Hiba Aouled Mhemed
email: hibaaouledmhemed@hotmail.com

Sana Kordoghli

Mylène Marin Gallego

Loubna Hadhoum

Jean François Largeau

Fethi Zagrouba

Mohand Tazerout

A novel low-cost material for thiophene and toluene removal : study of the tire pyrolysis volatiles

Keywords: UT, Used Tires, UTPO, Used Tires Pyrolysis Oil, CG, Spent Coffee grounds, BC_CG, waste tires, desulfurization, biochar, cracking, regeneration

It remains challenging to produce new and efficient adsorbents to remove sulfur compounds from waste tire pyrolysis oils. The present work investigated the ex-situ treatment by desulfurization and cracking of volatiles from the pyrolysis of waste tires using an innovative synthetic oil mixture. For this purpose, biochars from the pyrolysis of abundant biomasses (date seeds and spent coffee grounds) were used to improve the quality of these volatiles.

Furthermore, the purification efficiency of these biochars was compared to that of commercial activated carbon. Thus, the influence of the operating conditions was studied. Finally, to

Introduction

Nowadays, energy and environment are key consideration in discussions of sustainable development. An inevitable increase in used tires is a severe environmental problem. Around 1.5 billion tires are produced worldwide each year, which corresponds to approximately 17 million tons of used tires (UT). In 2013, used tires in European Union countries reached 3.6 million tons [START_REF] Czajczyńska | Use of pyrolytic gas from waste tire as a fuel: A review[END_REF]. The main routes for waste tires management are landfilling, energy recovery, material recovery, retreading, and reuse/export. Meanwhile, the composition of tires makes their energy recovery a profitable business. To minimize the negative impacts of this waste on the environment, pyrolysis can be considered the most promising practice.

Among the products formed through scrap tire pyrolysis, pyrolysis oil has been considered a promising alternative fuel since its fuel characteristics are similar to commercial diesel. Its composition consists of paraffin, alkenes, and aromatic compounds. Its high calorific value of around 41-44 MJ/kg attracts its use as a substitute for conventional liquid fuels [START_REF] Sathiskumar | Recycling of waste tires and its energy storage application of by-products-a review[END_REF]. However, despite its high energetic value, sulfur, nitrogen, and heavy hydrocarbons make this oil unsuitable for use as vehicles fuel [START_REF] Jadav | Investigation of nano catalyst to enhance fuel quality in waste tyre pyrolysis[END_REF]. Indeed, sulfur comes from the rubber vulcanization step in tire manufacturing: rubber monomers are crosslinked with sulfur during the vulcanization process to obtain mechanical strength and improve the thermostatic properties of rubber [START_REF] Al-Lal | Desulfurization of pyrolysis fuels obtained from waste: Lube oils, tires and plastics[END_REF]. Some authors have reported the sulfur content in used tires pyrolysis oil (UTPO) to be 0.3 wt% [START_REF] Aydın | Optimization of fuel production from waste vehicle tires by pyrolysis and resembling to diesel fuel by various desulfurization methods[END_REF], but the majority of reported values are around 1.4 wt% [START_REF] Hita | Opportunities and barriers for producing high quality fuels from the pyrolysis of scrap tires[END_REF]. Sulfur is usually converted to H2S, CH3SH, COS, and SO2 during pyrolysis [START_REF] Choi | Non-catalytic pyrolysis of scrap tires using a newly developed two-stage pyrolyzer for the production of a pyrolysis oil with a low sulfur content[END_REF]. Sulfides (CS2, H2S) and thiophene are the main sulfur species in solid pyrolytic residues, while thiophene and sulfites (SO2) are the predominant sulfur compounds in the UTPO [START_REF] Pan | Pyrolytic transformation behavior of hydrocarbons and heteroatom compounds of scrap tire volatiles[END_REF]. Hence, the choice of thiophene as the main sulfur compound of the UTPO can also serve as a model for other sulfur species such as CS2, COS, or mercaptans.

According to several studies, UTPO can be desulfurized using the conventional sulfur removal methods for petroleum fuels. Currently, the industrial process (Technology Readiness Level (TRL) 9) commonly used for the removal of sulfur from petroleum fractions is hydrodesulfurization (HDS). This catalytic process has been used in refineries since 1993 to remove sulfur compounds such as sulfides and disulfides from petroleum fractions. The temperature and hydrogen pressure of HDS are usually kept in the range of 300 to 400 °C and 5 to 13 MPa, respectively. The hydrodesulfurization catalysts are usually Co or Ni promoted Mo or W sulfide. γ-Al2O3 is the traditional support material for HDS catalysts as it provides a high surface area and high thermal and mechanical stabilities [START_REF] Saha | Review on recent advances in adsorptive desulfurization[END_REF]. Although refineries around the world commercially apply the HDS process, its requirements make it a costly operation.

In this regard, several desulfurization techniques have been developed. All methods other than HDS are still in the technological development phase (TRL4 and TRL5) [START_REF] Ahmad | Sulfur in petroleum: petroleum desulfurization techniques[END_REF]. The adsorption process has become a promising approach to ultra-deep desulfurization in this context. This economical method, environmentally friendly, can reduce the sulfur content to <1 ppm [START_REF] Zhou | Effects of oxidative modification of carbon surface on the adsorption of sulfur compounds in diesel fuel[END_REF].

Desulfurization by adsorption, however, faces two significant challenges :

1. develop high-capacity adsorbents ; 2. identify adsorbents that are more selective towards the adsorption of sulfur compounds (mainly aromatics) in a fuel containing several other aromatic and olefinic compounds.

Hence, there is a constant effort to invent new adsorbents to remove thiophenes from fuels [START_REF] Jeevanandam | Adsorption of thiophenes out of hydrocarbons using metal impregnated nanocrystalline aluminum oxide[END_REF]. The most widely used adsorbents in the industry are activated carbons, zeolites, silica gels, activated alumina, and metal-organic frameworks (MOFs). Although their efficiency, these materials have relatively high manufacturing costs. Whence, the interest to develop an economical adsorbent with a high absorption and regeneration capacity. A novel approach to using biomass waste as a source of adsorbents for toxic gases and various liquids has been developed [START_REF] Wallace | Efficient Air Desulfurization Catalysts Derived from Pig Manure Liquefaction Char[END_REF]. The term biochar has been used to refer to these increasingly popular and profitable materials. Biomass, abundantly available, is a promising source of biochar production. Among the most relevant substitution opportunities in Tunisia, the biomass generated annually by Tunisian phoenix culture and coffee grounds are renewable energy sources that can represent both environmental and economic benefits [START_REF] Motghare | Comparative study of different waste biomass for energy application[END_REF]. Currently, Tunisia ensured the production of around 250 000 tons of dates in 2016 [START_REF] Gifruits | Groupement Interprofessionnel des Fruits -Tunisie -www.gifruits.com[END_REF]. This significant production generates substantial solid waste, such as date seeds and branches, discarded or mixed with animal feed. On the other hand, coffee is the second most-consumed product worldwide. Therefore, a considerable amount of this residue is generated. The annual worldwide generation of spent coffee grounds through soluble coffee production industries is approximately 6 million tons which can pose several problems in terms of proper disposal [START_REF] Mussatto | A study on chemical constituents and sugars extraction from spent coffee grounds[END_REF].

Given the complexity and non-reproducibility of pyrolytic oils, a synthetic liquid representative of tire pyrolysis oils and sulfur pollutants is essential for understanding potential competition during desulfurization. Our findings presented in this study are very different from those described in the literature until now. In this paper, it is highlighted that there is no study concerning the use of biochar as an adsorbent for the hot dynamic adsorptive desulfurization of the volatile fraction resulting from the tires pyrolysis, which has been reported in the literature. This study was performed to obtain helpful information for designing industrial carbon-based sulfur and tar removal system. Moreover, it exposes novelty because it developed a promising eco-friendly alternative fuel from waste tires using highefficiency and low-cost adsorbents. These materials were tested in a synthetic liquid representative of tire pyrolysis oils. A commercial activated carbon was used as an adsorbent for comparison. The adsorption and cracking experiments were performed under an inert atmosphere. The data obtained were used to compare the performance of the different carbonaceous materials and to understand the relationship between their properties and their effectiveness in reducing sulfur and tars. The results provided valuable information on the deactivation mechanism and the interaction between aromatics and the surface of the biochar.

The regeneration of the best sulfur adsorbent by two methods was also evaluated.

Materials and Methods

Biochars and activated carbon

Two carbonaceous materials were tested in the setup. The feedstock for the production of biochars are wastes generated by Tunisian phoeniciculture (date seeds) and by soluble coffee production industries (spent coffee grounds). The powder of spent coffee grounds was pelletized before the production of biochar. The obtained pellets have a diameter of 4 mm and a length between 5 and 10 mm. The production of biochar was carried out by pyrolysis at 750 °C, and a heating rate of 5 °C/min, for 3 hours. This temperature is chosen to ensure that no additional pyrolysis reactions take place during the desulfurization tests. Fig. S1 represents the biochar production reactor. Pyrolysis-derived biochar will be referred as BC_GC (biochar derived from spent coffee grounds) and BC_DS (biochar derived from date seeds). The extruded commercial activated carbon (AC) has been provided by EHEIM AKTIV. The particle diameter of the carbonaceous materials is between 2 and 3 mm.

