M Clavel

X-RAYS FROM THE GALACTIC CENTER AND ACTIVITY OF THE CENTRAL SUPERMASSIVE BLACK HOLE

Keywords: Galaxy, center -X-rays, general

X-rays from the Galactic center and activity of the central supermassive black hole

M.

Introduction

At about 8 kpc, the Galactic center is the closest galactic nucleus and it is therefore an excellent laboratory to study the complexity of these inner regions and, in particular, the interactions between a supermassive black hole and its environment. Since their respective launch about two decades ago, XMM-Newton and Chandra have been revealing an increasing number of X-ray features in the central degrees of our Galaxy, including numerous point and extended sources, such as X-ray binaries, stellar clusters or supernova remnants, but also more diffuse emission having thermal or non-thermal origins (see e.g. Ponti et al. 2015). Radio and sub-millimeter observations of this exact same region show that it also contains dense molecular gas and this is why it has been named the Central Molecular Zone (CMZ, Morris & Serabyn 1996).

At the very center, there is the supermassive black hole, Sagittarius A , which is one of the very faint point sources seen in X-rays. Indeed, the 4 × 10 6 M black hole Sgr A is the least luminous known supermassive black hole. While its quiescent X-ray luminosity is around L X ∼ 10 33 erg s -1 , the X-ray counterpart of Sgr A also experiences daily flares having a timescale of about an hour and during which the X-ray luminosity can increase by up to two orders of magnitude (e.g. Neilsen et al. 2013;Haggard et al. 2019, and references therein). This means that, since the advent of X-ray astronomy, the emission of Sgr A has remained at least eight orders of magnitude lower than its Eddington luminosity: it is presently a dormant supermassive black hole.

Therefore, the current feedback from Sgr A onto the Galactic center environment is expected to be extremely weak. Nevertheless, detailed studies of its surroundings have revealed signatures of important feedback, attesting to a higher level of activity from the Galactic nucleus in the past, with different tracers probing timescales from centuries to million years ago. We review the observations of the corresponding features detected in X-rays, namely: light echoes tracing the most recent past (Section 2) and bipolar outflows generated over much longer timescales (Section 3). The constraints obtained on the past activity of Sgr A are then summarized in Section 4, along with relics seen at other wavelengths.

X-ray echoes tracing several short outbursts in the last centuries

The past X-ray emission from Sgr A is propagating in the inner region of our Galaxy, where it is reflected by the dense molecular clouds of the CMZ, creating light echoes. The X-ray spectrum of the reflecting clouds is then SF2A 2019 characterized by a continuum component created by the Compton scattering of the incident signal, absorbed at low energy, and on top of which there are fluorescent emission lines, including a strong Fe Kα line at 6.4 keV (see e.g. Sunyaev & Churazov 1998). This fluorescent line is therefore a powerful tool to trace the echoes of Sgr A 's past outbursts in the CMZ and the extent of this region combined with the sensitivity of the current instruments allows to probe the brightest events that occurred in the last thousand years.

The 6.4 keV emission correlated with molecular clouds has been detected across the whole CMZ and, in particular, towards the main molecular complexes: Sgr A, Sgr B and Sgr C (see Figure 1; Terrier et al. 2018, and references therein). Several attempts were made to constrain Sgr A 's past light curve from the Fe Kα flux measured in individual molecular clouds but, due to the lack of constraints on the line-of-sight position of these clouds, this technique led to divergent results in terms of age and duration of the corresponding events (e.g. Ponti et al. 2010;Capelli et al. 2012;Ryu et al. 2013). The next step was then to use the variability of the echoes, also detected in an increasing number of molecular structures (e.g. Inui et al. 2009;Ponti et al. 2010;Clavel et al. 2013Clavel et al. , 2014;;Zhang et al. 2015;Chuard et al. 2018;Terrier et al. 2018), to better constrain Sgr A 's past light curve. This alternative technique consists in studying the fine X-ray variations to spotlight the simultaneity in the illuminated cloud behaviors. Using Chandra high spatial resolution, two different time behaviors were identified among the brightest clouds of the Sgr A complex (Clavel et al. 2013). The constraints implied on both the density and the distribution of the clouds make these two behaviors no longer compatible with one single illuminating event. The conclusion of this work is that at least two distinct outbursts, having a luminosity of a few 10 39 erg s -1 and a duration of about 2 and more than 10 years, are currently propagating in the inner region of our Galaxy (Clavel et al. 2013). To have more information about these events, a similar systematic analysis was performed making use of the light curves obtained across the whole CMZ with XMM-Newton from 2000 to 2012. This study concludes that all bright clouds are significantly varying over the 12-year period (see Figure 1), so the existence of a putative bright event lasting for several decades and currently propagating in this region is excluded. Moreover, all variations observed are compatible with the two time behaviors described previously and can therefore be explained by the same short events (Terrier et al. 2018). Since the spectral shape of the reflected emission depends on the line-of-sight position of the reflecting clouds, dedicated spectral modeling can provide the age of the events identified through variability studies. The reflected spectra expected for different geometries were computed using Monte Carlo simulations and then used to fit the spectra observed in Sgr B and Sgr C (Walls et al. 2016;Chuard et al. 2018). This technique provided the line-of-sight position of several clumps. The 3D distribution obtained is only compatible with the presence of at least two different past events from Sgr A , in full agreement with the results described in the previous paragraph. Furthermore, all slow varying features are compatible with being illuminated by the older event occurring about 240 years ago, while the ones seeing faster variations could be illuminated by a more recent, and probably shorter event, occurring about 140 years ago (Chuard et al. 2018). These results rule out the single-event scenario assumed to derive the line-of-sight position of the reflecting features from alternative techniques (see e.g. Churazov et al. 2017), but it does not exclude the existence of a putative third event that would be propagating in more distant portions of the CMZ than the ones investigated by Chuard et al. (2018).

Galactic latitude

As a summary, the X-ray signal reflected within the CMZ attests than the illuminating source experienced more than one order of magnitude variations over few years only, and several short events producing hard X-rays that are at least a million times brighter than Sgr A 's quiescent state are needed to explain this variability. Possible alternatives to the supermassive black hole to produce such outbursts have been discussed and discarded based on the energetic involved and the spectral shape required for these events (see Terrier et al. 2010;Clavel et al. 2013;Mori et al. 2015;Zhang et al. 2015;Krivonos et al. 2017, for more details, including spectral constraints derived from INTEGRAL and NuSTAR hard X-ray observations). Therefore, there is strong evidence that Sgr A has been brighter in the last centuries but the origin of these past outbursts remains uncertain. Several scenarios have been proposed (e.g. involving partial or full tidal disruptions of stars, planets, asteroids or gas clouds, Yu et al. 2011;Zubovas et al. 2012;Czerny et al. 2013), but a theoretical model matching all the constraints recently derived from the observations, in terms of recurrence, duration, luminosity and spectrum of Sgr A 's past events has not yet been published.

3 X-ray lobes and chimneys carved by an enhanced level of activity in the last million years.

Periods of intense activity from the supermassive black hole are also expected to generate outflows that could possibly carve high-energy features in the Galactic nucleus environment, visible as an excess of thermal emission.

The first possible such signature has been detected close to Sgr A and is known as the X-ray lobes (Baganoff et al. 2003;Morris et al. 2003;Markoff 2010;Heard & Warwick 2013;Ponti et al. 2015). They are 15-pc bipolar lobes extending from Sgr A towards the Galactic north and south, with a plasma temperature kT ∼ 0.7-1 keV, and are showing a strong pressure gradient (see Figure 2, left). Due to their apparent symmetry about the Galactic plane, these structures have been interpreted as outflowing gas originating from the central few parsecs of our Galaxy. This region contains the supermassive black hole and a young cluster of massive stars, as well as dense molecular gas forming the so-called circumnuclear disk (CND, see Genzel et al. 2010, for a review). The location and shape of the CND is such that it could have collimated an isotropic outflow emanating from Sgr A itself or from its immediate vicinity, creating the bipolar lobes of hot plasma (see e.g. Morris et al. 2003). Assuming that the X-ray gas is filling the whole volume of the lobes, and considering an outflow velocity comparable to the speed of sound set by the gas temperature, the thermal energy of each lobe is ∼ 6 × 10 50 erg and they have a sound-crossing time t s ∼ 3 × 10 4 yr (Ponti et al. 2019). The average power of the outflow is therefore allowing for several possible origins.

Over the time scale needed for the X-ray lobes to form, two continuous processes can be considered. First, winds from massive stars in the central parsec could be an important contribution in terms of energetic since their velocities are high enough to produce X-rays when they interact with the interstellar medium. However, the stellar cluster being several million years old and the lobes only ∼ 3 × 10 4 years old, this scenario does not allow to reproduce the morphology of the X-ray lobes and in particular their pressure gradient (e.g. Markoff 2010). The second possible continuous process is a putative outflow generated by the current level of accretion onto Sgr A . Indeed, only a very small fraction of the material captured at the Bondi radius is expected to be effectively accreted onto Sgr A (Wang et al. 2013), so the rest could be converted into outflows sculpting the environment of the black hole. If the energy budget could match the one needed to produce the lobes, again, the shape of the X-ray features is not in favor of this second continuous scenario but is rather pointing to explosive or outbursting events (e.g. Ponti et al. 2015).

Therefore, episodic scenarios have also been proposed. For instance, the X-ray lobes could be a supernova remnant (SNR). Indeed, the central cluster is expected to produce between one and ten supernova explosions every 10 5 years, so these events could contribute to power the lobes (e.g. Ponti et al. 2015). In this case, the compact object created during this event could be one of the known X-ray sources in the vicinity of Sgr A . Several candidates have been proposed and a hydrodynamic simulation taking into account the complexity of In each image, the three colors are tracing different X-ray energies soft (red) to hard (blue, see the corresponding inserts in each panel). These two bipolar structures are roughly symmetric about the Galactic plane, with the X-ray lobes nested at the base of the X-ray chimneys (in the right panel, the 15-pc lobes are the white tiny features surrounding Sgr A ). The slight asymmetry in the shape of the X-ray lobes can be explained by foreground absorption (Ponti et al. 2015), while the differences between the northern and southern chimneys could result from asymmetries in the dense interstellar medium distribution (Ponti et al. 2019).

the Galactic center environment has been able to reproduce the morphology of the X-ray lobes with a SNR that would be associated with the young magnetar SGR J1745-2900 (Yalinewich et al. 2017). From the X-ray point of view, this scenario is robust. However, no non-thermal radio counterpart of the lobes has been detected and this would be a major puzzle if the young supernova origin is confirmed (e.g. Ponti et al. 2015). This is why past outbursts from Sgr A are also considered as a relevant hypothesis. The estimated age of the X-ray lobes is the same order of magnitude as the recurrence time computed for tidal disruptions of stars by the supermasssive black hole at the Galactic center (1-2×10 4 yr, Alexander 2005), so one tidal disruption event, with a luminosity close to the Eddington one, could be responsible for the corresponding outflow. Nevertheless, less luminous but more frequent outbursts, such as the ones characterized from the X-ray echoes (see Section 2), can also be considered as a viable scenario. If we further assume that more prominent outbursts happened every 5000 years or so, such a collection of past events would also provide an explanation for the brighter knots observed within both lobes (Markoff 2010;Ponti et al. 2015). So the 15-pc X-ray lobes could be tracing Sgr A 's past activity over the last ∼ 3 × 10 4 years.

