Pablo Carpio

Rolando T Candidato

Lech Pawłowski

María Dolores Salvador

Solution concentration effect on mechanical injection and deposition of YSZ coatings using the solution precursor plasma spraying

Keywords: Solution precursor plasma spraying, Yttria-stabilized zirconia, Thermal barrier coating

YSZ coatings were developed by solution precursor plasma spraying using from different solution concentrations. The solutions were characterized and correlated with the rheological properties (specific weight, viscosity and surface tension). The mechanical injection was used therefore a correct injection is essential to get a correct liquid jet break-up inside the plasma plume. The optimal injection pressure must be adjusted for each solution since the solution characteristics affect on the injection process. It The ideal injection pressure was experimentally found out by the observation of the splats' morphology. The particles were not melted at low pressures but solidified prior to the deposition when the injection pressure is high. It was also found out that the optimal static pressure varies with the concentration, but the dynamic pressure is equal for every solutions and slightly superior to the dynamic pressure of the plasma plume. After injection optimization, coatings using different solution concentrations were developed and their microstructures were investigated. The liquid jet fragmentations as well as the heat requirements were modified with the concentration so that coating microstructure was affected. The resulting coatings display a lamellar and porous microstructure with the presence of unmelted/unpyrolysed material. Coatings' porosity increased while coatings' grow-up was less intensive when diluted solutions were used as feedstock.

Introduction

Finely-structured materials have been extensively studied in the last years because they exhibit improved features comparing with their conventional counterparts. Atmospheric plasma spraying (APS) is a suitable technique to deposit thick ceramic coatings but one of its limitations is the deposition of fine particles. Coatings from fine particles improve the characteristics respect to their conventional counterparts but particles smaller than 20 µm cannot injected because of their low density and poor flowability [START_REF] Fauchais | Engineering a new class of thermal spray nano-based microstructures from agglomerated nanostructured particles, suspensions and solutions: an invited review[END_REF]. In order to solve this problem, some modifications has been realized and some new techniques such as suspension plasma spraying (SPS) and solution precursor plasma spraying (SPPS) were developed [START_REF] Fauchais | Suspension and solution plasma spraying[END_REF][START_REF] Pawlowski | Suspension and solution thermal spray coatings[END_REF][START_REF] Killinger | Review of new developments in suspension and solution precursor thermal spray processes[END_REF].

The main characteristic of these techniques consists of using the liquid feedstock injection (suspension or solution precursor) instead of a powder. Two possible mechanisms have been developed to inject liquids: spray atomization and mechanical injection. The atomization needs an external energy (centrifugal force, external pressure kinetic energy of another fluid, etc.) to break up the liquid jet, and the atomized liquid is injected inside the plasma plume where a secondary break up occurs due to the high drag forces between the droplet and the plasma. In a mechanical injection, the liquid feedstock is forced to drive through a nozzle by a pressurized system. In this case, the secondary break up which occurs inside the plasma plume is crucial in the formation of the droplets. Immediately after the fragmentation, a series of phenomena as the solvent evaporation, the pyrolysis in the case of SPPS deposition, and the solid melting occur inside the plasma jet. The droplet size, energy requirements and the heat transfer control the described phenomena.

SPPS deposition has been studied to develop different kinds of ceramic coatings application such as solid oxide fuel cells [START_REF] Wang | Solution precursor plasma spray of nickel-yittia stabilized zirconia anodes for solid oxide fuel cell application[END_REF], bioactive coatings [START_REF] Candidato | Preliminary study of hydroxyapatite coatings synthesis using solution precursor plasma spraying[END_REF][START_REF] Xiao | Bioactive glass-ceramic coatings synthesized by the liquid precursor plasma spraying process[END_REF], sensors [START_REF] Zhang | Microstructure and gas sensing properties of solution precursor plasma-sprayed zinc oxide coatings[END_REF][START_REF] Chien | Rapid and continuous deposition of porous nanocrystalline SnO2coating with interpenetrating pores for gas sensor applications[END_REF] and etc. It exhibits some advantages with respect to SPS, i.e. molecular level mixing of constituent chemicals, quick chemical formulation, or the control of amount of unpyrolized material [START_REF] Gell | Thermal barrier coatings made by the solution precursor plasma spray process[END_REF]. One of the most studied coating material has been the yttria-stabilized zirconia (YSZ) for thermal barrier applications because of its low thermal conductivity and good stability at high temperature [START_REF] Gell | Thermal barrier coatings made by the solution precursor plasma spray process[END_REF][START_REF] Gell | Highly durable thermal barrier coatings made by the solution precursor plasma spray process[END_REF][START_REF] Ganvir | Characterization of Thermal Barrier Coatings Produced by Various Thermal Spray Techniques Using Solid Powder, Suspension, and Solution Precursor Feedstock Material[END_REF]. Some aspects of the SPPS deposition have been treated in several works. Concretely, the effect of spraying parameters in the coating microstructure, and consequently the coating properties, has been extensively studied [START_REF] Xie | Processing parameter effects on solution precursor plasma spray process spray patterns[END_REF][START_REF] Wang | Optimization of solution precursor plasma spray process by statistical design of experiment[END_REF][START_REF] Fauchais | Parameters controlling liquid plasma spraying: Solutions, sols, or suspensions[END_REF]. Besides the influence of the feedstock (precursor chemistry, solvent type, concentration…) has been also studied since the liquid jet fragmentation, energy requirements or the heat transfer between feedstock and plasma plume are modified [START_REF] Chen | The solution precursor plasma spray coatings: Influence of solvent type[END_REF][START_REF] Chen | Effect of solution concentration on splat formation and coating microstructure using the solution precursor plasma spray process[END_REF][START_REF] Duarte | La2Zr2O7(LZ) Coatings by Liquid Feedstock Plasma Spraying: The Role of Precursors[END_REF]. However, the variation of the injection process in function of the change of feedstock properties has not been considered in the case of the mechanical injection. An appropriate correct solution injection is crucial element of deposition process since the liquid jet must enter inside the plume core. The liquid injection depends on the specific weight and speed of the plasma gases and the liquid jet. Hence the injection speed must be adjusted for each solution since its density and other properties changes as discussed by various authors such as e.g. [START_REF] Fauchais | Suspension and solution plasma spraying[END_REF][START_REF] Pawlowski | Suspension and solution thermal spray coatings[END_REF].

The liquid feedstock can be injected inside the plasma plume by two different modes: mechanical and atomization. A primary break-up of the liquid jet occurs outside the plasma plume in the case of atomization mode, and then the droplets are injected inside the plume and a secondary break-up can occur because of the high kinetic energy of the plasma. Although Chen et al. [START_REF] Chen | The solution precursor plasma spray coatings: Influence of solvent type[END_REF] addressed the effect of the concentration on the splat morphology and on the coating microstructure but the atomization was used as injection mechanism and the feedstock flow rate and the fragmentation jet was controlled by the atomizing gas. In the case of the mechanical injection, the flow rate and the jet diameter was adjusted by the static pressure and the nozzle diameter respectively. Besides, a primary fragmentation did not occur and consequently, the secondary fragmentation, which depends on the injection, became primary.

Hence, a correct injection displays a higher importance in the case of mechanical injection.

A study was performed in the present work to inject correctly the precursor solutions with different concentrations by means of mechanical injection system. It was realized by adjusting the feedstock flow rate. YSZ coatings obtained from solutions at different concentrations were compared and their microstructures were discussed considering the processes which occur inside the plasma jet.

Experimentation

Preparation of the precursors and its characterization

The precursors used as feedstocks were zirconium acetate (Zr[C 2 H 3 O 2 ] 4 ) and yttrium nitrate (Zr[NO 3 ] 4 ) Y[NO 3 ] 4 , both supplied by Sigma Aldrich, USA. They were mixed using a magnetic stirer for 1 hour in a stoichiometric ratio corresponding to 4 wt.% yttria-stabilized zirconia. The precursors were diluted at different ratios to study the effect of the feedstock concentration in the spraying process. The dilutions were performed with a mixture of ethanol and water 50/50 wt. %. The previous research work reported that this mixture is optimal [START_REF] Sokołowski | Characterization of microstructure and thermal properties of YCSZ coatings obtained by suspension plasma spraying[END_REF].

The solution precursor plasma spraying used in this work is a complex process, where the inflight phenomena experienced by solution droplets affect considerably the final microstructural characteristics of the coatings. Moreover, the phase transitions and possible formation of decomposition phases as well as other reaction by-products may occur and should be characterized and associated to the phase composition of the coatings. In order to observe the thermal behavior of the precursors vs. temperature, they were tested by thermogravimetric analysis and differential scanning analysis (TG-DTA) with the use of STA 449 F3 Jupiter (Netzsch, Germany). The dried precursor was placed in an alumina crucible and was heated from 25 ºC to 900 ºC with a heating rate of 10 ºC/min under argon atmosphere. The crystalline phases of the precursors treated at different temperatures were also found by X-ray diffraction (D8 Advance, Brucker, USA) under Bragg-Brentano configuration with Cu K α1 =0.15406 nm radiation. A step increment of 0.02° (2θ) in the scanning range of 20°-80° (2θ) was used and the resulting diffractogram was analyzed using Diffrac + EVA Software. Prior to thermal characterization of the precursors, they were dried at 150 ºC in an electrical furnace, being the weight loss around 5%, and a amorphous solid is obtained after the drying. obtaining an amorphous solid. The thermal treatments at different temperatures were realized in an electric furnace with a 1h of dwell time and slow cooling.

The rheological behavior of the precursors with different concentrations was determined using a rotational rheometer (RheolabQC, Anton Paar, Austria) operating at controlled shear rate (CR) by loading the shear rate from 0 to 1000 s -1 in 300 s, maintaining at 1000 s -1 for 60 s and downloading from 1000 to 0 s -1 in 300 s. The measurements were carried out at 25 °C using a double-gap cylinder system since this geometry is optimal for the measurement of low viscous liquids [START_REF] Cappi | Rheological properties of aqueous Si 3N 4 and MoSi 2 suspensions tailor-made for direct inkjet printing[END_REF]. Surface tension was measured by tensiometer (DCAT11, DataPhysics Instruments, Germany) using a platinum Wilhelmy plate system.