Adsorbents characterizations

Biochars and commercial activated carbon are complex materials whose physicochemical properties may be involved in the purifying mechanisms by adsorption. Therefore, a multiscale characterization was carried out (Fig. S2). The different carbonaceous materials were characterized in terms of composition and surface properties. The description of the elementary analysis, immediate analysis, potential hydrogen (pH), and the point of zero charge (pHpzc) have been developed in previous work [START_REF] Mhemed | Kinetic Study of Lignocellulosic Biomasses Pyrolysis Using Thermogravimetric Analysis[END_REF][START_REF] Mhemed | Gas adsorptive desulfurization of thiophene by spent coffee grounds-derived carbon optimized by response surface methodology: Isotherms and kinetics evaluation[END_REF]. Before elementary and mineral analysis, biochars were reduced to powder in a ball mill model Retsch MM400. Minerals were determined using X-ray fluorescence spectrometry model EDX-800HS2 (Shimadzu, USA). The characterization of textural properties was determined by the Brunauer, Emmett and Teller theory (BET) theory using the Micromeritics Smart VacPrep instrument. Morphological characterization of solid samples was performed using the scanning electron microscopy (SEM) technique (JEOL JSM 5800LV). Oxygen groups on the surface of biochars and AC were evaluated using the Boehm-titration method [START_REF] Mhemed | Gas adsorptive desulfurization of thiophene by spent coffee grounds-derived carbon optimized by response surface methodology: Isotherms and kinetics evaluation[END_REF].

Experimental setup

Fig. 1 shows the process diagram of synthetic pyrolytic liquid treatment by desulfurization and cracking. The experiments were carried out in a vertical fixed bed reactor. The dimensions of the bed are an internal diameter of 3.8 cm and a maximum height of 30 cm. The diameter of the carbonaceous material particle is approximately between 2 and 3 mm. The catalytic bed is perforated to ensure a good gas distribution and avoid preferential paths. Three thermocouples were used to measure the temperature at the core, the bottom, and the head of the catalytic bed (TE2, TE3, and TE1, respectively). In addition, two pressure sensors are placed upstream and downstream of the reactor to calculate the pressure drops (∆P) induced by the system and to detect any pressure drops during the tests. The nitrogen is continuously preheated to 200 °C before entering the reactor. The heating operation lasts 90 min until temperature stabilization (500 °C). The synthetic mixture was injected into the gas stream using a syringe pump when the system was stable. The ascending gas is purified by crossing the fixed bed. An electrical box exempts the control and regulation of the gas preheater and the heating of the cracker.

The acquisition interface permits the monitoring and recording of the internal (TE1, TE2, TE3, TE4), upstream, and downstream temperatures (TE5, TE6) of the reactor, as well as the differential pressure between the inlet and the outlet of the reactor (∆P). At the exit of the reactor, the gaseous effluent is analyzed online by a micro-GC (R-3000, SRA Instruments).

The reactor operating pressure is the atmospheric pressure. For safety reasons, ventilation is kept active throughout the experiments. The N2 flow was continued for all the test. The concentration of the injected toluene was set at 9800 ± 1% ppm in mixture 1. However, the concentrations of toluene and limonene in mixture 2 were calculated to be 3940 ± 1.20 % ppm and 5860 ± 1.35 % ppm, respectively.

The thiophene adsorption capacities of the different adsorbents were determined for an initially fixed concentration of 500 ± 0.5% ppm [START_REF] Saha | Green synthesis of nano silver on TiO2 catalyst for application in oxidation of thiophene[END_REF][START_REF] Saha | Comparative studies of extraction ability of organic solvents to extract thiophene from model fuel[END_REF][START_REF] Królikowski | Separation of thiophene, or benzothiophene from model fuel using glycols. Liquid-liquid phase equilibria and oxidative desulfurization study[END_REF]. The bed height (h= 20 cm), the reactor temperature (T=500 °C), and the duration of the experiment (60 min) are kept unchanged during the various tests.

Synthetic mixture

Mixture 1 X X X Mixture 2 X X X

Adsorption and cracking experiments

The breakthrough curves obtained from the micro-GC measurements were used to calculate the total adsorption capacity Ws (mg/g adsorbent) which is defined by [START_REF] Mccabe | Unit operations of chemical engineering[END_REF] :

0 0 0 0 ( ) 1 e t S C t U C dt C W h ρ   × × -     = × ∫ (1)
The adsorption capacity at the breakthrough point (mg/g) is :

0 0 0 0 ( ) 1 p t P C t U C dt C W h ρ   × × -     = × ∫ (2)
Where C0 and C(t) are the concentration of thiophene at the inlet and outlet of the reactor, respectively (g/cm 3 ), U0 is the inlet flow velocity (cm/s), h is the bed height (cm), ρ is the density of the adsorbent (g/cm 3 ), tp is the break point-time (min) where effluent concentration was about 5 % of the influent concentration while te (min) is the exhaustion time where the effluent concentration equals about 95 % of the influent concentration. The integration of the vacuum breakthrough curve was also performed to take into account the residence time of the gas flow in the whole vacuum system. The blank test residence time in the entire installation (pipes, reactor, and preheater) was deduced from the obtained breakthrough curves.

Integration was carried out by performing the trapezoid method on each breakthrough curve.

In industrial practice, a bed is operated until an effluent reaches an arbitrarily chosen low concentration value known as the breakpoint concentration. It is the maximum allowable concentration that the effluent can reach. At this breakpoint, the bed is not fully used for effluent uptake. Then, the term called Length of Unused Bed (LUB) is then calculated by the following equation:

The adsorption efficiency is given by the relationship below [START_REF] Ji | Dynamic adsorption of Cu(II) from aqueous solution by zeolite/cellulose acetate blend fiber in fixed-bed[END_REF] :

( ) 0 % 100 s adsorption V e W m C Q t η × = × × × (4)
Where Qv is the total volumetric inlet flow rate (cm 3 /h), m is the mass of adsorbent in the fixed bed (g).

Once the best thiophene adsorption operating conditions were identified, a gas chromatograph (GC, Perkin Elmer Clarus 680) coupled with a mass spectrometer (MS, Perkin Elmer Clarus 600 S) equipped with an FID was used to identify the cracking products detected by the micro-GC but not identified or quantified. Thus, gas samples were taken at the outlet of the reactor. The sample composition was processed in an SBLTM-5ms Capillary Column, 30 m long, 0.25 mm internal diameter, and 0.25 μm film thickness. The temperature of the GC-MS oven was maintained at 50 °C for 2 min. Then, the temperature was raised to 110 °C at 2 °C/min, followed by 100 °C/min to 280 °C, maintained for 10 min. The injector temperature

1 p s W LUB h W   = × -     (3)
is 200 °C. The split mode is splitless. On each test, 10 μl of the gas sample was injected. For the column, the flow rate of carrier gas (helium) was 1.0 ml/min [START_REF] Ma | Effects of pressure and residence time on limonene production in waste tires pyrolysis process[END_REF].

Adsorbent regeneration

A preliminary laboratory study allowed us to fix regeneration's most critical operating conditions. Indeed, the regeneration of the best adsorbent was studied at the predefined desorption temperature (720 °C) in two gas atmospheres :

a. 15 vol.% CO2 in N2 ;

b. Pure N2.

To ensure a non-brutal regeneration, a constant gas volume (0.08 ± 0.003 Nm 3 /h) was introduced in the reactor. Thus the regeneration cycle was carried out with various adsorption and desorption cycles according to the steps presented in Fig. S3.

Results and discussions

Sorbents characterization

The composition of the biochars and the commercial activated carbon are presented in Table 2. The BC_DS contains one of the highest amounts of carbon and one of the lowest ashes among the samples, while BC_GC contains the lowest amount of carbon and the highest of ashes. Ash is the inorganic part of biomass left as a solid residue after the thermal conversion.

Ash is primarily made of silica, aluminum, iron, and calcium compounds, but small amounts of magnesium, titanium, sodium, and potassium may be present [START_REF] Benedetti | Characterization of char from biomass gasification and its similarities with activated carbon in adsorption applications[END_REF]. Since the structure of ash compounds is crystalline, a significant presence of crystalline phases is expected in samples rich in ash, such as BC_GC. To confirm these findings and obtain further information on the possible crystalline phases present in the samples, XRD analysis is recommended in future work. The aromaticity and polarity of the adsorbents were evaluated by calculating the H/C and (O + N)/C molar ratios, respectively. The molar ratios of H/C for BC_CG, BC_DS and AC are 0.02, 0.02 and 0.01, respectively. Efficient carbon sorbents are characterized by strong carbonization and high aromaticity if the molar ratio of H/C is less than 0.5 [START_REF] Kołodyńska | Comparison of sorption and desorption studies of heavy metal ions from biochar and commercial active carbon[END_REF]. The polarity index [(O + N)/C] for BC_CG, BC_DS and AC were 0.10, 0.08 and 0.12, respectively. The higher polarity index for BC_CG, and AC compared to that for BC_DS implies that BC_CG, and AC are more hydrophilic and contain more polar functional groups for coordination with thiophene, which should be confirmed by Bohem-titration results.

Overall, the pH of all biochars is around 7. A decrease in acidity (biomass with a pH around 5) after pyrolysis (biochar with a pH ranging from 7 to 7.5) is noted. The analysis of the ash composition using X-ray fluorescence spectrometry is represented in Fig. S4. The potassium (K) is the main element in BC_CG and BC_DS (3.12 and 2.57 wt%, respectively), while AC is mainly rich in iron (4.73 wt%). The amount of sulfur and calcium in AC is not negligible. Minerals such as Ca, Mg, and K can improve the catalytic activities of carbonaceous materials [START_REF] Huang | Cracking of Model Tar Species from the Gasification of Municipal Solid Waste Using Commercial and Waste-Derived Catalysts[END_REF]. Transition metals (Fe, Al, Co, Ni, Zn) are essential for catalytic cracking reactions. Some of these species are present in AC with appreciable quantities (4.73 wt % of Fe) and lower amounts in biochars. Thus, AC is theoretically a good candidate for tar cracking. Nitrogen adsorption indicates that the microporous volume of AC is significantly higher than other materials used in this study (Table 2). The BC_CG has the lowest specific surface area (49.44 m²/g). Its microporous volume is about fifty times smaller than that of AC.