Furthermore, all the energetic processes, either continuous or episodic, that have been described in the previous paragraphs are also expected to occur over much longer time scales, and could therefore have shaped even more extended structures. The first hint of such signatures has been an excess of soft X-ray emission detected by Suzaku to the south of the Galactic center. Its X-ray spectrum is consistent with a recombining plasma, its absorption indicates that it is located in the inner regions of our Galaxy and a past starburst activity or an episode of enhanced activity from Sgr A have been proposed as most plausible origins (Nakashima et al. 2013). More recently, a similar excess interpreted as a magnetized hot gas outflow from the Galactic center was also detected by Suzaku towards the Galactic north (Nakashima et al. 2019). The high-latitude survey performed by XMM-Newton in 2016-2018 confirms the existence of these brighter regions and revealed that they are part of a 160-pc bipolar structure connected to the central region and extending on both sides of the Galactic plane (see Figure 2, right). These features, called the X-ray chimneys, have a plasma temperature kT ∼ 0.7 keV and no pressure gradient is detected within them, which means that they are likely close to hydrostatic equilibrium. Their thermal energy is about 4 × 10 52 erg and their sound crossing time t s ∼ 3 × 10 5 yr, so the average power of the corresponding outflow is possibly only about a factor five greater than the one derived for the 15-pc lobes (Ponti et al. 2019). The energetic events shaping the X-ray chimneys could therefore be very similar to the ones discussed as possible origins of the X-ray lobes, and the lobes could even trace the most recent episode of energy injection into the chimneys.

However, the base of the chimneys is likely not limited to the X-ray lobes but could instead correspond to a ∼ 50-pc region surrounding Sgr A . So, outflows from past energetic events, such as supernova explosions, occurring in this wider region could also be injected into the chimneys. Furthermore, Ponti et al. (2019) suggested that the X-ray chimneys could have transported powerful outflows from the Galactic center into the Halo, so the total energy conveyed by this channel could be greater than the one still radiating today. This scenario is supported by ROSAT, Suzaku, Chandra and XMM-Newton detections of X-ray emission at even higher latitudes, forming what could be the base of the Fermi bubbles (see Figure 2, right; Ponti et al. 2019, and references therein). If confirmed the power transported by the chimneys could be up to four orders of magnitude greater than the one computed from their X-ray emission (e.g. Su et al. 2010). Such powerful outflows could be associated with star formation processes at the Galactic center in the last few million years, and the contribution from Sgr A could be limited. Nevertheless, scenarios involving the most extreme events such as tidal disruptions of stars by the supermassive black hole or periods of intense accretion transforming the Galactic center into an active galactic nucleus (AGN) cannot be excluded (see also Section 4).

Conclusions

By studying the X-ray features present in the inner regions of our Galaxy, it is possible to constrain the duty cycle of our dormant supermassive black hole.

The X-ray light emitted by Sgr A in the last centuries is still propagating in the central molecular zone, generating X-ray echoes. The variability and the spectral properties of this reflected emission reveals the existence of several short outbursts from Sgr A over the corresponding period. Despite an increasing precision in the description of these past events, their exact nature is still unknown.

The outflows generated by the past activity of our Galactic nucleus are also expanding away from the center, carving X-ray thermal features. The lobes and chimneys detected in the central degrees suggest periods of enhanced activity from the central region over the last million years. However, the link with Sgr A 's past activity is uncertain as processes associated with stellar formation and supernova explosions have also been proposed. So, the energy levels derived from these X-ray structures can be considered as upper limits on the past luminosity of the supermassive black hole.

This review focused only on X-ray features and is therefore not exhaustive since relics of past activity from our Galactic nucleus have also been detected at other wavelengths. This is for instance the case of the Fermi bubbles already mentioned in Section 3. These large bubbles detected in γ-rays extend over ∼ 10 kpc towards the Galactic poles and were likely generated by an important injection of energy in the last ten million years, such as could have been produced by past accretion events onto Sgr A or a nuclear starburst (Su et al. 2010). If these two scenarios are able to reproduce the γ-ray emission, it is not the case of the Hα emission detected along the Magellanic Stream. The latter is too bright to be the result of stellar formation and favors an AGN-like activity of the supermassive black hole, with an intense UV radiation emitted few million years ago (Bland-Hawthorn et al. 2013). Therefore, it is likely that the gaseous accretion disk that triggered the formation of the central stellar cluster about six million years ago also significantly increased the accretion rate onto Sgr A , creating a period of intense past activity that likely contributes to the large X-ray and γ-ray relics detected in the Galactic environment.

Fig. 1 .

 1 Fig. 1. Background-and continuum-subtracted intensity maps of the central molecular zone measured by XMM-Newton at 6.4 keV in 2000-2001 (Top) and 2012 (Bottom). The dotted square regions highlights areas where significant variations were detected. Figure published by Terrier et al. (2018).

Fig. 2 .

 2 Fig. 2. Left: Chandra observation of the 15-pc X-ray lobes. Figure adapted from Muno et al. (2004). Right: XMM-Newton survey of the 160-pc X-ray chimneys. Figure published byPonti et al. (2019). Both panels are displayed in Galactic coordinates. In each image, the three colors are tracing different X-ray energies soft (red) to hard (blue, see the corresponding inserts in each panel). These two bipolar structures are roughly symmetric about the Galactic plane, with the X-ray lobes nested at the base of the X-ray chimneys (in the right panel, the 15-pc lobes are the white tiny features surrounding Sgr A ). The slight asymmetry in the shape of the X-ray lobes can be explained by foreground absorption(Ponti et al. 2015), while the differences between the northern and southern chimneys could result from asymmetries in the dense interstellar medium distribution(Ponti et al. 2019).

Univ. Grenoble Alpes, CNRS, IPAG, F-38000 Grenoble, France c Société Francaise d'Astronomie et d'Astrophysique (SF2A)

SF2A 2019

MC acknowledges financial support from the French National Research Agency in the framework of the "Investissements d'avenir" program (ANR-15-IDEX-02) and from CNES.


Anita Haeussler

Srirat Chuayboon

Stéphane Abanades
email: stephane.abanades@promes.cnrs.fr

Solar redox cycling of ceria in a monolithic reactor for two-step H 2 O/CO 2 splitting: Isothermal methane-induced reduction versus temperature-swing cycle

Solar thermochemical cycles provide an efficient route to convert solar energy into valuable chemical energy carriers, such as solar fuels. The splitting of CO2 and H2O using metal oxide redox pairs permits to produce clean synthetic fuels. Furthermore, CO2 is upgraded into a valuable product (CO) that can be further converted to liquid hydrocarbon fuels when combined with H2. Among the possible candidate materials for two-step redox cycling, ceria appears promising given the high oxygen mobility and exchange property in the crystal lattice offering large amounts of oxygen vacancies ( in CeO2-) while retaining fluorite structure, rapid and reversible transition between Ce 4+ and Ce 3+ oxidation states, and stable crystal structure during cycling. During a first step at high temperature, the metal oxide (CeO2) is reduced by releasing O2 thus creating oxygen vacancies in the ceria structure. In a second step at lower temperature, the non-stoichiometric oxide (CeO2-) is re-oxidized with CO2 or H2O, leading to the production of CO or H2 (solar fuels). This temperature-swing operating mode requires high temperatures during the reduction and a temperature gap between the two steps, which impacts the solar-to-fuel efficiency. The use of methane as reducing agent in the reduction step can be used to decrease the reduction temperature and allow isothermal operation. A comparison between the two operating modes, namely isothermal methane-induced reduction versus temperature-swing cycle, was performed in a monolithic solar reactor integrating ceria porous foams. The reduction of ceria using methane results in a higher reduction extent and fuel production with lower cycle temperatures (950-1050°C) at the expense of using a carbonaceous reducer. Thus, the temperature-swing operation appears as a more suitable long-term option for sustainable solar fuel production, but shows more stringent requirements on the reacting materials and solar reactor. The fuel production in the temperature-swing cycle was further increased by decreasing the pressure or increasing the reduction temperature, while the material performance stability was not altered after extended cycling.

INTRODUCTION

Conventional energy production pathways show adverse environmental impacts; therefore, alternative renewable energies should be developed to produce clean energy carriers. Thermochemical cycles using concentrated sunlight for H 2 O and CO 2 splitting into fuels is an elegant way to convert and store intermittent solar energy into chemical fuels [START_REF] Steinfeld | Solar Thermochemical Process Technology[END_REF][START_REF] Falter | [END_REF][3]. The solar-driven two-step ceria redox cycle for CO (or H 2 ) production from CO 2 (or H 2 O) dissociation offers clean sustainable renewable fuels. When coupled to the capture of CO 2 from atmospheric air or exhaust combustion gases, the CO production can be considered as carbon neutral. Ceria is considered as a state-of-art material for the thermochemical cycles due to its ability to maintain its crystallographic structure over a large range of nonstoichiometry and operating conditions. On top of that, it shows a thermodynamically favorable and rapid oxidation [4][5][6][7][8][9][START_REF] Koepf | Liquid fuels from concentrated sunlight[END_REF].

SolarPACES 2019 AIP Conf. Proc. 2303, 170009-1-170009-11; https://doi.org/10.1063/5.0028582

Published by AIP Publishing. 978-0-7354-4037-1/$30.00

The first step of the redox cycle is an endothermic reduction at high temperature (Eq. 1) using concentrated solar energy as external heat source. The second step is an exothermic oxidation (Eq. 2) with an oxidant gas (CO 2 or H 2 O) at lower temperatures:

(1) [START_REF] Falter | [END_REF] The production of CO and H 2 in a solar reactor with ceria as reactive material has been demonstrated [START_REF] Chueh | [END_REF][12][13][14]. Currently, the highest energy efficiency of 5.25% was reached in a monolithic reactor with ceria reticulated foam [15].

However, the major drawbacks of this process are: the high operating temperature requirement that exceeds 1400°C during the reduction step [16], and the temperature swing between reduction and oxidation steps, which in turn adversely results in high sensible heat losses, thereby decreasing solar-to-fuel efficiency.

Alternatively, employing hydrocarbon as reducing agent such as methane (CH 4 ) in the reduction step can lower the reduction temperature until the cycle can be operated isothermally (both reduction and oxidation steps at ~1000°C), thereby avoiding sensible heat losses [17]. In this case, the reduction reaction can be represented by:

(3)

The cycle consists of both endothermic ceria reduction and simultaneous partial methane oxidation (Eq. 3), followed by exothermic reduced ceria oxidation with H 2 O/CO 2 (Eq. 2). The main advantages are the isothermal operation, the production of a syngas suitable for methanol synthesis during the reduction, and the absence of costly catalysts for methane reforming.

This study aims to present the influence of operating parameters on the thermochemical performance of a monolithic solar reactor. A comparison between isothermal (using CH 4 for ceria reduction) and temperature-swing cycles is performed.

EXPERIMENTAL SETUP AND METHODS

The experiments were performed using a 1.5 kW th vertical-axis solar facility (composed of a sun-tracking heliostat reflecting sunlight to a facedown parabolic concentrator) at PROMES-CNRS (France). The solar monolithic reactor for the two-step H 2 O/CO 2 splitting either performed with reduction step using methane (isothermal) or with temperature-swing operation driven by highly concentrated solar radiation is depicted in Figure 1. The reactive foam (~ 50 g) was placed in a ceramic cavity (alumina) with inner diameter of 50 mm and 80 mm height for temperatureswing cycles. The cavity aperture was positioned at the focal point of the solar concentrator. The incident concentrated solar power was thus absorbed by the cavity after passing through a transparent hemispherical window. Reticulated porous ceria foams with 87% porosity and cylindrical geometry were prepared as reactive oxygen carrier materials via a replication technique using polymer templates [START_REF] Karl | United States[END_REF]. Two foams were tested with different pore size densities: 10 and 20 pores per inch (ppi), designated in the following as 10 ppi and 20 ppi foams. The temperature was monitored at different locations of the reactor thanks to two B-type thermocouples (T1 and T2 as represented on Figure 1) and a solar-blind pyrometer (temperature range: 500-2500°C, wavelength: 5.14 m) [START_REF] Hernandez | [END_REF], pointing into the cavity through a fluorine window. The cavity pressure was also measured by a pressure sensor. During temperature-swing cycles, the reactor was solar-heated to the desired temperature in pure Ar (1.2 NL/min) to release O 2 (1400-1450°C during thermal reduction) and cooled during oxidation with CO 2 (or H 2 O) to produce CO (or H 2 ) in the 800-1100°C range. The CO 2 flow rate was 0.4 NL/min (25% mole fraction) and H 2 O flow rate was 200 mg/min (17% mole fraction). In the reduction step at low pressure, a pump at the reactor outlet was used to decrease the total pressure in the reactor chamber swept continuously with inert gas flow. Concerning isothermal CH 4 -induced reduction, another reactive foam with the same porosity was used and experiments were conducted at 950, 1000, and 1050°C while alternating the flow between CH 4 (0.2 NL/min) along with Ar (0.2 NL/min) for reduction step (50% inlet CH 4 mole fraction) and H 2 O (200 mg/min) for subsequent oxidation step (55% steam mole fraction at inlet). The produced gases (O 2 , H 2 , CO for temperature-swing redox cycle and H 2 , CO, CO 2 , CH 4 for methane-promoted redox cycle) were analyzed at the output thanks to online gas analyzers (electrochemical sensor for O 2 , thermal conductivity detector for H 2 , and infrared sensors for CO, CO 2 , CH 4 , calibrated with standard gases). The averaged oxygen nonstoichiometry ( ), fuel yields, gas production rates, and reactor performance were experimentally studied and compared.