Plasma spray process

Coatings were deposited using a single cathode plasma torch (SG-100, Praxair, USA) mounted on a 5-axis robot (IRB-6, ABB, Switzerland). The liquid feedstock was injected by mechanical system which consists of two pressurized vessels (one for the feedstock and the other one for the water), connected to one nozzle of 0.3 mm diameter located radially close to the plasma plume at 8 mm from the exit of the plasma plume and with an angle of 30º. The static pressure inside the vessels can be controlled and it results is crucial in the injection optimization as well as the control of the feedstock flow rate. The injection was external since corrosion problems in the torch were observed using the internal injection.

The splats morphology at different static pressures and solution molarities was evaluated. For this purpose, polished stainless steel (AISI 316L) substrates were used and one scan with a high torch scanning speed (1000 mm/s) was deposited to observe the splats. Then, thick coatings from different solution were developed using the optimal static pressure optimized thanks to the splats observation. Sand-blasted substrates were used for this purpose since a rough surface is necessary to improve the adherence. The number of spraying passes was adapted in each coating necessary to achieve a thickness around 100 µm. The spraying parameters were kept constants for every coating: electrical power of 40 kW, argon and hydrogen as plasma gases with flow rate of 45 slpm and 5 slpm respectively, spraying distance of 60 mm and a torch scan speed of 500 mm/s. The spraying parameters were selected from previous works [START_REF] Candidato | Development of hydroxyapatite coatings by solution precursor plasma spray process and their microstructural characterization[END_REF].

Coating characterization

Splats morphology and coating microstructure (topography and cross-section were observed by scanning electron microscopy (JEOL IT 300 LV, Jeol Cop, Japan) at low vacuum mode coupled with electron dispersion spectrometer (EDS). The samples were embeeded in an epoxy resin and polished up to 0.1 µm to examine the coating cross-section. Besides, the porosity and thickness was calculated by image analysis (Image-Pro Plus, Media Cybernetics, USA) from 10 SEM micrographs at 2000x and 500x magnification respectively. Then, the crystalline phases of the coatings were detected by XRD. Finally, thermal diffusivity of these coatings at different temperatures (up to 1000 ºC) were measured by laser flash method equipment (LFA 427, Neztsch, Germany) using a double layer model including in the model the heat loss and the contact resistance considerations. Then, thermal conductivity was calculated from determined thermal diffusivity and porosity values as well as the specific heat values addressed of the YSZ addressed in previous studies [START_REF] Sokołowski | Characterization of microstructure and thermal properties of YCSZ coatings obtained by suspension plasma spraying[END_REF].

Results

Precursor characterization

Thermal behavior examination of the precursors at equilibrium conditions was conducted to understand the processes which might take place during spraying. Fig. 1 shows the thermogram of the dried precursor in which 60% of weight loss was observed over the entire temperature cycle. Two small endothermic points at 125 and 225 ºC were observed and attributed to the physically adsorbed water evaporation which corresponds to a loss weight of 16%. However, the main weight loss is attributed to the chemical reaction decomposition to volatile gases, formation amorphous ZrO 2 and Y 2 O 3 as reflected in the exothermic peak at 350ºC. The most important exothermic peak is reflected at 500ºC which corresponds to the formation of the former ZrO 2 crystals while boarder exothermic peak at 820ºC corresponds to the crystalline grown of tetragonal ZrO 2 . These findings agree with the literature data [START_REF] Govindarajan | In situ particle generation and splat formation during solution precursor plasma spraying of yttria-stabilized zirconia coatings[END_REF]. It is important to remark that the YSZ formation is an exothermic process therefore, it means an energetic contribution inside the plasma plume which is beneficial especially in conventional torches where the power is limited.

The crystalline phases formed at different temperatures were confirmed by XRD, as shown in Fig. 2. The dried precursor and precursor treated at low temperatures was amorphous since no crystalline lattice was formed. Some peaks were distinguished at 450 º and 700 ºC of treatment and attributed to YSZ pattern but they are small and broad. This mean that small crystalline nucleus could have been formed but the crystalline lattice is formed at higher temperatures at which the YSZ pattern is clearly observed. Precursor treated at 850ºC and 1000ºC was tetragonal YSZ (pdf Card No.50-1089) being more crystalline at 1000ºC. The rheological behavior of the precursors diluted at different ratios can be observed in the flow curves (Fig. 3). The relationship between shear rate and shear stress was linear with a correlation R 2 >0.99. It means that the solutions exhibited a Newtonian behavior with constant viscosity. This result is important in the injection of liquid feedstocks since shear rate in the injector is huge so the viscosity should be low even at high shear rates [START_REF] Carpio | Alumina-zirconia coatings obtained by suspension plasma spraying from highly concentrated aqueous suspensions[END_REF]. The variation of the surface tension and the apparent viscosity at different solution concentration, which affects in the injection and the coating microstructure, is plotted in Fig. 4. It is observable that both surface tension and viscosity are increased when the solution concentration was higher.

Splats morphology

Single torch scans at different static injection pressures were performed with the aim of finding the optimal injection. For this purpose, the splats morphology for each scan was observed by SEM. It was concluded that the splats morphology depends on the injection pressure rather than the solution concentration. Fig. 5 displays huge difference in the deposits at several static pressures in the case of 4 mol/L solution. At low pressures (60 kPa for 4 mol/l solution), no splats were found and only bulk fragmented solids were observed. These fragmented solids could have corresponded to the big solution droplets whose solvent was evaporated during the spraying but the solid was unpyrolized and unmelted. Rounded and flattened particles with the typical spat morphology appeared in the center of the sprayed zone when the static pressure is increased (80 kPa for 4 mol/l solution), although no melted material remains in lower proportion in the edge of the sprayed zone. In the case of very high pressures (higher than 1.0 bar/l solution), rounded and no flatten particles can be observed which seems to be badly adhered. These particles consist pyrolyzed and melted material which solidified before impacting onto the substrate [START_REF] Fauchais | Suspension and solution plasma spraying[END_REF][START_REF] Chen | Effect of solution concentration on splat formation and coating microstructure using the solution precursor plasma spray process[END_REF].

Unmelted material flattened and rounded particles can be observed when different solution concentrations were sprayed at different static pressures. However, the different kind of deposits were observed at different pressures. The pressure necessary to observe the splats changes so that typical splats and resolidified particles were appreciated at lower pressures when the solution concentration was reduced. The static pressure for which the splats were observed without unmelted material was selected as the optimal one [START_REF] Carpio | Effect of particle size distribution of suspension feedstock on the microstructure and mechanical properties of suspension plasma spraying YSZ coatings[END_REF]. Figure 6 display single scan images of solutions with different n molarity using the optimal pressure. Splats with similar morphology and different sizes (between 2 and 40 µm) were observed and average splat size seems to be reduced at lower concentrations. As the splat size was lower than that obtained at conventional APS coatings and slightly lower than SPS coatings, it can be deduced that the microstructure of these SPPS coatings is finer [START_REF] Fauchais | Engineering a new class of thermal spray nano-based microstructures from agglomerated nanostructured particles, suspensions and solutions: an invited review[END_REF].

Coating microstructure

The cross-sections micrographs of the coatings (Fig. 7) reveals a bimodal microstructure observed in suspension plasma sprayed coatings [START_REF] Carpio | Effect of particle size distribution of suspension feedstock on the microstructure and mechanical properties of suspension plasma spraying YSZ coatings[END_REF]. It consists of unmelted or unpyrolized material surrounded by a melted matrix having a lamellar microstructure. The porosity of these coatings is high because the unmelted zones retains a high amount of micropores (diameter < 1 µm) although some macropores (diameter > 1 µm) can be found inside the melted matrix [START_REF] Carpio | Effect of particle size distribution of suspension feedstock on the microstructure and mechanical properties of suspension plasma spraying YSZ coatings[END_REF]. Transversal cracks were clearly observed just in 4 mol/L coating and some of them in 2 mol/L coating. These cracks are caused by the thermal stresses created during the deposition. Besides, these cracks can improve the thermal fatigue resistance by stress discharge [START_REF] Ganvir | Characterization of Thermal Barrier Coatings Produced by Various Thermal Spray Techniques Using Solid Powder, Suspension, and Solution Precursor Feedstock Material[END_REF].

The porosity and thickness of these coatings are shown in Table 1. The porosity values are high but the coatings are even more porous when the starting solution was diluted. The thickness growth by pass was smaller for low concentration solution. The weight of material sprayed was directly proportional to the solution concentration. It can be deduced that that a higher amount of material gets lost when diluted solutions are sprayed.

Figure 8 shows different coatings' topography. Coatings from concentrated solutions exhibited an almost planar surface with some small clusters. However, bigger clusters appeared at lower molarities, with a cauliflower-type microstructure typical in columnar coatings [START_REF] Sokołowski | The key process parameters influencing formation of columnar microstructure in suspension plasma sprayed zirconia coatings[END_REF]. Although the columnar structure is not really present in these coatings (some columns were visible in coating sprayed using 0.5 mol/L solution), the formation of columnar microstructure by reduction of the solution concentration confirms the results of previous research [START_REF] Sokołowski | The key process parameters influencing formation of columnar microstructure in suspension plasma sprayed zirconia coatings[END_REF]. Micrographs at higher magnification revealed that coatings are formed by particles smaller than 1 µm (which correspond to the splats) but big agglomerates are also present (which correspond to resolidified material).

XRD patterns of coatings obtained from different solutions are shown in Fig. 9. The patterns are quite similar and just tetragonal phase is observed. Besides, the chemical composition of the coatings, as well as of the splats, is homogeny since it was corroborated by EDS. Thus, the crystalline phases and chemical composition are independent on solution concentration.

