prospects[END_REF][START_REF]Additive manufacturing -General principles -Part 2: Overview of process categories and feedstock[END_REF] give new possibilities to manufacture and test optimised micro-geometries. Moreover, new acoustic materials (or their prototypes) are already being developed using AM technologies, e.g.: 3D-printed periodic foams [START_REF] Zieliński | Pore-size effects in sound absorbing foams with periodic microstructure: modelling and experimental verification using 3D printed specimens[END_REF], optimally graded porous materials [START_REF] Boulvert | Optimally graded porous material for broadband perfect absorption of sound[END_REF], adaptable sound absorbers [START_REF] Opiela | Microstructural design, manufacturing and modelling of an adaptable porous composite sound absorber[END_REF], acoustic metamaterials based on the Kelvin cell [START_REF] Rice | Design of a Kelvin cell acoustic metamaterial[END_REF], periodic acoustic structures composed of rigid micro-rods [START_REF] Fotsing | Acoustic properties of periodic micro-structures obtained by additive manufacturing[END_REF] or micro-bars [START_REF] Ring | Design, experimental and numerical characterization of 3D-printed porous absorbers[END_REF], and even micro-perforated panels [START_REF] Liu | Acoustic properties of multilayer sound absorbers with a 3D printed micro-perforated panel[END_REF][START_REF] Liu | Acoustic measurement of a 3D printed micro-perforated panel combined with a porous material[END_REF] and plates with complex patterns of micro-slits [START_REF] Zieliński | Sound absorption of plates with micro-slits backed with air cavities: Analytical estimations, numerical calculations and experimental validations[END_REF]. Most of this research indicates great potential for the development of new acoustic materials using AM technology.

However, 3D-printing technology should be evaluated regarding the quality of the reproduced microgeometry and its influence on sound dissipation. The purpose of this work is to evaluate the influence of several additive manufacturing technologies on the sound absorption of 3D printed samples. Following well-established traditions in acoustic [START_REF] Horoshenkov | Reproducibility experiments on measuring acoustical properties of rigid-frame porous media (round-robin tests)[END_REF][START_REF] Bonfiglio | How reproducible are methods to measure the dynamic viscoelastic properties of poroelastic media?[END_REF] and non-acoustic [START_REF] Du Plessis | Standardized X-ray tomography testing of additively manufactured parts: A round robin test[END_REF][START_REF] Townsend | An interlaboratory comparison of X-ray computed tomography measurement for texture and dimensional characterisation of additively manufactured parts[END_REF][START_REF] Du Plessis | Laboratory X-ray tomography for metal additive manufacturing: Round robin test[END_REF] testing of materials, this work is a round robin test, i.e. an inter-laboratory test, where manufacturing of samples and their acoustical measurements are performed independently by various laboratories. The only thing that is shared at the beginning of such independent investigations is the agreed periodic micro-geometry of the porous materials developed in that way.

The influence of additive manufacturing processes on the sound absorption of a few materials with designed porosity has already been reported [START_REF] Kennedy | The influence of additive manufacturing processes on the performance of a periodic acoustic metamaterial[END_REF][START_REF] Zieliński | Differences in sound absorption of samples with periodic porosity produced using various Additive Manufacturing Technologies[END_REF]. The current paper presents most of the results obtained from the measurements of over 90 samples manufactured in 7 different AM technologies on [START_REF] Fotsing | Acoustic properties of periodic micro-structures obtained by additive manufacturing[END_REF] AM devices (15 different) in six laboratories and acoustically tested by the sample manufacturers and also in three other laboratories to ensure the correctness of the measurements. For most samples of the same periodic micro-geometry, good and sometimes very good consistency of the measurement results was obtained, which together with their microscopic examination allows to indicate technologies, materials and devices capable of the best reproduction of the designed microstructure. It seems that such comparative acoustic tests can also be used to assess the quality of reproduction of designed materials with fine details. The outline of this paper is as follows. The periodic geometries of the porous samples are described in Section 2. Then, the manufacturing techniques are briefly discussed and samples produced using them are presented in Section 3. The whole scheme of the round robin study including sample production and acoustic tests in the impedance tube using the transfer function method with two microphones [START_REF] Dalmont | Acoustic impedance measurement, Part I: A review[END_REF][START_REF] Dalmont | Acoustic impedance measurement, Part II: A new calibration method[END_REF][START_REF] Boonen | Calibration of the two microphone transfer function method with hard wall impedance measurements at different reference sections[END_REF] is discussed in Section 4. Finally, the measurement results are compared in Section 5. The first periodic cell is very simple: it contains a single spherical pore that is connected to identical pores of neighbouring cells through very short (horizontal or vertical) channels of the same size. The pore diameter is 0.9L c and the channel diameter is 0.4L c , where L c is the size (length of the edge) of the cubic cell. This One-Pore Cell (OPC) is depicted in Figure 1 Cylindrical samples with this periodic microstructure will be designated as OPC-3 samples. For the second CAD geometry the cubic cell size was set to L c = 5 mm and a vertical cylinder with a diameter of 29 mm was cut from the array of 6 × 6 × 12 such cells so that its height is 60 mm, see Figure 1(b,c).

Periodic porosity designs

These higher cylindrical samples with larger pores will be designated as OPC-5 samples.