RESULTS AND DISCUSSION

Temperature-Swing Cycles

Concerning the temperature-swing mode, different parameters, such as the reduction and oxidation temperatures, and the total pressure applied during the reduction, have been investigated in order to optimize the fuel production yield. The influence of the reduction temperature was studied by determining its impact on the O 2 released during reduction and subsequent fuel production during oxidation. Figure 2 presents two cycles performed with different reduction temperatures (1400°C and 1450°C for T1) followed by a non-isothermal re-oxidation with CO 2 (oxidation during natural cooling without any solar power input, CO 2 injection at about 1000°C). O 2 and CO are produced during reduction and oxidation respectively. Their amounts is calculated by integrating the production peaks. Given the slight withdrawal of T2 thermocouple from the cavity to avoid its direct exposure to solar flux, T2 is lower than T1, even if it is located in the upper cavity region where the solar input is maximum. The temperature measured by the pyrometer can be 70°C higher than T1. This highlights a temperature gradient in the foam between the upper part of the foam directly exposed to the solar flux and the bottom part distant from the solar input. A temperature gradient in the foam leads to a non-uniform reduction extent reached in the foam during the reduction step. Therefore the reported nonstoichiometries ( ) calculated in the following are the foam-averaged non-stoichiometries. T1 was considered as the reference temperature in the following to ensure that the whole foam is heated above this temperature. The temperature increase of 50°C during the reduction step (from 1400 to 1450°C) enhanced the O 2 produced by 44 % (103.3 to 139.0 mol/g, =0.036 to 0.048) then leading to a 15 % increase of the CO amount produced. The peak rate of CO production also increased accordingly. Increasing the reduction temperature thus permits to improve the reduction step, in turn leading to an improved fuel production yield. However it should be kept in mind that the maximum temperature is limited by the reactor materials and the reactive material itself. Higher temperatures may result in materials sintering, losses by sublimation and increased heat losses. Furthermore the increase of the reduction temperature leads to an increase of the temperature swing between both cycle steps, thus favoring heat losses through materials heating/cooling. The other parameter investigated during the reduction step is the pressure. According to thermodynamics, the oxygen partial pressure directly affects the reduction extent reached at equilibrium by the non-stoichiometric material. The oxygen partial pressure was varied by modifying the total pressure. Two cycles were performed at atmospheric and reduced pressure, respectively, as represented in Figure 3. The decrease of the pressure from 865 hPa to 110 hPa during the reduction leads to an improvement of the reduction extent by an increase of 48% in the oxygen amount produced (86 to 125 mol/g, =0.030 to 0.043). Consequently, the increase of the reduction extent is followed by an improvement of the CO production yield from 160 mol/g to 258 mol/g. However, the energy consumption by pumping leads to energy penalties that must be balanced with the gains in the fuel productivity. A decreased pressure during the reduction step has a beneficial effect on the thermochemical cycling performance as it permits to increase both the oxygen amount produced and the fuel production yield. The impact of the oxidation temperature on the fuel production yield was also studied. Figure 4 depicts the O2 and CO production rates during three cycles with different oxidation temperatures (1160°C, 1050°C and 950°C respectively). During these cycles, the oxidation was performed at a constant temperature, thus requiring solar power input to maintain the temperature. The decrease of the oxidation temperature leads to an increase of the amount of fuel produced along with an increase of the fuel production rate. The CO amount produced increases from 81 to 197 mol/g when the oxidation temperature decreases by ~200°C, because the oxidation reaction is thermodynamically favored when decreasing the temperature. Similarly, the CO maximal production rate shows a 6fold increase (0.33 to 2.05 mL.g-1.min-1) when decreasing the re-oxidation temperature from 1160°C to 950°C. The decrease of the oxidation temperature has a beneficial impact on the oxidation step by improving both the CO yield and the production rate. However the decrease of the oxidation temperature also leads to an increase of the temperature swing between each step, which thus induces additional sensible heat losses. In the case of non-isothermal oxidation (oxidation during cooling), there is no solar energy input in the reactor and the temperature freely decreases. On the contrary during isothermal oxidation, the reactor is heated to hold constant the temperature. Figure 5 shows the influence of isothermal oxidation (~1050°C) in comparison with non-isothermal oxidation (from 996°C to 847°C) on the fuel production rate. The CO production yield is higher in the non-isothermal mode (260 mol/g) compared with the isothermal mode (121 mol/g). The maximal fuel production rate is almost 3 times higher than the one in isothermal mode. The isothermal oxidation is slow whereas the non-isothermal oxidation reaches completion rapidly. The non-isothermal mode allows a higher fuel production and rate in comparison with isothermal mode without any solar energy input required during the oxidation step. However, after a non-isothermal oxidation with temperature decreasing freely, the temperature swing needed to reach the next reduction step is higher than for isothermal oxidation. Consequently the associated energy input required for heating to the next reduction step is also higher. In order to evaluate the impact of the oxidant gas (CO2 or H2O) on the reactor performance, different cycles have been performed with CO2 and H2O as oxidant gas (Figure 6). For both temperatures presented (1050°C and 900°C), the CO shows higher peak production rate and shorter duration than H2 production. At ~900°C, the H2 and CO production are similar (221 and 210 mol/g, respectively) whereas at ~1050°C the amount of H2 produced (128 mol/g) is lower than CO (194 mol/g). Above ~1000°C, the fuel production is favored with CO2 as oxidant gas (oxidation with H2O is less favorable). Under ~1000°C, the fuel production is similar whatever the oxidant gas. However, the CO2 splitting reaction is faster than water splitting reaction. Therefore water splitting requires a lower oxidation temperature than CO2 splitting. Decreasing the oxidation temperature to favor the hydrogen production implies further heat losses. However in both cases, the oxidant gas conversion is not complete. The output gas is thus an oxidant/fuel mix which would require being separated in industrial process. Hydrogen can be easily separated from water at room temperature whereas the separation of CO from CO2 requires additional energy or advanced technologies [7]. In summary, concerning the temperature-swing cycle, the increase of the reduction temperature or the decrease of the oxygen partial pressure (by pumping) enhanced the ceria reduction extent, in agreement with thermodynamics. During the oxidation step with CO 2 (or H 2 O), the fuel production rate increased when decreasing the oxidation temperature at the expense of a larger temperature swing (inducing heat losses). The highest oxygen non-stoichiometry achieved during reduction (at ~1425°C and pressure of 110 hPa) was =0.059 . Maximum fuel production during one cycle was 0.302 mmol/g CeO2 (35 cycles performed, 20 ppi foam) with peak fuel production rate of 3.3305 ml.g -1 .min - 1 . The fuel productions from the 10 and 20 ppi foams are summarized in Table 1. In the case of the 10 ppi foam, cycles 3, 4, 10, 14 and 17 were performed with similar operatory conditions (T red ~1400°C, T ox ~1040°C), and the fuel production showed good stability over cycles. The pore size of the foam does not show any significant impact on the fuel production amount. For example, the CO amount produced in average per cycle for the 10 and 20 ppi foams are 4.33 L/kg CeO2 and 4.97 L/kg CeO2 respectively. In total, the 10 ppi and 20 ppi foams produced 3.31 L and 2.78 L of CO respectively (during 14 and 10 cycles, respectively). The H 2 produced by the 10 and 20 ppi foams reached 0.68 L and 1.10 L respectively (during 3 and 4 cycles respectively). Each foam successfully underwent ~30h of continuous on sun operation without any significant performance decline, demonstrating their good thermal stability.

Isothermal Cycles with CH 4 -Induced Reduction

On-sun experiments were performed with another reactive foam (18.37g, 10 ppi) for twelve consecutive cycles at 950 °C (1 cycle), 1000 °C (10 cycles) and 1050 °C (1 cycle) in order to investigate the isothermal cycles with CH 4induced reduction. When employing CH 4 as reducing agent, the cycle was operated isothermally, avoiding the heat losses associated with temperature swing between reduction and oxidation steps as described before. Figure . 7 shows the representative transient production rates (from cycle #2) of syngas along with nominal reactor temperature (corresponding to T1) during ceria endothermic reduction with methane, followed by exothermic ceria oxidation with H 2 O at 1000 °C. During reduction step (Figure 7a), at the beginning of reaction the formation of both CO 2 and H 2 O (not measured) is attributed to the excess of oxygen at ceria surface reacting with CH 4 , and the reactor temperature drop is due to the endothermic reaction. H 2 and CO production rates increased significantly (H 2 /CO molar ratio approaching two), while CH 4 was concomitantly consumed. The syngas then decreased progressively after completing ceria reduction. When the CO production rate approached zero, the CH 4 flow was subsequently stopped. During subsequent oxidation (Figure 7b), at initial stage the temperature increased slightly as a result of exothermal reaction; meanwhile, H 2 increased sharply while a small CO production rate was first observed, followed by CO 2 , arising from the gasification of carbon deposition (stemming from CH 4 cracking in the first step). The final obtained H 2 , CO, and CO 2 yields (calculated by time-integration of the measured syngas production rates produced per gram of CeO 2 ) were 3.83, 1.74, and 0.10 mmol/g CeO in the reduction step and 2.24, 0.13, and 0.02 mmol/g CeO in the oxidation step, respectively. The at both the reduction and oxidation step was found to be very similar ( red =0.37 and ox =0.36), thereby confirming complete ceria oxidation. In this test, the methane conversion, solar-to-fuel energy conversion efficiency, and energy upgrade factor (their expressions were defined in [17]) of 56.8%, 3.9%, and 1.0, respectively, were accomplished. This thus outperforms the temperature-swing cycle in terms of higher solar-to-fuel thanks to employing methane that dramatically lowers the reduction temperature and associated heat losses, in turn improving the solar-to-fuel . In order to demonstrate ceria cycling stability during isothermal cycles with CH 4 -induced reduction, the syngas yields during ceria reduction with methane (Figure. 8a), followed by ceria oxidation with H 2 O (Figure . 8b) is plotted for twelve successive runs. These syngas yields were quantified separately, segregating the gases produced from the main reactions regarding both partial reduction of ceria by methane (Eq. 3) and ceria oxidation (Eq. 2), and from the side reactions regarding both the H 2 produced by methane cracking (CH 4 2H 2 +C) and the gases produced from C deposit gasification with steam during oxidation (C+H 2 O CO+H 2 , and C+2H 2 O CO 2 +2H 2 ). A stable pattern in the H 2 (CeO 2 +CH 4 ), CO(CeO 2 +CH 4 ), and CO 2 (CeO 2 +CH 4 ) yields over the ten cycles at 1000 °C was achieved, in the ranges 3.48-3.64 mmol/g CeO2 for H 2 (CeO 2 +CH 4 ), 1.74-1.82 mmol/g CeO2 for CO(CeO 2 +CH 4 ), and 0.07-0.10 mmol/g CeO2 for CO 2 (CeO 2 +CH 4 ) (Figure 8a), in turn resulting in stable total syngas yield (5.67-6.80 mmol/g CeO2 ), red (0.35-0.38), and X CH4 evolution profile (46.9-60.9%, Figure 9). The H 2 (CH 4 cracking) yield at 1000 °C fluctuated slightly (in the range 0.35-1.27 mmol/g CeO2 ) because of a small difference in the CH 4 injection duration [20]. Note that the H 2 (CH 4 cracking) was quantified by the total H 2 yield measured by gas analysis minus the H 2 yield produced by the reaction of ceria with methane, which is equivalent to twice the quantity of produced CO, according to Eq. 3. When either increasing or decreasing the temperature (1050 °C at cycle #5 or 950 °C at cycle #6), the total syngas yield varied significantly because of a change in the reaction kinetics. The maximum total syngas yield (7.48 mmol/g CeO2 ) was consequently found at the maximum temperature (1050 °C), demonstrating kinetic rate improvement (Figure 10). The logarithm evolution of the reaction rates versus inverse temperature ( ) was plotted in Figure 10 to determine the activation energy (E a ) of the ceria reduction process. The reaction rate constants (k) were quantified from the peak production rates of H 2 and CO at 950 °C (cycle #6), 1000 °C (cycle #4), and 1050 °C (cycle #5). As a result, the slope of ln k for both H 2 and CO production rates increased linearly with the inverse temperature. The E a values were 114.2 kJ/mol for H 2 and 93.4 kJ/mol for CO. The E a value related to H 2 production rates was slightly higher compared to that of CO as a result of the side reaction effect attributed to CH 4 cracking. This side reaction produces additional H 2 and thus modifies the global H 2 production rate arising only from the reaction with ceria. In contrast, CO is only produced from methane reforming (Eq. 3) and better represents the kinetics of ceria reduction reaction. Moreover, the H 2 (CH 4 cracking) yield was negligible at 900 °C, in agreement with the lowest X CH (20%); in contrast, it was maximal (2.0 mmol/g CeO2 ) at the highest temperature (1050 °C) in agreement with the

