Peter Podhorec

Imen Grzegorz Gosiewski

Ben Ammar

Miroslawa Sokolowska-Mikolajczyk

Sylvain Milla

Sergii Boryshpolets

Marek Rodina

Zuzana Linhartova

Daniel Biro

Vlastimil Stejskal

Imen Ben

Mirosława Sokołowska-Mikolajczyk

Jarosław Chyb

Sylvain Milla D D

Sergii Boryshpolets C C

Marek Rodina C C

Daniel Biro E E

Jan Kouril

The effect of GnRHa with or without dopamine inhibitor on reproductive hormone levels and sperm quality in tench Tinca tinca

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

Introduction __________________________________________________________________

European pond aquaculture is dominated by culture of common carp supplemented with that of other cyprinids, including tench Tinca tinca. Tench is distinguished by high quality meat and reaches market size of 300-500 g at 2-3 years [START_REF] Gela | Tench (Tinca tinca) broodstock management inbreeding station under conditions of pond culture: a review[END_REF]. Its ability to tolerate high water temperature with low oxygen content makes it a popular and widely cultivated fish [START_REF] Rodriguez | Egg production of tench (Tinca tinca L.) kept in semi-intensive culture conditions[END_REF]. Tench is a multiple spawner, beginning the reproductive season when the mean daily water temperature reaches approximately 20 °C (Podhorec et al., 2012b).

The ongoing need for tench fingerlings has spurred substantial research into optimization of tench artificial reproduction [START_REF] Kouril | Hormonal induction of tench (Tinca tinca L.) ovulation with the same treatments over two consecutive reproductive seasons[END_REF]. The negative impact of culture conditions on the reproductive axis of fishes is well known and is primarily the result of diminishing luteinizing hormone (LH) release from the pituitary with consequent lack of final gamete maturation [START_REF] Zohar | Endocrine manipulations of spawning in cultured fish: from hormones to genes[END_REF]. Due to the strong inhibitory impact of captivity on tench final oocyte maturation and ovulation, most research effort has been concentrated on developing protocols focusing on tench females [START_REF] Kouril | Induced ovulation in tench (Tinca tinca L) by various LH-RH synthetic analogseffect of site of administration and temperature[END_REF][START_REF] Podhorec | Effective dose of mGnRHa for induction of ovulation in tench (Tinca tinca L.)[END_REF]Podhorec et al., 2012b). However, a major concern is also declining quality and quantity of sperm produced in captivity (Linhart et al., 2006), compounded by contamination of sperm by urine [START_REF] Linhart | Insemination, fertilization and gamete management in tench, Tinca tinca (L.)[END_REF]. Studies primarily address development of immobilizing solutions to counteract activation of sperm with urine [START_REF] Rodina | Kurokura solution as immobilizing medium for spermatozoa of tench (Tinca tinca L.)[END_REF] or improvement in spermiation rate, without evaluation of the endocrine system response to hormone stimulation (Caille et al., 2006).

Stimulation with dried carp pituitary and synthetic analogues of gonadotropin releasing hormone (GnRHa) have both led to significant enhancement of spermiation, more pronounced with the latter (Caille et al., 2006). GnRH alone is a prominent stimulator of synthesis and release of LH from the pituitary, a prerequisite for steroid release [START_REF] Zohar | Neuroendocrinology of reproduction in teleost fish[END_REF]. High sensitivity to GnRHa alone stimulation is an unique feature of tench [START_REF] Kouril | Induced ovulation in tench (Tinca tinca L) by various LH-RH synthetic analogseffect of site of administration and temperature[END_REF] compared to other cyprinids [START_REF] Peter | Induced ovulation and spawning of cultured fresh-water fish in Chinaadvances in application of GnRH analogs and dopamine antagonists[END_REF], which typically exhibit strong dopamine inhibition of LH release. For this reason the combined treatment comprising GnRHa and a dopamine inhibitor (e.g. Ovaprim, Ovopel) is commonly used in cyprinid aquaculture. However our recent study [START_REF] Podhorec | The effects of GnRHa with and without dopamine antagonist on reproductive hormone levels and ovum viability in tench Tinca tinca[END_REF] showed negative effect of combined treatment on tench egg viability.

There is currently no available information on the effect of such combined treatment on tench milt production, spermatozoa concentration, and reproductive hormone release.

The aim of the study was to investigate LH and gonadal androgens: testosterone (T) and 11-ketotestosterone (11-KT) level changes after GnRHa with or without a dopamine inhibitor administration, and their effects on the quality and quantity of expressible milt in tench.

Material and Methods __________________________________________________________

Broodstock and treatments

Mature male tench, 295 ± 24 g body weight (BW), were collected in spring from ponds located in the southern Czech Republic when water temperature reached 16 °C. No significant differences in BW were found between the treatment groups using one-way ANOVA (p N 0.05). Fish were housed at the research facility of South Bohemia University in Vodnany and randomly divided into four 700 l flow-through tanks supplied with water from the Blanice River. After a three day acclimatization period and gradual temperature increase necessary for the stimulation of gametogenesis (Podhorec et al., 2012b), the water conditions were temperature 20.0 ± 0.5 °C, natural photoperiod (16:8L:D) and dissolved oxygen 7.8 ± 0.4 mg l1. Following acclimatization, fish (7 per group) were divided into four tanks and subjected to a single, intraperitoneal injection of: (i) [D-Tle6, Pro9, NEt]-mGnRH (mGnRHa) at 20 Ng kg1, Bachem AG, Germany); (ii) [D-Tle6, Pro9, NEt]-mGnRH (mGnRHa) at 20 Ng kg1, combined with 20 mg kg1 of metoclopramide (Sigma-Aldrich, USA); (iii) 20 mg kg1 metoclopramide (Sigma-Aldrich, USA) or (iv) 0.9% NaCl (Braun Melsungen AG, Germany). All injected substances were dissolved in 0.9% NaCl. Heparinized needles and 1 ml syringes were used to collect serial blood samples (900-1000 Nl) by caudal venipuncture prior to injection (0 h) and at 12 and 24 h postinjection. Blood samples were centrifuged (4000 × g for 10 min at 8 °C) and plasma was stored at 80 °C until hormone analysis. Fish were anesthetized with 0.03 ml l1 clove oil prior to manipulation.

Sperm analysis

The urogenital pore was dried with paper towel before direct sperm collection into 5 ml syringes containing 2.5 ml of immobilizing solution Kurokura 180 [START_REF] Rodina | Kurokura solution as immobilizing medium for spermatozoa of tench (Tinca tinca L.)[END_REF]. After measurement of total sperm volume, syringes were immediately placed on ice and transported to the laboratory for analysis.

