Shiqi Wang

Ahmad Batikh
email: ahmad.batikh@insa-toulouse.fr

Lucien Baldas

Azeddine Kourta

Nicolas Mazellier

Stéphane Colin

Stéphane Orieux

On the modelling of the switching mechanisms of a Coanda fluidic oscillator

Keywords: Fluidic oscillator, Coanda effect, Active flow control, CFD, Hot-wire anemometry 1

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

For many decades, fluidic actuators have attracted the attention of researchers for their high potential for a wide range of applications, such as reducing drag on bluff bodies [START_REF] Aubrun | Separation flow control on a generic ground vehicle using steady microjet arrays[END_REF], increasing lift of airfoils [START_REF] Melton | Active Flow Control at Low Reynolds Numbers on a NACA 0015 Airfoil[END_REF][START_REF] Lin | Experimental Study of Flow Separation over NACA633018 Wing with Synthetic Jet Control at Low Reynolds Numbers[END_REF], enhancing mixing in combustion chambers [START_REF] Sun | Air flow control by fluidic diverter for low nox jet engine combustion[END_REF][START_REF] Guyot | Active Combustion Control Using a Fluidic Oscillator for Asymmetric Fuel Flow Modulation[END_REF] or enhancing heat transfer [START_REF] Camci | Forced Convection Heat Transfer Enhancement Using a Self-Oscillating Impinging Planar Jet[END_REF][START_REF] Trávníček | Impingement heat/mass transfer to hybrid synthetic jets and other reversible pulsating jets[END_REF]. Unsteady actuation appears as a much more attractive strategy than steady actuation to enhance the efficiency of the system, for instance by increasing the entrainment of momentum in active flow control applications (see e.g. [START_REF] Greenblatt | The control of flow separation by periodic excitation[END_REF][START_REF] Berk | Entrainment effects in periodic forcing of the flow over a backward-facing step[END_REF]) or heat transfer rate of an impinging jet [START_REF] Alimohammadi | Thermal management using pulsating jet cooling technology[END_REF]. These periodic fluidic disturbances can be provided by various kinds of actuators [START_REF] Cattafesta | Actuators for Active Flow Control[END_REF] such as zero net mass flux (ZNMF) actuators, plasma actuators or micro-electro-mechanical-systems (MEMS). Among them, fluidic oscillators can emit oscillating jets in a large operating frequency and velocity range when supplied with a pressurized fluid. They do not require any moving part, since their oscillations are totally self-induced and self-sustained and only depend on the internal flow dynamics. All in all, these properties yield highly reliable and robust actuation system, which are mandatory from industrial scope [START_REF] Cerretelli | An Experimental and Numerical Investigation on Fluidic Oscillators For Flow Control[END_REF][START_REF] Gregory | Characterization of the Microfluidic Oscillator[END_REF][START_REF] Bobusch | Experimental study of the internal flow structures inside a fluidic oscillator[END_REF].

According to Gregory [START_REF] Gregory | A Review of Fluidic Oscillator Development and Application for Flow Control[END_REF], fluidic oscillators can be classified into two main categories based on their underlying oscillation mechanism. The first kind of oscillator is the wall-attachment fluidic oscillator, the principle of which is based on the Coanda effect; it is thus also called Coanda oscillator. The second kind of oscillator is the jet-interaction fluidic oscillator with a principle based on complex interaction dynamics inside a cavity. Raghu proposed in his work [START_REF] Raghu | Fluidic oscillators for flow control[END_REF] a slightly different classification by discriminating the oscillators between switching (or pulsing) jet fluidic oscillators and sweeping jet fluidic oscillators based on the different patterns of the generated jets. On the one hand, a sweeping jet oscillator has only one outlet through which an undulant jet issues, regardless of its feedback mechanism [START_REF] Raman | Cavity Resonance Suppression Using Miniature Fluidic Oscillators[END_REF] or jet-interaction mechanism [START_REF] Raghu | Feedback-free fluidic oscillator and method[END_REF]. On the other hand, pulsing jet fluidic oscillators (Fig. 1) usually consist of one inlet supplied at constant pressure and two outlets, between which the flow switches alternatively. Even though a number of studies have been devoted to the pulsing jet oscillator for the purpose of different applications [START_REF] Cerretelli | Boundary Layer Separation Control With Fluidic Oscillators[END_REF][START_REF] Tesař | Strangely behaving fluidic oscillator[END_REF][START_REF] Simões | Microfluidic oscillator for gas flow control and measurement[END_REF], the physical mechanisms responsible for the jet switching are still misunderstood.

Accordingly, the design of such devices remains based on trial and error procedures. One of the reasons for that is the difficulty to access the inner flow by means of experimental methods. A comprehensive understanding of the switching mechanism is however essential to design properly this kind of actuator.

The behavior of the pulsing jet fluidic oscillator is based on the Coanda effect as shown in Fig. 1: the jet issuing from nozzle N attaches one of the two walls, W1 or W2, depending for instance on the initial conditions. If there was no feedback loops and if the outlet sections were large, the attachment to wall W1 or wall W2 would be stable and the flow would exit through the corresponding outlet, O1 or O2, respectively. With feedback loops, when the jet is attached to wall W1, part of the flow fills in the feedback loop F1 due to the hydraulic restriction at outlet O1. This yields a pressure increase in the left side of the device (in this particular example), which eventually promotes the jet switching. In an equivalent manner, the same phenomenon develops in the right side of the oscillator and results in a self-sustained oscillating behavior, with a pulsed flow alternatively exiting outlets O1 and O2. [START_REF] Gaylord | FLUERIC TEMPERATURE-SENSING OSCILLATOR DESIGN[END_REF] One of the main features of the pulsing jet is the oscillation frequency. Several studies, such as [START_REF] Tippetts | A fluidic flowmeter[END_REF][START_REF] Viets | Flip-Flop Jet Nozzle[END_REF][START_REF] Hayashi | A study on mechanism of oscillation in sonic oscillators : 2nd Report, Mathetmatical model of oscillators operated by air[END_REF][START_REF] Raman | Evaluation of Flip-Flop Jet Nozzles for Use as Practical Excitation Devices[END_REF][START_REF] Khelfaoui | Numerical and experimental analysis of monostable mini-and micro-oscillators[END_REF],

Fig. 1. Illustration of the basic configuration of a pulsing jet Coanda fluidic oscillator, adapted from

showed that the feedback loop length and diameter, the size and design of control ports P1 and P2, as well as the properties of the operating fluid play an important role in determining the oscillation frequency. In addition, there exists a pressure difference threshold between the control ports, above which the jet switching happens [START_REF] Arwatz | Suction and oscillatory blowing actuator[END_REF]. With fixed geometry and operating fluid, the frequency increases as the inlet pressure increases. This adds some complexity for analyzing the effectiveness of certain applications, as active flow control, since the frequency correlates with the injection momentum, which is dependent on the inlet pressure.

However, in a pulsing jet fluidic oscillator, it is possible to have a quasi-constant frequency response, independent to the inlet pressure. In the studies of Cerretelli et al. [START_REF] Cerretelli | An Experimental and Numerical Investigation on Fluidic Oscillators For Flow Control[END_REF][START_REF] Cerretelli | Boundary Layer Separation Control With Fluidic Oscillators[END_REF], two oscillators are exploited: the first one is almost pressure-controlled and its frequency varies with the inlet pressure, while the second one is almost pressure-insensitive and operates at a constant frequency under the same inlet pressure range. Nevertheless, their different frequency response behaviors have not been clearly explained. In another study by Tesař [START_REF] Tesař | Strangely behaving fluidic oscillator[END_REF] about this kind of oscillator, it is also demonstrated that the oscillation frequency is neither proportional to the inlet flow rate as in the case of a sonic fluidic oscillator [START_REF] Tesař | Strangely behaving fluidic oscillator[END_REF], nor kept constant as in the study of Cerretelli and Gharaibah [START_REF] Cerretelli | An Experimental and Numerical Investigation on Fluidic Oscillators For Flow Control[END_REF]. No complete physical explanation has been proposed yet to the different frequency response patterns of this type of oscillator. Furthermore, Simões et al. [START_REF] Simões | Microfluidic oscillator for gas flow control and measurement[END_REF] experimentally and numerically investigated a fluidic oscillator with both liquid and gas as working fluids. They proposed a correlation law to express the relationship between the switching frequency and the length of the feedback loop as:

0 1 / [2( )] 2[( / ) ( / ) ] t s f f L C h U τ ξ τ + + = = (1)
where t τ , s τ are the transmission time of pressure wave through the feedback loop and the switching time respectively, 0 C the sound velocity, f L the feedback loop length, h the nozzle-to-splitter distance, U the jet velocity and ξ an empirical constant. However, it is not clear how this law was established, and thereby if it can be extrapolated to different designs of Coanda fluidic oscillators (for instance when varying the distance between control ports and nozzle).

In the present study, the behavior of a Coanda fluidic oscillator is investigated over a large range of operating conditions in order to bring a better understanding of the physical mechanisms at the root of this kind of pulsing jet oscillator. The three above mentioned frequency response patterns (frequency proportional to the inlet flow rate, constant frequency, or more complex behavior) were reproduced with the same oscillator core just by modifying its feedback loops volume. A detailed computational study of the internal flow pattern was conducted and the switching mechanism was described in a fully detailed way. The frequency response of the oscillator was explored and a new switching model was proposed and validated by a complete experimental study. Following this analysis, a new and simple function to estimate the operating frequency is proposed, as a guiding tool for the design of pulsing jets oscillators.