Discussion

Solution injection

Before spraying, the injection is optimized by selection of different static pressures (p 0 , kPa. However, the dynamic

= • (1)
Where ρ is the solution density (kg/m 3 ) and v is the feedstock injection speed (m/s). The speed was calculated from the feed rate which was experimentally determined:

= = ( 2
)
Where Q is the volumetric flow rate (m 3 /s), M is the feed rate (kg/s), A is the injector section (m 2 ) and D is the injector diameter (m). Table 2 shows the flow rate, speed and dynamic pressure of a solution injected at different static pressures. It is clearly seen that these parameters change with static pressure.

According the literature, the liquid jet can penetrate plasma plume when the dynamic pressure of the liquid jet ( , ) is equal or higher than the dynamic pressure of the plasma plume [START_REF] Fauchais | Engineering a new class of thermal spray nano-based microstructures from agglomerated nanostructured particles, suspensions and solutions: an invited review[END_REF][START_REF] Fauchais | Suspension and solution plasma spraying[END_REF].

, ≥ , (3)
Maps of the plasma gas speed and specific weight, and consequently the dynamic pressure, could be calculated using the software T&TWinner [START_REF] Pateyron | Modeling of phenomena occurring in plasma jet during suspension spraying of hydroxyapatite coatings[END_REF]. It was found that the maximum dynamic pressure of the plume was 11 kPa. As seen in Table 3, the dynamic pressure of the solution was found to be ~15 kPa in all the cases, which equates to a slightly superior pressure than the dynamic pressure of the plasma, which was calculated to be 11 kPa as stated before.

These findings agree with the literature which addressed that dynamic pressure of the feedstock must be equal or higher that dynamic pressure of the plume [START_REF] Fauchais | Suspension and solution plasma spraying[END_REF]. By comparing the dynamic pressure of plasma gases and of solution liquid jet (Table 3), the optimal injection was found to occur when dynamic pressure of the liquid jet is slightly superior to the dynamic pressure of the plasma gases. Consequently, the liquid jet cannot penetrate inside the plume at lower pressures. On the other hand, it may happen that for too high liquid jet pressure, the jet goes radially through the plume instead of holding inside the center of plume. This may correspond to the situation of solid which gets pyrolysed and melted inside the plume but resolidifying before impacting onto the substrate. It agrees with the results observed in the Fig. 5. An illustrative diagram of the injection at different pressures is shown in Fig. 10.

The optimal injection pressure is different for each solution, as it was demonstrated in section 3.4. Since the specific weight changes therefore the liquid jet speed must be adjusted. Hence, the feedstock flow rate is modified changing the injection pressure and it must be taken into account in the heat transfer between plasma plume and the feedstock.

Droplet fragmentation

Coating microstructure depends on the phenomena which occur inside the plasma plume, such us: (i) jet fragmentation; (ii), evaporation solvent; (iii), pyrolysis; and, (iv) solid melting.

Thus, it is important to understand the effect of the solution concentration (molarity) in each one of these phenomena. The liquid jet fragmentation is hard to estimate. Nevertheless, the literature correlates the fragmentation with two dimensionless numbers: Weber (We) and Ohnersorge (Oh):

= ! "# ! $ ! = % &' % (' ) * + ) , ) (4)
-ℎ = / $! #$ ! "# ! $ ! = 0 ) 1 ) + ) , ) (5)
Where is the density (kg/m 3 ), v is the speed (m/s), 2 is the surface tension (N/m), 3 is the viscosity (Pa•s) and 4 is the jet diameter (m) considering it is 1.5 times higher nozzle diameter [START_REF] Fauchais | Suspension and solution plasma spraying[END_REF]. Index g and l refers the plasma plume and liquid feedstock respectively. It can be seen in Table 3 that Oh number is almost constant but the Weber number depends on the concentration via surface tension. A stripping break-up occurs at We numbers between 100 -350, but the fragmentation increases with the We values. It means that smaller droplets are formed at lower concentration, therefore, the deposited splats had a reduced size, as can be seen in Fig. 10 6.

Heat transfer

The plasma must provide the heat necessary to evaporate the solvent and to melt the solid in the case of suspensions but the energy delivery or absorbed in the pyrolysis must be also considered. These requirements depend on the feedstock and on the degree of unmelted or unpyrolysed particles in the coatings. As the feedstocks were composed of the precursors such as zirconium acetate and yttrium nitrate delivered heat during its pyrolysis (see Fig. 1). On the other hand, the feedstock contains solvent which must be evaporated prior to the pyrolysis.

The specific enthalpy of solvent evaporations (1.78 MJ/kg in the present case) is one order higher than the solid melting. Consequently, the heat needed for evaporation is the greatest one (enthalpy of the solvents is as high as1.78 MJ/kg). Thus, diluted solutions need more heat because of the higher content of solvent. It can be recalled pointed that 4 mol/L solution is almost pure precursor while 0.5 mol/L solution just contain only 12 mass% of it. ) and needs less heat for pyrolysis.

Other factor which affects in the required thermal energy is the feedstock flow rate which was smaller at lower concentrations. It reduced the thermal energy needed by 20%, which is relatively small comparing to of the energy needed by solvent to evaporate. The higher thermal energy needed by solution results in the presence of unmelted/unpyrolysed material, as can be observed in Fig. 7 and Table1

Also, heat transfer between the liquid feedstock and the plasma plume must be considered.

The transfer depends on the feedstock and on plasma jet properties, as well as, on the droplet size. The heat arrives rapidly to the center of small droplet. At big droplets a solid shell can be formed on the external part since a evaporation occurs before the heat necessary to pyrolyze the center of the droplet. The presence of shells in the droplets of concentrated solutions can be correlated with the presence of unpyrolysed material in some coatings [START_REF] Pawlowski | Suspension and solution thermal spray coatings[END_REF].

Finally, an effect phenomenon related to the liquid jet fragmentation is the Knudsen effect.

The effect results in small droplets moving parallel to the substrate. It may lead to the formation of columns and higher roughness of coatings sprayed using diluted solutions [START_REF] Fauchais | Suspension and solution plasma spraying[END_REF],

which is demonstrated in the topographic microstructures of Figure 6.

Conclusion

YSZ coatings were successfully developed by SPPS process. Firstly, the feedstock characterizations were performed and it was corroborated that rheological properties depends on solution concentration (viscosity and surface tension increase at higher concentrations).

Then, the optimal injection pressure was selected from the splat observation. It was important because the feedstock cannot be well-injected at low pressures and mostly unmelted material is deposited. On the other hand, resolidified material (spherical particles instead of flatten drops) can be found at high pressures since jet cross the plasma plume. Thus, the static pressure must be adjusted for each solution since the optimal static pressure depends on the solution characteristics. Finally, solutions of different concentrations were applied to obtain coatings characterized microstructurally. The coatings sprayed using diluted solution exhibited higher porosity and greater content of unmelted zones/unpyrolized because of the higher energetic requirement to evaporate the solvent. However, some coatings sprayed using concentrated solution also displayed high porosity since the droplet fragmentation is lower (higher surface tension therefore lower Weber number). Big droplets mean that a shell is formed in the external part of the droplets during the pyrolysis and it contribute in the porosity retention. This resulted from the formation of large droplets because of high viscosity and low surface tension of such solution. Finally, It can be concluded that the injection must be adjusted to spray solutions with different concentration in order to obtain desired coatings' microstructure. Besides, it is corroborated that coating microstructure depends on feedstock characteristics as well as the injection process.

Acknowledgment

  This work has been supported by the Spanish Ministry of Economy and Competitiveness (project MAT2015-67586-C3-R). P. Carpio acknowledges the Valencia Government (APOSTD/2016/040) and the Spanish Ministry of Science, Innovation and Universities for his current post-doc contract (FJCI-2016-27822) for his post-doc contract.

Figure 1 .

 1 Figure captions

Figure 2 .

 2 Figure 2. XRD pattern of the precursor used for the synthesis of 4 mol% YSZ by SPPS

Figure 3 .

 3 Figure 3. Flow curves (shear stress vs shear rate) of precursors diluted at different ratios.

Figure 4 .

 4 Figure 4. Apparent viscosity (stripped line) and surface tension (continuous line) of the

Figure 5 .

 5 Figure 5. Single scan depositions of 4 mol/L solution sprayed at different static pressures: a)

Figure 6 .

 6 Figure 6. Single scan depositions at optimal static pressure (marked between parenthesis) of

Figure 7 .

 7 Figure 7. Cross-section micrographs of coatings obtained at different solution concentration:

Figure 8 .

 8 Figure 8. Surface micrographs of coatings obtained at different solution concentration: left

Figure 9 .

 9 Figure 9. XRD pattern of the coatings obtained from solutions with different concentrations.

Figure 10 .

 10 Figure 10. Sketch of the injection and the liquid jet trajectory at different static pressures.

Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9

 3456789 Figure 3

Figure 10

 10 Figure 10

Table list Table 1 .

 list1 Porosity, thickness and thickness by pass of coatings sprayed at different solution concentrations.

	Concentration	Porosity	Thickness	Thickness/pass
	(mol/L)	(%)	(µm)	(µm)
	4	25.1±4	114±5	14.2
	2	26.2±4	106±4	6.6
	1	35.6±5	96±4	3.0
	0.5	34.4±7	32±5	1.6

Table 2 .

 2 Flow rate (M: in mass, Q: in volume), speed (v) and dynamic pressure (p k ) of 4 mol/L solution injected at different static pressures (p 0 ).

		M
	p 0 (kPa)
		(g/min)
	60	20.1	14.1	3.33	7.9
	80	28.0	19.6	4.63	15.3
	100	34.1	23.9	5.64	22.7

Q (ml/min) v (m/s) P k (kPa)

Table 3 .

 3 Flow rate (M: in mass, Q: in volume), speed (v), dynamic pressure (p k ), Weber number (We) and Ohnesorge number (Oh) of different solutions (different molarity and specific weight) injected at optimal static pressure (p 0 )..