CONCLUSION

The thermochemical performance of a solar monolithic reactor integrating ceria oxygen carrier for H 2 O or CO 2 splitting has been experimentally investigated. In the temperature-swing mode, the reduction extent, and the fuel production yield can be improved by increasing the reduction temperature and/or decreasing the total pressure (oxygen partial pressure) during the reduction step. A decrease of the oxidation temperature improves the oxidation rate. Hence, both the fuel yield and production rate can be enhanced with a decrease of the oxidation temperature or with a nonisothermal oxidation (free cooling with no solar input), although also increasing the temperature swing and inducing heat losses. In total the solar reactor was run over 60h during continuous on sun operation with total production of 1.77 L and 6.09 L of H 2 and CO, respectively and good performance stability.

Furthermore, a comparison between temperature-swing and isothermal cycle (using methane as reducer) has been successfully performed. The reduction extent reached by ceria is higher ( =0.33-0.37) when methane is used during the reduction. Ceria reduction extent and fuel production yields are about 6 to 10 times higher when employing CH 4 as reducer at 1000°C, when compared to thermal reduction in Ar at 1400°C. Employing CH 4 in the reduction step eliminates the sensible heat losses due to reactor cooling, thus in turn enhancing solar-to-fuel efficiency. However, it comes at the expense of CH 4 cracking side reaction leading to carbon deposition that can be gasified with H 2 O/CO 2 in the oxidation step. A temperature of 1000 °C is recommended for isothermal cycles with CH 4 -induced reduction to limit methane cracking reaction while enabling fast enough reaction kinetics. The CH 4 -induced reduction represents an attractive option for the short-term solar process implementation, while temperature-swing cycle eliminates the need for carbonaceous reducer and thus represents the most promising option in the long-term for sustainable fuel production.

FIGURE 1 .

 1 FIGURE 1. a) photography and b) schematic illustration of the ceria redox cycling in a 1.5 kWth monolithic solar reactor driven by real high-flux concentrated sunlight

FIGURE 2 .

 2 FIGURE 2. O2 and CO production rates along with reactor temperatures, during consecutives cycles for ceria foam with temperature-swing mode for the 20 ppi foam

FIGURE 3 .

 3 FIGURE 3. O2 and CO production rates along with reactor temperatures at different pressures during reduction: (a) atmospheric pressure and (b) reduced pressure during the reduction step for the 20 ppi foam

FIGURE 4 .

 4 FIGURE 4. Influence of the oxidation temperature during three thermochemical cycles with a reduction temperature of 1400°C followed by an oxidation at 1160°C, 1050°C and 900°C respectively, 20 ppi foam

FIGURE 5 .

 5 FIGURE 5. CO production rate after a reduction at ~1400°C in isothermal (1050°C) and non-isothermal (996°C to 847°C) oxidation along with the T1 temperature (dotted lines) for the 20 ppi foam

FIGURE 6 .

 6 FIGURE 6. CO and H2 production rates during isothermal oxidation at 1050°C (dotted lines) and 900°C (solid lines)

FIGURE 7 .

 7 FIGURE 7. Syngas production rate along with reactor temperature T1 for (a) reduction with CH4 and (b) oxidation with H2O of ceria foam cycled isothermally at 1000 °C.

FIGURE 9 .FIGURE 10 .

 910 FIGURE 9. Comparison of red and ox in ceria along with CH4 conversion during 12 consecutive redox cycles performed isothermally.

TABLE 1 .

 1 Summary of the fuel production with operatory conditions for 10 ppi and 20 ppi foams, * indicates a low pressure of ~110 hPa during the reduction step

	Cycle Foam	Reduction /	Oxidant	O2	Fuel
			oxidation	gas	production	production
			temperatures
			(°C)		( mol/g)	( mol/g)
	1	10 ppi 1378 / 967	H 2 O	105	208
	2	10 ppi 1417 / 1006	H 2 O	109	216
	3	10 ppi 1404 / 1034	CO 2	106	213
	4	10 ppi 1407 / 1032	CO 2	97	188
	5	10 ppi 1400 / 1079	CO 2	107	212
	6	10 ppi 1412 / 1114	CO 2	88	120
	7	10 ppi 1407 / 1121	CO 2	84	161
	8	10 ppi 1380* / 907	CO 2	102	203
	9	10 ppi 1408 / 1213	CO 2	115	136
	10	10 ppi 1408 / 1059	CO 2	98	194
	11	10 ppi 1453 / 1050	CO 2	116	226
	12	10 ppi 1404 / 869	CO 2	98	197
	13	10 ppi 1403 / 1011	CO 2	78	151
	14	10 ppi 1405 / 1055	CO 2	86	152
	15	10 ppi 1402 / 954	CO 2	96	191
	16	10 ppi 1458 / 1061	CO 2	119	237
	17	10 ppi 1405 / 1057	H 2 O	77	129
	1	20 ppi 1410 / 1064	CO 2	110	186
	2	20 ppi 1408 / 1162	CO 2	67	81
	3	20 ppi 1411 / 1056	CO 2	60	121
	4	20 ppi 1409 / 955	CO 2	87	197
	5	20 ppi 1412 / 1053	H 2 O	95	118
	6	20 ppi 1401 / 870	H 2 O	90	250
	7	20 ppi 1410 / 903	CO 2	106	210
	8	20 ppi 1410 / 705	CO 2	118	273
	9	20 ppi 1424* / 1048	CO 2	174	296
	10	20 ppi 1407 / 858	H 2 O	108	222
	11	20 ppi 1409 / 1000 844	CO 2	123	261
	12	20 ppi 1407 / 996 847	CO 2	126	260
	13	20 ppi 1456 / 997 845	CO 2	149	302
	14	20 ppi 1454 / 1004 673	H 2 O	135	266

highest X CH4 (65.9%), thereby indicating that the extent of CH 4 cracking reaction is strongly dependent on the temperature [21]. For isothermal cycles with CH 4 -induced reduction, a temperature trade-off at 1000 °C is recommended to hasten the kinetic rate of ceria reduction while alleviating the side reaction associated with CH 4 cracking.

Concerning oxidation (Figure 8b), the H 2 (CeO 2-+H 2 O) yield was stable at 1000 °C (in the range 2.04-2.17 mmol/g CeO2 ), except for cycles #5 and #6 caused by the different temperature during the reduction step and its impact on the reduction extent. Likewise, the H 2 (C+H 2 O), H 2 (C+2H 2 O), CO(C+H 2 O), and CO 2 (C+H 2 O) yields at 1000 °C were fairly constant in negligible amounts (0.11-0.18, 0.04-0.10, 0.11-0.18, and 0.02-0.05 mmol/g CeO2 , respectively). Note that the H 2 (C+H 2 O) yield is equal to the CO yield measured by gas analysis (C+H 2 O CO+H 2 ), while the H 2 (C+2H 2 O) yield is equal to twice the CO 2 yield measured by gas analysis (C+2H 2 O CO 2 +2H 2 ). The total syngas yield was in the range 2.38-2.57 mmol/g CeO2 , and ox was in the range 0.35-0.37 (consistent with red ) (Figure 9). Thus, the cycling stability of ceria in the isothermal cycles with CH 4 -induced reduction was validated with respect to stable patterns in produced syngas, , and X CH4 .


Hiroyuki Kimura
email: kimura3438@ihi-g.comb

Stéphane Abanades
email: stephane.abanades@promes.cnrs.frc

Satoshi Seo

Production of C 2 -hydrocarbons and Hydrogen from Methane Pyrolysis with Concentrated Solar Energy

A novel concept of CH4 decomposition into H2 and carbon materials via a two-step thermochemical and thermo-catalytic conversion process with concentrated solar power is proposed. Solar experiments were conducted in order to investigate the first high-temperature step of the proposed two-step thermochemical conversion process. This step consists of CH4 decomposition into C2-hydrocarbons and hydrogen at high temperatures. A significant effect of H2 addition to CH4 and a beneficial effect of the use of tungsten strips as catalyst on the augmentation of C2H2 and C2H4 yields were observed.

INTRODUCTION

For realizing a sustained energy society as envisioned in the Paris Agreement on Climate Change, implementations of various renewable technologies for reducing carbon dioxide (CO 2 ) emissions are necessary. One candidate of such technologies is the direct solar conversion of methane (CH 4 ) to hydrogen fuel (H 2 ) and solid carbon materials (C) such as carbon nanotubes with concentrated solar energy and thus without co-producing carbon dioxide [1,2]. However, some technological problems still remain to be solved for the practical use of the direct solar conversion; e.g. the rapid and significant pyrolytic carbon deposition on the reactor walls and soot deposition on the windows at high temperatures above 1,300 °C.

Aiming to tackle these problems, a new concept for the co-production of hydrogen and valuable carbon materials from methane is proposed in this work. In the new concept, a novel two-step conversion process for decomposing CH 4 into H 2 and C, synthesizing C 2 -hydrocarbons as intermediate products, is introduced. A series of solar tests has been conducted for investigating the first step of the proposed conversion process.

CO-PRODUCTION OF HYDROGEN AND CARBON MATERIALS BY DECOMPOSING METHANE

Several series of solar experiments to decompose CH 4 into H 2 and C have been carried out in the collaboration between PROMES-CNRS and IHI Corporation in the last decade. One is the test with a drop-tube-type solar reactor in which carbon black particles are continuously entrained into the flow of CH 4 as catalysts. The entrainment of particles into the gas flow improves the heat transfer from the internal surface of the heated tube to the flow of CH 4 via the entrained particles absorbing the heat from the tube surface by radiation and then releasing the heat to the CH 4 gas around the particles by convection [3]. Another test series involved an indirectly-heated packed-bed type solar reactor, containing a reactor tube which is filled with carbon black powders as catalyst [4]. The catalytic activity of different carbon catalysts (carbon black, activated charcoal) has also been studied between 900 and 1,200°C in an in-house solar thermogravimetric device [5]. Finally, contrary to the above, tests with an empty tube reactor, i.e. without any catalyst, have been carried out at temperatures up to 1,550°C [START_REF] Kimura | Industry Perspective -IHI Corporation: solar thermal cracking of methane for CO2-free production of hydrogen in collaboration with PROMES-CNRS" as an oral presentation[END_REF]. In the above-mentioned series of tests, very small amounts of acetylene (C 2 H 2 ), ethylene (C 2 H 4 ), and ethane (C 2 H 6 ) were produced in addition to H 2 and C as products of the decomposition of CH 4 . This observation inspired the participating members to devise and propose a new two-step approach for the co-production of H 2 and C materials from CH 4 with concentrated solar power.