The paper is organized as follows. The geometry of the model oscillator used in this study, the numerical simulations settings and the experimental set-up are described in Section 3. The computational study of the internal flow pattern and the effect of the supply pressure on the oscillator behavior are reported in Sections 4 and 5, respectively, while the experimental results are presented and analyzed in Section 6. The main conclusions from this study and some perspectives for future work are given in the last section.

Oscillator model description

For getting a better understanding of the oscillator flow dynamics, both numerical and experimental studies were performed in this work in a very complementary approach. On the one hand, the numerical model allows a detailed analysis of the internal flow pattern, which is difficult to realize experimentally, but essential to study the internal oscillation mechanisms of the device. On the other hand the experimental approach allows to validate numerical results and to complete the study to unveil the key parameters of the Coanda fluidic oscillator. For these reasons, a fully 2-D shape oscillator is proposed with large depth over transversal lengths ratios. The use of two-dimensional geometric shapes not only reduces the cost of a numerical calculation, but also makes it easy to compare the results of the computational study with those of the experimental study. Fig. 2 shows a sketch of the proposed oscillator with key dimensions and a more detailed view of the central part (switching zone) is proposed in Fig. 3. This design is basically composed of a nozzle N, 200μm in width, two feedback loops F1 and F2 connected to two control ports P1 and P2 and two outputs O1 and O2, 500µm in width. It is worth noticing that there is no offset along the y-axis between control ports P1 and P2 and the nozzle inlet, unlike many other actuator designs found in the literature (e.g. [START_REF] Gaylord | FLUERIC TEMPERATURE-SENSING OSCILLATOR DESIGN[END_REF], see Fig. 1). The 2-D simulations were performed on the geometry previously detailed in Fig. 2. A pressure condition was set for both the inlet and the outlet limits of the computational domain; with a limited inlet absolute total pressure up to 0.25 MPa, and an outlet absolute static pressure of 0.1 MPa in sections O1 and O2. The working fluid was air considered as a perfect gas with the following properties: molar mass M = 28.9 g/mol, specific heat capacity at constant pressure cp = 1005 J/kg K, dynamic viscosity µ = 1.8 10 -05 Pa s, Prandtl number Pr = 0.7.

The open source CFD solver OpenFOAM has been chosen to implement the numerical simulations for its ability to deal with complex geometries and its high parallel computation capability. After a detailed study of the sensitivity of various numerical schemes, including the turbulence model, mesh density, temporal and spatial discretization schemes, Courant number choices, and considering both the numerical precision and the calculation cost, the schemes chosen in the following series of simulations are the following: sonicFoam solver, realizable k-epsilon turbulence model, 2 nd order upwind schemes in spatial terms, backward 2 nd order scheme in temporal term, and a maximum local Courant Number limited to 0.3 with a time step of 4×10 -9 s. The grid density at the wall permits to obtain an average value of the dimensionless wall distance y + of the order of 10, compatible with the used standard wall function. The mesh consists of quadrangle cells only in order to get high mesh quality. The total cell number is 1.2×10 5 , with 20 nodes in the throat part and 15 nodes in both outlet sections. In order to reach a regular periodic behavior (a statistically stable behavior), more than 20 periods were simulated.

Experimental setup

For the experimental study, two oscillators, called Osc.1 and Osc.2, have been designed, manufactured and tested (Fig. 4). Both designs have an identical central part (switching zone) which is exactly the same as the one used in the numerical work. However, Osc.1 has a complete "two-dimensional" design (including the feedback loops in the same plane) with a uniform depth of 10 mm (i.e. much larger than the width of the internal channels of the device); the measured frequencies and outlet velocities can then be compared to 2D simulation results. On the other hand, feedback loops F1 and F2 of Osc.2 can be changed by linking ports α and β (see Fig. 2) using plastic tubes with various lengths (200, 300, 400, 500, 600, and 766 mm). The extremities of these tubes are connected to ports α and β , perpendicularly to the plane. Tubes with an internal diameter of 4 mm are used in Osc.2 to have the same cross-sectional area than the feedback loops of Osc.1.

Fig. 4. Osc.1 (left) and Osc.2 (right) with the same central part used in the experimental study

In order to obtain working frequency and outlet velocity profiles of the oscillator, two measurement techniques were employed: a hot wire anemometry and a transient pressure measurement with a transducer. The hot wire anemometry can measure the outlet velocity profile while providing the oscillator working frequency. The hot wire, however, is so fragile that it was only employed in low inlet pressure conditions, while the pressure transducer can be used in a wide range of inlet pressures. Thus, the velocity in the axis of each outlet of the oscillator was measured by a TSI 1210-T1.5 hot wire for inlet total pressure ranging only from 0.1 to 0.3 MPa, while the frequency responses were provided by an Endevco 8510B-200 pressure sensor for a wider inlet pressure range, from 0.1 to 0.7 MPa. The sampling frequency was 25 kHz and the signal was recorded during 10 s.

Computational study of the flow patterns inside the fluidic oscillator

The main goal of this computational study was to provide a better understanding of the inner flow to get new insights about the physical mechanisms underlying the switching process. A detailed analysis of the jet switching process inside the oscillator has been performed by comparing the pressure and velocity contours at key instants and by examining the evolution with time of the area-weighted average pressure and velocity magnitude in various sections along the feedback loops and at the exit of the oscillator [START_REF] Wang | Experimental and numerical study of the frequency response of a fluidic oscillator for active flow control[END_REF]. It has been found that the jet oscillation frequency has a direct relationship with the forth and back propagation of the pressure wave in the feedback loop.

The switching of the jet would then result from both the pressure difference between the control ports at the jet base, By considering several reference times to illustrate the oscillation process, the periodic behavior can be observed on the evolution of area-averaged velocities in y-direction (see Fig. 2) at both oscillator outlets presented in Fig. 5, and on the evolution of the pressure differences 2 for the location of sections B) presented in Fig. 6, together with snapshots of velocity contours at four characteristic times of the first half period in order to illustrate the jet switching mechanism. In addition, a video showing the evolution with time of the pressure and the velocity in the oscillator over one period has been provided as supplementary material (video.SM01) for a better visualization of the jet switching phenomenon. 1-At time 0 t , which is defined as the beginning of a period, the main flow jet (MFJ) in the interaction zone is switching from the right side to the left side since the area-averaged velocity y U at the left outlet is increasing while it is dramatically decreasing at the right outlet, as it can be seen in Fig. 5. At this time, the average value of 6-After this time, the second half-period begins with a symmetric behavior and the whole period is completed at time 3 t , as shown in Fig. 5 and Fig. 6. The duration 2 2 1 ( )

t t t ∆ = -, is about 0.27 T.
The main conclusion of the above analysis is that the pressure difference between the control ports at the jet base is not sufficient to cause the jet switching, which only occurs when, in addition, the counter effect of the pressure difference between the two branches is reduced enough, due to the reflected propagation of the pressure waves in the feedback loops. It can also be observed from Fig. 5 and Fig. 6 of the reflected wave. From 0 t to 1 t , the wave propagation direction is the same as the air flow direction while they are in opposite directions from 1 t to 2 t , leading to a lower wave propagation velocity in the later period, resulting in 1 2 t t ∆ < ∆ . Thus, period T can be written as:

1 2 1 2( ) 2 f f t s s o o L L T f C u C u τ τ τ   = = + = + +   + -   (2)
where t τ and τ s are the transmission time and switching time respectively. Lf is the feedback loop length from section A to section P. Introducing the Mach number in front of the wave in the first step of the period / Ma u C = , Eq. 2 can be recast as follows:

1 2 2 1 1 2 1 1 f s o L T C Ma Ma τ   = + +   + -   (3)
In general, 1 u and 2 u are small but not negligible compared to the sound velocity ( 2 1 0.2 Ma Ma < ≤ according to the numerical simulations), especially when the supply pressure of the oscillator is low. Then, Eq. 3 could be approximated by: ( )

1 2 2 1 1 2 f s o L T Ma Ma C τ ≈ - + + + , (4)
which can be expressed as:

4 1 2 2 f s o L Ma T C τ ∆   ≈ + +     , (5)
where Ma are not accessible in practice. Anyway, assuming that the influence of the wave propagation asymmetry is small in comparison to the leading order term and that s τ is negligible compared to the whole period T , Eq. 5 reduces to

4 1 = = f o L T f C (6)
Interestingly, this relationship differs by a factor 2 from that proposed by Simões et al (2005), which is induced by the back and forth wave propagation in the feedback loop.

Computational study of the effects of the pressure differences between the two branches and the two control ports

The above-obtained results suggest that the jet switching is not only controlled by the pressure difference at the base of the jet, but also by the pressure difference between the two branches. In order to confirm this assumption, a simplified geometry, which represents the central part of a typical oscillator, has been chosen, as shown in Fig. 3b.

This simplified oscillator has the same dimensions as the above simulated oscillator but without feedback loops.

Pressures at both branch outlets (sections G1 and G2) and loop outlets (sections E1 and E2) can be set independently. Four series of unsteady simulations were thus carried out with different pressure differences, as detailed in this section, to study the effect of the pressure difference at control ports only (series 1), the effect of the pressure difference at branches only (series 2) or the combined effect of the two pressure differences (series 3 and 4)

on the switching process.

In series 1, the inlet total pressure was set to 0.25 MPa, and a static pressure of 0.1 MPa was imposed in sections Gl, G2, and E2 (case 0), thus E 2 E1 2 1

0

G G P P - - ∆ = ∆ = .
The numerical settings described in section 3 were used for these simulations.