CAD geometry models for OPC-3 and OPC-5 samples with larger diameters are constructed in a similar way only the cell array must be larger in both horizontal directions, for example, for a diameter of 40 mm the cell array must have 14 × 14 × 12 cells for OPC-3, and (at least) 8 × 8 × 12 cells for OPC-5.

If a sample is to be 3D printed with a diameter slightly larger than 40 mm (for example, to cut it to fit later), then the number of cells in both horizontal directions must be even larger but always even (not odd). Although the CAD geometry for OPC samples can be easily built with any CAD tool, we provide

OpenSCAD code for OPC samples in Table A.1 in the Appendix. This code automatically adheres to the above rules when the user sets only the cell size and sample diameter.

Four-Pore-Cell geometry

The second periodic cell with open porosity is more complex than the previous one because it was built randomly. A periodic arrangement of four spherical pores of different sizes in a cubic cell was randomly generated using the algorithm proposed in [START_REF] Zieliński | Generation of random microstructures and prediction of sound velocity and absorption for open foams with spherical pores[END_REF]. The central point of the cubic cell was moved to the centre of the smallest pore, and the whole system was scaled (normalised), so that the size (edge)

of the cubic cell is 1. Then, the normalised positions (centres) and pore diameters were rounded to two decimal places. The normalised values of pore diameters, i.e. 0.59, 0.64, 0.67, and 0.76, respectively, will be reduced by 10% as explained below. Because the pore diameter is smaller than the size of a cubic cell, only one instance of central pore is in the cubic cell. On the other hand, for each of the other three pores, four periodically shifted pore instances are partly in the same cubic cell, which means that together there are 13 instances of 4 different pores in such a Fore-Pore Cell (FPC). This initial FPC design is shown in the top frame in Figure 2 for three different orientations of the periodic cell.

The bottom row in this frame shows the results of analyses carried out by the Z-suite slicing software (dedicated for Zortrax 3D printers) for the periodic cell scaled to the size of 5 mm. These analyses show that the wide fragments of the edges around the windows connecting the pores are too thin to be properly 3D printed using Zortrax M200 3D printer (or other FDM devices with a standard nozzle size of 0.4 mm), and un-designed windows (holes) will also be created, because some parts of the wall between the pores are too thin. Therefore, the diameters of all the pores were reduced by 10% and rounded again to two decimal places (see the middle frame in Figure 2), and then cylindrical channels were set between each pair of pores that were originally connected by windows. To accurately re-create these connections, the normalised diameters of the cylindrical channels were set to the values found for the windows in the initial design (rounded to two significant digits). In this way, the edges of the windows are now thick enough so that they can be manufactured correctly using FDM devices (see the final design in the bottom frame in Figure 2). This and other crucial issues relevant to the FDM technology are studied, e.g., in [START_REF] García Galicia | Improving printing orientation for Fused Deposition Modeling printers by analyzing connected components[END_REF].

All the necessary data for the final FPC design is given in Table 1 for the cell orientation designated FPC-Z, because the axis of the porous cylinder of the corresponding FPC-Z sample is along the Z-axis.

Two other orientations will also be used, namely: FPC-X when the axis of the porous cylinder of the respective sample is along the X-axis, and FPC-Y when it is along the Y-axis. The porous cylinders are The normalised values are given in Table 1 and they must be scaled by L c = 5 mm, which is the size

FPC-X FPC-Y FPC-Z

H:50 mm (length of the edge) of the cubic cell. CAD models for FPC samples are obtained in a similar way as for OPC-5 samples, cf. Figure 1(b), but the cylinder height is 50 mm (not 60 mm), so for example, the CAD models for samples with a diameter of 29 mm shown in Figure 3 were cut from three-dimensional arrays with 6 × 6 × 10 (not 12) cells. We provide OpenSCAD code for FPC samples in Tables A.2 and A.3 in the Appendix. The cell size for FPC samples is fixed to 5 mm, so the user sets only the cell orientation and sample diameter.

Manufacturing of samples

Additive Manufacturing technologies and devices

Periodic porous samples with microstructure designs proposed in Section 2 were fabricated in seven AM technologies that are briefly presented below [START_REF] Kumar | A comparison of additive manufacturing technologies[END_REF][START_REF] Ngo | Additive manufacturing (3D printing): A review of materials, methods, applications and challenges[END_REF][START_REF] Mitchell | Additive manufacturing -A review of 4D printing and future applications[END_REF][START_REF] Jiménez | Additive manufacturing technologies: An overview about 3D printing methods and future prospects[END_REF][START_REF] Yang | Additive manufacturing of metals: The technology, materials, design and production, 1st Edition[END_REF][START_REF] Dowling | A review of critical repeatability and reproducibility issues in powder bed fusion[END_REF]. The brand names of 15 different 3D printing devices used in this study are also provided.

FDM: Fused Deposition Modelling, a.k.a. Fused Filament Fabrication (FFF) or Layer Plastic Deposition (LPD), is probably the cheapest, easiest in service, and the most common method. In it a solid polymer is heated up in the extruder to the melting temperature and injected through a nozzle onto a platform, where it solidifies again when cooled. Once the cross-section pattern of a given layer is formed, the nozzle moves up by a certain distance (called the layer thickness) and starts to build the next layer by pushing a specific amount of the thermoplastic material to particular places. LCD: In this technology, sometimes called LCD-based SLA or UV LCD, the liquid photopolymer is also cured by light. However, despite generally the same principle of operation, LCD 3D printers do not have any laser in them but cure the entire layer of photopolymer resin at once using a liquidcrystal display (LCD) screen with ultraviolet light-emitting diode backlighting. This approach is much faster than SLA, and independent of the dimensions of fabricated objects. LCD 3D printers used in this study: Zortrax Inkspire, Anycubic Photon.