Its mesoporous volume is about five times smaller than commercial activated carbon. The average pore diameters of the three materials are more significant than 2 nm. So these adsorbents have more pores in the mesoporous region.

Fig. S5 illustrates the slow nitrogen diffusion inside the pores. Also, a sharp increase in the amount of nitrogen adsorbed near saturation was observed. This may be associated with the condensation in the inter-particle voids [START_REF] Sotomayor | Characterization of micro/mesoporous materials by physisorption: concepts and case studies[END_REF]. The hysteresis phenomenon is generally associated with the capillary condensation in mesoporous structures [START_REF] Qi | Pore characterization of different types of coal from coal and gas outburst disaster sites using low temperature nitrogen adsorption approach[END_REF]. Fig. S5 (a) shows that the BC_CG has the greatest hysteresis, followed by BC_DS (Fig. S5 (b)). The hysteresis loop of these two samples appears to belong to type-H4. This type of deformation is often attributed to narrow, slit-shaped pores and the presence of mesopores. Thus, imprisoned nitrogen cannot be released due to its affinity with the biochar heterogeneous surface properties [START_REF] Tang | Adsorption Affinity of Different Types of Coal: Mean Isosteric Heat of Adsorption[END_REF]. Most probably, the pore wall of biochar traps nitrogen molecules. Moreover, the isotherms of BC_CG and BC_DS are type I (according to the IUPAC classification). This indicated a certain microporosity of these biochars, with a progressive saturation of the substantially equivalent adsorption sites. The nitrogen adsorption isotherm of AC (Fig. S5 (c))

is also type I: it has a slightly horizontal plateau, indicating the presence of very narrow micropores [START_REF] Li | Preparation of activated carbons from polycarbonate with chemical activation using response surface methodology[END_REF].

Density Functional Theory (DFT) has also been applied to N2 adsorption data. The DFT model is based on the thermodynamics equation, which calculates the amount of specific adsorption in a range of pores by solving the distribution function of gas density in a particular pore space [START_REF] Li | Application of low pressure gas adsorption to the characterization of pore size distribution of shales: An example from Southeastern Chongqing area, China[END_REF]. Fig. 2 illustrates the pore size distributions obtained by DFT. All three materials have a distribution towards pore sizes greater than 2 nm, with a more marked distribution for AC. The size distribution of pores confirms that they are predominantly mesoporous for the three materials. The AC has much narrower and smaller ultra-micropores (~0.6 nm). The BC_CG and BC_DS reveal a relatively identical mesoporous volume (0.002-0.0035 cm 3 /g), while AC is more extensive (0.015 cm 3 /g). All samples have a micropore size range from 0.4 nm to 2 nm with multiple peaks and a predominant maximum of ≤1.27 nm. In addition, BC_DS presents a non-negligible distribution of the macropores. This change in the pore size distribution may be due to a structural deformation that transforms mesopores into macropores [START_REF] Yang | Shale-gas accumulation and pore structure characteristics in the lower Cambrian Niutitang shales, Cen-gong Block, South China[END_REF]. The surface topology of the tested materials and their macrostructure were described using SEM analysis (Fig. S6). The morphology of BC_CG indicates its heterogeneous surface and porous nature. It shows the development of a macro-porosity between 2 and 5 µm. On the other hand, the BC_DS has a structure with very few pores on the smooth surface. However, AC seems more homogeneous with a low macro-porosity.

The oxygenated groups present on the surface of carbonaceous materials are lactones, carboxylic or phenol acids, and basic groups such as carbonyl, quinone, and pyronic structures. They can affect the catalytic properties of adsorbents [START_REF] Coughlin | Role of surface acidity in the adsorption of organic pollutants on the surface of carbon[END_REF]. The characterization of the various used materials functional groups by Boehm titration is given in Table 3. Based on these results, BC_CG had the most significant amount of groups with a strong acidic character (carboxylic groups : 0.35 mmol/g). These functional groups can interact with the gaseous matrix and then increase the adsorption capacity and the conversion of the aromatic ratio [START_REF] Huang | Cracking of Model Tar Species from the Gasification of Municipal Solid Waste Using Commercial and Waste-Derived Catalysts[END_REF].

On the other hand, the different materials (BC_DS and AC) are richer in compounds with a weak acid character, such as phenolic and lactonic groups (case of BC_DS). Though AC had high O content to the biochars (Table 2), much fewer oxygen, functional groups were observed in AC than in BC_CG. It may probably be because oxygen in AC exists in quinines or alcoholic hydroxyl, which could not be obtained by Boehm titration.

Effect of the mixture composition on thiophene removal efficiency

The breakthrough curves of BC_CG, BC_DS, and AC obtained with the different gas matrix are shown in Fig. 3. The matrix composition had no notable impact on the removal capacity of BC_DS, which was almost stable at 0.80-0.83 ± 0.01 mg thiophene.g -1 . Compared to mixture 1, the efficiency of BC_CG was practically improved in the second mixture (1.38 versus 0.97 ± 0.02 mg thiophene.g -1 , respectively). Thus, at the break point, 49 % of the bed capacity is used which approximately corresponds to an unused section of length (LUB) of 9.80 cm. Within experimental error, for such a front, LUB can be used to design the full-scale adsorber from the breakthrough curves obtained in small-scale laboratory experiments by adding LUB to the length of bed needed to achieve the required capacity.

This improvement may be due to the complexity of the second matrix and, subsequently, the growth of the competitive adsorption between toluene and limonene, whose critical adsorption diameters are 0.67 nm and 0.68 nm, respectively [START_REF] Cabrera-Codony | Competitive siloxane adsorption in multicomponent gas streams for biogas upgrading[END_REF]. Conversely, this competition becomes less pronounced for thiophene, whose required diameter is 0.53 nm. The breakthrough time of the AC was longer, indicating better retention of thiophene compared with biochar in the second matrix. The decline in biochar's breakthrough times reduces their adsorption capacities at the breakthrough point and consequently increase in the unused bed length. The AC's adsorption capacity has dramatically improved from a less complex mixture to a more complex blend (1.06 versus 0.60 ± 0.01 mg thiophene.g -1 , respectively). Its developed morphology may explain this comportment. The adsorption of organic compounds onto natural adsorbents has been described as a complex process, mainly controlled by chemical interactions and physical factors. The properties of the adsorbate molecules and the adsorbent, such as surface chemistry, specific surface areas, and porosity, often play critical roles. Studies on the part of physical factors, especially molecular size, aromatic compounds' structure, and adsorbents' pore structure, are relatively limited. Thus, the competitive effect of toluene and limonene with thiophene during adsorption was further investigated. Despite the variation in feed composition, BC_CG still has the best desulfurizing capacities. The large BC_CG surface acid groups may explain this result. Indeed, these groups contribute to a better selective adsorption of thiophene owing to the stronger acid-base interaction. Similar results were found by Shang et al. [START_REF] Shang | Kinetics and mechanisms of hydrogen sulfide adsorption by biochars[END_REF] comparing biochars with low specific surface area to activated carbon for H2S elimination.

Effect of the residence time

Fig. 4 shows the effect of residence time on the thiophene breakthrough curves in different synthetic mixtures. It is noted that the increase in residence time negatively influences the breakthrough time of BC_DS in both synthetic mixtures against an improvement in breakthrough time of BC_CG and AC. BC_DS has the lowest breakthrough time and therefore the smallest bed capacity used (38.60 %), which is equivalent to the most extended LUB of 12.30 cm. Thus, the decrease in adsorption capacity can be explained by a predominance of the axial dispersion phenomenon over the mass transfer and consequently a reduction of the adsorbate diffusion in the adsorbent [START_REF] Mohan | Breakthrough of toluene vapours in granular activated carbon filled packed bed reactor[END_REF]. AC has the longest breakthrough time (approximately 19 min, mix 2). Therefore, it provides the best-utilized capacity of the bed (53%), suggesting better resistance to mass transfer [START_REF] Yang | Synthesis of ceria nanorods as adsorbent for the adsorption desulfurization of gasoline fuel[END_REF]. However, no noticeable impact is perceived on the desulfurizing capacities of AC, which remained almost stable at 1.08-1.07 ± 0.02 mg thiophene.g -1 (Fig. 4a and Fig. 4b). This is because as the flow rate decreases, the contact time in the catalyst bed is longer, and intraparticle diffusion becomes more efficient.

Thus, the pollutant has more time to diffuse in the middle of the carbon particles, and hence a better adsorption capacity. Based on the amounts of thiophene adsorbed at breakthrough time and saturation, it is clear that increasing the residence time has a positive effect on the diffusion of thiophene and subsequently on its adsorption by the BC_CG. Indeed, this adsorbent shows the better thiophene removal by 1.44 ± 0.02 mg thiophene.g -1 (see Fig. 4b) which corresponds to a desulfurization efficiency of 77.10 ± 0.5 % (mixture 2; ts2 = 2.50 s).

Several authors have suggested that :

(i) the adsorption of a solute on solid takes place in two steps which are the surface fixation and then the diffusion through the internal layers of the adsorbent;

(ii) the limiting factor is the contact time between the two phases (gas and solid). While thiophene adsorption has been studied for deep desulfurization in liquid phase for transport fuels [START_REF] Delitala | Liquid-phase thiophene adsorption on MCM-22 zeolite and activated carbon[END_REF][START_REF] Patil | Investigating role of sulphur specific carbon adsorbents in deep desulphurization[END_REF], the literature on gas phase adsorption remains very scarce. A comparison of thiophene adsorption capacities with the literature is given in Table S1.

Adsorption is by nature an exothermic reaction. However, despite the severe operating conditions used (T= 500 °C), the adsorption capacities of thiophene by BC_CG are promising compared to other adsorbents reported in the literature.