Yanjun Wang

Mariasole Cipolletta

Lamiae Vernières-Hassimi

Valeria Casson-Moreno

Sébastien Leveneur
email: sebastien.leveneur@insa-rouen.fr

Application of the concept of Linear Free Energy Relationships to the Hydrogenation of Levulinic acid and its corresponding esters

Keywords: hydrogenation, levulinic acid and esters, γ-valerolactone, kinetic modeling, Taft equation, Linear Free Energy Relationships

Biomass valorization to chemicals, biofuels or materials will be more and more important during this century. Production of γ-valerolactone (GVL) from the hydrogenation of levulinic acid is a good illustration of this tendency. GVL can also be produced from alkyl levulinates hydrogenation.

Can we find a relationship between the structure and the kinetics of this reaction? Can we predict the kinetics of any alkyl levulinates by knowing the kinetics of another alkyl levulinate? This paper has studied these two questions by developing a kinetic model including the effect of gas-liquid mass transfer. We have demonstrated that the kinetics of hydrogenation of levulinic acid, methyl, ethyl and butyl levulinates to GVL using Ru/C follow the Taft equation, which is derived from Linear Free Energy Relationships. This equation measures the effects of polar and steric on a reaction series. We have demonstrated that polar effect of the reaction series is the most significant effect. This relationship can predict the values of kinetic constants just by knowing their structure.

INTRODUCTION

The use of biomass as feedstock for chemical industries is seen as a credible alternative to substitute fossil-derived feedstock. A well-known example is the use of vegetable oils for the production of biodiesel or the use of sugar cane and corn for the production of bioethanol. Such feedstock are also involved in food-production supply chains, leading to possible ethical conflicts related to the increase of the price of biomass, as happened in Mexico in 2007 during the so-called "tortilla crisis", when the grain-price increased up to 400% in one year [START_REF]The Mexican tortilla crisis of 2007: the impacts of grain-price increases on food-production chains Alder Keleman & Hugo García[END_REF]. To overcome the dilemma of food versus fuel, the academic and industrial research have put their efforts to valorize biomass of 2 nd generation as, for example, lignocellulosic biomass.

The 2 nd generation biomass feedstock is more complicated to valorize because a pretreatment is needed to fractionate/separate biomass components, i.e. lignin, cellulose and hemicellulose.

Different industrial processes exist like the production of cellulosic bioethanol or the Biofine process for the production of levulinic acid [START_REF] Moreno | A consequences-based approach for the selection of relevant accident scenarios in emerging technologies[END_REF]. According to the US department of energy, levulinic acid (LA) is considered as a top twelve building-block molecules from biomass. In addition, levulinic acid can be upgraded to γ-valerolactone (GVL) which is considered as an important platform molecule [START_REF] Yan | Conversion of levulinic acid and alkyl levulinates into biofuels and high-value chemicals[END_REF][START_REF] Huang | Towards Production of γvalerolactone via Hydrogenation of Aqueous Levulinic Acid[END_REF].

As discussed, in the previous articles of our group [START_REF] Yan | Conversion of levulinic acid and alkyl levulinates into biofuels and high-value chemicals[END_REF][START_REF] Huang | Towards Production of γvalerolactone via Hydrogenation of Aqueous Levulinic Acid[END_REF], biomass structure is complex and diverse, but it is possible to find a relationship between their structure and reactivity.

During the pretreatment of lignocellulosic biomass (i.e. acid catalysis), different solvents could be used such as water, methanol, ethanol or butanol leading to molecules with different substituents such as levulinic acid, methyl levulinate (ML), ethyl levulinate (EL) or butyl levulinate (BL) [START_REF] Geilen | Selective and Flexible Transformation of Biomass-Derived Platform Chemicals by a Multifunctional Catalytic System[END_REF][START_REF] Tukacs | Efficient catalytic hydrogenation of levulinic acid: a key step in biomass conversion[END_REF][START_REF] Tukacs | An improved catalytic system for the reduction of levulinic acid to γ-valerolactone[END_REF][START_REF] Tukacs | Direct asymmetric reduction of levulinic acid to gamma-valerolactone: synthesis of a chiral platform molecule[END_REF][START_REF] Li | Highly efficient hydrogenation of biomass-derived levulinic acid to γ-valerolactone catalyzed by iridium pincer complexes[END_REF][START_REF] Deng | Conversion of Carbohydrate Biomass to γ-Valerolactone by using Water-Soluble and Reusable Iridium Complexes in Acidic Aqueous Media[END_REF][START_REF] Ding | Production of methyl levulinate from cellulose: selectivity and mechanism study[END_REF]. Some articles have studied the effect of the substituent alkyl on the production of GVL by using different hydrogen donors such as molecular hydrogen [START_REF] Heeres | Combined dehydration/(transfer)-hydrogenation of C6-sugars (D-glucose and D-fructose) to γ-valerolactone using ruthenium catalysts[END_REF] or other chemical species (e.g. formic acid and alcohols [START_REF] Hu | Levulinic esters from the acid-catalysed reactions of sugars and alcohols as part of a bio-refinery[END_REF][START_REF] Kang | From lignocellulosic biomass to levulinic acid: A review on acidcatalyzed hydrolysis[END_REF]. Nevertheless, there is no study in literature trying to establish a correlation between the structure of the alkyl groups and the reactivity of the system.

The production of GVL by hydrogenation of LA or alkyl levulinate has been widely studied by different research groups, some being focused on process intensification [15][START_REF] Al-Shaal | Microwave-assisted reduction of levulinic acid with alcohols producing γ-valerolactone in the presence of a Ru/C catalyst[END_REF], some on catalytic aspects [START_REF] Tang | Conversion of biomass to γ-valerolactone by catalytic transfer hydrogenation of ethyl levulinate over metal hydroxides[END_REF][START_REF] Wojciechowska | Enhanced Production of γ-Valerolactone with an Internal Source of Hydrogen on Ca-Modified TiO2 Supported Ru Catalysts[END_REF] and some on process safety issues [START_REF] Wang | Thermal Risk Assessment of Levulinic Acid Hydrogenation to γ-Valerolactone[END_REF]. The most common catalyst is Ru/C and one should mention the work of Piskun et al. and Negahdar el al. [START_REF] Piskun | Kinetic modeling of levulinic acid hydrogenation to γ-valerolactone in water using a carbon supported Ru catalyst[END_REF][START_REF] Negahdar | Kinetic analysis of the catalytic hydrogenation of alkyl levulinates to γ-valerolactone[END_REF] who have developed different kinetic models for the production of GVL by molecular hydrogen.

The corrosive behavior of levulinic acid can be a limit to the industrial-scale implementation of the upgrading process to GVL. For this reason, the study of alkyl levulinate hydrogenation is becoming more and more interesting.

In this work, hydrogenation of LA, EL, ML and BL over Ru/C was studied. GVL was used as a solvent to solubilize all these substrates, whereas water cannot do and alcohol can lead to the side reaction of esterification. Furthermore, GVL as solvent has shown significant advantage in biomass valorization [START_REF] Bai | Catalytic Conversion of Carbohydrates into 5-Ethoxymethylfurfural by a Magnetic Solid Acid Using γ-Valerolactone as a Co-Solvent[END_REF][START_REF] Mellmer | Effects of γvalerolactone in hydrolysis of lignocellulosic biomass to monosaccharides[END_REF]. In addition, using GVL as a solvent could simplify the layout of the plant, minimizing the downstream processing.

Herein, the main objective is to link the kinetics of these reactions to the structure of the reactant.

As shown in Fig. 1, only the alkyl group R varies between these four compounds. For that, the kinetic constants were tested towards Taft equation, which is derived from Linear Free Energy Relationships.

To verify if hydrogenation of LA and the corresponding levulinates follow the Taft equation, a kinetic model was developed by taking into account gas-liquid mass transfer and the variation of physicochemical properties of the solvent with temperature.

To the best of our knowledge, there are no such studies in the literature. Usually, Taft equation is tested in the presence of only one reaction center and by using apparent kinetic constants [START_REF] Vojtko | A Method for Esterification Reaction Rate Prediction of Aliphatic Monocarboxylic Acids with Primary Alcohols in 1,4-Dioxane Based on Two Parametrical Taft Equation[END_REF]. Fig. 1. Structure of LA, ML, EL and BL.

MATERIALS AND METHODS

Chemicals

Levulinic acid (wt% ≥ 97%), γ-valerolactone (wt% ≥ 99%) and methyl levulinate (wt% ≥ 98%) were purchased from Sigma-Aldrich. Ethyl levulinate (wt% ≥ 98%) and furfural (wt% ≥ 99%) were obtained from Acros Organics. Ru/C (5 wt % ruthenium on activated carbon powder, reduced and 50% water wet) and n-Butyl levulinate (wt% ≥ 98%) were provided by the Alfa Aesar. H2 (>99.999%) was supplied by Linde. Acetone (Analytical grade) was bought from VWR. All the chemicals were used without further treatment.

Gas-liquid mass transfer and kinetic experiment of hydrogenation

To determine Henry's constant of hydrogen gas in pure γ-valerolactone, a 300ml reactor equipped with efficient gas entrainment impeller, gas reservoir and recording system was used (Fig. 2).

Firstly, valve V1 was opened and a desired amount of hydrogen gas was purged into the reservoir from the gas storage bottle through the pressure regulator R1. Secondly, valve V1 was closed and GVL solvent was poured into the reactor and vacuumed to make sure there is no air in the reactor.

Thirdly, the reactor was heated to the desired temperature. The reactor temperature was kept constant. Fourthly, valve V2 was opened and the outlet pressure was set to 20 bar by adjusting the pressure regulator R2. Fifthly, the valve V3 was opened and the reactor was purged with hydrogen.

Then, the stirring was set at 1000 rpm and valves V2, V3 and regulator R2 were kept open until the end of the experiment to make sure the experiment was performed in isobaric conditions. The experiment lasted for 30 min (to reach the equilibrium), then all the valves were closed and the reactor was cooled down. The pressure and temperature of the reservoir and reactor were recorded online during the experiment. To evaluate the value of gas-liquid mass transfer coefficients, four experiments in pure GVL were carried out at 20 bars at four temperatures: 373.15K, 393.15K, 413. 15 K and 423.15 K. For the kinetic experiments of hydrogenation of levulinic acid and its esters, the same procedure above described was employed. The desired amount of γ-valerolactone, substrates and Ru/C catalyst were introduced into the reactor. During the reaction, the samples were obtained from valve Vs at different times and reserved for further treatment and analysis. Experimental matrix for the hydrogenation is shown in Table 1. Fig. 2. Reactor scheme.