Spermatozoa concentration was counted under a light microscope (400×) using a Burker cell haemocytometer. Ten microliters of sperm were diluted with 0.9% NaCl (dilution rate 1000×), placed onto the haemocytometer (0.1 mm depth and 0.2 mm × 0.2 mm square), and left for 10 min to allow sedimentation before the mean number of spermatozoa per 16 squares was calculated according to Caille et al. (2006). Sperm volume and spermatozoa concentration were expressed in ml kg1 BW and billions kg1 BW, respectively. Evaluation of sperm velocity and motility rate was based on recordings of fresh sperm collected 24 h post-injection. One Nl of sperm was mixed with 50 Nl of distilled water containing 0.1% bovine serum albumin (BSA) added to prevent the spermatozoa heads sticking to the glass slide. Within 10 s of mixing, spermatozoon motility and velocity was recorded using a CCD video camera (Sony DXC-970MD; Sony, Tokyo, Japan) mounted on a dark-field microscope (Olympus BX50, magnification ×200) and illuminated with stroboscopic light controlled by Exposure Scope (University of South Bohemia, Faculty of Fisheries and Protection of Waters, Czech Republic). The focal plane was positioned on the bottom position of the open water drop. Sperm motility was recorded three times per male. Video recordings were analysed by a CASA plugin for ImageJ software [START_REF] Wilson-Leedy | Development of a novel CASA system based on open source software for characterization of zebrafish sperm motility parameters[END_REF] to calculate: (i) motility rate (percent of motile spermatozoa); (ii) the curvilinear velocity (VCL; Nm s1): the time-average velocity of a sperm head along its actual curvilinear path, as perceived in two dimensions in the microscope; (iii) straight-line velocity (VSL; Nm s1): the time-average velocity of a sperm head along the straight line between its first detected position and its last; (iv) the average path velocity (VAP; Nm s1): the time-average velocity of a sperm of a sperm head along its average path. This path is computed by smoothing the actual path according to algorithms in the CASA instrument: these algorithms vary between instruments. Mean velocity was based on motile spermatozoa only.

Hormone analysis

Luteinizing hormone concentrations in plasma (50 Nl) were assayed by heterologous enzyme-linked immunosorbent assay [START_REF] Kah | Development of an enzyme-linked immunosorbent-assay for goldfish gonadotropin[END_REF]. The use of the heterologous assay was validated by demonstrating parallel displacement curve of serially diluted tench plasma with standard common carp curve. Sensitivity of the assay was in the range of 0.6-200 ng ml1 with intra-and inter-assay coefficients of variance (CV) of 5 and 9%, respectively.

Testosterone (ng ml1) was assayed in 25 Nl of blood plasma using the DIAsource Testosterone ELISA Kit (DIAsource, KAPD1559). When necessary, plasma samples were diluted 1:2 to 1:4. The detection limit of the assay was 0.083 ng ml1, CV of the intra-assay ranged from 1.5 to 9.5%, and inter-assay CVs ranged from 7.6 to 13.9% for low and high levels, respectively. 11-Ketotestosterone (ng ml1) was assayed in 50 μl of plasma diluted 1:50 in ELISA buffer using the Cayman 11ketotestosterone EIA Kit (Cayman, 582751). Sensitivity was 1.3 ng ml1; CVs of the intra-assay ranged from 2.6 to 8.4%, and CVs of inter-assays from 9.4 to 12.1% for low and high levels, respectively.

Statistical analysis

Differences in spermatozoon concentration, motility data (after arc-sine transformation), LH and sex steroid levels were analysed by one-way ANOVA at each sampling time, followed by Tukey's HSD test. Data were tested for normality of residues and homogeneity of variances (Levene's test), to comply with the prerequisites for ANOVA. VCL, VAP and VSL data were analysed by Kruskal-Wallis test followed by Dunn's test. Differences were considered statistically significant when P < 0.05.

Results and Discussion ________________________________________________________

This work presents the first report of the effects of GnRHa treatment combined with the dopamine inhibitor metoclopramide on LH, T, and 11-KT secretion, milt production, and spermatozoa concentration in tench. Both the combined treatment and mGnRHa alone significantly increased LH concentration, milt production, spermatozoa concentration, with no detected negative impact on sperm viability (Table 1). Significant decrease of VLC and VAP was detected after treatment with metoclopramide alone compared to all other groups. However VSL was not negatively affected by metoclopramide treatment.

Equivalent potency of GnRHa treatment with and without metoclopramide on the quantity and quality of sperm could imply a lack of functional dopamine inhibition of LH release in tench males, however the assessment of blood LH concentration changes showed the presence of dopamine inhibition. A rapid surge in LH release (already after 12 h PI) was observed with the combined treatment versus gradual increase with mGnRHa alone (P < 0.05) (Fig. 1).Also significantly higher LH levels (P < 0.05) in metoclopramide group in comparison with control one were noted at that sampling point showing that this dopamine antagonist alone is sufficient to partially prevents from dopamine action in its blockage of LH release from the pituitary. At 24 h PI, continued increase was detected in LH values for the mGnRHa group, not differing (P > 0.05) from persistent high LH levels with the combined treatment. (Fig. 1). Based on these results we can conclude that dopamine inhibition in male tench represents the low-level suppression of LH release rather than strong inhibition of release. Results are in accordance with the varying pre-ovulatory LH profiles and ovulation rate seen with mGnRHa and combined treatment stimulation of tench females (Podhorec et al., 2012a;Podhorec et al., 2012b). Billard et al. (1984) and [START_REF] Gillet | Effects of GnRHa and pimozide treatments on the timing of ovulation and on egg quality in Arctic charr (Salvelinus alpinus) at 5 and 10 °C[END_REF] suggested that the rapid pre-ovulatory LH surge induced by combined treatment might adversely affect egg viability in rainbow trout (Oncorhynchus mykiss). This hypothesis has been recently supported by our group [START_REF] Podhorec | The effects of GnRHa with and without dopamine antagonist on reproductive hormone levels and ovum viability in tench Tinca tinca[END_REF], who observed a significantly decreased fertilization rate and hatchability in tench eggs after combined treatment compared to mGnRHa alone. Contrary to that, we did not detect adverse effects of combined treatment on sperm volume or sperm density, motility, or velocity (VLC, VAP, VSL), parameters generally considered indicative of sperm quality. The mode of LH action on final stages of sperm maturation might bean explanation for the lack of detrimental effects of the combined treatment. Release of LH from the pituitary induces a shift in the steroidogenic pathway in the testes leading to the production of maturation inducing steroid (MIS), which regulates sperm maturation. The MIS does not act directly on sperm, but via activation of specific enzymes that increase seminal plasma pH [START_REF] Mañanós | Reproduction and control of ovulation, spermiation and spawning in cultured fish[END_REF], which in turn increases cAMP in sperm allowing the acquisition of motility [START_REF] Nagahama | Endocrine regulation of gametogenesis in fish[END_REF]. No trials of sperm fertilizing ability were included in current study, but the increase of LH secretion and the augmented volume and the number of sperm cells, as a result of combined treatment, with unchanged sperm motility and velocity may suggest that also sperm fertilizing ability was not deteriorated.