Aromatic cracking

During experiments carried out under the best adsorption operating conditions (QV = 0.1 Nm 3 /h; ts = 2.50 s; T= 500 °C; synthetic mixture 2), the concentration of both toluene and limonene at the outlet of the system evolved differently. It is worth pointing out that in mixture 1 no gases other than N2 were detected with micro-GC in the exit gas. Only under these conditions, small amounts of H2 and CO2 were detected at the outlet of the system. H2 concentration in the exit gas exhibited different trends over time with BC_CG, and AC, peaking around 0.13 vol% during the beginning of the experiment for AC, to decrease gradually afterward and stabilizing around 0.03-0.04 vol%. However, hydrogen was stable for BC_CG (around 0.01 vol%) with no traces for BC_DS. The detection of H2 as the main reaction product and the lack of light hydrocarbons in the exit gas confirm the mechanism of decomposition of aromatics over biochar with dehydrogenation and coke formation through condensation/polymerization reactions. The CO2 is weakly produced (0.04 vol% for BC_CG; 0.01 vol% for BC_DS; and 0.07 vol% for AC). As a reminder, CO2 is the product of catalytic cracking resulting from the reaction between the deposited coke and the oxygenated functions of the carbonaceous surface. In these conditions, the toluene and limonene conversion values for BC_CG and BC_DS remained lower than for AC indicating a mild activity of these biochars surfaces (see Fig. 5 (a,b)). It is also interesting to notice the trend of both BC_CG and AC with a toluene curve stabilizing at α above 5 % for more than an hour (see Fig. 5 a).

These trends indicate that in these two materials, carbon deposition could support the decomposition of toluene. BC_DS exhibited a fast deactivation, as it converged with the toluene conversion value in about 37 min. For limonene, complete conversion was maintained for AC all along with the test. Both BC_CG and BC_DS sustained complete limonene conversion for a shorter time (about 35 and 15 min, respectively) (see Fig. 5 b). It appears as limonene was converted more efficiently than toluene. This observation agrees with the increased reactivity of chars towards compounds with increased aromaticity. GC-MS analysis of limonene cracking products indicates the production of a variety of compounds such as βterpinene, γ-terpinene, α-phellandrene, camphene, and α-terpinolene. Hydrogen displacement can occur in α-terpinolene, leading to a displacement of hydrogen within the ring to obtain γterpinene as a second isomer. However, α-phellandrene can be formed from the isomerization of γ-terpinene [START_REF] Tavera Ruiz | Transformation of dl Limonene into Aromatic Compounds Using Supported Heteropolyacid Catalysts[END_REF]. Thereby, limonene was decomposed into terpenes through isomerization reactions. A quantification study of the cracking products is recommended in future work.

Hence, the performance of toluene and limonene cracking over the various materials can be classified as follows: AC> BC_CG > BC_DS. The results of toluene and limonene cracking obtained in this work are compared with those in the literature (Table S2). It can be noted that the biochar from wood chips (pyrolyzed at 700 °C) used for toluene cracking tests leads to a cracking yield of 60.30 % at 900 °C [START_REF] Cordioli | Thermal and Catalytic Cracking of Toluene Using Char from Commercial Gasification Systems[END_REF]. In addition, the study of X. Liu et al. [START_REF] Liu | Low-temperature catalytic steam reforming of toluene over activated carbon supported nickel catalysts[END_REF] on the catalytic reforming of toluene in the presence of activated carbon impregnated with nickel (10 % Ni/CA) at 500 °C indicates a toluene conversion rate of 24 %. These conversion rates are still much lower than that achieved by BC_CG. Hence, this biochar can be considered as an economical and promising material in the cracking of toluene present in the volatiles derived from the pyrolysis of used tires.

Analysis of the spent materials

The post-desulfurization characterization focuses on the two best performing materials used under the best operating conditions (T= 500°C, ts= 2.50 s, mixture 2): BC_CG and AC. The exhausted materials are named with the prefix "Spent". The characterization of these materials (with three repeated analyses) is summarized in Table S3. It is noted that Spent BC_CG and Spent AC show a mass loss of 2 % and 5 %, respectively. This mass loss can be explained by the reaction of limonene with the carbon surface of these materials. In addition, the H, N and S content also decreased slightly after use due to possible volatilization. This volatilization may also contribute to the observed mass losses. Furthermore, textural properties also play an important role in the catalytic activity of carbonaceous materials. Therefore, the porosity was studied by nitrogen adsorption at 77 K. It is noticed that the specific surface of Spent AC has improved. This slight increase reveals that the coke deposited during the cracking step has a porous character. It increased the specific surface area of the AC instead of blocking its original porous structure. This increase in specific surface area is followed by a decrease in mesopore volume, confirming that the mesopores are more involved in the adsorption/cracking mechanism. A similar observation is described by the recent work of Wang et al [START_REF] Wang | Investigation of hot char catalytic role in the pyrolysis of waste tires in a two-step process[END_REF]. On the other hand, the specific surface area of BC_CG decreased against an increase in its micropore volume after use. This may result from the low activation of the biochar in the presence of CO2 produced during the cracking tests.

Discussion : mechanisms of thiophene removal by biochar

Preliminary characterization of the biochar revealed that the thiophene adsorption process on its surface consists of a complex reaction mechanism involving the physicochemical properties of the biochar and the gas composition. In our case, it is still difficult to discriminate the individual contribution of each property to the thiophene removal capacity of BC_CG. However, the results showed that the critical biochar properties for good thiophene removal are: mesoporous volume, acidic pH surface, presence of mineral species on the surface (especially P, K, Ca), nitrogen content, O-containing groups (especially acid groups), and the disorganized carbonaceous structure. The selectivity of adsorbents towards sulfur compounds depends on either physical (Vander Waals force) or chemical interaction (sigma and π bonds) between the adsorbate or target molecules (sulfur) and active sites on the adsorbent surface. Chemisorption follows mainly the combination or any one of the following mechanisms : π complexation, acid-base interaction, and direct sulfur-metal interaction.

Physical adsorption via van der Waals forces does not control the selectivity of sulfur compounds [START_REF] Zhang | Biochar for volatile organic compound (VOC) removal: Sorption performance and governing mechanisms[END_REF]. The π-complexation interaction is stronger than the van der Waals interaction and aids higher selectivity to sulfur compounds than other compounds.

For AC, the large surface area overweighed other properties such as O-containing functional groups and aromaticity degree, making it different from biochar. Although the surface area (BET) of AC was almost sixteen times larger than BC_CG, its sorption capacity for thiophene was lower than that of BC_CG. This revealed that extremely high surface area and large pore volume of sorbent might not be the only criteria for efficient removal of thiophene. Previous work indicated that most thiophene sulfur compounds are Lewis base, which is easy to be adsorbed onto acid sites [START_REF] Saleh | Influence of acidic and basic treatments of activated carbon derived from waste rubber tires on adsorptive desulfurization of thiophenes[END_REF]. According to the X-ray spectroscopy (EDX, Fig. S 4) and CHNSO analysis results, the prominent proportions of P, K, and N can affect the acidic and basic properties of the carbon surface and thus influence the adsorption behavior [START_REF] Sorokhaibam | Development of Newer Adsorbents: Activated Carbons Derived from Carbonized Cassia fistula[END_REF]. Indeed, recently, Seredych and Bandosz. [START_REF] Seredych | Investigation of the enhancing effects of sulfur and/or oxygen functional groups of nanoporous carbons on adsorption of dibenzothiophenes[END_REF][START_REF] Seredych | Role of phosphorus in carbon matrix in desulfurization of diesel fuel using adsorption process[END_REF] reported that in addition of oxygen and other heteroatoms, such as sulfur and phosphorus in the activated carbon matrix, had a positive effect on its capacity and selectivity to separate DBT from model diesel fuel. This heteroatom (P) is usually bonded in the forms of phosphates. In this respect, it was revealed that phosphorus naturally contained in the carbon matrix of BC_CG can increase its acidity, enhancing the attraction of thiophene compounds, whose character is slightly basic.

Furthermore, in thiophene, S and C atoms have negative net charges, while H atoms have positive net charges [START_REF] Xiang | Model construction of the macromolecular structure of Yanzhou Coal and its molecular simulation[END_REF]. In contrast, P and N atoms have positive and negative net charges, respectively. Thus, it is deduced that the S site with a negative net charge in thiophene tends to be adsorbed at P sites with positive net charges in π-conjugated carbon. A relevant study by Shimoyama and al. [START_REF] Shimoyama | Thiophene adsorption on phosphorus-and nitrogen-doped graphites: Control of desulfurization properties of carbon materials by heteroatom doping[END_REF] dealing with the improvement of adsorptive desulfurization by P-and N-doped carbon materials, stated that phosphorus has a 10-20 times greater doping effect than nitrogen on thiophene adsorption. Therefore, these heteroatoms are important for improving the adsorptive desulfurization properties of carbon materials. To date, the mechanism of this effect is not well defined; we therefore suggest that the undoped BC_CG sample, naturally rich in N and P, may attract more of the thiophene molecule to its surface. In order to better study this effect, an X-ray photoelectron spectroscopy (XPS) analysis is recommended in future work. In addition, oxygen functional groups on the surface of the carbon material are also important for the removal of sulfur compounds in diesel fuel [START_REF] Saleh | Adsorptive desulfurization of thiophene, benzothiophene and dibenzothiophene over activated carbon manganese oxide nanocomposite: with column system evaluation[END_REF]. The sulfur atom has a pair of electrons on the six-electron π system, and another pair lies in the plane of the ring.