Analytical methods

The evolution of density and viscosity with temperature for γ-valerolactone at atmospheric pressure were obtained by using DMA 4100 M and LOVIS 2000 ME microviscometer (Anton Paar, Austria). Temperature range from 283K-363K with 10K step was employed for the measurement with temperature accuracy of 0.02°C, density accuracy of 0.05 kg• m -3 ,viscosity accuracy of <0.5%.

To identify and quantify the chemical compounds from the hydrogenation kinetic experiments, at first, samples obtained were immediately filtered to separate the Ru/C catalyst in the solution.

Then, the colorless samples were diluted in acetone by using furfural as internal standard. Later the diluted solutions were prepared in vials for the further qualification and quantification analysis.

The identification of intermediate products from hydrogenation of levulinic acid and its esters, such as HPA, MHP, EHP, BHP, was performed by using GC-MS analysis. Gas chromatography Varian 3900 with Varian Saturn 2000 were applied with a capillary column (ZB-5ms, 30 m × 0.32 mm internal diameter × 0.25 μm film thickness). Helium (99.99%) was used as carrier gas at a constant flow rate of 1.0 mL•min -1 . The temperature of the injector and the detector was set at 270 °C. The oven temperature was programmed as 35 °C (3 min)-15 °C• min -1 -300°C. The injection volume was 5 μL, and the split ratio was 30:1.

The concentration of levulinic acid and its esters, γ-valerolactone and intermediate products was obtained from GC analysis. Bruker Scion GC436 gas chromatography (GC) equipped with FID detector (flame ionization detector), an autosampler and capillary column (Rxi-5ms, 30 m × 0.32 mm internal diameter × 0.25 μm film thickness) were used. Helium (99.99%) was used as carrier gas at a constant flow rate of 1.2 mL•min -1 . Other configurations of GC methods were the same with the GC-MS analysis. The standard deviation of the measured concentrations was found to be lower than 0.70 % showing the high repeatability of the analysis.

RESULTS AND DISCUSSION

Kinetics

As described in several articles [START_REF] Wang | Thermal Risk Assessment of Levulinic Acid Hydrogenation to γ-Valerolactone[END_REF][START_REF] Piskun | Kinetic modeling of levulinic acid hydrogenation to γ-valerolactone in water using a carbon supported Ru catalyst[END_REF][START_REF] Negahdar | Kinetic analysis of the catalytic hydrogenation of alkyl levulinates to γ-valerolactone[END_REF], hydrogenation of levulinic acid or its corresponding esters occurs in two reaction steps. The first step is the hydrogenation of the ketone group producing the The second reaction is often described as reversible for the case of hydrogenation in aqueous solvent [START_REF] Piskun | Kinetic modeling of levulinic acid hydrogenation to γ-valerolactone in water using a carbon supported Ru catalyst[END_REF][START_REF] Piskun | Hydrogenation of Levulinic Acid to γ-Valerolactone in Water Using Millimeter Sized Supported Ru Catalysts in a Packed Bed Reactor[END_REF]. Based on our experimental observation, the reaction was found to be irreversible when using GVL as a solvent. As mentioned in the work of Piskun et al. [START_REF] Piskun | Kinetic modeling of levulinic acid hydrogenation to γ-valerolactone in water using a carbon supported Ru catalyst[END_REF], the second reaction was assumed to occur in the liquid bulk phase.

The rate of hydrogenation of LA/ML/EL/BL to HPA/MHP/EHP/BHP on the catalyst can be described as

= * * * . (1)
where, . is the catalyst loading (kg.m -3 ).

The rate equation for the second reaction (ring-closure) can be expressed as

= * (2)
The first reaction can be described by a more complex reaction mechanism such as Langmuir-Hinshelwood. Nevertheless, the description of such mechanism needs the adsorption coefficients that are cumbersome to estimate. In this study, we have deemed that the use of equations ( 1) and

(2) can perfectly describe the kinetic rates.

The Taft equation is expressed as follows:

log " # $ %&' # ()*)()+,) %&' -= . * * 0 * + 2 * 3 + 4 (3)
where, the left side of the equation represents the ratio between the reaction rate of the substituted substrate with respect to the reference. On the right side of the Taft equation, there are three contributions. The first describes the polar effects of the substituent, multiplying the sensitivity factor of the reaction to polar effects (. * ) by the polar substituent constant ( 0 * ), which is a nearquantitative measure of the polar effect of the substituent. The second term describes the steric effects, multiplying the sensitivity factor of the reaction to steric effects (2) by the steric substituent constant (3 ), which is a near-quantitative measure of the steric effect of the substituent. The term 4 represents the resonance effect between the substituent and the reaction center. This term is equal to zero for the studied system because there is no resonance.

The Taft equation uses the substrate with the substituent methyl as the reference, that is the rate constant for the hydrogenation of methyl levulinate for the first reaction ( ,67 %8') and the rate constant of methyl 4-hydroxypentanoate ring closure for the second reaction ( ,69: %8'). The term (T) was added to highlight the fact that these rate constants depend on the reaction temperature.

By introducing Taft equation in equations ( 1) and ( 2), we obtain:

,;<=> . = ,67 %8' * 10

,LM . = ,69: %8' * 10 A N * %&' * C O+H. * IJ N %&' * K> O+H. *

where, Subst. and Int. are the suffix for substrate and intermediate, respectively.

The values of 0 * and 3 are available from literature [START_REF] Leveneur | Thermal Safety Assessment through the Concept of Structure-Reactivity: Application to Vegetable Oil Valorization[END_REF] for each substituent as summarized in From a previous study of our group [START_REF] Leveneur | Thermal Safety Assessment through the Concept of Structure-Reactivity: Application to Vegetable Oil Valorization[END_REF], we have noticed that Taft parameters . * %8' and 2%8' are temperature dependent. For that reason, the term (T) was added for these parameters. In this study, a linear relationship between . * %8' and 2%8' and the reaction temperature was assumed.

Therefore, one can describe them by the following equations:

. * %8' = P + Q * 8%R' (6)

2 %8' = S + T * 8%R' (7)
. * %8' = P + Q * 8%R' (8)

2 %8' = S + T * 8%R' (9)
During the modeling stage, the rate constants ,67 %8' and ,69: %8', and the parameters A1, B1, C1, D1, A2, B2, C2 and D2 were estimated.

Mass balance

Experiments were performed under isobaric and isothermal conditions.

Mass balance in the liquid phase

Mass balance for the different compounds present in the liquid phase can be expressed as:

U DEFGH(VH) U = - (10)
U 9 N X$Y U = 7 . * Z * - [ - (11)
U O+H)(\)]$VH) U = - (12)
U ^_Ù = ( 13
)
U abc U = (14)
where, * is the concentration of hydrogen at the gas-liquid interface, that was determined using Henry's constant %8' = 9 N X$Y * : `N,^)V,Hd( (mol.m -3 .bar -1 ), 7 . is the volumetric gas to liquid mass transfer coefficient for hydrogen (s -1 ). The detailed description of the mass transfer study is given in the following paragraph.

Mass transfer study

This is a gas-liquid-solid reaction system, thus, mass transfer plays an important role. A two-film theory was used to describe the mass transfer of hydrogen from the gas to the liquid phase [START_REF] Charpentier | Mass-transfer rates in gas-liquid absorbers and reactors[END_REF][START_REF] Frikha | Methodology of multiphase reaction kinetics and hydrodynamics identification: Application to catalyzed nitrobenzene hydrogenation[END_REF][START_REF] Gemo | Mass transfer and kinetics of H2O2 direct synthesis in a batch slurry reactor[END_REF][START_REF] Cai | Influence of gas-liquid mass transfer on kinetic modeling: Carbonation of epoxidized vegetable oils[END_REF].

The resistance from the gas side was neglected.

The effect of external mass transfer (from the bulk of the liquid phase to the surface of the solid catalyst) was experimentally verified for levulinic acid and butyl levulinate hydrogenation by varying the stirring rate. It was found that the external mass transfer resistance can be neglected above 1000 rpm (see Supplementary Material).

The effect of internal mass transfer (diffusion in the pores of the solid catalyst) was evaluated by using the Weisz-Prater criterion as used in Piskun et al. [START_REF] Piskun | Kinetic modeling of levulinic acid hydrogenation to γ-valerolactone in water using a carbon supported Ru catalyst[END_REF]. It was found that internal mass transfer can be assumed to be negligible because this criterion was found to be lower than 0.01.

In order to have an accurate description of gas to liquid mass transfer for hydrogen, an expression for the volumetric mass transfer coefficient for hydrogen kL.a taking into account the temperature, viscosity and density of the system was developed [START_REF] Cai | Influence of gas-liquid mass transfer on kinetic modeling: Carbonation of epoxidized vegetable oils[END_REF][START_REF] Cai | Structure-Reactivity: Comparison between the Carbonation of Epoxidized Vegetable Oils and the Corresponding Epoxidized Fatty Acid Methyl Ester[END_REF]. Due to the low concentration of the different substrates (< 20 % by weight), the evaluation was done considering pure GVL.

Kawase and Moo-Yong [START_REF] Kawase | Volumetric Mass Transfer Coefficients in Aerated Stirred Tank Reactors with Newtonian and Non-Newtonian Media[END_REF] have demonstrated that the mass transfer coefficient in an aerated tank reactor can be expressed as:

7 = √f * g T 9 N/c$Y * i j. k * " A c$Y l c$Y - j. k (15)
where T 9 N/c$Y is the coefficient for the diffusion of hydrogen in the liquid phase (m 2 .s -1 ), 7 is the gas-liquid mass transfer coefficient of hydrogen from liquid side (m.s -1 ), iis the energy dissipation rate per unit mass (W.kg -1 ), . 7 is the density of the liquid (kg.m -3 ) and m 7 is the dynamic viscosity (Pa.s).