To elucidate the potential impact of the treatments on steroid secretion, levels of Tand 11-KT were determined. Statistical analysis of T, the significant differences (P b 0.05) between the males treated with the combination (mGnRHa and metoclopramide) and control or metoclopramide injected fish were found at 24 h PI (Fig. 2). In case of 11-KT release showed nonsignificant changes in the course of the experiment (Fig. 3), due probably to the high individual differences among males. It is possible that 24 h, in the conditions of this experiment, was the start point of the increasing steroid secretion leading to the changes of other parameters of sperm quality.

Treatment with mGnRHa alone, metoclopramide alone, and their combination, significantly increased milt production and number of spermatozoa kg1 BW 24 h PI compared to controls (Table 1) with no differences among experimental groups (P N 0.05). The obtained results are in agreement with other studies (Caille et al., 2006;[START_REF] Linhart | Stimulation of spermiation in tench (Tinca tinca L.) by analogs Gnrh and carp hypophysis[END_REF] reporting significant positive impact of GnRHa treatment on milt volume and spermatozoa concentration in tench, with better results obtained with an acute dose as opposed to the slow release form of GnRHa treatment.

This study showed differing effects of treatment with GnRHa alone and GnRHa in combination with a dopamine inhibitor on LH release in tench males, with no observed adverse impact on sperm parameters. Data are expressed as the mean ± SEM. Different superscripts denote statistical differences between groups (P < 0.05).

____________________________________________________________________________

Figure 1 .

 1 Figure 1.Effects of 0.9% NaCl, mGnRHa, mGnRHa with metoclopramide, and metoclopramide on luteinizing hormone (LH) release in male tench Tinca tinca. Data are expressed as the mean ± SEM. Different superscript letters indicate significant differences (P b 0.05).

Figure 2 .

 2 Figure 2.Effects of 0.9% NaCl, mGnRHa, mGnRHa with metoclopramide and metoclopramide on testosterone release in male tench Tinca tinca. Data are expressed as themean ±SEM. Different superscript letters indicate significant differences (P b 0.05).

Figure 3 .

 3 Figure 3.Effect of 0.9% NaCl, mGnRHa, mGnRHa with metoclopramide and metoclopramide on 11-ketotestosterone release in male tench Tinca tinca. Data are expressed as the mean ± SEM. Different superscript letters indicate significant differences (P < 0.05).

Table 1 . Effect of 0.9% NaCl, mGnRHa, mGnRHa with metoclopramide and metoclopramide on tench males (Tinca tinca) reproductive parameters.

 1

																			Curvilinear	Average path
										Number of	Number of				velocity 10 s	velocity 10 s	Straight line
										spermatozoa	spermatozoa	Motility 10 s	post sperm	post sperm	velocity 10 s post
		Body		Sperm ml		Sperm		(× 10	9 ) per	(× 10	9 ) per kg	post sperm	activation	activation	sperm activation
	Treatment	weight (g)	per male		ml kg	-1 BW	male			BW			activation (%)	(μm s	-1 )	(μm s	-1 )	(μm s	-1 )
	0.9% NaCl	239 ± 59	a	0.22 ± 0.07	a	0.79 ± 0.17	a	1.80 ± 0.42	a	6.50 ± 1.71	a	90.2 ± 0.51	a	81.92 ± 0.55	a	71.06 ± 0.63	a	51.83 ± 0.57	a
	mGnRHa	308 ± 45	a	0.66 ± 0.08	b	2.32 ± 0.36	b	4.95 ± 0.51	a	15.86 ± 1.45	b	92.3 ± 0.82	a	80.51 ± 0.57	a	72.00 ± 0.63	a	52.42 ± 0.68	a
	mGnRHa + Met.	303 ± 44	a	0.82 ± 0.25	b	3.85 ± 0.65	b	5.98 ± 1.04	a	22.19 ± 4.29	b	89.6 ± 0.31	a	80.68 ± 0.65	a	73.01 ± 0.77	a	51.19 ± 0.86	a
	Metoclopramide	331 ± 68	a	0.75 ± 0.3	b	2.51 ± 0.52	b	5.21 ± 1.59	a	17.01 ± 3.26	b	88.8 ± 0.43	a	75.75 ± 0.55	b	67.32 ± 0.64	b	53.56 ± 0.62	a


A Aniello

T Poinsot

L Selle

T Schuller

Hydrogen substitution of natural-gas in premixed burners and implications for blow-off and flashback limits

Keywords: HYDROGEN SUBSTITUTION, PREMIXED COMBUSTION, LEAN BLOW-OFF, FLASHBACK

Two multi-perforated premixed burners, designed for natural gas, are fuelled with increasing hydrogen content to assess the limits of H 2 substitution and investigate potential risks associated to it. The burners feature a different design, which affects flame stabilization and heat exchange between the fresh mixture and the hot burner walls. First, results are presented by means of stability maps that were collected at constant power and over a wide range of equivalence ratio, from pure methane-air to pure hydrogen-air mixtures. The impact of hydrogen addition on blow-off and flashback limits is then analyzed. On one side, it is observed that hydrogen addition increases blow off resistance, extending the operating range towards ultra-lean conditions. On the other side, hydrogen raises the thermal load on the burner favoring flashback. It is shown that the competition between the bulk velocity at the burner outlet and the laminar burning velocity is not a reliable parameter to predict flashback occurrence, while the thermal state of the burner represents a determining factor. An analysis of the thermal transient reveals a strict correspondence between the onset of flashback for a given mixture composition and the burner surface temperature. Results highlight the challenges linked to the design of fuel-flexible systems, pointing out practical limits of H 2 substitution in burners designed for operation with natural gas.

Introduction

Hydrogen is a promising fuel to achieve decarbonization of combustion processes. It can be produced from renewable sources via electrolysis and distributed in the gas network upon request, following a Power-to-Gas pathway [START_REF] Gahleitner | Hydrogen from renewable electricity: An international review of power-to-gas pilot plants for stationary applications[END_REF]. Hydrogen enhanced combustion is also considered in off-grid strategies Preprint submitted to Preprint in which the local production is stored and later consumed as for example in isolated sites [START_REF] Cabezas | Optimization of self-regulated hydrogen production from photovoltaic energy[END_REF].

However, adding hydrogen to standard fuels poses challenges [START_REF] Levinsky | Why can't we just burn hydrogen? challenges when changing fuels in an existing infrastructure[END_REF][START_REF] Schiro | Modelling and analyzing the impact of hydrogen enriched natural gas on domestic gas boilers in a decarbonization perspective[END_REF], since it modifies fundamental combustion characteristics [START_REF] Sánchez | Recent advances in understanding of flammability characteristics of hydrogen[END_REF] that can compromise the fulfillment of safety and pollution standards.