New Approach for the Co-production of Hydrogen and Carbon Materials from Methane

A new methodology for co-producing H 2 and C materials from CH 4 is proposed in this study. The main concept of this approach is to divide the conventional process of CH 4 decomposition,

4 CH g "solar heat" 8 H g 4 C s (1)
illustrated in Fig. 1 (a), into the following two elementary thermochemical conversion processes

4 CH g 12 H g "solar heat" C H g C H g 17 H g (2)
and

C H g C H g 17 H g "solar heat" 20 H g 4 C s (3)
as illustrated in Fig. 1 (b) [START_REF] Kimura | [END_REF]8].

The first step (Eq. ( 2)) is devoted to the synthesis of C 2 -hydrocarbons such as C 2 H 2 and C 2 H 4 (and a very small amount of C 2 H 6 ) from a mixture of CH 4 and H 2 at high temperatures in the range from 1,200 to 1,500 °C. The second step (Eq. ( 3)) corresponds to the production of H 2 and C from C 2 -hydrocarbons at lower temperatures in the range from 500 to 1,000 °C. In the first step of the above two-step conversion process, some of the carbon bound in CH 4 should convert to produce C 2 -hydrocarbons, while the remaining carbon contributes to solid carbon production. The mole fractions of CH 4 and H 2 on the left-hand side of Eq. ( 2) should be adjusted for optimizing the yields of C 2 -hydrocarbons; the H 2 /CH 4 ratio of 3 (= 12n/4n) is assumed in the above in one instance. A rapid cooling process (quenching operation) is required ideally at the outlet of the reactor for increasing the C 2 -hydrocarbon yields. Rare metals such as molybdenum (Mo) and tungsten (W) may be used as catalysts for further yield augmentation [9].

In the second step, a part (12n) of the total amount of H 2 (20n) on the right-hand side of Eq. (3) has to be recycled for the left-hand side of Eq. ( 2) for the first step. Thus, the net production of H 2 in the above instance with the H 2 /CH 4 ratio of 3 is 8n. Metal catalysts such as iron (Fe) and nickel (Ni) should be used as catalysts for decomposing C 2 -hydrocarbons and remained CH 4 into H 2 and C materials. The yields of H 2 and valuable C materials increase significantly when one feeds C 2 -hydrocarbons instead of CH 4 in the second step.

Advantages Obtained by Introducing the New Approach

One of the advantages obtained by introducing the above two-step thermochemical process for the decomposition of CH 4 into H 2 and C materials is that the amount of solid carbons (such as carbon black and graphite) produced in the reactor tube at high temperatures decreases, because some amount of C from CH 4 forms gaseous C 2 -hydrocarbons instead of solid materials. Such characteristics can contribute to the inhibition of obstructions in the reactor tube due to the production of solid carbons in the tube at high temperatures.

Another advantage is a diversification of the types of carbon materials produced while decomposing CH 4 into H 2 and C. The proposed thermochemical process produces not only carbon black in the first step at high-temperature condition but also carbon nanotubes in the second step at low-temperature condition.

PRELIMINARY INVESTIGATION OF THE CO-PRODUCTION OF C 2 -HYDROCARBONS AND HYDROGEN FROM METHANE

At the first stage of this project, the first step (Eq. ( 2)) of the two-step conversion process for synthesizing C 2hydrocarbons from CH 4 at high temperatures was focused and investigated. The test facilities and setup used and the test method and conditions considered for the investigation are described below. The investigation was conducted by using a tubular packed-bed solar reactor developed in previous collaboration between PROMES-CNRS and IHI Corporation [4,5,[START_REF] Kimura | Industry Perspective -IHI Corporation: solar thermal cracking of methane for CO2-free production of hydrogen in collaboration with PROMES-CNRS" as an oral presentation[END_REF]10]. The reactor was installed at the focus of a 2 m-diameter parabolic dish concentrator at PROMES-CNRS in Odeillo, France (Fig. 2).

Solar Test Facilities and Setups Used for the Investigation

The solar reactor mainly consists of two distinct parts: i.e. the cavity-type solar receiver and the tubular chemical reaction zone separated from the surrounding atmosphere, as shown in Fig. 3. The reactor tube is inserted vertically into the cavity solar receiver (45 mm height, 40 mm width, and 55 mm depth) that is surrounded by outer graphite walls, carbon felt, and ceramic fiber insulators and closed at the front by the dome-shaped glass window. The insulation materials are used to minimize the heat losses from the cavity receiver. As a result, the cylindrical reactor shell made of stainless steel (254 mm outer diameter, 218 mm length) is maintained at low temperature, which renders an active water-cooling system unnecessary. The only water-cooled components containing sealing rings are the front face (made of aluminum alloy) used to maintain the glass window and the two flanges used to fix the vertical alumina tube at the two extremities. A 10 mm-thick insulating carbon felt buffer is wrapped over the graphite cavity walls to avoid the direct contact of ceramic wool with the hot cavity walls. A 15 mm-diameter aperture at the front graphite wall of the cavity is used to absorb the concentrated solar irradiation through the hemispherical glass window. An insulating layer of carbon felt with a corresponding hole is placed over this front wall to reduce thermal losses. Then, the vertical alumina tube (25 mm outer diameter, 20 mm inner diameter) placed inside the cavity is heated mainly by radiation from the surroundings cavity walls and serves as a heat transfer wall to transmit the solar power from the receiver to the reactor tube and the gases and catalysts inside the tube. A pyrometer placed at the front of the reactor (Heitronics KT15, 4.9-5.5 μm) serves as optical device for direct measurement of the outer surface temperature of the reactor tube (using CaF 2 optical window). A B-type thermocouple shielded with alumina is used to measure the temperature of the catalyst bed.

Procedures for the Investigation with the Solar Reactor

Argon (Ar) as the inert carrier gas, CH 4 , and H 2 are supplied to the reactor tube through the gas inlet in the upward direction. The mixture of Ar, CH 4 , and H 2 is heated in the reactor tube by indirect solar heating. The heat energy absorbed into the gas mixture mainly contributes to the temperature increase of the gas mixture and the subsequent chemical decomposition of CH 4 . Ar, CH 4 , and H 2 are supplied and the controlled by mass-flow controller (MFC, Brooks Instruments model 5850S), individually. The CH 4 mole fraction was controlled by adjusting the flow rate of Ar as the inert dilute gas and the H 2 /CH 4 molar ratio was controlled while generally maintaining a total volumetric flow rate of 0.5 NL/min. Pure Ar was used during the heating of the solar reactor in the solar test and the cooling of the reactor after the test. The cavity located at the center of the solar reactor was constantly swept by the flow of nitrogen (N 2 ) at the rate of 0.5 NL/min to provide inert atmosphere around the cavity made of graphite which is rapidly oxidized if oxygen (O 2 ) exists in its surroundings. The cavity and the reactor tube were preliminary purged to remove residual air by the means of successive vacuum pumping and inert gases were filled before starting the heating with concentrated solar power. The solar experiments were then performed slightly above atmospheric pressure (88 kPa whereas the atmospheric pressure of the surrounding air is at 85 kPa at the site, which is over 1,500 meters above sea level) because of the pressure drop caused by the particle bed and the other components such as filter and lines for gas analysis.

In solar tests for this packed-bed type reactor, the mixture of Ar, CH 4 , and H 2 to be heated in the reactor tube was injected from the bottom side of the tube. The reactive CH 4 was injected once the steady state at a targeted temperature (i.e. isothermal) condition was achieved. Although most part of the carbon formed by the decomposition of CH 4 was deposited on the catalyst surface and was thus trapped in the particle bed, a carbon filter was implemented at the outlet for gas cleaning in order to trap the remaining fine particulates and to protect the analytical systems. The gaseous products were finally quantified by online gas analysis using a gas chromatograph (Varian CP4900 micro-GC) equipped with two columns using Ar carrier gas and thermal conductivity detectors (MolSieve 5A PLOT for H 2 and CH 4 , PoraPLOT U for H 2 , CH 4 , and the light C 2 -hydrocarbons C 2 H 2 , C 2 H 4 , and C 2 H 6 ). The gas sampling interval was 90 s. The outlet gas was analyzed online by a specific analyzer (NGA 2000 MLT3) for continuous H 2 and CH 4 analysis (by thermal conductivity and nondispersive IR detection, respectively).

The whole experimental set-up was settled at the focus of a solar concentrating system consisting of a horizontalaxis medium size solar furnace (comprising a sun-tracking heliostat and a 2 m-diameter parabolic dish solar concentrator). The focal point of the concentrator (0.85 m ahead of the parabolic mirror) was located at the cavity aperture for maximum solar radiation absorption inside the cavity receiver. The absorbed solar power which determines the operation temperature can be adjusted by the means of an intermediate shutter placed between the heliostat and the solar concentrator.

Operating Conditions for the Solar Tests

Three types of bed materials were considered in this series of the solar test as illustrated in Fig. 4: ( ) noncatalytic silicon carbide (SiC) particles, ( ) W powder mixed with non-catalytic Al 2 O 3 particles, ( ) empty tube, ( ) W strips. The mean diameter of SiC particles is 1.4 mm. The diameter of W powder is 12 m. The diameter of Al 2 O 3 particles is 1.14 mm. The widths, lengths, and thickness of the W strips are 20 mm, 40 mm, and 0.2 mm, respectively. Figure 5 shows the appearance of these bet materials.

In the case of the former two packed particle bed reactor, two cylindrical porous ceramic foams (reticulated porous ceramic RPC made of -SiC, 20 mm diameter, 20 mm height, 30 ppi) were used to support the particles at the center of the heating section of the reactor tube. On the other hand, in the case of the use of W strips as the catalyst material, one ceramic foam was enough to support the strips around the center of heating section of the reactor tube. An inner supporting tube was placed below the foams inside the reactor tube for the purpose of supporting the foams from the bottom side.

Mass of the bed material, H 2 /CH 4 ratio, total volumetric flow rate of the gas mixture, and temperature of the bed material attained in the test are tabulated in Table 1. The H 2 /CH 4 ratio at the inlet of the reactor was varied as a main parameter at each temperature conditions. In most cases, the total volumetric flow rate was kept constant at 0.5 or 1.0 NL/min. In this case, the C 2 H 2 and C 2 H 4 yields show no apparent influence of the change in H 2 /CH 4 ratio. The C 2 H 2 yield decreased from 0.03 to 0.00 when the ratio was changed from 0 to 2, while the C 2 H 4 yield increased from 0.02 to 0.06 when the ratio was changed as well.

Case ( ), Solar Test Results with Empty Tube

Figure 8 shows a series of the results obtained for the decomposition of CH 4 with empty tube. As already observed in the above cases, the CH 4 conversion and the H 2 yield increase when the temperature is increased and the both yields decrease when the ratio is increased. In contrary to the results in the above, the rate of decrease becomes larger the higher the H 2 /CH 4 ratio. The CH 4 conversion decreased significantly from 0.95 to 0.69 at 1.400 °C when the H 2 /CH 4 ratio was changed from 0 to 3. The H 2 yield decreased more significantly from 0.86 to 0.39 at the same temperature when the ratio was changed as well.

Both the C 2 H 2 and C 2 H 4 yields increase significantly as increasing H 2 /CH 4 ratio in this case without any bed materials in the tube. The C 2 H 2 yield increased from 0.14 to 0.22 at 1,400 °C when the ratio was changed from 0 to 3. The C 2 H 4 yield also increased from 0.05 to 0.23 at the same temperature when the ratio was changed as well.

Case ( ), Solar Test Results with W Strips

Figure 9 shows a series of the results obtained for the decomposition of CH 4 with W strips. The CH 4 conversion and the H 2 yield show similar trends as already observed for Case ( ). The CH 4 conversion decreased significantly from 0.95 to 0.69 at 1,400 °C when the H 2 /CH 4 ratio was changed from 0 to 3; this trend is mostly same with Case ( ). On the other hand, the H 2 yield decreased from 0.86 to 0.33 at 1,400 °C when the ratio was changed as well, more significantly than Case ( ).