According to the previous analysis in Section 4, the jet switching in the complete oscillator occurs at t ≈117 ms (cf. Fig. 5), approximately 1ms after the maximum pressure difference between the control ports is reached (at t ≈ 116 ms, cf. Fig. 6). For that reason, the base case (case 0) of series 1 was initiated from t = 0 and simulation results obtained after 1ms were used as initial flow field for the four cases run for an additional duration of 1 ms. Fig. 7 shows the velocity magnitude iso-contours obtained from the performed unsteady simulations after 1ms

for the base case (case 0), and after 2ms for the other cases (i.e. 1 ms of base case simulation + 1 additional ms of simulation with modified pressure at control port E2).

According to Fig. 7, in the base case (case 0) the jet is attached to the right branch, despite the small recirculation zone visible between the jet and the wall.

In this first series, the isolated effect of pressure difference between the control ports, E 2 E1 E 2 E1

P P P

-∆ = -, was examined by gradually increasing the pressure in section E2 while keeping the pressures in other sections at their initial value. The initial flow field in each simulation of this series was that obtained in the base case 0 as mentioned previously. A total of five simulations, denoted as cases 1-1, 1-2, 1-3, 1-4 and 1-5, were carried out, corresponding to E2 E1 i / P P -∆ equal to 0.14, 0.24, 0.28, 0.36 and 0.4, respectively. According to Fig. 7, as E2 E1 P - ∆ is increasing gradually, the jet gets more and more bended with even an attachment of the jet base to the left wall. Nevertheless, the jet always flows out through the same outlet (right one), whatever E2 E1 P - ∆ is at the control ports, proving that the switching cannot occur if the pressures in the two branches are kept at their initial level. This was also confirmed by additional simulations, starting from the base case which was run for 1ms. In series 3, the combined effect of increasing P - ∆ values -Results from numerical simulations Fig. 8 shows the distribution of all the studied cases according to the pressure differences between the control ports and between the branches. The cases where the jet switches are represented by a circular point, the other ones by a diamond point. From this repartition, it is possible to draw the switchability chart in which two zones can be defined: the small zone in gray at low G2 G1 P - ∆ values corresponds to non-switching jet configurations, the large zone in yellow at higher values of G2 G1 P - ∆ corresponds to switching jet configurations.

In the case of no pressure difference between the branches ( G 2 G1 / i P P -∆ = 0), the switching of the main jet cannot be caused only by a pressure difference E 2 E1 P - ∆ between the control ports, whatever its value. On the other hand, the pressure difference between the branches must overcome a certain threshold ( G 2 G1 / i P P -∆ = 0.1) to provoke the jet switching when the pressure difference between the control ports is zero. This threshold decreases with increasing the pressure difference between the control ports and stabilizes on a small non-zero value for ( E 2 E1 P - ∆ > 0.25) and upper. As a conclusion, main jet switching cannot be obtained with only a pressure difference between the control ports but always needs a non-zero pressure difference between the branches. Finally, an interesting observation was made during this computational study: increasing the pressure difference either between the branches or the control ports ( G2 G1 P - ∆ or

E 2 E1 P - ∆
) resulted in significant reductions of the switching time, i.e. the time needed for the jet from the beginning of the simulation to switch to the other branch. This numerical switching time was indeed of the order of 0.4 ms for the lowest values of the pressure difference between the branches leading to the jet switching ( G 2 G1 / i P P -∆ = 0.1) and decreased down to about 0.02 ms for the highest tested values of G2 G1 P - ∆

. Additional numerical simulations are however needed for a better characterization of the jet switching dynamics in relation with the pressure conditions in branches and control ports.

Experimental data and analysis

The numerical studies presented in the previous sections revealed the internal dynamics of the oscillator as well as the detailed switching process and key control variables, which are very difficult to analyze experimentally. In order to validate and complete these numerical results, an experimental characterization of the effect of the oscillator's feedback loop length and of the supply pressure on the oscillator dynamic behavior have been performed, thanks to hot-wire anemometry and pressure measurements, as described in Section 3.

Validation of computational model with Osc.1

The velocity signals at the center of both outlets of Osc.1 (flat design) shown in Fig. 9-A were recorded using a hot wire anemometer (wire diameter 3.8µm, wire length 1.27mm) for an inlet pressure 0.2MPa

= i P
. Since the hot wire is insensitive to the velocity direction, the small peaks following each large peak are expected to correspond to a negative velocity, directed towards the oscillator, and revealing a suction step after each blowing step. This suction flow is sought in active flow control applications [START_REF] Seifert | Active Flow Separation Control on Wall-Mounted Hump at High Reynolds Numbers[END_REF]. The maximum velocities in both outlets are similar, with a mean value around 70 m/s. It can be observed, however, that the blowing duration, 2 T ∆ , is significantly longer at the left outlet than at the right one, which demonstrates that the internal geometry of the oscillator is not totally symmetrical and that the jet issuing from the nozzle spends more time in the left part than in the right one. These minor deviations between the prototype's actual internal dimensions and the model that is used for the numerical simulations are due to assembling constraints. In order to validate the numerical results obtained in section 4 for a totally symmetrical geometry, flow simulations based on the actual non-symmetrical geometry of the experimental oscillator, obtained by X-ray visualization, were carried out with the same numerical settings used in Section 4.

Comparing the results of both computational studies, it was found that the geometry deviations observed on the real device do not modify the internal flow structure but only the duration of the different phases observed during one oscillation period, with a deviation in frequency of less than 5%.

The evolutions with time of the velocity magnitude in the center of both left and right outlet slots for Osc.1, obtained from the numerical simulations on the scanned non-symmetrical geometry, are presented in Fig. 9-B, and are compared to data obtained from hot wire measurements (Fig. 9-A). The measured and simulated frequencies are very close with deviation of about 5%. The simulated maximum velocity for Osc.1 is about 110 m/s, which is however higher than the measured 70 m/s. This deviation of about 50% on the maximum velocity could be due to several factors: uncertainties on the measurement of the geometrical dimensions, uncertainties on the velocity measurements (in particular, due to the size of the hot wire which was non negligible compared to the width of the outlet slots) and possible 3D effects not taken into account in the numerical simulations. In both measured and simulated results, the suction velocity can be observed and the suction duration in one period in the left outlet slot is shorter than that in the right one, because of the asymmetry observed in the throat region. The simulated maximum suction velocity is about 25 m/s, which is also of the same order than the measured one. Comparing the measured period of Osc.1, which has a fixed loop length of 391 mm, and that of Osc.2 with a length of 386 mm, 10% deviation is found in resulted oscillation periods with only a 1% variation of f L . However, it is coherent as the loops of Osc.2 have additional volumes upstream from connecting ports α and downstream from connecting ports β, and in addition 3D effects are present in Osc.2 (see Fig. 4).

The oscillation period decreases with the inlet pressure, down to an almost constant and minimum value reached for inlet pressures / 1.7 > i atm P P for both oscillators and whatever the feedback loop length. This evolution of the oscillation period is similar to the cases of both sonic fluidic oscillators [START_REF] Hayashi | A study on mechanism of oscillation in sonic oscillators : 2nd Report, Mathetmatical model of oscillators operated by air[END_REF] and sweeping fluidic oscillators [33] [34] described in the literature. According to Eq. 5, it can be explained by the fact that ∆Ma = Ma1 -Ma2 increases with the inlet pressure when this pressure is low enough to keep the flow subsonic at the nozzle throat, and then becomes almost constant as the flow reaches sonic conditions at the throat, assuming thatτs remains constant.

For a given Pi, it can be observed that for Osc.2 the period monotonically increases as the feedback loop length increases. This is in accordance with the prediction by Eq. 6 in Section 4, as a result of longer propagation time.

With the purpose to further verify the applicability of Eq. 6, the oscillation periods T measured for different feedback loop lengths are drawn in Fig. 11 as a function of the period ( 0 4 / f L C ) estimated by Eq. 6 for five values of i P (0.115, 0.13, 0.15, 0.2 and 0.25 MPa). For each inlet pressure, the oscillation period increases linearly with the length of the feedback loops. However, the slope decreases with increasing i P . The experimental data fit fairly well the predictions given by Eq. 6 with deviations lower than 15%. It can be clearly observed that when Pi is higher than 0.13 MPa, period T can be approximated by Eq. 6 with a limited error (within 15%), as shown by the dotted lines.

In case of 0.115MPa = i P (empty circles in Fig. 12), the deviation is more obvious, i.e. more than 15% when 0.6

f L m < ( 0 4 / 7 f L C ms <
), but the slope is closer to the estimated one. Looking at Eq. 3, the larger deviation can be explained by the fact that for this low value of Pi, the pressure differences obtained in branches and control ports are low, leading to larger values of the switching time τs, which is neglected in the linear relation Eq. 6. On the other hand, when Pi is very low, much less fluid flows back to the jet base along the feedback loop. In that case, Additional numerical simulations focused on the jet switching zone revealed that the jet switching in the Coanda fluidic oscillator studied in the present work is not controlled by the sole pressure difference between the two control ports, but by a combination of this pressure difference and the one between the oscillator's branches.