The overall quality of some of the obtained porous samples was assessed on a series of measurements of crucial dimensions made with the aid of digital microscopes. It was observed that the designed microstructure was precisely reflected only for the case of the resin samples (SLA and LCD). Other technologies provided rough surfaces, indistinct edges, more or less degenerated pores (especially near the cylindrical surface of a sample), and were not able to capture important albeit intricate details. Furthermore, the spherical voids present in the microstructure turned out to have smaller diameters than anticipated and were distributed by unintended distances. The reason for this is twofold: the highest printing resolution offered by SLA, LCD, and also DLP (with the lower limit of 25 microns) and the usage of a liquid material. Nevertheless, resin samples must be carefully post-processed (cleaned and exposed to light for some time after fabrication to enhance solidification) in order to avoid shrinkage over time. Because the LCD screen is composed of a number of pixels, the layers generated in the LCD technique consist of so-called voxels, a spatial equivalent of pixels. This is why LCD prints are usually slightly more rough than the SLA ones, but still their quality is better when compared to samples fabricated in a low-resolution technique of FDM, or even in finer SLS and DLMS or SLM techniques. It should be also noted that obtaining FDM samples with small details of decent quality usually require some experience in the selection of materials and 3D printing parameters.

Samples manufactured for impedance tubes with a diameter of 29 mm

Periodic porous samples dedicated for 29 mm impedance tubes were manufactured by: (1) IPPT technology which allowed to observe that the quality and surface features of the samples obtained from the same AM device and material were virtually the same. On the other hand, one can easily notice differences between samples produced using different AM technologies and even different devices of the same technology FDM. This is well illustrated in Figure 5 where surfaces for the FPC samples manufactured using Flashforge Creator PRO are compared with the ones from Zortrax M200. A relatively high quality of samples produced with Flashforge Creator PRO was obtained after many tests involving experimenting with 3D-printing parameters and various high quality ABS (acrylonitrile butadiene styrene) materials from different producers (the final choice was Rigid.Ink), whereas a good quality from Zortrax M200 was achieved using a dedicated high quality ABS material Z-Ultrat and parameters set up for this material by the 3D-printer producer. The diameter of metal samples produced in DMLS technology was deliberately chosen so that these samples loosely fit the 29 mm impedance tube and they must be wrapped in a tape for a tight fit (see the green tape around these samples in Figure 4). This had to be done (even knowing that the presence of the tape would affect the measurements) to avoid scratching the inside of the tube. The SLS samples were printed on Sinterit Lisa with a layer thickness equal 0.075 mm (the thinnest possible layer). across the channels and pores. On the other hand, the shape and surface quality of SLA samples produced by UTC from photopolymer resin is excellent when compared with other technologies: the surfaces are very smooth and the shapes of circular channels and spherical pores are not distorted.

EMPA produced complete sets of high-quality OPC and FPC samples from photopolymer resin using DLP technology and metal samples from stainless steel powder in DMLS technology (see Figure 9). The metal samples were wrapped with tape to fit it securely into the impedance tube. however the shape mapping is rather rough. Their overall quality is estimated to be lower than the samples manufactured with Flashforge Creator PRO or Zortrax M200, comparable to the samples manufactured with Creality Ender 3, and higher than the samples manufactured with Makerbot Replicator Z18.

Periodic porous samples for testing in 30 mm impedance tubes were manufactured by: (1) MIT using Autodesk Ember in DLP technology, and (2) IPPT using Zortrax M200 in FDM technology and Sinterit

Lisa in SLS technology.

The OPC and FPC samples manufactured by MIT in DLP technology are shown in Figure 13. Their quality was good, comparable with the high quality of resin samples manufactured in SLA technology by UTC, EMPA, and TCD. Figure 14 presents the complete sets of OPC and FPC samples manufactured by IPPT in FDM and SLS technologies, dedicated for testing in other laboratories having impedance tubes with an inner diameter of 30 mm (LAUM) or 40 mm (KUL). During the round robin investigations more than 90 samples were acoustically tested in impedance tubes, the majority of them in (at least) two different laboratories. The complete test for one sample consisted in measurements of the surface acoustic impedance and acoustic absorption for the sample in three (or, in fact six) configurations. First, the sample was placed with its bottom face directly on the rigid termination of the impedance tube so that its upper face was impinged by acoustic plane waves at normal incidence. Then, an air gap of 20 mm was created between the sample and the rigid termination by moving the rigid plunger, and the measurement was repeated. A final measurement was taken for OPC (3) , FPC (3) 3D Systems ProJet 160 (1) A few complete sets of these samples were manufactured using two different FDM/FFF devices. (2) IPPT only manufactured samples of this diameter (i.e. acoustic tests were carried out by another laboratory). (3) These samples were acoustically tested and then impregnated with cyanoacrylate and tested again. the case when the air gap was increased to 40 mm. This three-measurement procedure was repeated for the inverted sample so that its bottom face was exposed to incident plane acoustic waves. Inverted sample measurements generally confirmed the previous results very well, although especially in the case of FDM technology and metal samples, the bottom surface of a sample (which is in one way or another connected to the platform during the 3D printing process) usually slightly differs from the upper one. It was observed, however, that this effect was rather negligible since all discrepancies between two measurements -in the standard and inverted position, respectively -were not significant and occurred at higher frequencies (above 5 kHz) where the other factors (i.e. material, manufacturing technology, etc.) are much more important and usually cause more discrepancies.