As result, burners designed for natural gas, can only sustain limited hydrogen concentrations, typically 5 to 20 %vol in the fuel blend (https://www.hydrogeneurope.eu, 29/01/2022). Even in the best scenarios, a hydrogen volume fraction of 20% would represent only 7% mass reduction of CO 2 per energy unit with respect to natural gas. In order to develop new burners able to accommodate higher hydrogen concentrations, there is a need for more fundamental studies to understand critical issues associated to its addition.

Commercial systems are not designed for research purposes and it is difficult to monitor and control separately basic variables to generalize results, such as thermal power P th and equivalence ratio ϕ g . Moreover, the variety of hardware and control systems adds further complexity to the evaluation of hydrogen effects. For example, several self-aspirating burners have been tested with increasing hydrogen content showing a variable H 2 tolerance with a burner temperature that was barely affected in one case and augmented by more than 60% in the other [START_REF] Zhao | Influence of hydrogen addition to pipeline natural gas on the combustion performance of a cooktop burner[END_REF][START_REF] Zhao | Experimental assessment of the combustion performance of an oven burner operated on pipeline natural gas mixed with hydrogen[END_REF][START_REF] Choudhury | Combustion performance of low-nox and conventional storage water heaters operated on hydrogen enriched natural gas[END_REF]. In self-aspirating burners the mixture composition depends on the air velocity at the fuel nozzle outlet, which creates a negative static pressure that sucks fuel into the oxidizer stream. As result, the equivalence ratio of the combustible mixture could not be controlled independently, so that the results are only valuable for the specific operating conditions tested.

Pioneering studies on unconfined laminar premixed Bunsen flames [START_REF] Elbe | Further studies of the structure and stability of burner flames[END_REF][START_REF] Lewis | Stability and structure of burner flames[END_REF][START_REF] Mallens | Modeling of confined and unconfined laminar premixed flames on slit and tube burners[END_REF] demonstrated that the flame fronts stabilize where the flow velocity component perpendicular to the reaction front equals the laminar burning velocity, leading to a conical shaped flame. In particular, lowering the bulk velocity U b in premixed system promotes flashbacks [START_REF] Plee | Review of flashback reported in prevaporizing/premixing combustors[END_REF][START_REF] Lee | A numerical analysis of flame flashback in a premixed laminar system[END_REF], while higher values favor blow-off [START_REF] Elbe | Further studies of the structure and stability of burner flames[END_REF][START_REF] Lewis | Stability and structure of burner flames[END_REF]. These fundamental features have been studied since the early 1940s but, in spite of the empirical and numerical models proposed, they are not completely understood in practical configurations.

Moreover, the flame structure is influenced by the local flow velocity, local flame curvature and thermodiffusive properties of the mixture [START_REF] Echekki | Flame speed measurements at the tip of a slot burner: Effects of flame curvature and hydrodynamic stretch[END_REF][START_REF] Sivashinsky | Structure of bunsen flames[END_REF][START_REF] Vu | Tip opening of premixed bunsen flames: Extinction with negative stretch and local karlovitz number[END_REF][START_REF] Duva | Unstretched unburned flame speed and burned gas markstein length of diluted hydrogen/air mixtures[END_REF], which are particularly important for hydrogen [START_REF] Ern | Thermal diffusion effects in hydrogen-air and methane-air flames[END_REF][START_REF] Pelce | Influence of hydrodynamics and diffusion upon the stability limits of laminar premixed flames[END_REF]. In addition, both measurements [START_REF] Altay | Modeling the dynamic response of a laminar perforated-plate stabilized flame[END_REF][START_REF] De Goey | ten Thije Boonkkamp, Propagation, dynamics and control of laminar premixed flames[END_REF][START_REF] Mejia | Wall-temperature effects on flame response to acoustic oscillations[END_REF] and numerical simulations [START_REF] Altay | Two-dimensional simulations of steady perforated-plate stabilized premixed flames[END_REF][START_REF] Kedia | Mechanisms of stabilization and blowoff of a premixed flame downstream of a heat-conducting perforated plate[END_REF] prove that heat losses towards the flame holder also affect flame stabilization and flame dynamics.

A direct dependency between flame stand-off distance η, laminar burning velocity S L , heat losses and flame temperature is demonstrated in [START_REF] Ferguson | Stand-off distances on a flat flame burner[END_REF][START_REF] De Goey | Stabilization of adiabatic premixed laminar flames on a flat flame burner[END_REF] for example. Heat recirculation has also been shown to alter flame stabilization by preheating the incoming gases [START_REF] Ferguson | Stand-off distances on a flat flame burner[END_REF][START_REF] Ding | Chemiluminescence of burner-stabilized premixed laminar flames[END_REF][START_REF] Dai | Combustion characteristics of a porous media burner with partial hydrogen injection[END_REF]. In order to explore the impact of all parameters, the gas mass flow rates, fuel composition and equivalence ratio must be controlled separately.

Several works performed on multi-perforated cylindrical burners, already outlined the effects of burner porosity σ and surface temperature T w [START_REF] Lee | An experimental study of a cylindrical multi-hole premixed burner for the development of a condensing gas boiler[END_REF][START_REF] Veetil | Numerical simulations of steady perforated-plate stabilized syngas air pre-mixed flames[END_REF]. Tests on the heat exchanger efficiency also demonstrated that the risk of flashback increases at lower thermal power [START_REF] Lee | Performances of a heat exchanger and pilot boiler for the development of a condensing gas boiler[END_REF], although experiments were not conducted at fixed equivalence ratio. Studies dedicated to full-scale burners highlighted that pollutant concentrations depend on the temperature distribution inside the combustion chamber [START_REF] Hinrichs | Numerical and experimental investigation of pollutant formation and emissions in a full-scale cylindrical heating unit of a condensing gas boiler[END_REF]. The critical role of flame heat losses on NO x production was analyzed in [START_REF] Schiro | Experimental investigation of emissions and flame stability for steel and metal fiber cylindrical premixed burners[END_REF]. Recent studies revealed the impact of hydrogen enrichment in commercial devices in terms of flame stabilization and pollutant emissions [START_REF] González-Espinosa | Effects of hydrogen and primary air in a commercial partially-premixed atmospheric gas burner by means of optical and supervised machine learning techniques[END_REF][START_REF] Schiro | Potentialities of hydrogen enriched natural gas for residential heating decarbonization and impact analysis on premixed boilers[END_REF][START_REF] Schiro | Gas fired boilers: Perspective for near future fuel composition and impact on burner design process[END_REF][START_REF] Büyükakın | Numerical investigation on hydrogen-enriched methane combustion in a domestic back-pressure boiler and non-premixed burner system from flame structure and pollutants aspect[END_REF], but they do not consider flashback and blow off issues.

In the present work, the physical parameters that drive flame stabilization are identified and controlled, so that the impact of hydrogen addition on the operability of the systems is investigated.