In order to verify and validate the dynamics revealed by the simulations, two oscillators were designed, manufactured and tested. For the first model close to a 2D configuration (i.e. large depth over width ratio for all the channels), the measured oscillation frequency has been well estimated by the simplified relation deduced from the numerical simulations. The effect of feedback loop length was explored by using the second oscillator prototype with changeable feedback loops made by tubes of same cross-section area than in the 2D model. It was also found that the proposed simplified relation was able to predict the oscillation frequency with a deviation lower than 15%, except in the case of very low inlet pressures. However, the numerical analysis performed in the first part of this work permitted to explain the origin of this limited deviations.

Fig. 2 .Fig. 3 . 3 . 1

 2331 Fig. 2. Sketch and key dimensions (in mm) of the oscillator

  pressure difference between the two main branches, . 3a for the location of sections A and P). In order to clarify this idea, an in-depth analysis coupling global (frequency response at the oscillator's outlets) and local (switching mechanisms at the jet nozzle) flow dynamics is required.

Fig. 5 .Fig. 6 .

 56 Fig. 5. Evolution of the area-averaged velocity magnitude y U at both outlets with time -Numerical

  that 2 t ∆ is a bit longer than 1 t ∆ , because the HPCW propagates back along the left feedback loop once it is reflected with a velocity 2 o C u -, where 2 u is the local fluid velocity in front

  number in front of the reflected wave in the second step of the period 2 2 0

Fig. 7 .

 7 Fig. 7. Iso-contours of velocity magnitude in the simplified oscillator model for different values of E2 E1 P - ∆ . 0.25MPa = i P

  [START_REF] Viets | Flip-Flop Jet Nozzle[END_REF] and 0.32, for which no jet switching was observed. Video.SM05 in the supplementary material illustrates the switching case E no jet switching was observed.

Fig. 8 .

 8 Fig. 8. Jet switchability chart according to

Fig. 9 . 6 . 2

 962 Fig.9. Experimental and numerical velocity magnitude evolution with time in the center of Osc.1's left and right outlet slots for 0.20MPa = i P

Fig. 10 .

 10 Fig. 10. Oscillation period versus inlet pressure for Osc.1 and Osc.2 with feedback loops 4mm in diameter and for various feedback loop lengths -results from hot wire measurements

  be approximated to 1, which leads to a slope close to unity in the linear relation between T and

Fig. 11 .of the dimensionless slope estimated from a linear fit of each dataset with respect to the dimensionless inlet pressure 7 Conclusions

 117 Fig. 11. Oscillation period as a function of the feedback loop length for several inlet pressures. Results from hot wire measurements are represented by circles. The black solid line represents the prediction

  is the local fluid velocity in front of the wave. This propagation results in a rapid increase of pressure B1 P just after time 0 t . Simultaneously, in the right side, there is a low pressure expansion wave (LPEW) that propagates along the right feedback loop leading to a rapid decrease of pressure B2 P . After time 1 t , the HPCW and LPEW are reflected back after impacting the base of the jet in sections P, and they continue propagating back along the feedback loops. When they reach sections A1 and A2, at time 2

	4-t ,
	the previously negative value of	∆	A1 A 2 P -	suddenly increases up to about 0 and provokes the MFJ switching
	A1 A2 P -∆ from left to right side, as indicated by the inversion of the velocities in both exits. As long as is about 10 kPa though it is very fluctuating, while the value of B1 B2 P -∆	increases A1 A 2 P -∆ is
	and becomes positive. negative, before time 2 t , this pressure difference has a counter effect that prevents the jet switching. Once
	2-Shortly after time 0 t , the MFJ is attached to the left branch, thus the left side velocity reaches a maximum the pressures between sections A1 and A2 are balanced, the pressure difference P1 P2 P -∆ at the base of the
	value and the right side a minimum, but still positive, value. A high pressure compression wave (HPCW) jet becomes able to provoke the jet switching.
	propagates along the left feedback loop with a velocity 5-Time 2	1 C u o + , where o C is the sound velocity (≈340 m/s),
	and 1 u Accordingly, the value of B1 B2 P -∆	increases up to its highest value. The value of	∆	A1 A2 P -	remains close to 0,
	which can be explained as sections A1 and A2 are close to each other and the flow in this region is quite
	steady since	∆	P1 P2 P -	is negative and its absolute value is large enough to perfectly attach the MFJ to the left
	branch although	B1 B2 P -∆	is positive. There is no major change up to time 1 t , except a slight decrease of
	B1 B2 P -∆	and a slight increase of	∆	P1 P2 P -	. The duration 1 1 0 ( t t t ∆ = -	)	of this step is about 0.23 T, where T is
	the simulated oscillation period. T = 4.3 ms in the present simulation.
	3-At time 1 t , the HPCW reaches section P1 and the LPEW reaches section P2 almost simultaneously; the
	value of P1 P2 P -∆	suddenly reverses from its large negative value just before t1 to a positive value. From this
	time, the values of y U in both exits become more fluctuating since the attachment of the MFJ to the left
	wall becomes unstable due to the perturbation generated by the positive pressure difference P1 P2 P -∆	.
	However, this pressure difference between the two sides of the jet base is not able to provoke the MFJ
	switching, though it makes it very unstable. The value of	∆	A1 A 2 P -	becomes negative, which may also
	prevent the switching.

t corresponds to the end of the first half-period.


L e s mon na ie s v i r t uel le s permettent de transférer de l'argent sans avoir besoin de support physique et, généralement, sans faire appel à un intermédiaire. Elles doivent avoir au moins les mêmes propriétés de sécurité que les monnaies réelles : permettre les échanges, empêcher la duplication ou encore garantir l'anonymat des transactions. Récemment, plusieurs monnaies virtuelles cryptographiques ont vu le jour, telles que Bitcoin, Peercoin, Primecoin ou Litecoin. Aujourd'hui, ce sont des monnaies alternatives, car elles n'ont de cours légal dans aucun pays, même si elles sont pour l'instant largement tolérées.

Principe des cryptomonnaies

Anonymat

Le modèle bancaire traditionnel g a ra nt it u n c er t a i n de g ré d'anonymat, en limitant l'accès aux informations de transactions aux seules parties intervenant dans la transaction et à leurs banques respectives. Au contraire, le modèle des crypto-monnaies révèle publiquement toutes les transactions passées. Toutefois, une forme différente d'anonymat est préservée, puisque l'identité des possesseurs des clés publiques n'est pas nécessaire, seule une preuve de possession de la clé privée associée (la signature électronique de la transaction) est demandée pour dépenser des devises. La perte ou le vol de cette clé revient à perdre les devises qui y sont associées. De plus, le monde entier peut voir qu'un montant est transféré d'une clé publique à une autre, mais sans lien avec des personnes physiques ou morales. Toutefois, toutes les transactions d'une clé donnée sont liées, donc au moment où une personne entre ou sort du système de la crypto-monnaie, l'anonymat doit être levé, au moins auprès de l'organisme d'échange, et l'ensemble des transactions associées à cette clé peut alors être tracé.

Un nouveau système monétaire ?

Le minage décentralisé, dans lequel n'importe quel agent économique peut créer des devises, et la circulation décentralisée de ces devises sur Internet, dans laquelle aucun acteur ne prélève de commission, revêtent l'apparence du libéralisme économique. En réalité, le danger est que la décroissance des rendements implique que les fermes de minage (groupement de mineurs qui permet de garantir un revenu moyen pour chacun) doivent se concentrer afin de rester rentables. À partir du moment où quelques entités privées possèdent une majorité du marché de la certification des transactions, elles détiennent alors en pratique la capacité d'émission monétaire et l'intérêt d'avoir un système distribué est perdu. En cas de succès des transactions, la limitation du nombre de Bitcoins émis provoque une hausse de sa valeur. Or, si la valeur des devises électroniques est vouée à monter, il est préférable de ne pas les utiliser pour faire des achats et de les épargner. En outre, plus une monnaie a de succès, moins elle est utile pour faire des transactions. Mais, ainsi, si elle devient moins utile, sa valeur peut baisser et éventuellement se réguler.

Pour éviter l'écueil de la concentration des fermes de minage une solution peut être de passer d'une crypto-monnaie à une autre. Toutefois, pour chaque monnaie électronique, sa diffusion et son utilisation croissante ne manqueront pas d'attirer les autorités publiques, qui voudront taxer les transactions virtuelles et demanderont la mise en place de points de contrôle officiels des transactions. La même chose arrive aux systèmes d'échanges, de troc ou de monnaies locales : tant qu'ils restent limités à une petite association d'utilisateurs, les pouvoirs publics ne s'en préoccupent pas. S'ils prennent trop d'ampleur et deviennent un moyen d'évasion fiscale, les autorités s'y intéressent et des tribunaux considèrent que les transactions doivent être taxées comme les autres.

Références bibliographiques

• J.-P. DELAHAYE -Les preuves de travail (Bitcoin, Spam, etc.)

  une transaction en Bitcoins, à gauche un portefeuille, à droite une partie de la chaîne. n

,

  Pour la science, avril 2014.

• J.-P. DELAHAYE -Le Bitcoin : la cryptomonnaie, Pour la science, décembre 2013. • J.-G. DUMAS, P. LAFOURCADE et P. REDON -Architectures PKI et communications sécurisées, Dunod, 2015.