Round robin investigations

OPC, FPC

The round robin tests allowed to create a database of acoustical measurements obtained for samples manufactured in different AM technologies and measured by independent laboratories. The results were analysed and compared, in particular, for the samples with the same periodic microstructure. Most The asterisk symbol "*" means that the result was measured by a laboratory that did not produce the sample.

representative results for most of the samples listed in Table 2 and described in Section 3 will be presented in Section 5, including even some of the results obtained for samples of evidently very poor quality to illustrate the resulting change in the nature of the sound absorption curves.

To facilitate quick identification of measurement results (presented in Section 5) and the corresponding samples the result labels used to denote the measurement curves are collected in Table 3 together with the corresponding tube (sample) diameter, AM device, material, manufacturer, etc. For example, the label " SLA-UTC " means that the result denoted in this way was measured in a 29 mm impedance tube for a resin sample manufactured by UTC in SLA technology using Formlabs Form 2, and the sample was tested by its manufacturer. A slightly modified label " SLA-UTC* " means that the result for this sample was measured by ECL, that is, a laboratory which did not produce the sample, but simply got it for independent testing from UTC. The sample micro-geometry (i.e. OPC-3, OPC-5, FPC-X, FPC-Y, or FPC-Z) is specified on each graph which compares the results obtained for samples with the same micro-geometry (and height). The only exception is the graphs in Figure 20 where sound absorption for samples with different cell orientation are compared, and the curve labels are complemented by a micro-geometry reference.

Measurement results

Sound absorption for OPC samples

Sound absorption measured for OPC-3 samples that were produced using six different AM technologies (viz.: FDM/FFF, SLS, DMLS, LCD, SLA, and DLP) and various devices are compared in Similarly, the results of acoustic absorption for OPC-5 samples (cell size: 5 mm, height: 60 mm) manufactured using various 3D printing devices are also compared for the mentioned configuration cases, i.e.: without air gap, or backed with an air gap of 20 mm or 40 mm (see Figure 16).

The results presented in Figures 15 and16 (see also Table 3 to decode the result labels) were measured in impedance tubes of two different sizes, namely, with a diameter of 29 mm and 30 mm. In the latter case, it was the samples manufactured by MIT in DLP technology and tested by MIT and also (independently) by LAUM. On the graphs we present the absorption curves obtained by LAUM according to the ISO standard [START_REF]Determination of sound absorption coefficient and impedance in impedance tubes[END_REF]. They were virtually the same (though less fuzzy) as the curves provided by the manufacturer of samples who measured them according to the ASTM standard [START_REF] Astm E | standard test method for impedance and absorption of acoustical materials using a tube, two microphones and[END_REF]. The absorption curves measured in 29 mm impedance tubes by IPPT and UTC (i.e. the manufacturers of samples) were very well confirmed with independent tests by ECL. Although the samples were measured in different laboratories at slightly different conditions of pressure, temperature and humidity, discrepancies were very small indeed, and here we present only the curves obtained by the manufacturers.

A very good agreement is found between all curves measured for OPC-5 samples and between almost all curves measured for OPC-3 samples. The frequencies of absorption peaks are consistent across all OPC-5 samples and the main discrepancies are in wide frequency ranges between the peaks where the resin samples (SLA, LCD, and also DLP) have the lowest absorption, which was rather expected since the surface of resin samples is smooth and without fibres or other important imperfections. for FPC-X, FPC-Y, and FPC-Z samples, respectively. The presented results (see Table 3 to decode the result labels) are obtained for samples (with a periodic cell size of 5 mm, and height of 50 mm) that were manufactured in four substantially different technologies (viz.: FDM/FFF, SLS, DMLS, and SLA) using various devices available in three laboratories (IPPT, UTC, and EMPA). For almost all of these samples, in addition to the standard configuration with a rigid backing, the acoustic absorption was also measured in two additional configuration cases, i.e. when an air gap of 20 mm or 40 mm, respectively, was added between the sample and the rigid end of the impedance tube. Most of these results are also shown in the graphs below. The following observations were made from a comparison of the corresponding absorption curves.

In general, a very good agreement is found between the absorption curves measured for the corresponding SLA samples (manufactured by UTC, and independently by EMPA), SLS samples (manufactured by IPPT), and high-quality FDM/FFF samples (by IPPT). Moreover, for most of these samples, rather unexpectedly, this consistency of measurement results applies throughout almost the whole frequency range considered (i.e. from 200 Hz up to nearly 6 kHz). Only for some of them, more significant discrepancies appear at higher frequencies above 4.5 kHz. This was, in fact, expected for FPC samples because their micro-geometry is more complex than in the case of OPC samples, and is therefore more prone to imperfections during the 3D-printing process. And small inconsistent imperfections in micro-geometry cause differences in measurements at higher frequencies.