Both the C 2 H 2 and C 2 H 4 yields are higher than those for Case ( ). The C 2 H 2 yield increased from 0.15 to 0.25 at 1,400 °C when the ratio was changed from 0 to 3, more significantly than Case ( ). The C 2 H 4 yield also increased from 0.05 to 0.28 at the same temperature when the ratio was changed as well, more significantly than Case ( ) as with the C 2 H 2 yield.

CONCLUSIONS

A novel concept for CH 4 decomposition into H 2 and carbon materials via a two-step thermochemical and thermo-catalytic conversion process with concentrated solar power was proposed. Its first step is devoted to the synthesis of C 2 -hydrocarbons (C 2 H 2 and C 2 H 4 ) from a mixture of CH 4 and H 2 and rare metal catalysts at high temperatures from 1,200 to 1,500 °C. In the second step, H 2 and valuable C materials such as carbon nanotubes are generated from the C 2 -hydrocarbons at low temperatures from 500 to 1,000 °C. Non-catalytic SiC particles, W powder mixed with Al 2 O 3 particles, and W strips were used as bed materials in the reactor tube for investigating some beneficial effects on the increase of the C 2 -hydrocarbon yields. A significant effect of the addition of H 2 to CH 4 on the increase in the C 2 -hydrocarbon yields was observed. The conversion of CH 4 was 81 % and the yields of C 2 H 2 and C 2 H 4 were 25 % and 11 %, respectively, when the H 2 /CH 4 ratio was 4 and the reactor was heated up to 1,400 °C with the non-catalytic SiC particles in the reactor tube. The conversion of CH 4 was 69 % but the yields of C 2 H 2 and C 2 H 4 were 22 % and 23 %, respectively, when the H 2 /CH 4 ratio was 3 and the reactor was at 1,400 °C without bed materials in the reactor tube. No apparent effect (rather a negative effect in some cases) of the use of W powder as catalyst on the C 2hydrocarbon yields was observed. A slight beneficial effect of W strips as catalyst on the increase of the C 2 -hydrocarbon yields was observed. The yield of C 2 H 2 increased from 22 % to 25 % and that of C 2 H 4 increased from 23 % to 28 % with the use of W strips.

FIGURE 1 .

 1 FIGURE 1. Illustrations of the conventional and the newly-proposed thermochemical processes for decomposing CH4 into H2 and C materials: (a) The conventional one-step process by which CH4 is directly decomposed into H2 and C materials; (b) The newly-proposed two-step process by which CH4 is decomposed into H2 and C materials synthesizing a few types of C2hydrocarbons as intermediate products.

FIGURE 2 .

 2 FIGURE 2. The 1.5kW horizontal axis solar furnace at PROMES-CNRS in Odeillo and a tubular-type packed-bed solar reactor co-developed in the collaboration between PROMES-CNRS and IHI Corporation.

FIGURE 3 .

 3 FIGURE 3. Schematic of the tubular solar reactor used in the investigation of the co-production of C2-hydrocarbons and H2 from CH4.

FIGURE 4 .FIGURE 5 .TABLE 1 .FIGURE 6 .

 4516 FIGURE 4. Schematic of the solar experiments for investigating the effect of bed material on CH4 decomposition at high temperatures: ( ) with SiC particles; ( ) with W powder mixed with Al2O3 particles; ( ) empty tube; ( ) W strips.

FIGURE 7 .FIGURE 8 .FIGURE 9 .

 789 FIGURE 7. Results of the solar tests with W powder and Al2O3 particles: (a) CH4 conversion; (b) H2 yield; (c) C2H2 yield; (d) C2H4 yield.

ACKNOWLEDGMENTS

The authors thank the technical support of O. Prevost and R. Garcia from PROMES for reactor design and construction.

Results of the Solar Tests

The main results of the solar experiments are described below. The following discussion focuses on the cases of ( ), ( ), ( ), and ( ) (cf. Table 1) in order to highlight the effect of H 2 addition to CH 4 and the use of rare metal powder or strips on the increase in C 2 -hydrocarbon yields.

The CH 4 conversion and the H 2 yield are respectively calculated by

, , ,

where

, ,

where, , , , , and , are the molar flow rates of C 2 H 2 , C 2 H 4 , and C 2 H 6 at the outlet of the reactor, respectively.

Case ( ), Solar Test Results with Non-catalytic SiC Particles

Figure 6 shows a series of the results obtained for the decomposition of CH 4 with non-catalytic SiC particles. The CH 4 conversion and H 2 yield increase as increasing the temperature at any H 2 /CH 4 ratio. However, they decrease as increasing the H 2 /CH 4 ratio at fixed temperature. The CH 4 conversion decreased from 0.83 to 0.52 and from 0.94 to 0.81 at 1,300 °C and 1,400 °C, respectively, when the ratio was changed from 0 to 4. The H 2 yield decreased from 0.74 to 0.36 and from 0.86 to 0.63 at 1,300 °C and 1,400 °C, respectively, when the ratio was changed as well.

On the other hand, the C 2 H 2 and C 2 H 4 yields increase slightly as increasing the H 2 /CH 4 ratio. An exception is the case of 1,200 °C at which almost no change or a slight decrease can be recognized. The C 2 H 2 yield increased from 0.16 to 0.24 and from 0.15 to 0.25 at 1,300 °C and 1,400 °C, respectively, when the ratio was changed from 0 to 4. The C 2 H 4 yield increased from 0.05 to 0.10 and from 0.05 to 0.11 at 1,300 °C and 1,400 °C, respectively, when the ratio was changed as well. The C 2 H 6 yield was 0.00 at any condition of the H 2 /CH 4 ratio and the temperature.

Case ( ), Solar Test Results with W Powder Mixed with Al 2 O 3 Particles

Figure 7 shows a series of the results obtained for the decomposition of CH 4 with W particles mixed with Al 2 O 3 particles. The CH 4 conversion and H 2 yield increase as increasing the temperature at any H 2 /CH 4 ratio. And, they decrease as increasing the H 2 /CH 4 ratio, as well as in Case ( ), but very rapidly. The CH 4 conversion decreased from 0.89 to 0.70 at 1,300 °C when the H 2 /CH 4 ratio was changed from 0 to 2. The H 2 yield decreased from 0.86 to 0.63 at the same temperature when the ratio was changed as well.


Arnaud Pêcher
email: pecher@labri.fr

Xuding Zhu
email: zhu@math.nsysu.edu.tw

On the Circular Chromatic Number of Circular Partitionable Graphs

Keywords: circular coloring, chromatic number, partitionable graph

This paper studies the circular chromatic number of a class of circular partitionable graphs. We prove that an infinite family of circular partitionable graphs G has χ c (G) = χ(G). A consequence of this result is that we obtain an infinite family of graphs G with the rare property that the deletion of each vertex decreases its circular chromatic number by exactly 1.

As a (k, 1)-coloring of G is equivalent to a k-coloring of G, we have χ c (G) ≤ χ(G). On the other hand, it is known [START_REF] Vince | Star chromatic number[END_REF][START_REF] Zhu | Circular chromatic number: a survey[END_REF] that χ c (G) > χ(G) -1. Therefore χ(G) = χ c (G) . So the parameter χ c (G) is a refinement of χ(G), and χ(G) is an approximation of χ c (G).

If G is vertex χ-critical (i.e., the deletion of any vertex decreases its chromatic number), then deleting any vertex from G decreases its chromatic number by 1. However, for a vertex χ c -critical graph G (i.e., the deletion of any vertex decreases its circular chromatic number), the decrease of the circular chromatic number by the deletion of one vertex could be any rational number 0 < r < 2. Indeed, it seems very rare that a graph G has the property that the deletion of each vertex decrease its circular chromatic number by exactly 1. The question of characterizing such graphs was raised in [START_REF] Zhu | Circular chromatic number: a survey[END_REF]. Currently, complete graphs, the direct sum of vertex χ-critical graphs and a few isolated example graphs are known to have this property. In this paper, we study the circular chromatic number of partitionable graphs. We prove that for an infinite family of partitionable graphs G we have χ c (G) = χ(G). Since partitionable graphs G have the property for any vertex v, χ(G -v) = ω(G -v) = χ(G) -1, which implies that χ c (G -v) = χ(G) -1, it follows that for these partitionable graphs G, the deletion of any vertex decreases its circular chromatic number by exactly 1.

Another motivation for the study of the circular chromatic number of partitionable graphs concerns circular perfect graphs. Given positive integers k and d, k ≥ 2d, the circular complete graph

K k/d has vertex set Z k in which ij is an edge if d ≤ |i -j| ≤ k -d. A homomorphism from a graph G to a graph H is a mapping f : V (G) → V (H) such that f (x)f (y) is an edge of H whenever xy is an edge of G. Then a k-coloring of a graph G is equivalent to a homomorphism from G to K k and a (k, d)-coloring of G is equivalent to a homomorphism from G to K k/d
. So in the study of circular chromatic number of graphs, the circular complete graphs K k/d play the role of complete graphs as in the study of chromatic number of graphs. The circular clique number of a graph G is defined as

ω c (G) = sup{k/d : K k/d admits a homomorphism to G}.
It was shown in [START_REF] Zhu | Circular perfect graphs[END_REF] that ω c (G) is equal to the maximum of those k/d for which

K k/d is an induced subgraph of G. A graph G is called circular perfect [13] if for every induced subgraph H of G, χ c (H) = ω c (H). Since ω(H) ≤ ω c (H) ≤ χ c (H) ≤ χ(H)
, every perfect graph is circular perfect. On the other hand, there are circular perfect graphs which are not perfect. In particular, odd cycles and the complement of odd cycles are circular perfect graphs. So the class of circular perfect graphs is strictly larger than the class of perfect graphs, but it is still a very restrictive class. A sufficient condition for a graph to be circular perfect is given in [START_REF] Zhu | Circular perfect graphs[END_REF], and this sufficient condition is used to prove an analogue of Hajós Theorem for circular chromatic number. We call a graph G minimal circular imperfect, if G is not circular perfect but every proper induced subgraph of G is circular perfect. As an analogue to the study of perfect graphs, it would be nice to have an appealing conjecture on the structure of minimal circular imperfect graphs. Since all the minimal imperfect graphs are circular perfect, we need to search a larger area for minimal circular imperfect graphs. The class of partitionable graphs is a natural candidate. By studying the circular chromatic number of a subclass of circular partitionable graphs, we prove that all the partitionable graphs in this subclass are circular imperfect. The question whether they are minimal circular imperfect remains an open question.

Circular partitionable graphs and the main result

Suppose p, q ≥ 2 are integers. A graph G is a (p, q)-partitionable graph if |V (G)| = pq + 1, and for each vertex v of G, G \ {v} admits a partition into p cliques of cardinality q as well as a partition into q stable sets of cardinality p. A graph is partitionable if it is a (p, q)-partitionable graph for some p, q ≥ 2. Partitionable graphs were introduced by Lovász [START_REF] Lovász | A characterization of perfect graphs[END_REF] and Padberg [START_REF] Padberg | Perfect zero-one matrices[END_REF] as a tool in the study of perfect graphs. A graph G is perfect if for every induced subgraph H of G, we have χ(H) = ω(H).