Bensu Tunca
email: btunca@sckcen.be

Thomas Lapauw

Olesia Karakulina

Maria Batuk

Thierry Cabioc'h

Joke Hadermann

Rémi Delville

Konstantina Lambrinou

Jozef Vleugels

Thierry Cabioc'

Synthesis of MAX Phases in the Zr-Ti-Al-C System

Keywords: MAX phase ceramics, X-ray diffraction, Reactive hot pressing, Hydrides, Solid solution MAX phases, Ordered MAX phases

établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

A class of fascinating ternary carbides and nitrides, later labelled MAX phases, was discovered in powder form in the 1960s 1 and gained renewed interest in the mid-1990s when manufacturing in the bulk form allowed the determination of their intrinsic properties. 1,2 MAX phases combine properties characteristic of metals, such as good thermal/electrical conductivity, thermal shock resistance, appreciable machinability, damage tolerance and relative softness, with properties characteristic of ceramics, such as good high-temperature mechanical properties and satisfactory resistance to oxidation and corrosion. 3 The stoichiometry of MAX phases is described by the Mn+1AXn general formula, where M is an early transition element, A is an A-group element, X is C or N, and n is usually 1, 2 or 3, but higher-order MAX phases with n values of 4, 5 and 6 have also been reported. [4][5][6] MAX phases are layered hexagonal solids, with space group P63/mmc (no. 194), the structure of which consists of near close-packed layers of M6X octahedra interleaved with layers of A-group elements. The M6X octahedra resemble the octahedra found in the cubic rock-salt structure of the corresponding binary MX carbides, and the number of M6X layers actually determines whether the overall structure is 211, 312, 413 or higher-order. [START_REF] Barsoum | MAX Phases properties of machinable ternary Carbides and Nitrides[END_REF] MAX phases in the Ti-Al-C system have already been extensively studied and are known for their outstanding oxidation resistance that results from the formation of a protective Al2O3 oxide scale. 8 Moreover, Zr-Al-C-based MAX phases, such as Zr3AlC2 and Zr2AlC, have been recently synthesized 9,10 and their properties are yet to be determined. The potential importance of the latter for the nuclear industry is associated the low neutron absorption cross-section of zirconium, which makes them promising candidate materials for fuel cladding applications.

Especially in the post-Fukushima era, the development of accident-tolerant fuel (ATF) cladding materials that can outperform the zircaloy clads used in the current fleet of light water reactors is to aqueous corrosion and high-temperature steam oxidation with adequate radiation tolerance.

Prior studies on the oxidation behaviour of Zr-Al-C compounds, such as Zr2[Al(Si)]4C5 and Zr3[Al(Si)]4C6, showed the formation of non-protective ZrO2 oxide scales. 11 Similar studies on Ti-doped Zr2(AlSi)4C5 ceramics revealed an improvement in oxidation resistance attributed to the addition of Ti. 12 Also, the addition of Zr to Ti3(Si0.95,Al0.05)C2 to form (Ti1x,Zrx)3(Si0.95,Al0.05)C2 up to x=0.07 resulted in the improvement of the high-temperature stiffness of the starting material. 13 Several prior studies on MAX phases have demonstrated property improvement for the ternary compounds by the partial substitution of the elements on the M, A or X sites. Such substitution leads to the formation of solid solutions or ordered structures where the added elements occupy a preferred atomic position. 14 Recent attempts to synthesize 211 MAX phases in the Zr-Ti-Al-C system were reported as unsuccessful 15 and no (Zr1-x,Tix)2AlC MAX phases have so far been experimentally made. On the other hand, (Zr1-x,Tix)3AlC2 MAX phases were recently claimed to have been successfully synthesized, but the results of that study have not yet been published. 16 Solid solutions reported to contain Ti, Zr, Al and C include: (Ti,M)2AlC with M = V, Nb, Ta, Cr, [17][18][19][20] (Nb,Zr)2AlC, 18,21 (Ti,V)3AlC2, 21 (Ti,Nb)4AlC3, 22 and (Nb,Zr)4AlC3. 23,24 In this study, Ti was added to the Zr-Al-C system in an attempt to synthesize (Zr1-x,Tix)3AlC2 (ZT312) and (Zr1-x,Tix)2AlC (ZT211) MAX phase solid solutions. Different Ti contents were added in order to cover the whole Zr-Ti solid solubility range.

EXPERIMENTAL SECTION

2.1. Synthesis. ZrH2 (<6 μm, >99% purity, Chemetall, Germany), TiH2 (<8 μm; >99% purity; Chemetall, Germany), Al (<5 μm, >99% purity, AEE, United States) and C (<5 μm, >99% purity, Asbury Graphite Mills, United States) powders were used as starting materials for MAX phase synthesis. The Zr:Ti:A:C ratio in the starting powder was [3(1-x)]:[3x]:[1.2]:[1.6] with x = 0, 0.1, 0.2, 0.5, 0.66, 0.7, 0.8, 0.9 and 1, so as to investigate a wide range of the Ti and Zr solid solubility on the M site of the produced MAX phases.

The initial stoichiometries were selected to target the 312 stoichiometry, using a slightly higher Al content to compensate for possible Al losses due to melting during processing, and a lower C content to compensate for C interdiffusion from the graphite die/punch setup of the hot press.

The selection of the 312 stoichiometry was based on previous work on Zr-Al-C MAX phases 9,10 that showed an off-stoichiometric MAX phase formation, i.e., the synthesis of 211 and 312 MAX phases started from 312 and 413 initial starting powder mixtures, respectively. Powders were dry mixed for 48 hours in air, on a Turbula type multidirectional mixer. The powder mixtures were cold pressed at 30 MPa in a 30-mm inner-diameter graphite die setup. The powder compacts were hot pressed (W100/150-2200-50 LAX, FCT Systeme, Frankenblick, Germany) in the same setup, using graphite punches in vacuum (10 Pa). The heating rate was 25°C/min and the dwell time at the targeted 1350-1700°C range was 30 min, during which period the initial load of 7 MPa increased to 30 MPa. Pressureless sintering involved cold pressing powder compacts at 30 MPa that were subjected to the same thermal excursions as their hot pressed equivalents.

Characterization.

Samples were produced for scanning electron microscopy (SEM) and transmission electron microscopy (TEM) examination and the remaining ceramics were ringmilled to obtain powders for X-ray diffraction (XRD). Backscattered electron (BSE) detector images of bulk samples were obtained on a FEI Nova NanoSEM, while high-resolution scanning transmission electron microscopy (HRSTEM) combined with energy dispersive spectroscopy (EDS) were employed to analyse thin foils on a FEI Titan 3 aberration-corrected transmission electron microscope (200 kV), using a high-angle annular dark-field (HAADF) detector. Thinfoil preparation for TEM analysis involved mechanical thinning with diamond lapping films and polishing down to 100 µm before final preparation to electron transparency with a JEOL Cryo Ion Slicer at 6 kV. The phase assembly in the produced MAX phase ceramics was determined by XRD (Seifert 3003), using Cu Kα radiation operated at 40 kV and 40 mA, and the XRD patterns were obtained from powder samples using a Bragg-Brentano geometry in the 5 to 75° 2θ range with a step size of 0.02° and a time of 2 s per step. Ordering in the produced MAX phases was studied by means of a Bruker D2 advanced diffractometer with Cu Kα radiation, operated at 30 kV and 30 mA in the Bragg-Brentano geometry, with a divergence slit of 0.6°, a 2θ step size of 0.02° and a time of 0.1 s, due to the higher peak to background ratio. Lattice parameters and weight fractions of phases were determined by Rietveld refinement of the powder XRD patterns, using the Materials Analysis Using Diffraction (MAUD) software. 25 XRD pattern simulations were performed using the PowderCell software. 26

RESULTS AND DISCUSSION

3.1. Pressure-assisted MAX phase synthesis. The use of powder mixtures with an initial (Zr1x,Tix)3Al1.2C1.6 stoichiometry and x varying between 0 and 1 resulted in the synthesis of both the 5 211 (ZT211) and 312 (ZT312) MAX phases. Furthermore, local 413 stackings were observed inside 312 grains examined by HAADF STEM, as will be demonstrated later in the text. Therefore, it appears that a small amount of Ti stabilizes the ZT312 phase, which was difficult to form at higher Ti contents. At x=0.7 and x=0.8, only trace amounts of ZT312 were detected by XRD in the ceramics hot-pressed at 1600°C; similarly, trace amounts of ZT312 were found in the ceramic with x=0.66 hot-pressed at 1550C. The maximum total amount of MAX phases (76 wt%) in the hot-pressed Zr-Ti-Al-C ceramics was observed for x=0.9. Powder XRD results acquired from a whole range of ceramics hot-pressed at 1550°C are compared in Figure 2. 3a) containing fine (Zr,Ti)C grains were observed inside a ZT211 matrix. At 1450°C, the amount of intermetallics decreased and grain growth of the mixed carbides was observed, while the ZT312 phase did not form. At 1500°C, elongated ZT312 grains enveloped by ZT211 grains were observed, and the (Zr,Ti)C grains were dispersed in a mixed ZT211/ZrAl2 matrix.

Pressureless MAX phase synthesis.

After hot pressing, solidified residues were observed on the die/punch setup, a fact indicating material loss during processing. Possible chemical reactions between intermetallics in the Zr-Al system (Zr5Al4, Zr2Al3, ZrAl2, ZrAl3) known to result in liquid phase formation are summarised below. [START_REF] Wang | Phase Equilibria Diffus[END_REF] During hot pressing, it is possible that liquid products of such reactions might be squeezed into the clearances of the die/punch setup, forming the observed residues.