A clear difference can be seen at medium and higher frequencies between the SLS samples manufactured by UTC and impregnated with cyanoacrylate, and the non-impregnated SLS samples manufactured by IPPT. Recall that SLS samples produced by UTC were sintered from a larger grain powder, which resulted in microporosity requiring impregnation [START_REF] Plessis | Effects of process parameters on porosity in laser powder bed fusion revealed by X-ray tomography[END_REF][START_REF] Williams | In situ thermography for laser powder bed fusion: Effects of layer temperature on porosity, microstructure and mechanical properties[END_REF]. For the impregnated samples, medium-and high-frequency absorption peaks are shifted to even higher frequencies. Although this discrepancy was not observed for OPC samples, it can be associated with impregnation in the case of FPC samples.

The absorption curves measured for FPC samples made of metal in DMLS technology are only slightly different from most of the other curves: their second and third absorption peaks are shifted to slightly lower frequencies, and for FPC-Y and FPC-Z samples the curves become rather flat above 4 kHz.

To some extent, these discrepancies may also be related to the fact that metal samples are wrapped with tape (to avoid scratching the tube).

In Figures 17(a), 18(a), and 19(a), we also intentionally show the results obtained for a poor quality FDM sample produced using Makerbot Replicator Z18 from an ABS material suitable for printing shapes with not so small details. Even for these samples, the first absorption peaks (which appear below 1 kHz for FPC samples) are very well represented. On the other hand, the second and third absorption peaks are shifted to significantly higher frequencies, and the total absorption between the peaks is increased in FPC-X, FPC-Y, and FPC-Z). The first measurement in each pair is from the sample manufacturer (i.e. UTC or IPPT), while the second result was measured by ECL. It is easy to see that the differences between the measurements in each pair are insignificant and actually similar to the differences that usually occur when repeating acoustic tests for the same sample (in the same laboratory). To decode the labels for specific results, see Table 3.

The general observation is that the discrepancy between the measurements of the corresponding 40 mm samples is greater than for most of the samples manufactured for tubes with a diameter of 29 mm. Also for 40 mm samples these discrepancies are already growing above 3.5 kHz. There are several reasons for this.

The quality of 40 mm FDM samples manufactured by TUe (and acoustically tested by KUL) is lower than that of the 29 mm FDM samples manufactured by IPPT using Flashforge Creator PRO or Zortrax M200 (though it is better than the quality of samples 3D-printed with Makerbot Replicator Z18). LCD samples are also of rather poor quality because some of the channels are clogged with resin that was not completely removed. Nevertheless, the agreement for the remaining resin samples (i.e. excluding the LCD ones) is rather good, as it is in the case of FDM samples produced by TCD. It should also be noted that the absorption curves for 30 mm DLP samples are very similar to the sound absorption measured for 29 mm metal samples wrapped in tape. The impedance curves were measured in 29 mm tubes by sample manufacturers and independently by ECL. In fact, they were used to calculate the corresponding sound absorption curves, which are presented in Section 5.2. One may notice that the poles (zeroes) of the imaginary part of the surface impedance correspond to the extrema (local minima and maxima) of its real part and to the extrema (but this time local maxima and minima, respectively) of the corresponding absorption curve. We present samples of the same (macro-and) micro-geometry are still very similar but perhaps with more noticeable discrepancies. Finally, it should be noted again that the results measured by IPPT and ECL for the same sample are practically the same or very similar, although the tests were carried out in different laboratories over a period of several months (in early springtime by IPPT, and in summertime by ECL).

Surface acoustic impedance for some of the FPC samples

Concluding remarks

Based on the examination of samples and comparison of measured sound absorption curves and surface acoustic impedance, we determined that there are three reasons for the discrepancy between the measurement results:

1. Overall manufacturing quality associated with AM technology and the material used for production. The main factor here is the correct (undistorted) reproduction of the designed micro-geometry related to the resolution of 3D-printing (however, it should be remembered that the pore and channel diameters will always be slightly different than in the case of the CAD models used for 3Dprinting). Other factors are small imperfections (fibres, completely or partially clogged channels, microporosity, etc.) that appear during the production process, but also in post-production treatment (due to difficulties in removing resin or powder, deformation during curing, problems with curing inner parts of resin porous samples), and sometimes even later (delayed shrinkage effect).

2. Matching samples to the diameter of the impedance tube. This is a well-known reason for possible discrepancies [START_REF] Horoshenkov | Reproducibility experiments on measuring acoustical properties of rigid-frame porous media (round-robin tests)[END_REF][START_REF] Cummings | Impedance tube measurements on porous media: The effects of air-gaps around the sample[END_REF]. When 3D printing samples for the impedance tube, one should proceed with caution as follows (to get a 'perfect' or exact fit, tight or only acceptable): (a) samples are 3D-printed with slightly larger diameters than the diameter of the tube, and then cut to fit ('perfect', exact, or tight fit depending on the cutting method); (b) diameter is determined by trial and error (for specific AM device and material), so the samples can be 3D-printed for a very tight fit (rather tight than exact, but can be sufficient if the cylindrical shape of the samples is not distorted);

(c) samples are 3D-printed with a slightly smaller diameter than the tube, and then wrapped with tape (acceptable, but should be avoided if possible).

3. The proposed periodic micro-geometries have relatively large features (i.e. large pores, wide channels), so the sample faces have large details around their circular edges, different for different sample diameters, because they depend on how the periodic cells are cut by the cylinder edge. This can affect the measurements.

Nevertheless, the overall consistency of the results measured for all good quality samples with the same periodic microstructure proved to be good and we can also recommend SLA technology as the most suitable for manufacturing samples. On the other hand, the presented results show that for such relatively large periodic microstructures sufficient quality is achievable even when using FDM technology, provided that one has some experience (or luck) in matching the selection of parameters and materials for 3D-printing.