Here ω(H) is the clique number of G, which is the cardinality of a maximum clique of G. A graph G is minimal imperfect if G is not perfect, but every proper induced subgraph of G is perfect. The Strong Perfect Graph Theorem, which was conjectured by Berge [START_REF] Berge | Färbung von graphen, deren sämtliche bzw. deren ungerade kreise starr sind[END_REF] in 1961, proved by Chudnovsky, Robertson, Seymour and Thomas [START_REF] Chudnovsky | The strong perfect graph theorem[END_REF] in 2002, says that odd cycles of length at least 5 and their complements are the only minimal imperfect graphs. Before the proof of Berge's conjecture, it was shown by Lovász [START_REF] Lovász | A characterization of perfect graphs[END_REF] and Padberg [START_REF] Padberg | Perfect zero-one matrices[END_REF] that every minimal imperfect graph is a partitionable graph. Thus to prove Berge's conjecture, it suffices to show that none of the partitionable graph is a counterexample. Although the final proof of Berge's conjecture given by Chudnovsky, Robertson, Seymour and Thomas [START_REF] Chudnovsky | The strong perfect graph theorem[END_REF] takes a different route, the class of partitionable graphs has been studied thoroughly in the literature, and this turns out to be an interesting class of graphs. The understanding of the structure of this class of graphs may be helpful in the study of other graph theory problems. It is known (cf. [START_REF] Bland | Graphical properties related to minimal imperfection[END_REF]) that every (p, q)-partitionable graph has the following properties:

(1) p is the maximum cardinality of a stable set of G, and q is the maximum cardinality of a clique of G;

(2) G has exactly n stable sets of cardinality p and exactly n cliques of cardinality q, where n is the number of vertices of G;

(3) For each maximum clique C of G, there is a unique maximum stable set S such that C ∩ S = ∅; and similarly, for each maximum stable set S there is a unique maximum clique C such that C ∩ S = ∅;

(4) Each vertex belongs to exactly q maximum cliques, and belongs to p maximum stable sets.

In the study of partitionable graphs, some recursive constructions of sub-families of partitionable graphs are discussed in the literature [START_REF] Chvátal | Combinatorial designs related to the perfect graph conjecture[END_REF][START_REF] Boros | Recursive generation of partitionable graphs[END_REF]. The class of circular partitionable graphs was introduced by Chvátal, Graham, Perold and Whitesides [START_REF] Chvátal | Combinatorial designs related to the perfect graph conjecture[END_REF].

For two sets of integers X, Y , let X + Y denote the set {x + y :

x ∈ X, y ∈ Y }. If X = {x} is a singleton, we write x + Y instead of {x} + Y . Suppose m i ≥ 2 (i = 1, 2, • • • , 2r) are integers. Define integers µ i (for i = 0, 1, • • • , 2r), sets M i (for i = 1, 2, • • • , 2r
), and sets C, S as follows:

µ i = m 1 m 2 • • • m i (µ 0 = 1), M i = {0, µ i-1 , 2µ i-1 , • • • , (m i -1)µ i-1 }, C = M 1 + M 3 + • • • + M 2r-1 , S = M 2 + M 4 + • • • + M 2r . Let n = m 1 m 2 • • • m 2r + 1. We denote by C[m 1 , m 2 , • • • , m 2r ] the circulant graph with vertex set Z n = {0, 1, • • • , n -1}
, where xy is an edge if and only if x = y and (x -y) modulo n is equal to the difference of two elements of C.

Note that

µ i > i-1 j=1 max M j . This implies that |C| = m 1 m 3 • • • m 2r-1 and |S| = m 2 m 4 • • • m 2r . Let ω = |C| and α = |S|. Then n = ωα + 1. Suppose X is a subset of Z n . A circular shift of X is a set of the form i + X = {i + x (mod n) : x ∈ X}. Theorem 1 [6] Suppose m i ≥ 2 are integers for i = 1, 2, • • • , 2r. Then G = C[m 1 , m 2 , • • • , m 2r
] is an (α, ω)-partitionable graph. Moreover, the n maximum cliques of G are the n circular shifts of C, and the n maximum stable sets of G are the n circular shifts of S.

As an example, we consider the graph C[2, 2, 2, 2]. Then

µ i = 2 i , i = 0, 1, 2, 3, 4; M 1 = {0, 1}, M 2 = {0, 2}, M 3 = {0, 4}, M 4 = {0, 8}; C = {0, 1, 4, 5}; S = {0, 2, 8, 10}. The vertex set of C[2, 2, 2, 2] is Z 17 , and ij is an edge if |i -j| ∈ {1, 3, 4, 5}. The graph is depicted in Figure 2.
The following is the main result of this paper.

Theorem 2 Let x 1 , . . . , x p (p ≥ 2) be integers such that x i ≥ 2 for every 1 ≤ i ≤

p. Let δ = max x i and let G = C[x 1 , 2, . . . , x p , 2]. If p = 2 or x 1 x 2 . . . x p ≥ 2 p+1 δ then χ c (G) = χ(G).
The proof of Theorem 2 is left to Section 4.

Observe that for

G = C[x 1 , 2, . . . , x p , 2], α = 2 p , ω = x 1 x 2 • • • x p and |V (G)| = n = 2 p x 1 x 2 . . . x p + 1. Therefore χ f (G) = n α = ω + 1 2 p < χ c (G) = χ(G) = ω + 1.
There are a few papers devoted to the study of the circular chromatic number of circulant graphs [START_REF] Gao | Star-extremal graphs and the lexicographic product[END_REF][START_REF] Lih | Star extremal circulant graphs[END_REF]. It is known [START_REF] Zhu | Circular chromatic number: a survey[END_REF] that for any graph

G, χ f (G) ≤ χ c (G) ≤ χ(G). A graph G is called star extremal if χ f (G) = χ c (G)
. In [START_REF] Gao | Star-extremal graphs and the lexicographic product[END_REF][START_REF] Lih | Star extremal circulant graphs[END_REF], classes of star extremal circulant graphs are investigated. Theorem 2 provides a class of circulant graphs of another kind of extremality, i.e., circulant graphs with χ c (G) = χ(G).

Corollary 3 Suppose G = C[x 1 , 2, . . . , x p , 2] satisfies the condition of Theorem 2. Then G is circular imperfect. PROOF. It is known [13] that ω(G) ≤ ω c (G) < ω(G) + 1. As χ c (G) = χ(G) = ω(G) + 1, it follows that χ c (G) > ω c (G).
= C[x 1 , 2, x 2 , 2, • • • , x p , 2] are partitionable, χ(G -x) = ω(G -x) = χ c (G -x) = χ(G) -1 for each vertex x.
Therefore we have the following corollary.

Corollary 4 Suppose G = C[x 1 , 2, . . . , x p , 2] satisfies the condition of Theorem 2. Then χ c (G -x) = χ c (G) -1 for each vertex x of G.
In other words, the circulant graphs satisfying the condition of Theorem 2 have the property that the deletion of each vertex decreases its circular chromatic number by exactly 1. Not many such graphs were known before, and the problem of characterizing and constructing such graphs was raised in [START_REF] Zhu | Circular chromatic number: a survey[END_REF].

Structural properties of G

In the remainder of this paper,

G = C[x 1 , 2, x 2 , . . . , x p , 2]. Let ω = x 1 . . . x p , α = 2 p , n = αω + 1.
In this section, we shall be interested in elements of Z n only. Let c 0 = 1, and for i = 1, . . . , p, let

c i = 2 i-1 x 1 x 2 . . . x i , and
d i = i j=1 c j .
First we derive an explicit expression for the maximum cliques and stable sets of G. It follows from the definition that µ 0 = 1 and for

1 ≤ i ≤ p, µ 2i-1 = 2 i-1 x 1 x 2 . . . x i = c i , and µ 2i = 2 i x 1 x 2 . . . x i = 2c i . Therefore for 1 ≤ i ≤ p, M 2i-1 = 2c i-1 [0, x i -1], and M 2i = {0, c i }. So C = p j=1 M 2j-1 = p j=1 2c j-1 [0, x j -1], S = p j=1 M 2j = p j=1 {0, c j }.
By Theorem 1, the maximum cliques of G are the n circular shifts C + i of C (i ∈ Z n ), and the maximum stable sets of G are the n circular shifts S + i of S (i ∈ Z n ).

In the following, we consider the intersection S ∩ (S + i) of two maximum stable sets of G.

Lemma 5 For 2 ≤ i ≤ p, c i ∈ [-2d i-1 , 2d i-1 ]. PROOF. Since c 2 = 2x 1 x 2 > 2x 1 = 2d 1 , and n -c 2 ≥ 4x 1 x 2 + 1 -2x 1 x 2 > 2x 1 x 2 > 2d 1 , we conclude that c 2 ∈ [-2d 1 , 2d 1 ]. Assume i ≥ 3 and c i-1 ∈ [-2d i-2 , 2d i-2 ]. Since c i = 2x i c i-1 and x i ≥ 2, we have c i ≥ 2c i-1 + 2c i-1 > 2(2d i-2 ) + 2c i-1 > 2d i-1 . Furthermore, n -c i = 2 p x 1 x 2 • • • x p + 1 -c i > 2c i -c i = c i > 2d i-1 . Therefore c i ∈ [-2d i-1 , 2d i-1 ]. 2 For i = 1, 2, • • • , p, let S i = i j=1 M 2j = i j=1 {0, c j }.
Lemma 6 For every x in Z n and for every 2 ≤ i ≤ p,

S i ∩ (S i + x) = (S i-1 ∩ (S i-1 + x)) ∪ ((S i-1 + c i ) ∩ (S i-1 + c i + x)) or S i ∩ (S i + x) = (S i-1 ∩ (S i-1 + c i + x)) ∪ ((S i-1 + c i ) ∩ (S i-1 + x)).

PROOF. By definition, S

i = S i-1 ∪ (S i-1 + c i ). Hence S i + x = (S i-1 + x) ∪ (S i-1 + c i + x). Therefore S i ∩ (S i + x) = (S i-1 ∩ (S i-1 + x)) ∪ ((S i-1 + c i ) ∩ (S i-1 + c i + x)) ∪(S i-1 ∩ (S i-1 + c i + x)) ∪ ((S i-1 + c i ) ∩ (S i-1 + x)). If S i-1 ∩(S i-1 +x) = ∅, then x ∈ [-d i-1 , d i-1 ]. By Lemma 5, c i ∈ [-2d i-1 , 2d i-1 ]. Therefore c i + x, c i -x ∈ [-d i-1 , d i-1 ]. This implies that S i-1 ∩ (S i-1 + c i + x) = ∅ and (S i-1 + c i ) ∩ (S i-1 + x) = ∅. Therefore S i ∩ (S i + x) = (S i-1 ∩ (S i-1 + x)) ∪ ((S i-1 + c i ) ∩ (S i-1 + c i + x))
.

If S i-1 ∩ (S i-1 + x) = ∅, then (S i-1 + c i ) ∩ (S i-1 + c i + x) = ∅, and hence
S i ∩ (S i + x) = (S i-1 ∩ (S i-1 + c i + x)) ∪ ((S i-1 + c i ) ∩ (S i-1 + x)).
2

For 1 ≤ j ≤ i ≤ p, let B i,j = t∈{1,2,••• ,i}\{j} M 2t = t∈{1,2,••• ,i}\{j} {0, c t }.
Then S i = B i,j + {0, c j } = B i,j ∪ (B i,j + c j ). For convenience, let B j = B p,j .

Lemma 7

For every 1 ≤ i ≤ p and for every x ∈ Z n , x = 0, we have

|S i ∩ (S i + x)| ≤ 2 i-1 . Moreover, if |S i ∩ (S i + x)| = 2 i-1
, then there is a unique index j ≤ i such that x = ±c j and S i ∩ (S i + x) = B i,j or B i,j + c j , depending on x = -c j or x = c j . In particular, for any x ∈ Z n , |S ∩ (S + x)| ≤ 2 p-1 , and if equality holds then there is a unique index j such that x = ±c j and S ∩ (S + x) = B j or B j + c j , depending on whether x = -c j or x = c j .

PROOF. We prove this lemma by induction on i. It is obvious that for every x ∈ Z n , x = 0, we have |S 1 ∩ (S 1 + x)| ≤ 1, and equality holds only if x = ±c 1 . Let i ≥ 2 and suppose the lemma is true for i ≤ i -1.

By Lemma 6,

S i ∩ (S i + x) = (S i-1 ∩ (S i-1 + x)) ∪ ((S i-1 + c i ) ∩ (S i-1 + c i + x)) or S i ∩ (S i + x) = (S i-1 ∩ (S i-1 + c i + x)) ∪ ((S i-1 + c i ) ∩ (S i-1 + x))
First we consider the case that

S i ∩ (S i + x) = (S i-1 ∩ (S i-1 + x)) ∪ ((S i-1 + c i ) ∩ (S i-1 + c i + x)).