𝑍𝑟 5 𝐴𝑙 4 + 𝑍𝑟 2 𝐴𝑙 3 ↔ 𝐿 (1477℃) 𝑍𝑟 2 𝐴𝑙 3 ↔ 𝑍𝑟𝐴𝑙 2 + 𝐿 (1592℃) 𝑍𝑟𝐴𝑙 2 ↔ 𝐿 (1658℃) 𝑍𝑟𝐴𝑙 2 + 𝑍𝑟𝐴𝑙 3 ↔ 𝐿 (1583℃) 𝑍𝑟𝐴𝑙 3 ↔ 𝐿 (1583℃) 𝐴𝑙(𝐹𝐶𝐶) ↔ 𝐿 + 𝑍𝑟𝐴𝑙 3 (660℃)
As aluminium and its intermetallics are the Al source for MAX phase formation, select powder mixtures, i.e., (Zr1-x,Tix)3Al1.2C1.6 with x=0.2, 0.5, 0.9, were pressurelessly sintered to eliminate liquid-phase losses during processing. Indeed, no significant losses were observed during these tests. Quite interestingly, the amount of secondary Zr-Ti carbides in all pressurelessly-sintered ceramics was lower than in their hot-pressed equivalents. Table 1 summarises the results of Rietveld refinement of the XRD patterns acquired from both hot-pressed and pressurelesslysintered ceramics having the same powder mixtures, and gives a comparison of the acquired phase assemblies. The lower carbide content can be explained by the more efficient containment of Al and its compounds, which favours potential reactions involving these carbides and leading to MAX phase formation. In support of this hypothesis, the amount of Zr2Al3 and/or ZrAl3 needed for MAX phase formation was also higher in the pressurelessly-sintered ceramics. Based on the above, pressureless sintering in the Zr-Ti-Al-C system and its sub-systems should be studied in more detail in the future to improve the phase purity of the produced MAX phase ceramics. 4 shows HAADF STEM images of Zr-Ti-Al-C MAX phases and typical structural anomalies; all STEM images were acquired along the < 112 ̅ 0 > zone axis.

Figures 4a and4c show the (Zr0.8,Ti0.2)3Al1.2C1.6 ceramic hot-pressed at 1500°C. ZT211 stackings (Figure 4a) and alternating 211 and 312 stackings that create a local 523 structure in the ZT312 phase (Figure 4c) were observed. 523 peaks could not be identified in the XRD spectra, since the 523 stackings are believed to represent a small material fraction that might have resulted from local transformations between 211 and 312 phases. Similar observations have been made for MAX phases in the Hf-Al-C system. [START_REF] Lapauw | [END_REF] Figures 4b, 4d and4e show the (Zr0.5,Ti0.5)3Al1.2C1.6 ceramic pressurelessly-sintered at 1700°C. This sample was selected for detailed TEM analysis, due to the relatively high (52 wt%) ZT312 content (Figure 4b). Apart from the observed mixed stackings, "zipper"-like stacking faults were also observed (Figures 4d-4e), where MX layers within the same MAX phase unit cell merge into a single band of similar or larger thickness at different locations. The "zipper"-like features can be a consequence of the transition from the 211 to the 312 phase by the outward diffusion of Al, the inward diffusion of phases. [START_REF] Lapauw | [END_REF] These structural anomalies were observed at the boundaries between areas with different Ti/Zr ratios, as shown in Figures 4d-4e, where Ti-rich (darker) and Zr-rich (brighter) regions appear to converge at the locations of the defect-rich boundaries. Variations in the local chemical composition were mapped by EDS and results are presented in Figure 5. Areas marked as 1 and 2 had 55 at% and 47 at% Ti, respectively. Scanning areas situated further apart showed that the composition could vary from 78 at% to 50 at% Ti within the same grain. Similar variations in composition were also identified within (Zr,Ti)C grains. from the ceramic with x=0.5. Within this particular ceramic, grains with a higher Ti content (x0.7) were found. This could explain the deviation from the linear trend at x=0.5 in Figure 6a;

the observed overall agreement with linearity confirms that the ZT211 ceramics obey Vegard's law for solid solutions. 29 On the other hand, when the lattice parameters of the ZT312 phase are plotted, larger deviations from the linear fit is seen. Linear fit was plotted by using literature data for lattice parameters of Zr3AlC2 and Ti3AlC2 (Figure 6b). Local deviations from linearity around 0.67 Ti, is attributed to the effect of Zr and Ti ordering in the crystal structure on the ZT312 lattice parameters, as proposed below.

Ordering of Zr and Ti in ZT312 and ZT413 MAX phases.

As possible cause for the deviation from Vegard's law, the potential Zr and Ti atomic ordering in ZT312 MAX phases at x=0.67 (i.e., (Zr0.33,Ti0.67)3AlC2 or ZrTi2AlC2 stoichiometry) was studied by simulating the unit cell and comparing the experimental XRD peak intensities to the simulated ones. The effect of preferred orientation on the peak intensities was neglected, since the XRD spectra were acquired from powder samples. In the ZT312 unit cell it is important to note that Zr and Ti elements can occupy two different crystallographic sites (M1 at position 4f (1/3,2/3,z~0.132), close to the Alayers, and M2 at position 2a(0,0,0)), in the middle of the octaedra layers. The unit cell was here modified to have different Ti occupancies in the M1 position site whereas M2 sites were occupied only by Zr atoms. Erreur ! Source du renvoi introuvable. 7 shows the evolution in the XRD peak intensities when the structure becomes progressively fully ordered (transition from black dot to blue dot), i.e., the evolution from Ti=0 (Zr3AlC2) to Ti=1 (fully-ordered ZrTi2AlC2) on the M1 site. Figure 8 Five ordered MAX phases have so far been experimentally observed in three quaternary systems: Cr2VAlC2 and Cr2V2AlC3 30 , Cr2TiAlC2 12,27 , Mo2TiAlC 31 and Mo2Ti2AlC3. 32 In these MAX phases, elements from group VI (Cr and Mo) were located in the M1-site that is connected to the A-layer, while elements from group IV & V (i.e., Ti and V, respectively) were located in the middle of the M6X layers.

Theoretical studies on ordering in MAX phases have been carried out using computational tools.

A first set of ordered, double transition metal carbides was studied in relation with MAX phasederived two-dimensional materials known as MXenes. Anasori et al. predicted a set of 24 stable ordered MXene structures based on DFT-calculations; 33 however, the Zr-Ti-Al-C system was not included in these calculations. More recently, Dahlqvist et al. evaluated order and disorder in TiMAlC, TiM2AlC2, MTi2AlC2, and Ti2M2AlC3 where Zr was included in the M elements. 34 The calculated formation enthalpy (∆Hcp) values for the relevant MAX phases (TiZrAlC, TiZr2AlC2, ZrTi2AlC2 and Ti2Zr2AlC3) were predicted by these authors to be more positive compared to the competing phases, suggesting that MAX phases were unlikely to form in these systems. However, our experimental data demonstrate the opposite. On the other hand, when looking at the most stable ordered configurations, certain agreements can be found between theory and experiments. The predicted as most stable structures, i.e., structures with the lowest ∆Hcp at 0 K, for TiZrAlC (disordered), ZrTi2AlC2 (ordered with Ti on M1-site) and Ti2Zr2AlC3 (ordered with Ti on M1-site) match with the experimentally observed atomic stackings. For the TiZr2AlC2 stoichiometry, no clear indication of atomic ordering was experimentally observed, but also here a preferred occupancy of the M1-site by Ti atoms is not excluded.

Two main factors responsible for ordering were already addressed by Hume-Rothery: a clear difference in electronegativity () and in covalent radii. 35 Both differences are significant for the Ti-Zr pair and could account for the particular ordering in the Zr-Ti-Al-C system, where Ti occupies preferentially the M1-site, while Ti occupies preferentially the M2-site in the other ordered systems. For both factors, the ordering is consistent in all ordered MAX phases so far reported. It appears that the M-element with the smallest atomic radius (Cr < V < Mo < Ti < Zr) and the highest electronegativity (Mo > Cr > V > Ti > Zr) is positioned in the outer-layers of the M1-M2-M1 'sandwich' (312-structure) or M1-M2-M2-M1 'sandwich' (413-structure) next to the A-element layer, whereas the larger and less electronegative M-element is positioned inside the 'sandwich' structure on the M2-sites. It is difficult to identify the most dominant factor of the two, as the trends for the electronegativity and the atomic radius are exactly opposite. It can, however, be mentioned that in some cases the difference in electronegativity is limited (i.e., ∆ for V and Cr is 0.03, corresponds to <2% change in Cr), while the difference in atomic radii is invariably significant (>8%). Considering this steric-based hypothesis, one might suggest that smaller M-atoms form more stable bonds with small A-atoms (e.g., Al tends to form more stable bonds with Ti than with Zr), thus forming more compact triangular prisms.

As opposed to other ordered MAX phases, such as Mo2Ti2AlC3 or Mo2TiAlC2, 32 (Zr,Ti)3AlC2 has a gradually changing lattice parameter as a function of the amount of Ti substituting Zr (Figure 6b). Formation of ZT312, not only for the ordered stoichiometry but also within a range of stoichiometries, indicates that ZT312 can be obtained in a solid solution form or as an ordered quaternary MAX phase.

Secondary phases.