Imperfections (roughness, small fibers inside voids, microporosity) tend to significantly increase sound absorption, which is usually desirable. We believe that this can also be used to assess the 3D printing quality using acoustic measurements, like it was done, e.g., in [START_REF] Ibrahim | Acoustic resonance testing of additive manufactured lattice structures[END_REF]. It should be noted that, for example, the sound absorption between the peaks is very low, even at higher frequencies for samples with smooth surfaces without noticeable imperfections.

2. 1 .Figure 1 :

 11 Figure 1: One-Pore Cell (OPC) geometry: (a) two alternative representations of the periodic cell, (b) arrays of cells (L c = 3 mm or 5 mm) for CAD models of samples with a diameter of 29 mm, (c) CAD models of the periodic cylindrical samples OPC-3 and OPC-5 (diameter about 29 mm and height 36 mm or 60 mm, respectively)

  Figure 1(b), from which a vertical cylinder with a diameter of 29 mm and a height of 36 mm was cut.

Figure 2 :

 2 Figure 2: Four-Pore Cell (FPC) geometry: a re-design procedure and three orientations

Figure 3 :

 3 Figure 3: CAD models for FPC samples (diameter about 29 mm and height 50 mm) -generated in FreeCAD (upper row) and analysed in Z-Suite (lower row)

Figure 4 :Figure 5 :

 45 Figure 4: Some of the OPC and FPC samples ( ≈ 29 mm) manufactured by IPPT using FDM/FFF, SLS, DMLS, and LCD technologies FPC-X FPC-Y FPC-Z

Figure 6 :

 6 Figure 6: OPC samples ( ≈ 29 mm) manufactured by UTC using FDM, SLA, and SLS technologies

Figure 7 :Figure 8 :Figure 9 :

 789 Figure 7: FPC samples ( ≈ 29 mm) manufactured by UTC using FDM and SLA technologies Figures 6-8 present the samples manufactured by UTC. This laboratory used three different AM technologies to produce complete sets of OPC (see Figure 6) and FPC samples (see Figures 7 and 8), however, in the case of SLS technology the samples were impregnated with cyanoacrylate to close the micro-pores that were present because of the relatively large grains of the polymer powder used in the sintering process, so that pictures of the SLS samples before and after impregnation are shown in Figures 6 and 8. Microscopic examinations of the samples (see enlarged photographs of their surfaces in Figures 6-8) allowed to assess the relative quality of workmanship using different technologies. In the case of FDM sample quality is generally poor (when compared to the FDM samples produced by IPPT): the channels and pores are distorted and the surface is rough with tiny polymer fibres which often run

3. 3 .Figure 10 :Figure 11 :Figure 12 :Figure 13 :Figure 14 :

 31011121314 Photon) technologies, (2) TUe using Ultimaker 2 in FDM technology, and (3) IPPT using Zortrax M200 in FDM technology and Sinterit Lisa in SLS technology.

ECL 29 mm

 29 acoustical testing of samples from UTC and 29 mm samples from IPPT LAUM 30 mm acoustical testing of samples from MIT and 30 mm samples from IPPT KUL 40 mm acoustical testing of samples from TUe and 40 mm samples from IPPT IPPT : Institute of Fundamental Technological Research of the Polish Academy of Sciences (Poland) UTC : Université de Technologie de Compiègne (France) EMPA : Swiss Federal Laboratories for Materials Science and Technology (Switzerland) MIT : Massachusetts Institute of Technology (USA) TCD : Trinity College Dublin (Ireland) TUe : Eindhoven University of Technology (the Netherlands) ECL : Ecole Centrale de Lyon (France) LAUM : Laboratoire d'Acoustique de l'Université du Mans (France) KUL : KU Leuven (Belgium)

Figure 15 .

 15 Figure 15. Recall that the size of the cubic periodic cell in these samples is 3 mm, while their height is 36 mm. However, instead of being only backed by the rigid wall, the samples were also tested with an air gap of 20 mm or 40 mm set between the sample and the rigid plunger closing the impedance tube. Representative results for some of the samples backed with such air cavities are presented in Figures15(b), and 15(c), respectively.

Figure 15 :Figure 16 :

 1516 Figure 15: Sound absorption measured for OPC-3 samples: (a) no air gap, (b) with an air gap of 20 mm, (c) with an air gap of 40 mm

Figure 17 :Figure 18 :Figure 19 :1Figure 20 :

 17181920 Figure 17: Sound absorption measured for FPC-X samples: (a) no air gap, (b) with an air gap of 20 mm, (c) with an air gap of 40 mm

5. 3 .

 3 Sound absorption for FPC samples measured in tubes with different diameters In Figure 21, acoustic absorption curves measured in 40 mm impedance tubes for FPC samples manufactured by TCD in FDM, SLM, SLA, and LCD technologies, and also for samples manufactured by TUe in FDM technology, are compared with each other and with the corresponding results measured in 30 mm tubes for FPC samples manufactured by MIT in DLP technology. In addition, several absorption curves measured in 29 mm impedance tubes for FPC samples manufactured by UTC in SLA technology and by IPPT in DMLS technology, and already shown in Section 5.2, are again plotted as reference in the relevant graphs in Figure 21, in the frequency range up to 4.5 kHz (valid for 40 mm impedance tubes).