The diffusion coefficient T 9 N/c$Y can be expressed by the correlation of Wilke-Chang [START_REF] Wilke | Correlation of diffusion coefficients in dilute solutions[END_REF]:

T 9 N/c$Y = n.o * j pq * Zr * 6 c$Y [ B N * & c$Y l c$Y * s `N t.u (16)
where, ϕ is the association factor (-), MLiq is the molar mass of the solvent (g.mol -1 ), 8 7 is the temperature of the liquid phase (K), m 7 is the viscosity (cP) and w 9 N is the normal molar volume of hydrogen equal to 14.3 (cm 3 .mol -1 ).

By combining equations ( 15) and ( 16), we obtain: To estimate the mass transfer coefficient % 7 . a' yzU { |U , different experiments with only GVL solution were performed in the absence of chemical reactions. It was assumed that the number of moles disappearing in the reservoir corresponds to the number of moles of hydrogen in the liquid phase. Ideal gas law was used to determine the number of moles based on the pressure. ODE [START_REF] Ding | Production of methyl levulinate from cellulose: selectivity and mechanism study[END_REF] was solved in the absence of chemical reactions, i.e., R1=R2= 0 mol.m -3 .s -1 .

7 . a = % 7 . a' yzU { |U * " & c$Y l c$Y - j.k * " A c$Y l c$Y - j. k (17)
Finally, in order to determine Henry's constant as a function of temperature, mass transfer experiments were carried out at four different temperatures for a long period, in order to reach the thermodynamic equilibrium. Van't Hoff equation was used to express Henry's temperature's dependence:

%8 € ' = Z8 €|{ = 373.15R[ * "… † ‡∆9 DdX. € * ‰ & ^-ŠnŠ. k ‹OE (18)
For the mass transfer study, the number of moles of hydrogen in the liquid phase was used as an observable. The objective function was defined according to Equation ( 21) and solved as described in paragraph 3.5 (Modeling).

Physicochemical properties

The same methodology developed by our group [START_REF] Cai | Investigation of the Physicochemical Properties for Vegetable Oils and Their Epoxidized and Carbonated Derivatives[END_REF] was used to measure the evolution of density and kinematic viscosity of GVL with temperature.

Density (kg.m -3 ) varies with temperature T (K) as

. = a • + b • * 8 (19)
Viscosity follows an Arrhenius law:

m = A × p' V ^ * ' (20)
where, μ is the dynamic viscosity (Pa.s), A is the pre-exponential factor, Ea is the activation energy (J.mol -1 ), R is the universal gas constant (J.K -1 .mol -1 ) and T is temperature (K).

Fig. 4 shows the evolution of the measured density versus temperature, as per Equation [START_REF] Wang | Thermal Risk Assessment of Levulinic Acid Hydrogenation to γ-Valerolactone[END_REF]. Fig. 5 shows that viscosity follows Equation [START_REF] Piskun | Kinetic modeling of levulinic acid hydrogenation to γ-valerolactone in water using a carbon supported Ru catalyst[END_REF].

Modeling

For the modeling stage, the software ModEst [START_REF] Haario | MODEST-User's Guide[END_REF] was used. Ordinary differential equations were solved using ODESSA algorithm.

For the parameter estimation, the concentrations of substrate, intermediate and GVL were used as observable variables. The objective function was defined as:

= ∑ %" -" • ' (21)
where, " is the experimental value and " • is the simulated one.

The objective function was minimized by Simplex algorithm, then by Levenberg-Marquardt algorithm.

From the mass transfer experiments, it was found that ∆ ;z .9 = 5936.8 J.mol -1 and Z8 €|{ = 373.15R[ = 1.86 mol.m -3 .bar -1 as illustrated in Fig. 6. Compared to the hydrogenation of levulinic acid in water [START_REF] Piskun | Hydrogenation of Levulinic Acid to γ-Valerolactone in Water Using Millimeter Sized Supported Ru Catalysts in a Packed Bed Reactor[END_REF], the absorption of hydrogen in GVL is an endothermic phenomenon. This endothermic behavior was also observed for other organic solvents [START_REF] Brunner | Solubility of hydrogen in 10 organic solvents at 298[END_REF][START_REF] Angelo | Gas-Liquid Solubility of Hydrogen in n-Alcohols (1 ⩽ n ⩽ 4) at Pressures from 3.6 MPa to 10 MPa and Temperatures from 298[END_REF][START_REF] Raeissi | Understanding temperature dependency of hydrogen solubility in ionic liquids, including experimental data in [bmim][Tf2N][END_REF].

This can be beneficial because, as the reaction temperature increases, the solubility of hydrogen and the kinetics of hydrogenation increase. Fig. 6. Van't Hoff plot for the absorption of hydrogen in GVL.

For mass transfer modeling, the coefficient of explanation, defined as

= 1 -‰ ™ $ ‡™ • $ ™ $ ‡™ š $ ‹ , was
found to be 95% showing the reliability of the fitting to the experimental data. Table 3 shows the estimated values of mass transfer. Fig. 7 shows the fitting of the model to the experimental data.

In general, one can say that the model fits well the experimental data. For the kinetic modeling, the value of % 7 . a' yzU { |U estimated previously was used. Rate constants of reactions 1 and 2 for the hydrogenation of ML were expressed by using modified Arrhenius equation:

%8 € ' = Z8 €|{ [ * "… † ‡K $ € * " & ^-& ^)* -OE (22)
ODEs ( 10)-( 14 For the kinetic modeling, the coefficient of explanation was found to be equal to 99.82%, showing the good fitting of the model to the experimental data. Table 4 displays the estimated parameters and the standard deviations. Table 4 shows that standard deviation for kinetic constants are low. Nevertheless, the standard deviation for the Taft parameters are not low. This could be explained by the fact that equations ( 6)-( 9) might not be the most appropriate to represent the evolution of . * and 2 with temperature.

Fig. 8 shows some fitting of the model to the experimental data. In general, the fitting is correct.

One can observe that the fitting of the model to the intermediate concentration is less accurate. This is due to the high reactivity of these species making their analysis less accurate, which is particularly pronounced for the intermediate 4-hydroxypentanoic acid. The fitting of the model to the experimental concentration of 4-hydroxypentanoic acid is lower compared to the other ones due to its higher reactivity. The parity plot (Fig. 9) shows that the developed model is reliable. The parity plot (Fig. 9) shows that the developed model is reliable. Based on Table 4, it is possible to determine the kinetic constants for the different substrates (Table 5). From Table 5, it is possible to notice that the rate constants of hydrogenation of substrates in GVL are not proportional to the steric hindrance of the alkyl groups. Fig. 10. Evolution of rate constants 1 for different substrates with temperature.

From Fig. 10, we can notice that rate constants of hydrogenation of LA, EL, ML and BL are affected by the nature of the substituent and the temperature. This observation could be surprising because the substituent is relatively far from the ketone group. For reaction temperature lower than 135°C, the rate constant for the hydrogenation of levulinic acid is the highest one. Whereas, for reaction temperature higher than 135°C, the rate constants increase in the following order: ,K7 > ,7ª > ,67 > ,«7 . This difference of reactivity is not in agreement with the steric hindrance induced by the substituent. From Fig. 11, the influence of the substituent groups and temperature is more significant on rate constant 2 than for rate constant 1. This observation seems to be logical because the substituents are closer to the reaction center. In the temperature range 90-150°C, the rate constants increase in the following order: ,9:ª >> ,69: > ,K9: > ,«9: . When the steric hindrance is lower, then the reaction rate is faster.

From Figs. 10 and 11, it is possible to notice that steric hindrance could not explain properly the kinetic behavior of the hydrogenation of LA, ML, EL and BL.

Fig. 12 shows the evolution of the Taft parameters (. * , . * ,2 and 2 ) with temperature. The influence of the polar effect (. * and . * ) on both reactions is higher than the steric effect, and this difference is more pronounced when the reaction temperature is higher than 110°C. Steric effect can be considered as negligible for reaction temperature lower than 140°C for both reactions, i.e., 2 and 2 are lower than 1.

Furthermore, polar effect of reaction 1 starts to be significant when reaction temperature is ca.

115°C and 110°C for reaction 2. For reaction 1, the value of . * is negative. From Taft definition, this means that the reaction is accelerated by electron donating group. As the temperature increases, the ethyl group increases the most the electron donor capacity of the group ROOC-CH2-CH2-on the ketone group. For reaction 2, the value of . * is positive. From Taft definition, this means that the reaction is accelerated by electron withdrawing group. From the four substrates, the hydrogen group is the most electron withdrawing group explaining the fact that reaction 2 with hydrogen substituent is the fastest one. From the estimated kinetic constants, it is possible to plot the kinetics of GVL production under the same operating conditions for LA, ML, EL and BL at two temperatures (Figs. 13 and14).

Conclusion

Hydrogenation of levulinic acid and methyl, ethyl, butyl esters using Ru/C as catalyst to γvalerolactone (GVL) was studied. Experiments were performed at isobaric and isothermal conditions and GVL was used as solvent to avoid liquid-liquid reaction system.

A mass transfer investigation was done to evaluate the mass transfer coefficient ( 7 . a ) for the transfer of hydrogen from the gas to the liquid phase. The influence of reaction temperature, solvent viscosity and density was taken into account to determine the value of 7 . a. Besides, it was found that Henry's constant for hydrogen absorption in GVL follows a van't Hoff law. This absorption was found to be endothermic meaning that temperature increase leads to increase the amount of absorbed hydrogen.

A kinetic model, including mass transfer parameters, was developed by varying the reactant concentration, hydrogen pressure, reaction temperature and catalyst loading. The originality of this model was the use of Taft equation to take into account the steric and polar effects of the substituents (H-, CH3-, CH3-CH2-, CH3-CH2-CH2-CH2-) for the hydrogenation and ring-closure step reactions. We have demonstrated that the Taft parameters . * %8' and 2%8' vary with temperature. It was found that rGVL from LA> rGVL from ML> rGVL from EL> rGVL from BL.