The adsorbed amount of thiophene by BC_CG involves the transfer of electrons from the oxygen-containing functional groups due to π-π electron-donor interaction. The carboxylic acid groups on the surface of the biochar underwent as electron acceptors, forming a π-π electron acceptor-donor interaction with aromatic molecules, thus facilitating the adsorption of aromatic molecules. Different types of hydroxyl and amine groups in biochar can also be used as π electron donor sites. In this study, coke produced by the cracking of aromatic groups can itself form new functional groups that can then be cracked into CO, H2, and CO2 at high temperatures. Other active sites, such as the defects in graphene-like sheets, in addition to π interactions, can participate in the adsorption process of thiophene by BC_CG.

Regeneration

Regeneration of the best desulfurization candidate was performed according to the protocol described in Section 2.5. During CO2 regeneration, multiple adsorption-desorption cycles were achieved (see Fig. S7 a). It's noted that the breakthrough time increased as the regeneration progressed and reached 28 min at the sixth adsorption with a maximum thiophene adsorption capacity of 1.84 mg/g. Beyond the sixth cycle, the breakthrough time (tp) and the exhaustion time (te) decrease. The adsorbent is completely exhausted after 15 cycles of adsorption. On the other hand, thermal regeneration under nitrogen was also tested.

No optimization of the operating conditions was performed due to resource constraints.

Hence, the same CO2 regeneration process was adopted, but using only pure nitrogen. The It is represented by the following expression :

(

Where qn : the amount adsorbed by the regenerated adsorbent n times at a given equilibrium concentration (mg.g -1 ); qoriginal : the quantity adsorbed by the fresh material at a given equilibrium concentration (mg.g -1 ). The regeneration rate (RE) calculated for the two regeneration methods is presented in Fig. 6.

The improvement of BC_CG desulfurization capacities during the first six regeneration cycles under oxidizing atmosphere (CO2/N2), indicates that this method has positively affected the surface porosity of BC_CG. Indeed, the CO2 burns the pore blocking structures created during the catalytic cracking phase, developing a more porous structure. Hence, the elimination of these residues unblocks the pore entrances. This phenomena generates regeneration rates even higher than 100 % [START_REF] Salvador | Regeneration of carbonaceous adsorbents. Part I: Thermal Regeneration[END_REF]. The performance of BC_CG reaches about 90 % after seven cycles.

On the other hand, according to the regeneration rate (RE) the thiophene adsorption improved during the first cycle and then decreased significantly to 38 % in the 5th cycle. This can be explained by possible alterations in the surface chemistry of BC_CG, followed by a reduction of its surface area and deterioration of the pore structure. Thus, thermal regeneration led to a rapid decline in the adsorbent's performance. A similar result was found by Zeng et al. [START_REF] Zeng | Iron-activated bermudagrass-derived biochar for adsorption of aqueous sulfamethoxazole: Effects of iron impregnation ratio on biochar properties, adsorption, and regeneration[END_REF].

Although the high regeneration rate, successive cycles of heating and cooling can lead to mass loss and, very often, a significant deterioration of the adsorbent pore structure, thus reducing the final adsorption capacity and regeneration efficiency. According to the literature, thermal regeneration is the most widely used regeneration method and one of the most economical and efficient methods. On the other hand, several references have also applied regeneration by gasification using either steam or CO2. The comparison of the results of this study with the literature is presented in Table S4. The results of the present study show that the number of regeneration cycles achieved with CO2 is relevant compared to other works. Thus, this research revealed that spent coffee grounds is an economic and accessible feedstock with great potential for biochar production, thiophene removal, good toluene cracking performance, and promising regeneration.

Characterization of the regenerated material

Biochars resulting from the last CO2 and N2 regeneration cycle were characterized. They are noted as follows: Spent BC_CG_CO2 and Spent BC_CG_N2, respectively. The elemental analysis, pH and mass loss of Spent BC_CG_CO2 and Spent BC_CG_N2 are listed in Table 4.

CO2 regeneration shows high O/C and (O+N)/C ratios, leading to low aromaticity and hydrophobicity compared to fresh BC_CG. In addition, an increase in the pH of Spent BC_CG_CO2 and Spent BC_CG_N2 is noted which confirms the adsorption process.

Furthermore, a mass loss of about 14 % is reported under CO2 regeneration against a decrease of 8 % during thermal regeneration. Indeed, in large-scale processes, the mass loss due to gasification must be added to other unavoidable losses, such as the loss occurring during loading and unloading of the adsorption beds. These handling losses can reach 15-20 % after six cycles [START_REF] Salvador | Regeneration of carbonaceous adsorbents. Part I: Thermal Regeneration[END_REF].

The structural analysis shows a substantial decrease of approximately 96 % in the specific surface of Spent BC_CG_N2 compared to that of fresh BC_CG with a reduction of the microporous volume. Therefore, it's assumed that the accumulation of coke on the biochar surface can lead to a loss of its activity [START_REF] Hosokai | Mechanism of decomposition of aromatics over charcoal and necessary condition for maintaining its activity[END_REF]. On the other hand, a significant growth in the specific surface of Spent BC_CG_CO2 is noted in comparison with fresh BC_CG. The specific surface area reaches a considerable value of 286.89 m 2 /g. In fact, in the presence of CO2, the release of volatiles becomes more pronounced, leading to an increase in the specific surface area, the porosity, and the pores volume of the BC_CG. This improvement of the pore contact with the active sites and thus promote the gasification reaction. Thus, during the CO2 regeneration, a cleavage of groups such as hydroxyl, methylene, methyl… etc., leads to a decrease in the mass of biochar [START_REF] Gao | Effect of CO2 on pyrolysis behaviors of lignite[END_REF]. Moreover, a more pronounced destruction of the mesoporosity is observed in the case of Spent BC_CG_N2 than to that of Spent BC_CG_CO2.

This can lead us to understand the effectiveness of regeneration using an oxidizing agent versus simple thermal regeneration. When conventional techniques carry out regeneration, the decrease in the volume of the micropores is accompanied by a displacement towards pores of narrower sizes [START_REF] Ania | Effect of microwave and conventional regeneration on the microporous and mesoporous network and on the adsorptive capacity of activated carbons[END_REF]. Vmicro (cm 3 /g) 0.119±0.00 0.002±0.00

Vmeso (cm 3/ g) 0.014±0.00 nd 1 nd : not detected; traces: concentration below 0.06 wt %.; Fig. 7 illustrates the DFT analysis of BC_CG before and after N2 and CO2 regeneration. In the case of Spent BC_CG_N2, the pore width became almost zero, indicating that the blocking of larger pores occurred to a large extent. This can be attributed to the deterioration of the pore structure due to the high regeneration temperature. On the other hand, a development of the microporosity of Spent BC_CG_CO2 was observed. The micropore size of about 0.59 nm is similar to that of fresh AC. This micropore size is advantageous for thiophene adsorption whose critical diameter is 0.53 nm [START_REF] Yu | Adsorptive removal of thiophenic compounds from oils by activated carbon modified with concentrated nitric acid[END_REF]. Hence, under a CO2 atmosphere, the carbon skeleton's partial gasification would better preserve the porous structure. In addition, the coke deposited on the biochar due to the catalytic decomposition could react with the CO2 so that the texture blocking effect would be mitigated. Thus, the presence of CO2 improves the textural properties of biochar and increases its lifespan. The structures of Spent BC_CG , Spent BC_CG_CO2, and Spent BC_CG_N2 are presented in Fig. 8. We note that the biochar surface had become smoother, and the pores have widened due to the pores filling during the successive adsorption and desorption steps. Furthermore, following CO2 regeneration, the surface seems to have a larger pore structure, and the micropores are easily visible (case of Spent BC_CG_CO2). On the other hand, a porous destruction is observed by enlargement of the pores and a loss of structure between the adjacent pores at the end of the thermal regeneration (case of Spent BC_CG_N2).

Conclusion

In the context of desulfurization of the volatiles derived from the waste tire pyrolysis (TDV), the present work aimed to identify a suitable biochar for the selective removal of thiophene and the tars cracking. For this purpose, two synthetic pyrolytic liquids were evaluated. Hence three materials were tested : biochar from spent coffee grounds (BC_CG), biochar from date seeds (BC_DS) and commercial activated carbon (AC). The effect of the operating conditions was also studied. The results showed that the spent coffee grounds biochar and commercial activated carbon present the best desulfurization results with adsorption values of 1.44 mg/g and 1.08 ±0.02 mg/g respectively. In addition, BC_CG and AC showed the best catalytic activities for toluene cracking at 500 °C with an average conversion rate of 71 % and 91 % respectively. These carbonaceous materials contain K, Ca and Fe as well as functional groups that can act as active sites for adsorption in the ex-situ cracking. The valorization of biomass via pyrolysis has been proven to be a promising route to product a suitable biochar for thiophene adsorption TDV. From an economical and environmental point of view, it is necessary to regenerate this adsorbent in order to be able to reuse it. Hence, two types of regeneration were tested : thermal regeneration and CO2 regeneration. The BC_CG showed a promising regeneration capacities in the presence of CO2 with 14 regeneration cycles.

Based on the above results, BC_CG exhibited better thiophene removal performance compared to the commercial activated carbon. Therefore, it can be concluded that BC_CG has a strong potential as an economical alternative adsorbent material for volatiles derived from used tires pyrolysis enhancement, which can also help decrease spent coffee grounds disposal problems.

Fig. 1 .

 1 Fig. 1. Representative diagram of the post-pyrolysis reactor.

Fig. 2 .

 2 Fig. 2. Pore size distribution of BC_CG, BC_DS, and AC.

Fig. 3 .

 3 Fig. 3. Breakthrough curves of thiophene at 500 °C; 0.20 Nm 3 /h; (a) synthetic mixture 1; (b) synthetic mixture 2.

Fig. 4 .

 4 Fig. 4. Thiophene breakthrough curves at T= 500 °C; Qv= 0.1 Nm 3 /h; (a) mixture 1; (b)

Fig. 5 .