Figures 22 -

 22  show the real and imaginary parts of the (normalised) surface acoustic impedance measured for some FPC samples. The surface acoustic impedance shown on the graphs is normalised, i.e. divided by the characteristic air impedance accurately determined during the acoustic tests on the basis of ambient mean pressure, temperature, and humidity conditions.

Figure 21 : 16 (

 2116 Figure 21: Sound absorption measured in tubes with different diameters for: (a) FPC-X samples, (b) FPC-Y samples, (c) FPC-Z samples

Figure 22 : 16 (

 2216 Figure 22: Normalised surface acoustic impedance independently measured by ECL and IPPT for some FPC-X samples manufactured by IPPT and UTC

Figure 23 : 16 (

 2316 Figure 23: Normalised surface acoustic impedance independently measured by ECL and IPPT for some FPC-Y samples manufactured by IPPT and UTC

Figure 24 :

 24 Figure 24: Normalised surface acoustic impedance independently measured by ECL and IPPT for some FPC-Z samples manufactured by IPPT and UTC

Table 1 :

 1 Data for FPC geometry with the cubic cell centred on [0,0,0] and edge length 1

	Diameters of pores and positions all relevant pore instances	Diameters of channels between relevant pore instances
	No.	Pore instance	Pore diameter	Coordinates of the pore centre [x, y, z]	Pore instance	Pore instance	Channel diameter
	1	P1-1	0.53	0.00	0.00	0.00	1 (P1-1)	6 (P3-1)	0.26
	2	P2-1	0.58	-0.47 -0.44	0.04	1 (P1-1) 10 (P4-1)	0.24
	3	P2-2	0.58	0.53 -0.44	0.04	1 (P1-1) 11 (P4-2)	0.27
	4	P2-3	0.58	-0.47	0.56	0.04	2 (P2-1)	7 (P3-2)	0.34
	5	P2-4	0.58	0.53	0.56	0.04	2 (P2-1) 10 (P4-1)	0.34
	6	P3-1	0.60	0.45 -0.05 -0.35	3 (P2-2)	6 (P3-1)	0.34
	7	P3-2	0.60	-0.55 -0.05 -0.35	3 (P2-2) 10 (P4-1)	0.37
	8	P3-3	0.60	0.45 -0.05	0.65	4 (P2-3) 11 (P4-2)	0.34
	9	P3-4	0.60	-0.55 -0.05	0.65	5 (P2-4) 11 (P4-2)	0.37
	10	P4-1	0.68	0.04 -0.51	0.37	6 (P3-1) 12 (P4-3)	0.23
	11	P4-2	0.68	0.04	0.49	0.37	8 (P3-3) 10 (P4-1)	0.23
	12	P4-3	0.68	0.04 -0.51 -0.63
	13	P4-4	0.68	0.04	0.49 -0.63

POROUS CYLINDERS for 3 FPC orientations

  In Digital Light Processing, like in SLA technology, the liquid photopolymer is solidified by light, although coming from different source. As opposed to SLA, in DLP technology the light is emitted from a projector and guided to the predefined points by a sequence of mirrors (a Digital Mirror Device). DLP 3D printers used in this study: Kudo3D Titan 1, Autodesk Ember.

	The process continues until the whole object is completed. If it contains overhanging features, the
	application of additional supports may be inevitable. FDM 3D printers used in this study: Zortrax
	M200, Flashforge Creator Pro, Makerbot Replicator Z18, Creality Ender 3, Ultimaker 2.
	DMLS: In Direct Metal Laser Sintering, small particles of metal are joined together by a high-power
	laser. During the production cycle, the working platform is repeatedly lowered by a single layer
	thickness, and the fresh powder is tipped into it from the source bed by the recoater. Then, the laser
	follows a predefined path such that the shape of the current layer is finally reflected. In contrast
	to the FDM technology, support material is not needed as this purpose is served by an unsintered
	material. DMLS 3D printer used in this study: EOS EOSINT M280, Sisma MYSINT100.
	SLM: Selective Laser Melting technique has many in common with DMLS. The distinction between the
	two is often far from clear and relates mostly to patent restrictions as well as innovative solutions
	implemented like the recoater material -rubber in SLM and steel or ceramic in DMLS. SLM
	3D printer used in this study: 3D Systems ProX.
	SLS: Selective Laser Sintering is very similar to DMLS or SLM in operation. The main difference is
	in the applied laser and material. SLS is intended to work with polymer (e.g. nylon PA12, etc.)
	powders of grain size about 30 to 40 micrometres. SLS 3D printers used in this study: Sinterit
	Lisa, 3D Systems ProJet 160.
	SLA: Stereolithography is based on exposing to light a liquid photopolymer resin and causing it to cure
	in a layer-by-layer fashion.

In this technology the curing process is done by a laser beam directed by galvanometers onto particular points on a build platform or an object partially immersed in the pool of resin. In principle, SLA requires dedicated supports to yield best quality parts. However, the porous samples of the proposed micro-geometry were printed without them, because they were not necessary. SLA 3D printers used in this study: two devices Formlabs Form 2.