For this reaction system, the steric effect was found to be negligible for both reactions.

Nevertheless, the polar effects were found to be important for the ring-closure ones. The rate of GVL production is faster by using LA, because the electron withdrawing effect of the group H- . * sensitivity factor of a reaction series to polar effects

following intermediates: 4 -Fig. 3 .

 43 Fig. 3. Reaction mechanism for the hydrogenation of levulinic acid or its esters to γ-

where % 7

 7 . a' yzU { |U = √f * } n.o * j pq * Z~ * 6 c$Y [ t.• s `N t.u * i j. k was assumed constant for all the experiments, considering GVL as the main chemical compound. The temperature dependence of the density and kinematic viscosity of GVL was measured, and the results are shown in the next paragraph.

Fig. 4 .Fig. 5 .

 45 Fig. 4. Evolution of GVL density with temperature.

Fig. 7 .

 7 Fig. 7. Fit of the model to the mass transfer experiments under a pressure of ca. 20 bars.

  ) were solved. Concentrations of substrate, intermediate and GVL were used as observables for the non-linear regression stage. The following parameters were estimated: constants ,67 Z8 €|{[, 3 ,67 ,69: Z8 €|{ [, 3 ,69: and the parameters A1, B1, C1, D1, A2, B2, C2 and D2.

Fig. 8 .Fig. 9 .

 89 Fig. 8. Fitting of the model to the experimental data for the different substrates: LA (panel a),

Fig. 11 .

 11 Fig. 11. Evolution of rate constants 2 for different substrates with temperature.

Fig. 12 .

 12 Fig. 12. Influence of temperature on Taft parameters.

Fig. 13 .Fig. 14 .Fig. 15 .Fig. 16 .

 13141516 Fig.[START_REF] Hu | Levulinic esters from the acid-catalysed reactions of sugars and alcohols as part of a bio-refinery[END_REF]. Kinetics of production of GVL from LA, ML, EL and BL at 140°C and 20 bar of H2:

increases the kinetics of reaction 2 .

 2 The reaction 2 is slower for the hydrogenation for the other alkyl because alkyl groups are electron-donating groups. This study opens new possibility in chemical reaction engineering, by knowing the Taft parameters, it is possible to predict the rate constants with other substrates. The continuation of this work is to test the Taft equation on other substrates and have a better understanding on the evolution of Taft parameters with temperature. Notation Dj molecular diffusion coefficient of j [m 2 .s -1 ] Eai activation energy of reaction i [J.mol -1 ] Esi near-quantitative measure of the steric effect of a substituent i He Henry's coefficient [mol.m -3 .bar -1 ] ΔHsol dissolution enthalpy [J.mol -1 ] ki Rate constant of reaction i kL.a volumetric mass transfer coefficient [s -1 ] (kL.a)modified modified volumetric mass transfer coefficient [‰ ²³.or disappearance of compound j [mol.m -3 .s -a reaction series to steric effects µ liquid viscosity [Pa.s] 0 * near-quantitative measure of the polar effect of a substituent i ξ energy dissipation rate per unit mass [W.kg -1 ].mass density [kg.m-3 ]

Table 1 .

 1 Experimental matrix for the kinetic study.

			Initial concentration	Initial liquid mass	Temperature	Catalyst amount (dry)	H2 pressure
	Substrate	Run
			mol/m 3	kg	°C	kg	bar
		1	984.7	0.1267	100	0.0014
		2	893.3	0.1267	130	0.0014
	LA
		3	1818.2	0.1279	110	0.0007
		4	1921.3	0.1279	140	0.0007
		5	929.5	0.1256	100	0.0014
		6	953.4	0.1256	120	0.0014
		7	1908.3	0.1257	100	0.0007
	ML
		8	1875.0	0.1257	140	0.0007
		9	2346.3	0.1257	110	0.0016
		10	2357.4	0.1257	150	0.0016
		11	942.2	0.1251	100	0.0014
	EL
		12	921.7	0.1251	130	0.0014

Table 2 .

 2 The substrate and the corresponding intermediate have the same substituent, thus the same values of Taft parameters.

Table 2

 2

	Substrates & Intermediates	0 *	3
	BL & BHP	-0.13	-0.39
	EL & EHP	-0.1	-0.07
	LA & HPA	0.49	1.24
	ML & MHP	0	0

. Taft parameters for the reference (ML) and substituents (BL, EL, LA)

[START_REF] Leveneur | Thermal Safety Assessment through the Concept of Structure-Reactivity: Application to Vegetable Oil Valorization[END_REF]

.

Table 3 .

 3 Results of the mass transfer constant.

	PARAMETERS		UNITS		VALUE Std error (%)
	% 7 . a' yzU { |U	" Pa. s K -j.k	. "	Pa. s ž. ‡Š -j. k	. ‡	2.22.10 -6	2.4

Table 4 .

 4 Estimated parameters and standard deviation at TRef= 403.15K

		Units	Estimated Parameters	Estimated Relative Standard Error (%)
	,67 Z8 €|{ [ m 6 .mol -1 .kg -1 .s -1	3.79E-06	4.6
	3 ,67	J.mol -1	15000.00	21.8
	,69: Z8 €|{ [	s -1	8.92E-04	7
	3 ,69:	J.mol -1	59300.00	6.4
	A1	-	25.00	54.2
	B1	K -1	-0.07	51.5
	C1	-	-10.00	54.6
	D1	K -1	0.03	51.3
	A2	-	-22.00	50.4
	B2	K -1	0.06	46.7
	C2	-	-8.40	49.2
	D2	K -1	0.02	44.9

Table 5 .

 5 Kinetic constants for the hydrogenation of LA, ML, EL and BL (TRef= 403.15K)

	ML	EL	BL	LA

Acknowledgments

This study has been done in the framework of Task 2: "Green process: 2nd generation of biomass" of AMED project. The authors thank the AMED project. The AMED project has been funded with the support from the European Union with the European Regional Development Fund (ERDF) and from the Regional Council of Normandie. The China Scholarship Council: Cooperation Program with the UTs and INSAs (France) is thanked by the authors. The authors thank the Maîtrise des Risques et Environementaux department, and the Erasmus programme to make the research project of Mariasole Cipolletta possible.


Atef Khedher

Kamel Benothman

Mohamed Benrejeb

Didier Maquin

State and unknown inputs estimation via a proportional integral observer with unknown inputs

This paper deals with the problem of fault detection and identification in noisy systems. A proportionnal integral observer with unknown inputs is used in order to estimate the state and the faults which are assumed as unknown inputs. The noise's effect on the system is also minimized. The obtained results can be extended to non-linear systems described by multiple model representations.

Introduction

The model's state estimation is an important field of research with numerous applications in control and diagnosis. Generally the whole system's state is not always measurable and the recourse to its estimation is a necessity. State estimation permits also to replace expensive sensors or those with a difficult maintenance. An observer is generally a dynamical system wich permit the system's state reconstruction from the system's model and the measurments of its inputs and ouputs [Lu]. The state estimation methods of linear systems are efficient [Ed]. There exist a class of systems which are more complex and the hypothesis of linearity is not available. for these non linear systems, many studies are interested in the synthesis of non linear observers which permit the reconstruction of non measurable system's states. For example we can cite sliding mode observres [ES] and the Thau-Luenberger observers [Th]. Approaches using Takagi-Sugeno model's (sometimes said multiple model) are the object of many works in varied contexts including the take into account of unknown inputs or parametrical uncertainties [AC,IM]. Various studies were published on the unknown inputs [SA,Ed,AC]. Some of them tried to reconstruct the system's state in spite of the unknown input existence. Those inputs can be faults or perturbations affecting the system. This reconstruction is assured via the partial ellimination of unknown inputs [SA]. Other works choose to estimate the unknown inputs and the faults in order to estimate the studied system's state [Ed,AC].

The idea behind the multiple model is to represent a non-linear system in an interpolation's form of many local linear models. Each one of them is a linear time invariant (LTI) system valid around an operation point. The transition between models is done using the local model's activation functions [Ch,JF]. The Takagi-Sugeno structure is the most used in multiple model class representation for the system's design and analysis [MJ]. A non-linear model state estimation using this formalism is assured via a multiple observer which is an extension of linear system's observers [TS,PC,BP]. The state estimations can be used to generate fault indicators.

Linear system case

The objective of this part is to estimate a fault as an unknown input affecting a linear system via an unknown input proportional integral state observer.

Problem's formulation

Consider the linear model affected by an actuator fault and measurement noise. Its equation is given as following :

ẋ(t) = Ax(t) + Bu(t) + Ef (t)
(1)

y(t) = Cx(t) + Dw(t) (2)
Where x(t) ∈ R n represents the system's state, y(t) ∈ R m is the measured output, u(t) ∈ R r is the system's input, f (t) represents the fault and w(t) is the measurement noise. A, B and C are a known coefficients matrices with appropriate dimensions. E and D are respectively the fault distribution and the noise matrices which are supposed to be known. The observer is choosen to be as following :

ẋ(t) = Ax(t) + Bu(t) + E f (t) + K(y(t) -ŷ(t)) (3) ḟ (t) = L(y(t) -ŷ(t)) (4) ŷ(t) = C x(t) (5)
Where x(t) ∈ R n is the estimated system's state, f (t) represents the estimated fault, ŷ(t) ∈ R m is the estimated output, K is the proportional observer's gain and L is the integral gain to be computed. It is suposed that the fault affecting the system is bounded. The expressions of the state reconstruction error x(t) and the fault reconstruction error are given by the equations ( 6) and ( 7) as following :

x(t) = x(t) -x(t) (6) f (t) = f (t) -f (t) (7)
The dynamics of the state reconstruction error is given by the computation of ẋ(t) which can be written :

ẋ(t) = ẋ(t) -ẋ(t) = (A -KC)x(t) + E f (t) -KDw(t)
The fault error estimation is given as following :

ḟ (t) = ḟ (t) -ḟ (t) = ḟ (t) -LC x(t) -LDw(t)
In order to simplify the notations, the time index (t) will be omitted henceforth.