 5 Fig. 5. Synthetic mixture 2, ts2 = 2.50 s; T= 500 °C : (a) Toluene conversion; (b) Limonene

  successive adsorption curves are shown in Fig. S7 b. It is clear that the breakthrough time doubled between the first adsorption and the second adsorption. However, after the first regeneration cycle, the breakthrough time gradually decreases and reaches about 1 min in the last thiophene adsorption cycle. The regeneration rate (RE) represents the adsorbed amount by the regenerated material n times the adsorbed amount of the same fresh material. It serves as an indicator of regeneration efficiency [53].

Fig. 6 .

 6 Fig. 6. Evolution of the regeneration rate according to the regeneration cycle/method.

Fig. 7 .

 7 Fig. 7. Pore size distribution of Fresh BC_CG , Spent BC_CG , Spent BC_CG_CO2, and

Fig. 8 .

 8 Fig. 8. SEM images of Spent BC_CG , Spent BC_CG_CO2, and Spent BC_CG_N2.

Table 1 summarizes the operating conditions. N 2 Syringe driver Electric preheater Micro-GC FID Synthetic liquid Acquisition system Extractor ∆P TE 1 TE 2 TE 3 TE 5 Condenser T TE 6 Fixed bed Adsorbent TE 4Table 1 .

 11 Overview of experimental conditions.

	Materials	BC_CG	BC_DS	AC
	Operations conditions
	h (cm)	20.00 ± 0.20	20.00 ± 0.20	20.00 ± 0.20
	m (g)	56.00 ± 0.13	95.00 ± 0.17	105.20 ± 0.12
	Flow rate (QV) (Nm 3 /h)	0.20 ± 0.005	0.20 ± 0.005	0.20 ± 0.005
		0.10 ± 0.001	0.10 ± 0.001	0.10 ± 0.001
	Residence time (ts) (s)	1.25 ± 0.02	1.25 ± 0.02	1.25 ± 0.02
		2.50 ± 0.07	2.50 ± 0.07	2.50 ± 0.07

Table 2 .

 2 Elemental analyzes, pH, and humidity of biochars and commercial activated carbon.

	Materials	CG	BC_CG	DS	BC_DS	AC
	Elemental and proximate
	analysis
	C	50.40±0.28 82.72±0.50 47.92±0.53 87.27±0.38 85.53±0.14
	H	6.90±0.15 1.61±0.03	6.60±0.22	1.51±0.01	0.70±0.00
	N	2.40±0.05 2.79±0.07	0.90±0.00	1.62±0.01	0.27±0.00
	S	nd 2	nd	nd	nd	nd
	O 1	38.48±0.67 5.88±0.12 43.38±0.27	5.60±0.02	9.90±0.06
	ash	1.80±0.03 7.00±0.24	1.20±0.02	4.00±0.06	3.60±0.02
	H/C	0.14	0.02	0.14	0.02	0.01
	(O+N)/C	0.81	0.10	0.92	0.08	0.12
	pH	5.12±0.16 7.45±0.15	4.32±0.06	7.40±0.21	7.15±0.11
	pHpzc	nm 3 7.50±0.23	nm	7.40±0.27	7.00±0.13
	Textural analysis
	SBET (m 2 /g)	nm 49.44±0.22	nm 225.63±3.14 816.67±2.50
	Vtotal (cm 3 /g)	nm 0.038±0.00	nm	0.14±0.00	0.46±0.00
	Vmicro (cm 3 /g)	nm 0.004±0.00	nm	0.07±0.00	0.20±0.00
	Vmeso (cm 3 /g)	nm 0.026±0.00	nm	0.06±0.00	0.14±0.00
	Average pore diameter (nm)	nm 3.098±0.45	nm 2.56±0.027 2.27±0.035
	1 O (%) = 100 % -C (%) -H (%) -N (%) -S (%) -ash (%);
	2 nd : not detected; traces: concentration below 0.06 wt %.;
	3 nm : not measured;
	Vmicro by t-plot;
	Vmeso by BJH.

Table 3 .

 3 Results of Boehm titration of various materials.

	Adsorbents	BC_CG	BC_DS	AC
	Surface acidity (mmol/g)	0.37±0.04	0.10±0.01	0.21±0.07
	Surface basicity (mmol/g)	0.16±0.06	0.33±0.07	0.48±0.02
	Carboxylic groups (mmol/g)	0.35±0.09	0.14±0.03	0.20±0.04
	Lactone group (mmol/g)	0.23±0.05	0.15±0.07	0.00±0.00
	Phenolic group (mmol/g)	0.12±0.03	0.22±0.08	0.42±0.06

Table 4 .

 4 Preliminary characterization of the regenerated carbonaceous material.

	Materials	Spent BC_CG_CO2	Spent BC_CG_N2
	Elemental and proximate analysis
	C	81.94±1.50	86.76±2.30
	H	1.32±0.02	1.50±0.01
	N	3.07±0.03	2.94±0.02
	S	nd 1	nd
	O	6.67±0.24	1.80±0.01
	H/C	0.016	0.017
	O/C	0.081	0.021
	(O+N)/C	0.119	0.055
	pH	11.15±0.32	9.95±0.22
	m (g)	48.12±0.50	51.56±0.35
	Textural analysis
	SBET (m 2 /g)	286.89±4.50	1.92±0.03

Acknowledgements

This work has been partly carried-out in the frame of the partnership Hubert Curien Program through the PHC UTIQUE 2017 (Project code : 17G1139) and the authors gratefully acknowledge the support of the Mixt Franco-Tunisian Comity (CMCU). The authors are also grateful to the University of Gabes (Tunisia) for its supplementary financial support.


Eloïse Teboul

Bruno Magalhaes

Christophe Geantet

Karine Faure

Off-Line Lc

Off-line LC x SFC -HRMS development for the characterization of the wastewater from algae hydrothermal liquefaction

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

Off-line LC x SFC -HRMS development for the characterization of the wastewater from algae hydrothermal liquefaction Eloïse Teboul 1,2 Bruno Magalhaes 2 , Christophe Geantet 2 , Karine Faure 1

1 Institut des Sciences Analytiques, CNRS UMR 5280 5, rue de la Doua, 69100 Villeurbanne 2 IRCELYON, 2 avenue Einstein, 69100 Villeurbanne

Corresponding author: karine.faure@isa-lyon.fr 1 st dimension RPLC Biofuels based on cultivated algae are in development and in order to limit the environmental impact of this production, the aqueous waste resulting from the hydrothermal conversion process can be used as a culture medium for these algae. Some species of algae such as Chlorella Sorokiniana are even more productive when they grow in this recycled water. However, the accumulation of toxins in the wastewater has to be monitored.

Peak spread: LC 89 %; SFC 96 % and overall occupation rate 75 % (asterisk method)

Numerous ions with same m/z (± 2 ppm)

Experimental peak capacity 600 Evidence that two-dimensional separation is compulsory

• Due to numerous isomers, 1D LC or SFC coupled to HRMS was not sufficient to identify compounds. In this work, an off-line LC x SFC-HRMS method was developed. • Optimization showed that a 10 min-gradient combined with the collection of a large number of fractions are the best conditions in 1 D LC, while 2 D SFC has to be as fast as possible. • Treatment of water-rich fractions was not necessary, providing that the 2 D injection volume was kept below 10 µL. • Orthogonality of the LCxSFC was high with no visible correlation and an occupation rate of 75 %. Peak capacity reached 600 for an overall analysis time of 3.5 h.

Target analysis Extracted Ion Chromatogram (EIC)

Non target analysis Base Peak Chromatogram (BPC)

  Shift in retention time for low-retained compounds Injection of water-rich diluent Peak broadening for compounds eluting later than water (t R 1.0 min) Method development LC x SFC -HRMS conditions Analysis of wastewater from hydrothermal liquefaction of Chlorella Sorokiniana


Martin Gjorgjevski

Nicolas Keriven

Simon Barthelmé

Yohann De Castro

The Graphical Nadaraya-Watson Estimator in Latent Position Models

Étant donné un graphe avec un sous-ensemble de noeuds étiquetés, nous sommes intéressés par la qualité de l'estimateur de la moyenne qui, pour un noeud non étiqueté, prédit la moyenne des observations de ses voisins étiquetés. Nous étudions rigoureusement les propriétés de concentration, les limites de variance et les limites de risque dans ce contexte. Bien que l'estimateur lui-même soit très simple, nous pensons que nos resultats contribueront à la compréhension théorique de méthodes plus sophistiquées telles que les réseaux de neurones de graphes.

Abstract -Given a graph with a subset of labeled nodes, we are interested in the quality of the averaging estimator which for an unlabeled node predicts the average of the observations of its labeled neighbours. We rigorously study concentration properties, variance bounds and risk bounds in this context. While the estimator itself is very simple we believe that our results will contribute towards the theoretical understanding of learning on graphs through more sophisticated methods such as Graph Neural Networks.

Introduction

Given a undirected graph on n + 1 vertices (e.g. nodes represent people) and adjacency matrix A = [a i,j ] (e.g. edges represent social relationships) where all but the (n + 1)-st node have measurementss y i (e.g. salary, living expenses, etc), the graph regression problem adresses prediction of the (continuous valued) measurement y n+1 of the remaining node. While there are various sophisticated designs of Graph Neural Networks [START_REF] Hamilton | Inductive representation learning on large graphs[END_REF][START_REF] Kipf | Semi-supervised classification with graph convolutional networks[END_REF][START_REF] Veličković | Graph attention networks[END_REF][START_REF] Xu | How powerful are graph neural networks[END_REF] which can tackle this problem, little has been done in terms of statistical analysis of any potential solution in the context where the graph is random. In this paper we will consider the simplest estimator, which for the missing label of node n + 1 is taking the average over all of its neighbours, i.e.