DLP:

Table 2

 2 and UTC were independently tested by ECL, the 30 mm samples produced by MIT and IPPT were tested by LAUM, and finally, the 40 mm samples (delivered by TUe and IPPT) were tested by KUL. It should be noted that although IPPT produced samples for all considered impedance tube diameters (i.e. 29 mm, 30 mm, and 40 mm), it tested only those that fit in a 29 mm tube.

lists the nine laboratories involved in the round robin studies on sound-absorbing media with periodic open porosity produced using various Additive Manufacturing technologies. Six of these laboratories manufactured a different number of samples with diameters suitable for the impedance tubes used by them to measure acoustical properties of materials, viz.: 29 mm (IPPT, UTC, EMPA), 30 mm (MIT), and 40 mm (TCD, TUe). In addition, IPPT manufactured also sets of samples for tubes with diameters of 30 mm and 40 mm.

Seven different AM technologies (FDM, SLS, DMLS, SLM, SLA, DLP, LCD) were used to produce the samples from four kinds of input materials, namely: ABS polymer filaments, polymer powders, metal powders (aluminium, stainless steel, or cobalt-chrome), and photopolymer resins. The types of samples produced by each of the six laboratories are listed in Table

2

together with the appropriate AM technology and 3D-printing device. Although the samples were acoustically tested by their manufacturers, in addition their acoustic properties were also measured in other laboratories, namely: the 29 mm samples produced by IPPT

Table 2 :

 2 Laboratories involved in round robin manufacturing and testing of periodic porous samples, the corresponding impedance tube sizes, AM technologies and devices, types of materials, and manufactured samples (OPC = 2 samples: OPC-3 and OPC-5; FPC = 3 samples: FPC-X, FPC-Y, and FPC-Z)

	Institution	Tube size	FDM/FFF	SLS	DMLS	SLM	SLA	DLP	LCD
	(laboratory) ≈ sample ABS polymer polyamid powder IPPT 29 mm OPC (1) , FPC (1) Zortax M200 Flashforge Creator PRO OPC, FPC Sinterit Lisa	metal powders OPC, FPC EOSINT M280	photopolymer resins OPC Zortrax Inkspire
		30 mm (2)	OPC, FPC Zortax M200	OPC, FPC Sinterit Lisa
		40 mm (2)	OPC, FPC Zortax M200	OPC, FPC Sinterit Lisa
	UTC	29 mm	OPC, FPC Makerbot Replicator Z18

Table 3 :

 3 Result labels used in Figures15-24, combined with the corresponding impedance tube diameters used for measurements, additive manufacturing devices and materials, and involved laboratories

	Result label	Tube size ≈ sample 3D-printing device [& material] Manufactured & tested by	Independent testing by
	FDM-IPPT		Zortrax M200 [polymer filament ABS (Z-Ultrat)]	IPPT	ECL
	FFF-IPPT FFF-IPPT*		Flashforge Creator PRO [polymer filament ABS]	IPPT	ECL
	SLS-IPPT SLS-IPPT*		Sinterit Lisa [polyamid powder]	IPPT	ECL
	DMLS-IPPT DMLS-IPPT*		EOS EOSINT M280 [aluminium powder]	IPPT	ECL
	LCD-IPPT	29 mm	Zortrax Inkspire [photopolymer resin]	IPPT
	FDM-UTC		Makerbot Replicator Z18 [polymer filament ABS]	UTC
	SLS-i-UTC		3D Systems ProJet 160 [polyamid powder] (impregnated)	UTC	ECL
	SLA-UTC SLA-UTC*		Formlabs Form 2 [photopolymer resin]	UTC	ECL
	DLP-EMPA		Kudo3D Titan 1 [photopolymer resin]	EMPA
	DLP-MIT DLP-MIT*	30 mm Autodesk Ember [photopolymer resin]	MIT	LAUM
	LCD-TCD		Anycubic Photon [photopolymer resin]	TCD
	SLA-TCD		Formlabs Form 2 [photopolymer resin]	TCD
	SLM-TCD	40 mm	3D Systems ProX [cobalt-chromium powder]	TCD
	FDM-TCD		Creality Ender 3 [polymer filament ABS]	TCD
	FDM-TUe*		Ultimaker 2 [polymer filament ABS]	TUe	KUL

Acknowledgements

This work is based upon collaboration between the authors supported by European Cooperation in Science and Technology (COST) through the COST Action CA15125 -DENORMS: "Designs for Noise Reducing Materials and Structures". T. G. Zieliński and K. C. Opiela would also like to acknowledge the financial support from the National Science Centre (NCN), Poland, under Grant Agreement No. 2015/19/B/ST8/03979. The research of A. Carvalho de Sousa is funded by an Early Stage Researcher grant within the European Project ACOUTECT Marie Curie Initial Training Network (GA 721536). The research of E. Deckers is funded by a postdoctoral grant from the Research Foundation -Flanders (FWO).

Appendix A. Code listings for the CAD models of OPC and FPC samples Table A.1 contains the code for the CAD models of OPC samples. The code for the CAD models of FPC samples is given in Tables A. [START_REF] Boulvert | Optimally graded porous material for broadband perfect absorption of sound[END_REF] and A.3. These codes were provided to all laboratories that independently used them to generate STL files and G-codes for their 3D printers. They are in the OpenSCAD language [START_REF]OpenSCAD -The Programmers Solid 3D CAD Modeller[END_REF][START_REF]OpenSCAD Language[END_REF] and we strongly recommend to use them with FreeCAD [START_REF]FreeCAD -Open-source 3D parametric CAD modeler[END_REF][START_REF]Import OpenSCAD code[END_REF].