The following matrices are introduced :

ϕ = x f , ǫ = w ḟ (8)
From the equations ( 8) and ( 8), one can obtain :

φ = A 0 ϕ + B 0 ǫ (9)
with :

A 0 = A -KC E -LC 0 , B 0 = -KD 0 -LD I (10)
The matrix I is the identity matrix with appropriate dimensions. A Lyapunov function V is given as following :

V = ϕ T P ϕ (11)
Where P indicates a defined positive matrix. The state reconstruction error x(t) and the fault reconstruction error f (t) tend towards zero if V (t) < 0. In addition to that, the effect of the vector ǫ is minimal if the inequality ( 12) is solved with minimizing the positif scalar µ.

V (t) + ϕ T Q ϕ ϕ -µ 2 ǫ T Q ǫ ǫ < 0 (12)
The matrices Q ϕ and Q ǫ are two defined positive matrices. The inequality (12) can be written :

ψ T Ωψ < 0 (13)
with :

ψ = ϕ ǫ , Ω = A T 0 P + P A 0 + Q ϕ P B 0 B T 0 P -µ 2 Q ǫ (14)
The inequality ( 12) is verified if Ω < 0. The matrix A 0 can be written as following :

A 0 = Ã -K C ( 15
)
with :

à = A E 0 0 , K = K L , C = C 0 (16)
The matrix B 0 can be written on the following form :

B 0 = -K D + Ĩ (17)
The matrices Ĩ and D are as following :

Ĩ = 0 0 0 I , D = D 0 (18)
Let's pose G = P K and m = µ 2 . The matrix Ω can be written as :

Ω = P Ã + ÃT P -G C -CT G T + Q ϕ -G D + P Ĩ ĨT P -DT G T -mQ ǫ (19)
The resolution of the inequality Ω < 0 leads to find P and G. K is determined via the resolution of K = P -1 G

Example

In order to validate the obtained results of the previous section, a linear system described by the matrices A, B, C and D is choosen as following :

A =    
-0.3 -3 -0.5 0.1 -0.7 -5 2 4 2 -0.5 -5 -0.9 -0.7 -2 1 -0.9

    , B =     1 2 5 1 4 -3 1 2     , C =   1 0 0 0 0 1 0 0 0 0 0 1   , D =   0.5 0.5 0.2 0.2 0 0.1   E = B
The system's input u(t) is defined as following :

u(t) = u 1 (t) u 2 (t) T
The signal u 1 (t) is a telegraph type signal varying between zero and one. The signal u 2 (t) is computed by the following equation u 2 (t) = 0.3 + 0.1 sin(πt). The fault f (t) is written :

f (t) = f 1 (t) f 2 (t)
T with :

f 1 = 0, t ≤ 0.6sec sin(0.5πt), t > 0.6sec , f 2 = 0, t ≤ 1sec 0.4, t > 1sec (20)
The µ, K and L computation gives the following results : µ = 0.8367, The simulation results are shown in the figures ( 6) and ( 7). This method allows to estimate the unknown inputs well even in the case of non constant input.

K =     28

Extension to multiple model representation

The objective of this part is to estimate a fault as an unknown input affecting a non-linear system represented by multiple model via an unknown input proportional integral state multiple observer.

Problem's formulation

The goal of this section is to extend the obtained results in paragraph 1 for the multiple model structure. Considering a non-linear system affected by an actuator fault, its equation is given as following :

ẋ(t) = M i=1 µ i (u(t))(A i x(t) + B i u(t) + E i f (t)) (21a) y(t) = Cx(t) + Dw(t) (21b)
Where x(t) ∈ R n represents the system's state, y(t) ∈ R m is the measured output, u(t) ∈ R r is the system's input, f (t) represents the fault and w(t) is the measurement noise. A i , B i and C are a known coefficients matrices with appropriate dimensions. E i and D are respectively the fault distribution and the noise matrices which are supposed to be known. The scalar M represents the number of local models. The observer is choosen to be as following :

ẋ(t) = M i=1 µ i (u(t))(A i x(t) + B i u(t) + K i (y(t) -ŷ(t)) + E i f (t)) (22a) ḟ (t) = M i=1 µ i (u(t))L i (y(t) -ŷ(t)) (22b) ŷ(t) = C x(t) (22c)
Where x(t) ∈ R n is the estimated system's state, f (t) represents the estimated fault, ŷ(t) ∈ R m is the estimated output, K i are the local model's proportional observer's gains and L i are the local model's integral gains to be computed. Like in paragraph 1, It is suposed that the fault affecting the system is bounded.

Using the same expression of x(t) and f (t) given by the equation ( 6) and ( 7), the dynamics of the state reconstruction error is given by the computation of ẋ(t) which is written :

ẋ(t) = ẋ(t) -ẋ(t) = M i=1 µ i (u(t))(A i -K i C x(t) + E i f (t) + K i Dw(t)) (23)
The fault estimation error can be written :

ḟ (t) = ḟ (t) -ḟ (t) = ḟ (t) - M i=1 µ i (u(t))(L i C x(t) -L i Dw(t)) (24)
Like the linear case and in order to simplify the notations, the time index (t) will be omitted henceforth. By the use of the defintions of ϕ and ǫ gives in (8), the equations ( 23) and ( 24) can be written :

φ = A m ϕ + B m ǫ (25)
with :

A m = M i=1 µ i (u) Ãi , and B m = M i=1 µ i (u) Bi (26)
where :

Ãi = A i -K i C E i -L i C 0 , Bi = -K i D 0 -L i D I (27)
The matrix I is the identity matrix with appropriate dimensions. By considering the same expression of th Lyapunov function V given in (11) the inequality ( 12) can be written as following :

ψ T Ωψ < 0 (28)
with :

ψ = ϕ ǫ , Ω = A T m P + P A m + Q ϕ P B m B T m P -µ 2 Q ǫ (29)
The inequality ( 28) is verified if Ω < 0.

The matrix A m can be put on the following form :

A m = Ãm -Km C (30)
The expressions of Ãm , Km and C are given as following :

Ãm = M i=1 µ i (u(t)) Ãmi , Km = M i=1 µ i (u(t)) Kmi , C = C 0 (31) with : Kmi
= K i L i , Ãmi = A i E i 0 0 (32)
In the same way, the matrix B m can be formulated as following :

B m = -Km D + Ĩ (33)
with :

Ĩ = 0 0 0 I , D = D 0 (34)
Posing G m = P Km and m = µ 2 ,the matrix Ω can be put in the following form :

Ω = P Ãm + ÃT m P -G m C -CT G T m + Q ϕ -G m D + P Ĩ ĨT P -DT G T m -mQ ǫ (35)
the following equations can be solved for i = 1...M :

Ω i < 0 (36)
with :

Ω i = P Ãmi + ÃT mi P -G i C -CT G T i + Q ϕ -G i D + P Ĩ ĨT P -DT G T i -mQ ǫ ( 37
)
and G i = P Kmi . The matrices P and G i are computed via the resolution of the inequality Ω i < 0. The matrix Kmi is determined as following : Kmi = P -1 G i .

Example

Considering the non-linear system described by a multiple model structure with two local models, four states and three outputs. Its structure is given by the following equation :

ẋ(t) = 2 i=1 µ i (u)(A i x(t) + B i u(t) + E i f (t)) (38a) y(t) = Cx(t) + Dw(t) (38b)
The system matrices are defined as below :

A 1 =    
-0.3 -3 -0.5 0.1 -0.7 -5 2 4 2 -0.5 -5 -0.9 -0.7 -2 1 -0.9

    , A 2 =     -0.2 -3 -0.6 0.3 -0.6 -4 1 -0.6 3 -0.9 -7 -0.22 -0.5 -1 -2 -0.8     , B 1 =     1 2 5 1 4 -3 1 2     B 2 =     4 6 0 0 -4 2 7 6     , C =   1 0 0 0 0 1 0 0 0 0 0 1   , D =   0.5 0.5 0.2 0.2 0 0.1   , E 1 = B 1 , E 2 = B 2
The system inputs are :

u(t) = u 1 (t) u 2 (t)
T .

The signal u 1 (t) is a telegraph type signal whose amplitude is included in [0, 0.5].

The signal u 2 (t) is computed by the following equation u 2 (t) = 0.3 + 0.1 sin(πt). The faultf (t) is written :

f (t) = f 1 (t) f 2 (t)
T with : The simulated results are shown in the figures (3,4,5,6 and 7). Even in the case of multiple model, the method presented allows to estimate well the system's state and the unknown inputs.

f 1 = 0, t ≤ 0.6sec sin(0.5πt), t > 0.6sec , f 2 = 0, t ≤ 1sec 0.4, t > 1sec (39) Choosing Q ϕ = Q ǫ = I, The µ, K 1 , K 2 ,

Conclusion

This paper presents an estimation method of faults considered as unknown inputs in the linear and non linear case. The class of the considered non-linear systems are those described by multiple model structure. This method uses a proportionnal integral observer with unknown inputs and its advantage is the take into account of the non-null fault dynamics. The validation of the method is done on an academic example.

Fig. 2 .

 2 Fig. 1. fault f1 and its estimate

Fig. 7 .

 7 Fig. 3. first output reconstruction error

  L 1 and L 2 computation gives the following results : µ = 4.4721,

			18.5771 13.6432 130.4662			0.0448 -0.1260 0.0304	
	K 1 =	  	-6.5165 42.3315 67.5280 77.6976 -81.7903 -211.2738	   , K 2 = 10 5 *   	0.0172 -0.0509 0.0177 -0.4144 1.0144 0.0562	  
			0.1124 -5.8306 43.0292			0.0411 -0.1029 -0.0005
	L 1 =		-92.3284 208.4753 185.6276 -22.2858 -23.9227 363.7863	, L 2 = 10 5 *		0.2031 -0.5101 0.0028 0.5230 -1.2936 -0.0427