To our knowledge, the statistical properties of this estimator have not been studied in the statistical or machine learning literature. In this article we seek to understand the properties of this estimator in a context when the graph is treated as random.

To conduct our analysis we will work with a random graph model known as Latent Position Model (LPM) [START_REF] Peter D Hoff | Latent space approaches to social network analysis[END_REF], where to each node one associates a latent position in space, and typically nodes that lie closer together in space the latent space are more likely to be linked. Due to the nature of this model, the estimator (1) will resemble the Nadaraya-Watson (NW) estimator, a popular regression estimator in the nonparametric estimation literature [START_REF] Alexandre | Introduction to Nonparametric Estimation[END_REF], which just averages measurements in a window centered at the location where signal prediction is to take place. For this reason we decide to title it the Graphical Nadaraya-Watson (GNW) estimator. There are two major differences between the NW and the GNW estimator. First, NW is more expressive in the sense that it comes with a tunable parameter (bandwidth) h n , which sets the size of the neighbourhood over which to average. For the GNW there is no tunable parameter, because neighbourhood size is imposed by the graph. Therefore, the performance of GNW depends on a latent bandwidth h n that is not user-chosen and determines connectivity patterns in the random graph.

We show that the GNW estimator is strongly related to a NW estimator with bandwidth h n . Surprisingly, despite the added randomness in the GNW estimator, its error is within a multiplicative constant of the NW error. In addition, our analysis holds in a regime where the graph is asymptotically almost sparse. Whereas most of the methods discussed in the literature require degree of logarithmic order i.e. d n ≥ C log(n) [START_REF] Lei | Consistency of spectral clustering in stochastic block models[END_REF][START_REF] Imbuzeiro | Concentration of the adjacency matrix and of the laplacian in random graphs with independent edges[END_REF], we show that the variance of GNW will converge to zero as soon as d n → ∞, covering almost all regimes of sparsity (excluding the bounded degree regime d n ≤ D).

Regression in the Latent Position Model 2.1 Background on the LPM

The Latent Position Model (LPM) [START_REF] Peter D Hoff | Latent space approaches to social network analysis[END_REF] is a generative model which samples a random graph on n nodes in two stages. First, a sample of n i.i.d. latent variables X i ∈ R d with density p is drawn. Second, for each pair of nodes i, j a Bernoulli variable with parameter k(X i , X j ) determines if there is an edge between nodes i and j. Here, k is a symmetric kernel on R d , taking values in [0, 1]. The edge generating Bernoulli variables are conditionally independent given the latent variables. Intuitively we are more likely to observe an edge between two nodes with positions that are similar with respect to k.

When k is a similarity kernel (e.g. radial basis function) as in the NW estimator (4), edges are likely to occur between nodes whose latent positions are nearby in the latent space. As an example, when k(x, z) = I(||x -z|| ≤ h), NW and GNW coincide. This is the random geometric graph [START_REF] Penrose | Random Geometric Graphs[END_REF]. By allowing for discontinous kernels, LPMs can instantiate a model with intrinsic community structure known as Stochastic Block Model (SBM) [START_REF] Holland | Stochastic blockmodels: First steps[END_REF]. Despite lack of attention to the graph regression problem, classification has been adressed in the context of LPMs [START_REF] Tang | Universally consistent vertex classification for latent positions graphs[END_REF], albeit using a spectral method while we use local averaging approach.

As large graphs in the real world tend to be sparse [START_REF] Albert | Statistical mechanics of complex networks[END_REF], a significant effort in the community detection literature is dedicated to understanding statistical properties of graphs with low expected degrees [START_REF] Peter | The method of moments and degree distributions for network models[END_REF][START_REF] Can | Sparse random graphs: regularization and concentration of the laplacian[END_REF][START_REF] Lei | Consistency of spectral clustering in stochastic block models[END_REF][START_REF] Imbuzeiro | Concentration of the adjacency matrix and of the laplacian in random graphs with independent edges[END_REF]. In such frameworks one considers asymptotic regimes where k n (x, z) = α n K(x, z) with K being a fixed kernel and α n → 0 as n → ∞. The scaling α n determines the sparsity of the graph where nα n is the expected degree of the graph, up to a multiplicative constant. The parameter α n is not user-chosen (and not userknown). In this sense, we will instead consider asymptotic regimes with

k n (x, z) = α n K x -z h n (2)
where K : R d → [0, 1] is compactly supported, 0 < α n ≤ 1 and h n > 0, with α n , h n being parameters that are not userchosen. We emphasize again that the main difference between the setup for NW (4) and GNW (1) is the freedom to choose the bandwidth h n : this choice is up to the user for NW, for GNW it is not. As k n (x, z) ≤ α n , we see that smaller values of α n will generate sparser graphs. The benefit of Latent Position Models is that they allow us to perform graph analysis in the familiar setting of Euclidean geometry. We will use that correspondence to relate signal prediction of graphs to classical nonparametric regression.

Nonparametric Regression

The regression problem can be stated as estimating a regression function f : R d → R given noisy measurements

Y i = f (X i ) + ϵ i (3)
where f : R d → R with ||f || ∞ ≤ B, ϵ i additive centered noise with finite variance. One is also given the data points X 1 , ..., X n which can be either deterministic (fixed design) or random i.i.d. samples from a distribution with density p (random design). A classical approach is the weighted average Nadaraya-Watson estimator [START_REF] Luc | The uniform convergence of the nadaraya-watson regression function estimate[END_REF][START_REF] Györfi | A Distribution-Free Theory of Nonparametric Regression[END_REF][START_REF] Alexandre | Introduction to Nonparametric Estimation[END_REF]]

fNW (x) = n i=1 Yik(x,Xi) n i=1 k(x,Xi) if n i=1 k(x, X i ) ̸ = 0 0 otherwise (4)
Here, k(x, z) = K x-z h depends on function K : R d → R and the bandwidth h which controls the scale on which the data is being averaged. This parameter needs to be chosen carefully, as too small values of h produce estimates of high variance, while too large values of h give highly biased estimators, an instance of the Bias-Variance tradeoff, a well known phenomenon in statistics. There are two main measures of statistical performance for NW (4), the pointwise and integrated risk. For a given point x ∈ R d , the pointwise risk is given by

R fNW (x), f (x) = E fNW (x) -f (x) 2 (5)
where the expectation is taken over the noise and the data points X 1 , ..., X n for the random design setting (only over the noise for the fixed design). It is also known as mean squared error (MSE). This metric is local in the sense that it only captures statistical information for a particular point. A metric that captures global statistical information is the integrated risk given by

R fNW , f = R fNW (x), f (x) p(x)dx (6)
The integrated risk is also known as mean integrated squared error (MISE) and can be interpreted as the risk for a new random variable X with density p, independent from the data X 1 , ..., X n .

Framework and Outline

We observe a random graph with n + 1 nodes sampled according to a LPM and assume that for all nodes but the last there is a label of the form (3). Conditionally on node n + 1 having latent position x ∈ R d , we write a(x, X i ) for the indicator of an edge between the node n + 1 and node i. With this notation, the GNW estimator (1) becomes

fGNW (x) = n i=1 Yia(x,Xi) n i=1 a(x,Xi) if n i=1 a(x, X i ) ̸ = 0 0 otherwise (7)
For x ∈ R d , we introduce the local expected degree d n (x) by

d n (x) = n R d k n (x, z)p(z)dz (8)
which acts as local sample size. Indeed, d n (x) is the expected degree of node n + 1 given that its latent position is x. Finally, for x ∈ R d and f : R d → R bounded, we introduce the local averaging operator

b n (f, x) = f (z)kn(x,z)p(z)dz kn(x,z)p(z)dz if d n (x) > 0 0 otherwise (9)
which can be considered as a biased version of f (x) (the sequence of operators b n (f, x) resembles an approximation of identity and under mild conditions on K and f discussed in Section 4 one has b n (f, x) → f (x) if h n → 0). Note that b n (f, x) does not depend on α n in (2). Similarly to the pointwise risk (5) of NW (4), we will consider the pointwise risk for GNW (7)

R fGNW (x), f (x) = E fGNW (x) -f (x) 2 (10)
where the expectation is taken over all random variables appearing in the model (edge randomness, latent positions and noise), as well as the integrated risk

R fGNW , f = R fGNW (x), f (x) p(x)dx (11)
Our main result is a bound on the integrated risk [START_REF] Lei | Consistency of spectral clustering in stochastic block models[END_REF]. The approach taken in this paper is to bound [START_REF] Can | Sparse random graphs: regularization and concentration of the laplacian[END_REF] over the support of p and then to integrate the result. To this goal, we follow a bias-variance decomposition inspired approach by introducing the bias proxy

b n (x) = b n (f, x) -f (x) (12)
and the variance proxy

v n (x) = E fGNW (x) -b n (f, x) 2 (13)
We emphasize that the bias and variance proxies do not correspond to the standard notions of bias and variance of GNW and the standard bias-variance decomposition does not hold. Indeed, in Section 3, Proposition (1) we compute E[ fGNW (x)] explicitly. For simplicity we choose to work with the bias [START_REF] Imbuzeiro | Concentration of the adjacency matrix and of the laplacian in random graphs with independent edges[END_REF] and variance [START_REF] Penrose | Random Geometric Graphs[END_REF] proxies instead of the standard bias-variance decomposition. It can be shown that the bias and variance proxies are within O(e -2dn(x) ) of the true bias and variance [START_REF] Gjorgjevski | The graphical nadaraya watson estimator on latent position models[END_REF]. Moreover, for bounded functions f with ||f || ∞ ≤ B, we can replace the bias-variance decomposition by the inequality

R fGNW (x), f (x) ≤ min 2 b 2 n (x) + v n (x) , 4B 2 (14)
Hence it suffices to provide bounds on the bias [START_REF] Imbuzeiro | Concentration of the adjacency matrix and of the laplacian in random graphs with independent edges[END_REF] and variance proxies [START_REF] Penrose | Random Geometric Graphs[END_REF]. As the variance proxy ( 13) can be treated in greater generality, we tackle it first in Section 3. The bias proxy [START_REF] Imbuzeiro | Concentration of the adjacency matrix and of the laplacian in random graphs with independent edges[END_REF] is treated in Section 4, along with the pointwise [START_REF] Can | Sparse random graphs: regularization and concentration of the laplacian[END_REF] and integrated [START_REF] Lei | Consistency of spectral clustering in stochastic block models[END_REF] risks.