Two cylindrical premixed burners featuring different designs and used in condensing boilers are considered. The analysis is limited to forced draft burners, in which fresh gases are maintained above atmospheric pressure. Only the low power range is investigated, since low volumetric flow rates favor flashback and increase the thermal stress on the burner, which are two main concerns related to H 2 enrichment. First, the hydrogen and methane concentrations in the global mixtures are defined in Sec. 2. The description of the two burners and of the methodologies used to analyze the effect of hydrogen enrichment are provided in Sec. 3. The burners response to dual fuel methanehydrogen mixtures is illustrated in Sec. 4 for two different powers via stability maps collected in steady state conditions. The effects of H 2 on blow-off and burner surface temperature is evaluated in Secs. 5 and 6, respectively. Finally, section 7 focuses on flashback limits.

Definition of hydrogen enriched admixtures

In this work, methane is used as a surrogate for natural gas. Methane and hydrogen are perfectly premixed before injection and the fresh gases are defined by the equivalence ratio ϕ g and by the molar fraction of hydrogen X H2 in the fuel blend. Considering the global balance equation for lean combustion of this multi-component fuel:

ϕ g (X CH4 CH 4 + X H2 H 2 ) + (2 - 3 2 X H2 )(O 2 + aN 2 ) → → ϕ g (X CH4 CO 2 + (2 -X H2 )H 2 O) + (2 - 3 2 X H2 )((1 -ϕ g )O 2 + aN 2 ) (1)
The global equivalence ratio ϕ g is defined as α/α s in which α is the ratio between the actual mass flow rates of fuel and air, while α s represents its value at stoichiometry:

α s = X CH4 W CH4 + X H2 W H2 (2 -3 2 X H2 )(W O2 + aW N2 ) (2)
in which a = 3.76 and W k is the molar weight of species k. The molar fraction of H 2 in the fuel mixture is given by:

X H2 = ṅH2 ṅH2 + ṅCH4 (3)
where ṅH2 and ṅCH4 are the molar flow rates of hydrogen and methane. Since the fuel is a binary mixture, the methane molar fraction corresponds to X CH4 = 1 -X H2 .

It is also useful to define the amount of hydrogen in the fuel considering the percentage of the total thermal power P th provided by hydrogen oxidation:

P H2 = X H2 Q H2 X CH4 Q CH4 + X H2 Q H2 ( 4
)
where Q H2 and Q CH4 are the molar lower heating values of hydrogen and methane. On one hand, this definition allows to compare operating conditions at constant power, which is a critical design parameter. On the other hand, P H2 is equal to the reduction in CO 2 emission associated to hybridization. The nonlinear relations between P H2 and X H2 , as well as the reduction percentage in CO 2 mass emission per energy unit caused by hydrogen enrichment EF / EF CO2 , are plotted in Fig. 1.

Materials and methods

Experimental setup

Air and CH 4 and H 2 mix directly into the supply lines, ensuring the injection of a fully premixed charge at the bottom of the bench displayed in Fig. 2. Mass flow rates are individually controlled using three mass flow meters and regulators: Vögtlin instruments GSC-B for air and H 2 , Bronkhorst F-201CV for CH 4 . This allows to control independently the hydrogen substitution in the fuel P H2 , the equivalence ratio ϕ g and the total thermal power P th over a wide range of operating conditions.

Once injected, the mixture is guided via a diverging duct into a cylindrical plenum with a diameter of 90 mm. It passes through a SiC porous structure characterized by 80% porosity and 40 mm thickness which breaks down the large flow structures and acts like a flame arrestor in case of flashback. For safety, a K-type thermocouple is also installed inside the plenum to measure the incoming gas temperature. Finally, the laminar flow feeds the cylindrical burner as shown in Fig. 2.

The burner is tested without the confinement of the combustion chamber and heat exchanger, which are instead part of the commercial system. This open configuration ensures optical access to the combustion region, but it may lead to results that deviate quantitatively from practical operations.

Wall and gas temperature measurements

A two-color infrared pyrometer (FLUKE Endurance series) with spectral response between 1.5 µm and 1.6 µm is used to measure the surface temperature T w of the burner to avoid overheating and verify convergence towards steady state condition. With the double wavelength approach the temperature measurements can be performed irrespective of the surface emissivity, which depends on material surface properties and temperature. The pyrometer operates between 250 • C and 1200 • C, with a precision of ± 0.3%.

In order to account for the preheating of the fresh gases by the hot burner wall, flow temperature measurements in front of the small burner holes are performed following the technique presented in [START_REF] Ding | Chemiluminescence of burner-stabilized premixed laminar flames[END_REF]: the burner operates in steady state condition when the fuel rate is shut forcing flame quenching; this instant corresponds to the initial time of data acquisition and a movable K-type thermocouple is placed near the external multi-perforated surface of the burner to measure the gas flow temperature exhausting the burner; after a certain time delay, related to the thermal inertia of the thermocouple itself, temperature decreases exponentially following the gas temperature reduction and data are collected with 5 seconds interval; finally an extrapolation between the time corresponding to the first valid signal acquisition and initial instant t = 0 is performed to retrieve the preheating temperature of the gas at the instant preceding quenching. Figure 3 shows the result of the extrapolation obtained from three independent gas temperature T u measurements performed with the burner B1 at 4 kW for P H2 = 40 % and ϕ g = 0.6. It shows that the method yields repeatable results with a uncertainty of about ± 10 K.

Mixture injection

Velocity measurements

A hot wire (MiniCTA -Dantec Dynamics) is used to determine the velocity component in the radial direction, perpendicular to the burner holes. The probe is positioned at a distance d = 0.5 mm from the burner wall, while the hot wire is moved along the burner longitudinal direction (see Fig. 10(b)). Velocity measurements are taken with a spatial resolution of 0.1 mm, ensuring at least 4 points inside the diameter of a single burner hole. Each value is the average of measurements performed with a sampling frequency of 1 Hz over a duration of 5 seconds. Table 1: Dimensions and external surface porosity values of burners B1 and B2 shown in Fig. 4.

Burners description

The forced draft burners B1 and B2 are representative of two technologies used in residential condensing boilers and designed to operate between 2 and 30 kW with natural gas. diameter and two lateral hollow slots. This pattern is reproduced periodically along the burner tangential and axial directions with a step of 6.7 mm and 7.2 mm, respectively. Since the apertures of the hollow slots point towards the circular holes, the structure of the flow at the flame base is more complex for the burner B2 than burner B1. Also, a swirler is used at the inlet of the burner that helps to homogenize the flow at the burner hole outlets. Over the conditions investigated the pressure losses characterizing the two burners are comparable and limited between 20 Pa and 50

Pa.