By induction hypothesis, |S

i-1 ∩ (S i-1 + x)| ≤ 2 i-1 and |((S i-1 + c i ) ∩ (S i-1 + c i + x))| ≤ 2 i-1 . Therefore |S i ∩ (S i + x)| ≤ 2 i-1 + 2 i-1 = 2 i . Moreover, if |S i ∩ (S i + x)| = 2 i , then |S i-1 ∩ (S i-1 + x)| = 2 i-1 and hence there is a unique index j ≤ i -1 such that x = ±c j and S i-1 ∩ (S i-1 + x) = B i-1,j or B i-1,j + c j . Then S i ∩ (S i + x) = B i-1,j + {0, c i } = B i,j or S i ∩ (S i + x) = B i-1,j + c j + {0, c i } = B i,j + c j Next we assume that S i ∩ (S i + x) = (S i-1 ∩ (S i-1 + c i + x)) ∪ ((S i-1 + c i ) ∩ (S i-1 + x)).
If x = ±c i then the same argument as in the previous case works. Assume x = c i .

Then |S i ∩ (S i + x)| = |S i-1 + c i | + |S i-1 ∩ (S i-1 + 2c i )|.
Note that if i ≤ p -1, then 2c i ∈ C. If i = p, then 2c p = -1. In any case, 2c i is equal to the difference of two integers of C. Therefore {0, 2c i } is an edge of G. Hence 2c i ∈ S i-1 -S i-1 (as S i-1 is a stable set of G). This implies that

S i-1 ∩ (S i-1 + 2c i ) = ∅. Hence |S i ∩ (S i + x)| = |S i-1 | = 2 i-1 ,
and

S i ∩ (S i + x) = B i,i + c i as B i,i ∩ (B i,i + 2c i ) = ∅ ( B i,i is a stable set and {0, 2c i } is an edge).
If x = -c i , then the same argument shows that [START_REF] Zhu | Circular chromatic number: a survey[END_REF]) that for each i, X i = ∅ and moreover, for each i, there is a vertex x ∈ X i and a vertex y ∈ X i+d such that xy is an edge of G. We need to prove that d = 1.

S i ∩ (S i + x) = B i,i . 2
4 Proof of Theorem 2 Assume that χ c (G) = k/d, where gcd(k, d) = 1. Let f be a (k, d)-coloring of G, which is viewed as a homomorphism from G to K k/d . For i ∈ Z k , let X i = f -1 (i) be the set of vertices of G of color i. Let Y i = X i ∪ X i+1 ∪ • • • ∪ X i+d-1 . Then Y i is a stable set. It is known (see Lemma 1.3 of
Lemma 8 If χ c (G) = k/d and gcd(k, d) = 1, then d ≤ 2.
PROOF. Assume to the contrary that d ≥ 3. We consider two cases.

• Case 1: p = 2 Note that in this case α = 4.

Assume that there is an index i for which This completes the proof of Case 1.

|X i | ≥ 2. Since d ≥ 3, X i ∪ X i+1 ∪ X i+2 ⊆ Y i and X i-1 ∪ X i ∪ X i+1 ⊆ Y i-1 . However, |Y i-1 |, |Y i | ≤ α = 4, and |X i-1 ∪ X i ∪ X i+1 |, |X i ∪ X i+1 ∪ X i+2 | ≥ 4 (as |X i | ≥ 2,
• Case 2: ω ≥ 2 p+1 δ If there exists i ∈ Z k such that |Y i | + |Y i+1 | + |Y i+2 | = 3α then |Y i | = |Y i+1 | = |Y i+2 | = α. By Claim 7, we have |Y i ∩ Y i+1 | ≤ α 2 . Hence |Y i+1 \ Y i | ≥ α 2 . As Y i+1 \ Y i = X i+d , we get |X i+d | ≥ α 2 . As d ≥ 3 and X i+2 = ∅, we conclude that |X i+2 ∪ X i+d | > α 2 . However X i+2 ∪ X i+d ⊆ Y i+1 ∩ Y i+2 . So Y i+1 and Y i+2 are two distinct maximum stable sets with |Y i+1 ∩ Y i+2 | > α 2 , contrary to Lemma 7. Thus for every i ∈ Z k , we have |Y i | + |Y i+1 | + |Y i+2 | < 3α. Since i∈Z k |Y i | = dn, we have i∈Z k (|Y i | + |Y i+1 | + |Y i+2 |) = 3dn ≤ (3α -1)k.
As n = αω + 1 and k d < ω + 1, we have 3n = 3(αω + 1) < (3α -1)(ω + 1).

It follows that ω < 3α -4. Hence

2 p+1 δ ≤ ω < 3α -4,
which is a contradiction, as α = 2 p and δ ≥ 2.

Lemma 9

There exists an i ∈ Z k such that Y i , Y i+1 , . . . , Y i+2δ are all of size α.

PROOF.

• Case 1: p = 2 If there exists i ∈ Z k such that |X i | ≥ 3 then Y i and Y i-1 are two maximum stable sets sharing |X i | > α/2 vertices, contrary to Lemma 7. Thus, for every i ∈ Z k , we have

|X i | ≤ 2. Since i∈Z k |X i | = n = 4x 1 x 2 +1 = 2k-1, it follows that
Let a be the element in Z n such that Y i = S + a.

By definition, for any

1 ≤ t ≤ p, S = B t + {0, c t } = B t ∪ (B t + c t ). By Lemma 10, |Y i ∩ Y i+1 | = α
2 . By Lemma 7, Y i+1 = Y i ± c t for some 1 ≤ t ≤ p. First we consider the case that Y i+1 = Y i + c t . Since Y i = S + a = (B t + a) ∪ (B t + a + c t ), we have By Lemma 7, j -a-(j-2)c t = c t . Thus we conclude that Y i+j-1 = S+a+(j-1)c t and X i+j = B t + a + jc t , as X i+j = Y i+j-1 \ X i+j-1 .

X i+1 = (Y i ∩ Y i+1 ) = (S ∩ (S + c t )) + a = B t + c t + a.
If Y i+1 = Y i -c t , then the same argument shows that for j = 1, 2, • • • , 2δ, X i+j = B t + a -jc t and Y i+j-1 = S + a -(j -1)c t . 2

Now we derive the final contradiction. Let t be the index given in Lemma 11. Without loss of generality, we assume that for j = 1, 2, • • • , 2δ, X i+j = B t +a+jc t . If t < p, then 2x t+1 c t = c t+1 . Then X i+2x t+1 = X i + 2x t+1 c t = B t + a + c t+1 . By definition, c t+1 ∈ B t . Thus a + c t+1 ∈ B t + a = X i . On the other hand, 0 ∈ B t , and hence a + c t+1 ∈ X i+2x t+1 . This is a contradiction, as 2x t+1 ≤ 2δ < ω < k, which implies that X i ∩ X i+2x t+1 = ∅. If t = p, then

2x 1 c t = (2 p x 1 x 2 • • • x p )x 1 = (n -1)x 1 = -x 1 .
Then X i+2x 1 = X i + 2x 1 c t = B t + a -x 1 . By definition, x 1 ∈ B t . Thus a ∈ B t + a -x 1 = X i+2x 1 . On the other hand, 0 ∈ B t , and hence a ∈ B t + a = X i . This is a contradiction, as 2x 1 < k and hence X i ∩ X i+2x 1 = ∅. This completes the proof of Theorem 2.

1 Introduction

 1 Suppose G is a graph and k and d, k ≥ 2d, are positive integers.A (k, d)-coloring of G is a mapping f : V (G) → Z k such that for each edge xy of G, d ≤ |f (x)f (y)| ≤ k -d. The circular chromatic number of G is defined as χ c (G) = inf{k/d : G has a (k, d)-coloring}.

Fig. 1 .

 1 Fig. 1. The circular partitionable graph C[2, 2, 2, 2]

  If a ≡ b (mod n), then a, b are treated as the same. However, by an abuse of notation, we also use the natural order of integers in the following sense: If a < b, we denote [a, b] the set of integers a ≤ x ≤ b. Note that it is possible that a, b ∈ {0, 1, • • • , n -1}. However, [a, b] always denote a subset of {0, 1, • • • , n -1}, by means of taking modulo n. For example, [-1, 1] = {-1, 0, 1} = {n -1, 0, 1}. For a set A of integers, let aA = {ax : x ∈ A}, a + A = {a + x : x ∈ A}. Again the multiplications and additions are modulo n.

  and each |X j | ≥ 1).Therefore |Y i-1 | = |Y i | = 4, and hence Y i-1 , Y i are maximum stable sets of G. However, |Y i-1 ∩ Y i | = |X i | + |X i+1 | ≥ 3 > α/2, contrary to Lemma 7.Hence, for every i ∈ Z k , we have|X i | = 1. In particular, n = k = 4ω + 1. From k d ≤ χ = ω +1 and ω > 2, we get d ≥ 4. Since Y 0 is a stable set of size d, we have d = 4. Thus Y 0 and Y 1 are two maximum stable sets of G sharing d -1 vertices, contrary again to Lemma 7.

2 By Lemma 8 ,

 28 we have d = 1 or 2. If d = 1, we are done. Thus we assume d = 2, and we shall derive a contradiction. As ω < χ c (G) ≤ χ(G) = ω + 1 and d = 2, we have k = 2ω + 1.

X

  i+2 = (Y i+1 \ Y i ) = ((S + c t ) \ S) + a = B t + 2c t + a. Assume 3 ≤ j ≤ 2δ, X i+j-1 = B t + a + (j -1)c t and Y i+j-2 = S + a + (j -2)c t . Assume Y i+j-1 = S + j for some j ∈ Z n . Since Y i+j-2 = S + a + (j -2)c t and Y i+j-1 ∩ Y i+j-2 = X i+j-1 = B t + a + (j -1)c t , it follows that S ∩ (S + (j -a -(j -2)c t )) = B t + c t .

2

  It is unknown if any of the graphsG = C[x 1 , 2, . . . , x p , 2] are minimal circular imperfect. A computer search shows that C[2, 2, 2, 2] is not minimal circular imperfect. The subgraph of C[2, 2, 2, 2]induced by the set X = {2, 3, 4, 5, 6, 9, 11, 12, 13, 14, 16} has circular chromatic number 4 and circular clique number 3, and hence is circular imperfect.

	Since the graphs G

 [START_REF] Bacsó | On minimal imperfect graphs with circular symmetry[END_REF]Supported in part by the National Science Council of ROC under grant NSC91-2115-M-110-004

there exists a unique j ∈ Z k , such that |X j | = 1. Thus Y j+1 , Y j+2 , . . . , Y j+n-2 are n -2 stable sets of size α = 4. As n -2 = 4x 1 x 2 -1 > 2δ + 1, we are done.

• Case 2: ω ≥ 2 p+1 δ Assume to the contrary that for every i ∈ Z k , there exists one stable set in Y i , Y i+1 , . . . , Y i+2δ of size strictly less than α, then k-1 i=0

On the other hand, as k-1 i=0 |Y i | = 2n, we have k i=0

As ω ≥ 2 p+1 δ and α = 2 p , easy calculation derives a contradiction.

2

In the remainder of this section, let i be an index such that Y i , Y i+1 , . . . , Y i+2δ are all of size α.

Lemma 10

For every j = i + 1, i + 2, . . . , i + 2δ, we have

PROOF. By Lemma 7,

Lemma 11 There exists an index t ∈ {1, 2, • • • , p} and a ∈ Z n , such that either for all j = 1, 2, . . . , 2δ,

or for all j = 1, 2, . . . , 2δ,

PROOF.

Open question

Theorem 2 gives the circular chromatic number of some circular partitionable graphs such that their stability number is a power of two (these graphs are said to be of type 1 or 2 in [START_REF] Bacsó | On minimal imperfect graphs with circular symmetry[END_REF]).

However, we believe that our result is likely to hold for most of the circular partitionable graphs: e.g., is it true that every graph C[m 1 , m 2 , . . . , m 2r ] with r ≥ 2 has its circular chromatic number equal to its chromatic number?