Statistical properties of GNW

The main result of this section is a bound on the variance proxy (13) of order 1/d n (x). Additionally, we show that under bounded noise, fGNW (x) [START_REF] Peter D Hoff | Latent space approaches to social network analysis[END_REF] concentrates around b n (f, x) (9) with a rate that is exponentially decaying in d n (x) [START_REF] Holland | Stochastic blockmodels: First steps[END_REF]. Finally, we derive an explicit formula for E[ fGNW (x)] in terms of b n (f, x) and d n (x). Our key insight is the decoupling trick, a novel technique specialised for Bernoulli variables which introduces independence into the weights of GNW that otherwise are ratios of dependent variables. For proofs we refer to the extended version of this paper [START_REF] Gjorgjevski | The graphical nadaraya watson estimator on latent position models[END_REF].

Theorem 3.1 Suppose that f : R d → R with ||f || ∞ ≤ B and E[ϵ 2 1 ] = σ 2 . Then σ 2 1 -e -dn(x) 2 d n (x) ≤ v n (x) ≤ 261B 2 + 65σ 2 d n (x)
For the asymptotic analysis, it states that the variance proxy [START_REF] Penrose | Random Geometric Graphs[END_REF] tends to zero as soon as d n (x) → ∞. Conversely, for bounded degree graphs, in the presence of noise the variance proxy (13) stays bounded away from 0. With stronger assumptions on the distribution on the noise one can prove stronger concentration results, as given by the following Theorem.

P | fGNW (x) -b n (f, x)| ≥ δ ≤ 6 exp(-Cd n (x))
Here, C > 0 depends on δ,B and σ, but not on the sample size n.

We conclude this section by a formula for E[ fGNW (x)]. Interestingly, for smooth kernels, an analogous explicit formula for E[ fNW (x)] is not available in the kernel regression literature.

Proposition 1 Suppose that ||f || ∞ ≤ B. Then E fGNW (x) = b n (f, x) 1 -(1 - d n (x) n ) n
4 Risk of GNW So far we have bounded the variance proxy [START_REF] Penrose | Random Geometric Graphs[END_REF] in terms of the local expected degree (8) (Theorem 3.1). In this section we bound the pointwise [START_REF] Can | Sparse random graphs: regularization and concentration of the laplacian[END_REF] and integrated [START_REF] Lei | Consistency of spectral clustering in stochastic block models[END_REF] risk. In view of the bound ( 14) to bound the risk [START_REF] Can | Sparse random graphs: regularization and concentration of the laplacian[END_REF] we need to bound the bias proxy [START_REF] Imbuzeiro | Concentration of the adjacency matrix and of the laplacian in random graphs with independent edges[END_REF], and we also need to understand the relationship between the degree d n (x) and the parameters α n and h n . These problems are addressed in Subsection 4.1. In order to bound the integrated risk [START_REF] Lei | Consistency of spectral clustering in stochastic block models[END_REF], we need to control the pointwise risk [START_REF] Can | Sparse random graphs: regularization and concentration of the laplacian[END_REF] uniformly over the support of the data distribution p. This problem is addressed in Subsection 4.2.

Bias bound and Pointwise risk of GNW

In the NW literature, it is standard to assume compactly supported kernels [START_REF] Györfi | A Distribution-Free Theory of Nonparametric Regression[END_REF][START_REF] Alexandre | Introduction to Nonparametric Estimation[END_REF], an assumption that we adopt as well.

Assumption 1 There exists M 1 , M 2 > 0 such that for all z ∈ R d we have 1 2

I (||z|| ≤ M 1 ) ≤ K(z) ≤ I (||z|| ≤ M 2 )
We denote the support of the measure induced by p with Q. In order to control the bias proxy [START_REF] Imbuzeiro | Concentration of the adjacency matrix and of the laplacian in random graphs with independent edges[END_REF], one needs a regularity assumption on f . We work under the assumption of Hölder continuity. For 0 < a ≤ 1 and L > 0, we say that f belongs in the Hölder class Σ (a, L) on Q if for all x, z ∈ Q, we have

|f (x) -f (z)| ≤ L||x -z|| a .
Assumption 2 There exist 0 < a ≤ 1, L > 0 and B > 0 such that f ∈ Σ (a, L) on Q and sup

R fGNW (x), f (x) ≤ C 1 h 2α n + C 2 nα n h d n p 0 (x) (16)
where C 1 , C 2 depend on the various parameters appearing in the assumptions, and are explicitly given in the extended version of this paper [START_REF] Gjorgjevski | The graphical nadaraya watson estimator on latent position models[END_REF].

Integrated risk of GNW

Finally, to bound the integrated risk (11) using Equation ( 16), one needs to deal with the quantity p 0 (x) [START_REF] Alexandre | Introduction to Nonparametric Estimation[END_REF] appearing in the expression. The simplest way to do this is to assume that for all x ∈ Q, we have p(x) ≥ p 0 . In that case one can take p 0 (x) ≡ p 0 in Equation ( 16), giving

R fGNW , f ≤ C 1 h 2α n + C 2 nα n h d n ( 17
)
This rate is classical for the NW estimator (4) under uniform density assumption [START_REF] Györfi | A Distribution-Free Theory of Nonparametric Regression[END_REF][START_REF] Alexandre | Introduction to Nonparametric Estimation[END_REF]. One can get a slightly weaker rate for noncompactly supported densities p.

Assumption 4 There exist 0 < β ≤ 1 and L > 0 such that p ∈ Σ(β, L) and p 1/2 (x)dx < ∞ Theorem 4.3 Suppose that Assumptions 1, 2, 3, 4 hold. If

M 1 h n < r 0 , R fGNW , f ≤ C 1 nα n h d+β n + C 2 h min{2α,β/2} n where C 1 , C 2 depend on B, σ 2 , c 0 , d, α, β, M 1 , M 2 and L.
Typically one optimizes the rates in Equation ( 17) or Theorem 4.3. However, as the user has no control of h n , we provide a range of values of h n depending on nα n such that for Theorem 4.3 the integrated risk ( 11) is uniformly controlled over the class of regression functions and densities apperaing in the assumptions of 4.3. Set γ = min (2a, β). There exist values C 3 , C 4 > 0 depending on the various parameters apperaing in the assumptions (explicitly given in [START_REF] Gjorgjevski | The graphical nadaraya watson estimator on latent position models[END_REF]) such that if r ≤ γ d+β+γ , h n < min ( r0 M1 , 1) and

(nα n ) -1-r d+β ≤ C 3 h n ≤ (nα n ) -r γ (18) then R fGNW , f ≤ C 4 (nα n ) r (19)
Note that for r = γ d+β+γ the bound in (19) is the strongest and the interval in (18) shrinks to a point.

Conclusion and perspectives

We showed that both the pointwise and integrated risk bounds of the risk of fGNW are similar to ones of the classical NW estimator. If α n and h n fall into the suitable range of values (i.e. h n → 0 and nα n h d n → ∞) then GNW will perform well. As GNW uses only one-hop neighbourhood information, it does not take advantage of the global graph structure, it would be interesting to compare it with graph spectral based regression estimators (such as graphical Kernel Ridge Regression). It would also be interesting to understand if estimating the latent positions could be statistically beneficial for estimation.

Figure 1 -

 1 Figure 1 -Sampling a LPM: Left -generating uniformly 300 latent positions on [0, 1] 2 . Right: generating a graph with gaussian kernel k(x, y) = exp -||x-y|| 2 h 2

Theorem 3 . 2

 32 Suppose that f : R d → R is bounded with ||f || ∞ ≤ B and the noise variables satisfy |ϵ i | ≤ σ. Then

  x∈Q |f (x)| ≤ B Lemma 4.1 Suppose Assumptions 1 and 2 hold.

Assumption 3

 3 sup x∈Q |b n (f, x) -f (x)| ≤ LM a 2 h a nThe classical NW estimator (4) is known to perform poorly near the boundary of the support and in regions where the density function p is low. The following definition describes sets for which the boundary issue can be mitigated. We say that G ⊆ R d has (r 0 , c 0 )-measure-retaining property if for all x ∈ G and all r ≤ r 0 , m(G ∩ B r (x)) ≥ c 0 m (B r (x)),where m denotes Lebesgue measure on R d . There exist r 0 , c 0 > 0 such that Q has the (r 0 , c 0 )-measure-retaining property.

Lemma 4 . 2

 42 Suppose that Assumption 1, 3 hold.If M 1 h n < r 0 and x ∈ Q is such that inf z∈Q |x-z|≤M1hn p(z) ≥ p 0 (x) > 0 (15) Then 1 d n (x) ≤ C nα n h d n p 0 (x)where C depends on c 0 , M 1 and d.Combining Theorem 3.1 and Lemma 4.2, we see that if Assumptions 1 and 3 hold, then v n (x) ≤ C2 nαnh d n p0(x) . If, in addition Assumption 2 holds then, combining the last bound with Equation (14) and Lemma 4.1, we get

Acknowledgement

This work was partly supported by ANR GRandMa (ANR-21-CE23-0006); MIAI@Grenoble Alpes (ANR-19-P3IA-0003) and the Ampère Scholarship of Excellence of ENS de Lyon.