These two designs result in different burner porosities σ B1 = 8% and σ B2 = 3% which, considering constant volumetric flow rate and gas inlet temperature, leads to a strong difference in bulk flow velocities U B1 b and U B2 b through the holes:

U B2 b = σ B1 σ B2 U B1 b ≃ 2.7U B1 b ( 5
)
This definition of the bulk flow velocity does not account for the non-uniformity of the flow field along the burner or for the specific geometry of the outlets and must be considered as an average value.

Figure 5(a-b) displays direct views of flames in both burners at ϕ g = 0.6 and for burners B1 and B2 in Fig. 5(c).

P H2 = 40% showing B1 B2 ϕ g -P H2 T max [K] ϕ g -P H2 T max [K] 0.
that the surface of burner B1 produces a stronger thermal radiation than B2, since flames stabilize closer to the metallic surface of B1 leading to higher temperatures of the burner walls. Figure 5(c) shows the mean normalized temperature distribution along the vertical direction for both burners.

Temperature profiles were normalized by the maximum wall temperature T w for each operating conditions synthesized in Tab. 2 and then averaged separately for burner B1 and B2. For each burner, the normalised temperature distribution remains roughly the same for all conditions explored.

Measurements taken at locations 1 to 6 as indicated in Fig. 5(a-b) also show that the temperature is more uniform along burner B1. For both burners the highest temperature occurs between locations 4 and 5. At the top and at the bottom of the burner, the wall temperature decreases due to heat losses. The relative temperature reduction at the bottom of burner B2 is much stronger than in burner B1. Moreover, the internal structure itself also influences the gas residence time inside the burner, affecting the temperature T u at which the unburned gases exit the burner holes. As sketched in Fig. 4, burner B1 favors preheating, since the fresh mixture makes a U-turn at the top of the burner resulting in a longer residence time of the reactants. Inside B2, instead, the mixture is directly distributed over the external surface limiting the heat transfer to the flow. This effect is illustrated in Tab. 3. The preheating temperature of the fresh gases is measured at given wall temperature T w for burners B1 (T B1 u ) and B2 (T B2 u ). The wall temperature T w is adjusted tuning the hydrogen content P H2 at constant equivalence ratio, minimizing the variation on the total mass flow rate. Table 3 shows that burner B1 promotes larger heat transfer to the reactants than burner B2. The same wall temperature results in greater preheating of the reactants with burner B1. In the rest of the study, T w must be considered as the wall temperature measured at location 4 in Figs. 5(a-b).

ϕ g P H B1 2 P H B2 2 T w (K) T B1 u (K) T B2 u (K) 0.

H 2 hybridization impact on stabilization maps

Figures 6(a-d) present burners stability maps performed at constant power of 2 kW (at the bottom) and 4 kW (at the top) for burner B1 (left) and burner B2 (right) as a function of the hydrogen enrichment P H2 and global equivalence ratio ϕ g . At constant hydrogen content P H2 the equivalence ratio ϕ g is changed by modulating the air mass flow rate and, due to limitations of the air mass flow controllers, its minimum value is ϕ g = 0.45 at 4 kW and ϕ g = 0.30 at 2 kW. Three isolines representative of the ratio U b /S l = 1, calculated for mixture temperatures T u = 300 K (red), 500 K (green) and 700 K (blue), are superposed on the maps and will be discussed in Sec. 7. The maps illustrate the combinations of P H2 and ϕ g that allow to operate the burners without facing undesirable events, namely blow-off, overheating and flashback:

• Blow-off is defined here as the condition at which the flames detach from the lower side of the burner, leading to incomplete combustion.

• Overheating occurs when the burner surface temperature T w overcomes 1050 K, which is the one measured for B1 at the reference condition of P H2 = 0%, ϕ g = 0.8 and P th = 4 kW.

• Flashback happens when the flame front propagates upstream through the injection system [START_REF] Kalantari | Boundary layer flashback of non-swirling premixed flames: Mechanisms, fundamental research, and recent advances[END_REF].

Figures 6(a-d) show that stable operating conditions, regardless of the burner and the power investigated, are limited by two main boundaries: one at the bottom-left side of the maps and the other at the top-right side. The first corresponds to the blow-off ( ). For a given hydrogen content P H2 , the reduction of the equivalence ratio ϕ g decreases the reactivity of the mixture and the flames blow-out. Interestingly, increasing the hydrogen content P H2 the blow-off limit occurs at lower equivalence ratios and it is possible to sustain leaner combustion for both burners. For ). The red, green and blue curves on each plot identify the isolines at U b /S L = 1 for fresh gas temperature of Tu =300 K, 500 K and 700 K respectively.

instance, Fig. 6(a) shows that, increasing P H2 from 0% to 40%, it is possible to reduce ϕ g from 0.60 to 0.45. The other limits are overheating ( ) and flashback ( ). Figures 6(a-d) indicate that the highest achievable hydrogen enrichment P H2 is reduced by increasing the equivalence ratio ϕ g , independently of the burner and thermal power. Closer to stoichiometry, the limiting factor is the thermal resistance of the material because, thanks to the greater reactivity of the mixture, flames stabilize closer to the burner surface promoting heat transfer to the wall. At leaner conditions, instead, the maximum hydrogen enrichment P H2 is dictated by flashback, which occurs for burner surface temperatures below 1050 K. Nevertheless, Figs. 6(a-d) show that H 2 addition does not widen the range of achievable operating conditions: an improvement of the lean blow-off limit is systematically accompanied by a reduction of the highest achievable equivalence ratio due to overheating or flashback.

The aforementioned conclusions apply qualitatively to both burners. However, from a quantitative standpoint, differences are found between burners B1 and B2 for a constant power. On one hand, B2 exhibits a greater flashback resistance, allowing for higher hydrogen content at the same equivalence ratio. For example, at P th = 4 kW and ϕ g = 0.6, flashback appears at P H2 = 50% and P H2 = 80%

for B1 and B2, respectively. On the other hand, B2 is more prone to blow-off and a series of low-ϕ g low-P H2 operating conditions are only achievable with B1, as for example P H2 = 0% and ϕ g = 0.6.

To sum up, there are operating conditions that are stable for one burner and not achievable with the other, independently of power. Since the range of operating conditions is the same for both burners, these differences must be related to their design. Burner B2 generates higher bulk flow velocities U b at the hole outlets and features a lower influence on the unburned gas temperature T u than burner B1 (Sec. 3.4). As a matter of fact these aspects facilitate flame detachment from the wall, hindering flashback and eventually favoring blow-off. Thus, the hole pattern of the multi-perforated surface influences the performances of the burner. However, improving flashback tolerance worsens the blow-off resistance, limiting the effective fuel flexibility at fixed equivalence ratio.