A bimodal grain size distribution was observed for the (Zr,Ti)C carbides, i.e., a combination of significantly larger (>2 µm) and submicrometre-size grains (see Figure 3). It should be noted that the grain size of these mixed carbides is well below the grain size of the initial TiH2 and ZrH2 powders. The angular morphology of the (Zr,Ti)C carbides indicates that they might have precipitated from a liquid phase rather than formed by solid state diffusion. Fine-grained ZrC synthesis was also reported during a self-propagating reaction in elemental Al-Zr-C powder mixtures. 36 Formation of secondary phases was also observed in interrupted pressurelessly-sintering ceramics made from a (Zr0.8,Ti0.2)3Al1.2C1.6 starting powder that was heated at the same rate as the hot-pressed ceramics. When heated to 625°C, i.e., below the melting temperature of Al (660°C), for 5 minutes and immediately furnace cooled, the ceramic contained only about 3 wt% elemental Al (Table 3). The amount of elemental Al was comparable after a similar treatment at 675°C, while the Al content in the initial powder mixture was 10.6 wt%. This indicates that Al reacts to form intermetallics even below its melting point. The presence of substantial amounts of ZrAl3 and especially ZrAl2 (Erreur ! Source du renvoi introuvable.3) was confirmed by XRD and Rietveld refinement results. No Ti was seen in the XRD patterns, while the lattice parameter of Zr and its intermetallics with Al were refined to be smaller than in their original pure phases, suggesting that Ti was in solid solution with both Zr and/or its intermetallics with Al. The solubility of Ti in ZrAl3 (crystal structure D023, ordered tetragonal I4/mmm) was reported to be 15 at%, while the Zr solubility in TiAl3 (crystal structure D022, ordered tetragonal I4/mmm) was up to 2 at%. 37 Isostructural aluminides Ti5Al11 and ZrAl3 (both D023) have a continuous solid solubility, while ZrAl3 and TiAl3 with the same structure (D023) also show a continuous solid solubility. 37 It was also reported that Zr substitutes Ti in TiAl and Ti3Al. 38 As a result, a certain amount of Ti is also expected to be found in the Zr-Al intermetallics. However, no detailed analysis on these intermetallics was performed within this study. Formation of ZrC was observed in the ceramic heated to 675°C, but not in the one heated to 625°C. These results indicate that the coarse (Zr,Ti)C grains observed in the hot-pressed ceramics might have already formed during the heating stage between 625-675°C. The lattice parameter of ZrC was close to the literature value of 4.683 Å, 39 indicating no (or a very small amount) of Ti in solid solution at such low temperatures. The ZrH2 and TiH2 starting powders dehydrogenated during heating and no residual peaks of these phases were identified. With a Ti content in the starting powder of ≥25 at% (≥ x=0.5), two distinct sets of peaks were observed in the XRD patterns of the hot-pressed discs, belonging to two ternary mixed (Zr,Ti)C carbides, one rich in Zr and the other rich in Ti. The lattice parameters of the Zr-rich and Ti-rich carbides are plotted in Figure10, indicating the phase separation. Taking into account the miscibility gap in the ZrC-TiC system, 40 the decomposition of mixed carbides is not surprising.

Although the calculated temperatures for the miscibility gap are very high (~2000 K), simulation studies suggested that carbon vacancies in the mixed (Zr,Ti)C carbides account for significant deviations in the onset temperature of phase separation. 41 The onset of immiscibility was shown to occur at lower temperatures as the amount of C vacancies in the mixed carbide increased. The separation temperatures in the performed simulations for 20% C vacancies are very similar to the experimental processing temperatures in this work, but the extent of the C sub-stoichiometry was not systematically analysed. This decomposition was noticed for (Zr1-x,Tix)3Al1.2C1.6 stoichiometries with x≥0.5. Pressureless sintering was found to be an alternative method of MAX phase synthesis in the Zr-Ti-Al-C system that resulted in a higher MAX phase content and less secondary carbides as compared to the hot-pressed ceramic equivalents, probably due to the reduced loss of liquid phase material.

Figure 1

 1 Figure 1 shows the evolution of the different types of MAX phases (211 and 312) as a function of the hot-pressing temperature and atomic fraction, x, of Ti in the (Zr1-x,Tix)3Al1.2C1.6 starting powder. Each circle indicates a hot-pressed ceramic and the numbers indicate the type of MAX phases identified by XRD analysis. All ceramics contained Zr-and/or Ti-rich ternary (Zr,Ti)C carbides, whereas only the Zr-and Ti-based end members contained binary ZrC and TiC, respectively. The secondary carbides are addressed in a dedicated section below. ZrAl2, ZrAl3 or Zr2Al3 intermetallics were also observed in some of the ceramics hot-pressed below 1550°C. In order to avoid confusion, secondary carbides and intermetallics are not included in Figure 1.

Figure 1 .

 1 Figure 1. Map of MAX phases synthesized in the Zr-Ti-Al-C system as a function of the hotpressing temperature and the Ti content, x, in the (Zr1-x,Tix)3Al1.2C1.6 starting powder. Only the 312 (red) and 211 (blue) MAX phases are indicated. Green indicates trace MAX phase amounts.

Figure 2 .

 2 Figure 2. XRD patterns and phase assembly in ceramics hot-pressed at 1550°C as a function of the starting powder composition. Numbers in parenthesis indicate uncertainty in the last digit.

Figure 3 .

 3 Figure 3. BSE images of (Zr0.8,Ti0.2)3Al1.2C1.6 ceramics hot-pressed at (a) 1400°C, (b) 1450°C and (c) 1500°C.

Figure 4 .

 4 Figure 4. HAADF STEM images along the < 112 ̅ 0 > zone axis of MAX phase grains. (a) 211 and (c) 523 stackings in the (Zr0.8,Ti0.2)3Al1.2C1.6 ceramic hot-pressed at 1500°C. (b) 312 stackings and (d-e) "zipper"-like stacking faults in the ZT312 phase in the (Zr0.5,Ti0.5)3Al1.2C1.6 ceramic pressurelessly-sintered at 1700°C. Stacking faults outline the boundaries between areas with different Ti/Zr ratios, as indicated by differences in the local compositional contrast.

Figure 5 .

 5 Figure 5. HAADF STEM image and EDS maps for Zr, Ti and Al showing compositional nonuniformities within a ZT312 MAX phase grain.

Figure 6 .

 6 Figure 6. Lattice parameters of the ZT211 (a) and ZT312 (b) MAX phases obtained by Rietveld refinement of the XRD spectra of the produced ceramics, as a function of the Ti content, x, in the starting powder.

  compares simulated and experimental XRD intensities for the ZT312 phase in the (Zr0.5,Ti0.5)3Al1.2C1.6 ceramic hot-pressed at 1700°C. Only the refined ZT312 intensities are plotted from the experimental XRD data, while the theoretical peak intensities of Zr3AlC2, Ti3AlC2 and disordered (Zr0.5,Ti0.5)3AlC2 phases are added for comparison. The comparison of XRD peak intensities was done by normalising intensities to the one of the (014) diffraction peak.. In order to evaluate only the evolution of the XRD peak intensities, the lattice parameters were kept constant during simulation even if 2θ values are also expected to shift when the element occupancy changes. The intensity trend, indicated by arrows for the main peaks in Figure 7, agreed with the changes occurring in the simulated XRD patterns (Figure 8), implying that ordering presumably occurred in the ZT312 phase.

Figure 7 .

 7 Figure 7. Simulated XRD patterns of the ZT312 structure with changing Ti occupancies in the M1 site (1/3,2/3,0.1324). The arrows show the change in XRD peak intensity as the structure

Figure 8 .

 8 Figure 8. Theoretical XRD peak intensities of Zr3AlC2, Ti3AlC2, and (Zr0.33,Ti0.67)3AlC2 with Ti occupancy of 0.5 at both M1 and M2 sites, as compared with the refined peak intensities of the ZT312 phase in the (Zr0.5,Ti0.5)2Al1.2C1.6 ceramic hot-pressed at 1700°C.

Figure 9 .

 9 Figure 9. (a-b) STEM/EDS results acquired from a (Zr0.5,Ti0.5)3Al1.2C1.6 ceramic pressurelesslysintered at 1700°C indicate Zr enrichment in the central M-site of the ZT312 phase. (c) An average intensity profile reveals ordering in the vertical atomic rows, where a frame highlights a local 413 stacking.

Figure 10 .

 10 Figure 10. Cubic lattice parameter of mixed Zr-Ti carbides as a function of the starting powder composition of the hot-pressed ceramics.

  MAX phase solid solutions have been synthesized in the Zr-Ti-Al-C system by reactive hot pressing, as well as by pressureless sintering of ZrH2, TiH2, Al and C powder mixtures. (Zr1-x,Tix)3Al1.2C1.6 starting powder stoichiometries with x=0-1 were investigated. Ti was found to substitute Zr in Zr2AlC and Zr3AlC2 phases to form (Zr,Ti)2AlC and (Zr,Ti)3AlC2 solid solutions over the complete compositional range. HAADF STEM imaging combined with EDS revealed ordering of the Ti and Zr atoms in ZT312, with the Zr atoms 'sandwiched' between two Ti layers in the M6C structure. Atomic ordering similar to that observed in ZT312 was also observed in ZT413 bands, with the two middle layers (M2 position) showing a higher Zr content 'sandwiched' again between two Ti layers (M1 position). The ordering was also supported by the comparison of simulated and experimental XRD spectra. HRSTEM examination of ZT211 and ZT312 showed that a mixture of 211-312 (523) and 413 stackings can be present inside a single grain, suggesting a transition from one MAX phase to another. Decomposition of the mixed (Zr,Ti)C secondary phases into Zr-rich and Ti-rich carbide mixtures was observed upon sintering in the 1350-1700°C range, in agreement with the existing miscibility gap in the ZrC-TiC system.

Table 1 .

 1 Phase assembly of the pressurelessly-sintered and hot-pressed ceramics, as obtained by Rietveld refinement of the acquired XRD patterns.

	Powder Mixture	T (C) / P (MPa)	ZT211 ZT312 (Zr,Ti)C (Ti,Zr)C Zr2Al3	ZrAl3	Rwp (%)	Rexp (%)
	(Zr0.8,Ti0.2)3Al1.2C1.6 1550/30 12(1)	34(1)	54(1)	-	-	-	9.49	2.90
	(Zr0.8,Ti0.2)3Al1.2C1.6 1550/0	55(1)	12(1)	30(1)	-	3(1)	-	11.55	3.09
	(Zr0.5,Ti0.5)3Al1.2C1.6 1700/30	-	14(1)	46(1)	40(1)	-	-	8.46	3.28
	(Zr0.5,Ti0.5)3Al1.2C1.6 1700/0	-	52(1)	27(1)	5(1)	-	16(1) 10.69	3.41
	(Zr0.1,Ti0.9)3Al1.2C1.6 1500/30 43(1)	34(2)	-	23(2)	-	-	12.13	8.84
	(Zr0.1,Ti0.9)3Al1.2C1.6 1500/0	51(1)	39(1)	-	10(1)	-	-	12.87	3.92

3.3. STEM results. Figure

Table 2 .

 2 . Refined lattice parameters of the ZT211 and ZT312 MAX phases as determined by

	Rietveld analysis XRD data. Refinement was done using the P63/mmc space group with the
	following atomic positions: for ZT211 (Zr1-xTix) at 4f(2/3,1/3,z~0.08), Al at 2c(1/3, 2/3, 1/4), and
	C at 2a(0,0,0); for ZT312, (Zr1-xTix)1 at 4f(1/3,2/3,z~0.132), (Zr1-xTix)2 at 2a(0,0,0), Al at
	2b(0,0,1/4), and C at 4f(2/3,1/3,z~0.07);
	x	0 9,10	0 †	0.1 †	0.2 †	0.5 †	0.67 †	0.7 †	0.8 †	0.9 †	1 †	1 1,7
	312 a (Å) 3.33308(6) 3.3263(4) 3.3106(3) 3.2938(3) 3.220(1) 3.206(1)			3.0897(3) 3.0624(6) 3.07
	312 c (Å) 19.9507(3) 19.947(3) 19.836(2) 19.755(3) 19.325(9) 19.209(2)			18.641(3) 18.544(2) 18.58
	211 a (Å) 3.3237(2) 3.323(2) 3.314(3) 3.2656(5) 3.1813(1) 3.1627(2) 3.1583(1) 3.1314(2) 3.0919(2) 3.0596(4) 3.04
	211 c (Å) 14.5705(5) 14.570(3) 14.507(7) 14.363(3) 14.0204(9) 13.9534(9) 13.9307(9) 13.8381(8) 13.750(1) 13.647(2) 13.60
	Rwp (%)		16.8	11.42	9.5	8.86	10.82	8.79	12.63	10.26	15.44
	Rexp (%)		2.96	2.93	2.89	3.5	3.56	3.41	3.51	4.58	4.13
	† These stoichiometries were studied in the framework of this work.
	A hexagonal crystal structure with space group P63/mmc was assumed for both ZT211 and ZT312 MAX phases.
	Numbers in parenthesis indicate uncertainty in the last digit.

Table 3 .

 3 Rietveld refinement results of XRD data from interrupted pressureless sintering runs.

ACKNOWLEDGEMENTS

The presented research was partly funded by the European Atomic Energy Community's (Euratom) Seventh Framework Programme FP7/2007-2013 under Grant Agreement No. 604862 (FP7 MatISSE) and falls within the framework of the EERA (European Energy Research Alliance) Joint Programme on Nuclear Materials (JPNM). T. Lapauw thanks the Agency for Innovation by Science and Technology (IWT), Flanders, Belgium, for PhD Grant No. 131081. B. Tunca acknowledges the financial support of the SCK•CEN Academy for Nuclear Science and Technology. The authors acknowledge the Fund for Scientific Research Flanders (FWO-Vlaanderen) under Grant G.0431.10N.F. and the Hercules Foundation under Project ZW09-09.


Eva Quéheille

Anne Ventura

Lauredan Le Guen

Eric Lefebvre

Laury Barnes-Davin

Laury Barnes

Application of Transition LCA method on CO2 capture and utilization in a cement plant

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

WHAT IS TRANSITION LIFE CYCLE ASSESSMENT (LCA) ?

Each year, cement accounts for 8% of worldwide emissions. The main cause: at high temperature, the decarbonation of calcium carbonate from limestone during clinker production.

CCS enables long-term capture of emitted CO 2 , while CCU enables the use of captured CO 2 in various products. CCS and CCU, along with other actions, should be part of a joint effort for the cement industry to reach carbon neutrality in the future.

Transition LCA is a new LCA approach developed by University Gustave Eiffel. The objective is to compare an existing situation at system level (several products and functions can be considered) with different possible futures, then to select the scenarios which would provide the greatest improvements.

Key features: long-term time horizon, geographical context, involvement of stakeholders.

STEP-BY-STEP PROCEDURE TO APPLY TRANSITION LCA

1 Investigated transition solution CCS and CCU with hydrogen production in a cement plant of the French group Vicat

Existing reference situation

The cement plant does not have a carbon capture system. Products likely to be substituted (e.g. hydrogen) are produced in a conventional, carbon-based way.

Goal and scope

Offer prioritization: treat as much CO 2 from the plant as possible. Time scope : by 2030.

Reference flows

Total amounts of the products likely to be substituted in the target territory. Total amounts of CO 2 that can be geologically stored in the target territory.

Territories Source

Where transitions solutions are implemented: the cement plant.

Target

Where the solutions may induce changes: boundaries determined by market demand. The situation to which the scenarios are compared is defined. Only the scenarios whose impacts are lower than the reference situation are considered relevant.

In progress for the study case: the LCA model of the existing cement plant, coupled with the conventional production of the products likely to be substituted.

Assess the most relevant scenarios with the stakeholders and co-decide for the most preferable scenario for the study case.

Proceed with a sensitivity analysis to identify the key levers for the scenario to be efficiently implemented.

Iterate from step 2 if necessary.

CONCLUSION

Transition LCA is a recent LCA method which aims to study possible transition solutions for an existing situation to reduce its environmental impacts in the future. It should enable local stakeholders to identify the most interesting scenarios, then the conditions and action levers to efficiently implement the transition. Transition LCA results can be paired with economic or social indicators for in-depth decision-making.

2

  Model to collectively identifyand describe all the scenarios that would enable to achieve the transition according to the source territory. On the right, a simplified version of the scenarios flowchart for the study case (CCS and CCU in a cement plant of the Vicat group). But a large set of pathways and technologies for CCUS exists. How to identify the best scenarios for a specific cement plant? Study case: transition of a cement plant of the Vicat group, in France.The current global approach is to model each feasible technology as a unit product LCA, then create the scenarios' model by linking accordingly the unit systems and adapting it to the territorial and industrial flows.

Recommended LCA software: Define transition scenarios with the stakeholders Identify goal and scope 11

th International Conference on Industrial Ecology (ISIE), 02-05 July 2023 WHY CARBON CAPTURE,

UTILIZATION AND STORAGE (CCUS) IN A CEMENT PLANT ? APPLICATION OF TRANSITION LIFE CYCLE ASSESSMENT METHOD ON CARBON CAPTURE, UTILIZATION AND STORAGE IN A CEMENT PLANT Eva Quéheille 1 , Anne Ventura 2 , Lauredan Le Guen 2 , Eric Lefebvre 3 , Laury Barnes-Davin 3

  1 University Gustave Eiffel, AME/SPLOTT, Campus of Lyon, Bron, France 2 University Gustave Eiffel, MAST/GPEM, Campus of Nantes, Bouguenais, France

	Sources	While	Applying	Various	Energy	Scenarios."	ACS	Sustainable	Chemistry	&	Engineering	9	(30):	10123-41.
	https://doi.org/10.1021/acssuschemeng.1c02002.

3 Vicat Group, L'Isle-d'Abeau, France Perspective: Complete the application of Transition LCA on the study case and eventually improve the method with the learnings from this work. REFERENCES Gerloff, Niklas. 2021. "Comparative Life-Cycle Assessment Analysis of Power-to-Methane Plants Including Different Water Electrolysis Technologies and CO 2

Lehne, Johanna, and Felix Preston. 2018. Making

  Concrete Change. Innovation in Low-Carbon Cement and Concrete. Chatham House Report. The Royal Institute of International Affairs.

Mutel, Chris. 2017. "Brightway: An Open Source Framework for Life Cycle Assessment." The Journal of Open Source Software 2 (12): 236. https://doi.org/10.21105/joss.00236.

Sacchi, R., T. Terlouw, K. Siala, A. Dirnaichner, C. Bauer, B. Cox, C. Mutel, V. Daioglou, and

G. Luderer. 2022. "PRospective EnvironMental Impact AsSEment (Premise): A Streamlined Approach to Producing Databases for Prospective Life Cycle Assessment Using Integrated Assessment Models." Renewable and Sustainable Energy Reviews 160 (May): 112311. https://doi.org/10.1016/j.rser.2022.112311. Ventura, Anne. 2022. "Transition Life Cycle Assessment: A New Method to Face Ecological Transition." Frontiers in Sustainability 3 (August): 21. https://doi.org/10.3389/frsus.2022.801668.