Blow-off limit description

Hydrogen addition is known for improving the blow-off resistance [START_REF] Schefer | Combustion of hydrogen-enriched methane in a lean premixed swirl-stabilized burner[END_REF][START_REF] Ghoniem | Stability and emissions control using air injection and h2 addition in premixed combustion[END_REF]. In this section, flame images are used to investigate the differences in the mechanisms leading to blow-off for different fuel blends in both burners. Images are taken with a NIKON D7500 camera, equipped with Micro-Nikkor 105 mm, from a direction tangent to the cylindrical surface of the burner (see Fig. First, the effect of H 2 substitution is considered and, secondly, the impact of the ports distribution is discussed. Figures 7(b-d the white-dotted lines represent the axis of two adjacent burner holes.

On one side, Figures 7(b-d) show pure-CH 4 flames for equivalence ratios ϕ g = 0.70, 0.65 and 0.55.

In Fig. 7(b), flames are attached to the burner surface at ϕ g = 0.70 and a series of independent conical reaction fronts develop beyond each slit hole. Reducing ϕ g results in a simultaneous increase of the bulk flow velocity U b and a reduction of the adiabatic laminar burning velocity S L , which concomitantly lead to a gradual increase in the flame length and a coalescence of adjacent conical flame fronts as shown in Fig. 7(c). This phenomenon is exacerbated in Fig. 7(d) where the entire reaction layer detaches from the burner surface, resulting in a severe blow-off.

On the other side, Figs. 8(b-d) show for the same burner B1 H 2 -enriched flame images (P H2 = 50%) at ϕ g = 0.50, 0.45 and 0.40, respectively. Figure 8(b), similar to Fig. 7(b), illustrates well-anchored flames and stable combustion. However, a reduction of ϕ g to 0.45 marks a strong difference with respect to pure methane case: the flames lengthen and their curvatures increase, leading to flame tip-opening as in Fig. 8(c). This effect is due to the low Le number of molecular H 2 [START_REF] Vu | Tip opening of premixed bunsen flames: Extinction with negative stretch and local karlovitz number[END_REF][START_REF] Mizomoto | Effects of preferential diffusion on the burning intensity of curved flames[END_REF][START_REF] Mizobuchi | Numerical study of tip opening of hydrogen/air bunsen flame[END_REF], which alters the properties of the mixture leading to a preferential diffusion within the flame front.

According to [START_REF] Bouvet | On the effective Lewis number formulations for lean hydrogen/hydrocarbon/air mixtures[END_REF] and considering the Le number of CH 4 (H 2 ) equal to 1.0 (0.3), the characteristic Le number of the fuel blend at P H2 = 50% drops to 0.46, justifying the observation of thermodiffusive effects [START_REF] Howarth | An empirical characteristic scaling model for freely-propagating lean premixed hydrogen flames[END_REF][START_REF] Berger | Intrinsic instabilities in premixed hydrogen flames: Parametric variation of pressure, equivalence ratio, and temperature. part 1 -dispersion relations in the linear regime[END_REF]. As a consequence, the flame tip is quenched while the reaction rate strengthens at the base of the conical flames. For that reason the reaction front shifts towards the periphery of the burner outlets promoting interactions among adjacent flames. Fig. 8(d) shows that this trend is intensified at blow-off conditions, in which the reaction zones lie entirely between burner ports.

To sum up, Le effects due to hydrogen addition promotes a different flame stabilization at very lean conditions (i.e. during blow-off). In case of CH 4 -air mixtures the flames detach from the burner surface as a continuous reactive layer. In case of hydrogen enrichment, instead, single flames stabilize in the low velocity region among the burner outlets, where combustion is sustained thanks to the interaction among several holes. This mechanism strengthens by increasing the hydrogen substitution and it is reasonable to expect that it also depends on the ports distribution over the burner surface. This is further discussed in Fig. 9(b-d), in which the burner B2 is considered.

This figure shows the blow-off transition for P H2 = 50% and equivalence ratios ϕ g = 0.60, 0.55, 0.50. Figure 9(b) shows M flames stabilized along the burner wall, where each lobe corresponds to a single outlet of the burner. In analogy with Fig. 8, when the equivalence ratio is reduced, Le effects make the reaction rate stronger at the flame base and weaker at the flame tip (Fig. 9(c)). A further reduction of the equivalence ratio to 0.50 (Fig. 9(d)) produces a stabilization of the flame fronts between the two holes that are close to each other, while the reaction is quenched elsewhere.

On one side, this observation confirms that the molecular diffusivity of hydrogen addition might promote a change of flame stabilization that increases the blow off resistance. On the other side, Figs. 9(d) shows that this mechanism is promoted by the proximity of the the burner holes. For burner B2, in fact, the distance between burner holes varies and H 2 -enriched flames stabilize only if two holes are sufficiently close to each other. In the vicinity of the holes, hydrogen diffuses laterally encountering a relatively high-temperature and low-velocity region that enables flame stabilization.

Moreover, limiting the distance between holes inhibits dilution by external air engulfment, which normally acts at the base of isolated Bunsen flames favoring local quenching [START_REF] Bollinger | Effect of burner-tip temperature on flash back of turbulent hydrogenoxygen flames[END_REF][START_REF] Dugger | Flame stability of preheated propane-air mixtures[END_REF]. This interpretation requires the local flow velocity between the holes to be comparable to the hydrogen diffusion velocity in air at these thermodynamic conditions. This is verified considering the burner B1. In this setup, the hydrogen diffusion velocity V d,H2 into air can be evaluated using Fick's law, under the hypothesis of constant mass fraction gradient of H 2 over the distance l = 2.5 mm between the hole axis and the mid point between two consecutive holes (see Fig. 10(b)). If the mass fraction of H 2 is considered to be equal to the value of the incoming premixed gases at the burner hole axis and linearly decreases to zero at the mid point between two consecutive holes, then V d,H2 is of the order of 0.25 m/s. The flow velocity component U r along the burner radial direction is measured in non-reactive conditions (see sec. Overall, the discussion above suggests that either H 2 -enrichment and a specific hole pattern must be considered to interpret flame stabilization, since both are necessary but not sufficient conditions to allow different flame stabilizations. These experiments also suggest that, in addition to the increased reactivity of H 2 -enriched mixtures, a specific distribution of the burner holes can reinforce the blow-off resistance when H 2 is involved due to its high diffusion velocity.

Effect of flame stabilization on the burner thermal load

The burner surface temperature T w controls the lifetime of materials and it affects the mixture reactivity, since fresh gases are heated by the hot walls. The impact of the wall temperature T w on flame stabilization is explored in this section. Figures 11(a-b) display the contour maps of surface temperature T w at 4 kW for both burners. Data are retrieved by interpolating the temperature measurements collected with the pyrometer at steady state conditions over the operating points covered in Figs. 6(a-b). Here T w represents the temperature at location 4 displayed in Fig. 5(a-b).

The wall temperature T w depends on the adiabatic flame temperature T ad and on the stand-off distance η between the flame root and the flame holder. As demonstrated for example in case of flat flames [START_REF] Ferguson | Stand-off distances on a flat flame burner[END_REF][START_REF] De Goey | Stabilization of adiabatic premixed laminar flames on a flat flame burner[END_REF], these two parameters generally control the heat load on the burner. However, data listed in Tab. 4 demonstrate that T ad does not correlate to T w , at least for the conditions investigated. For instance, for operating conditions 3 and 5 T w reaches comparable values but the corresponding adiabatic flame temperature T ad differs by 130 K. An additional example is given considering operating conditions 1 and 5. In this case the adiabatic flame temperature is similar for both conditions, while T w differs by more than 150 K. Interestingly, Table 4 shows that the wall temperature T w results in a better correlation with the laminar burning velocity S L Tu , rather than with T ad . In fact, as S L increases with both equivalence ratio ϕ g and fraction of hydrogen P H2 in the fuel blend, the burner surface temperature T w increases consistently. Here S L Tu is retrieved T w, adiabatic flame temperature T ad and laminar burning velocity S L , which is calculated considering fresh gases at Ta = 300 K and at T u. The laminar burning velocity is highly affected by the fresh gas temperature Tu. T w results in good correlation with S Tu L , rather than T Tu ad .

ϕ g -P H2 T u (K) S L Ta (cm/s) S L Tu (cm/s) T w (K) T ad Tu (K)
from CANTERA 1D simulations imposing T u as unburnt gas temperature.

Flame stabilization is subjected to a positive feedback loop. An increase in T w preheats the reactants at T u resulting in larger S Tu L and a flame closer to the wall, thus increasing T w . Table 4 shows the impact of the fresh gas temperature T u on the laminar burning velocity S L Tu that is between 2 to 4 times higher than considering ambient conditions S L Ta . This also means that the mixture composition and the thermal state of the burner are strictly coupled, which needs to be taken into account in view of the fact that this interaction can affect flashback directly [START_REF] Bollinger | Effect of burner-tip temperature on flash back of turbulent hydrogenoxygen flames[END_REF][START_REF] Dugger | Flame stability of preheated propane-air mixtures[END_REF][START_REF] Khitrin | Peculiarities of laminar-and turbulent-flame flashbacks[END_REF].

To highlight the actual variation of the stand-off distance η with increasing H 2 substitution, Figs. 12 and 13 illustrate the evolution of the flame front position at constant equivalence ratio for burners B1 and B2 at 4 kW. Figures 12(a) and 13(a) depict the windows of investigation of the burners (as in section 5). Figures 12(b-d) refer to burner B1 at 4 kW and ϕ g = 0.6, varying the hydrogen power content P H2 from 0% to 46%. In case of pure methane flames, the reaction front is detached from the burner surface (Fig. 12(b)) because this operating condition is close to the blow-off limit (Fig. 6(a)). An increment of P H2 to 25% results in shorter flames, confirming the H 2 stabilization effect against blow-off. In Fig. 12(d), the hydrogen power content P H2 is augmented to 46% and the H 2 -enhanced flame reactivity drives the flame closer to the burner surface. The conical laminar flames in Fig. 12(c) transform into independent nearly flat flames anchored at the edges of the burner holes in Fig. 12(d) and the heat load on the burner reaches its maximum. Similar observations are made in Figs. 13(b-d) with the burner B2, when it is fueled with a growing hydrogen content at ϕ g = 0.7. The flame shapes differ with respect to those shown for burner B1 because of the distribution of the exit holes. Flames of burner B2 exhibit a "M" shape, where each lobe forms in the wake of a single burner hole. Boosting the hydrogen content P H2 from 20% to 55%, the flames shorten continuously and the burner surface temperature T w raises from 750 K to 1020 K. In both burners the reaction front approaches dangerously the burner hole outlets when the hydrogen content is increased (Figs. 12(d) and 13(d)). An exacerbation of this phenomenon leads to flashback.

Flashback limit description

Flashback occurs in premixed burners when the flame propagates upstream through the injection system [START_REF] Kalantari | Boundary layer flashback of non-swirling premixed flames: Mechanisms, fundamental research, and recent advances[END_REF], which can be the result of different mechanisms. The simplest criterion to interpret flashback is based on premixed flame kinematics. Since a premixed flame lies where the component of the flow velocity perpendicular to the flame front matches the local flame speed, flashback is expected when the laminar burning velocity S L overcomes the incoming bulk flow velocity U b . This can be seen as a particular case of the general theory of the velocity gradient [START_REF] Elbe | Further studies of the structure and stability of burner flames[END_REF][START_REF] Lewis | Stability and structure of burner flames[END_REF]. According to that and considering a laminar flow with a given velocity profile, flashback occurs when the flow velocity gradient at the wall equals the ratio between the laminar flame speed and the flame thickness. However, if a flame stabilizes above small holes or narrow tubes, this theory tends to lose its validity. In this case, heat losses increase lowering the flame reactivity and ultimately hindering its passage towards the injection system. In these conditions, the main governing parameters become the wall temperature T w and the flame holder diameter [START_REF] Bollinger | Effect of burner-tip temperature on flash back of turbulent hydrogenoxygen flames[END_REF][START_REF] Dugger | Flame stability of preheated propane-air mixtures[END_REF][START_REF] Jung | Direct prediction of laminar burning velocity and quenching distance of hydrogen-air flames using an annular stepwise diverging tube (asdt)[END_REF]. Because of the high temperature reached by the burner walls and the substantial preheating of the fresh mixture, autoignition close to the hot metallic burner surfaces inside the burner should also be taken into consideration. In fact, depending on the mixture composition and flow field at the exit holes, the burner wall temperature may potentially serve as source of energy to ignite the mixture [START_REF] Boeck | Hot surface ignition dynamics in premixed hydrogen-air near the lean flammability limit[END_REF], triggering flashback.

Considering now the present study, Figs. 6(a-d) show three isolines representative of U b /S l = 1 calculated for three different gas temperatures T u =300 K (red), 500 K (green) and 700 K (blue). While the slope of the isolines is in good agreement with the one of the experimental flashback limits, the measured gas temperature T u for the points that belong to the flashback lines (Figs. 6(a-d)) is not constant and the ratio between the bulk flow velocity through a burner hole U b and the laminar burning velocity S L varies substantially as indicated in Tab. 5. This compared to three conditions representative of flashback events. The ratio U b /S L is calculated considering flow and laminar burning velocity at the measured gas temperature T u , showing that (U b /S L ) T u increases considerably with the hydrogen content P H2 at different flashback conditions.

Hence, the H 2 -induced flashback cannot be simply predicted in view of the competition between the bulk velocity U b and the laminar burning velocity S L . With the goal of broadening the number of operating conditions subjected to flashback and investigate this phenomenon, a different measurement procedure is adopted. At the beginning, the burner is in thermal equilibrium with ambient. Then, the targeted mixture composition is imposed. After few seconds, the burner is ignited and the evolution of wall temperature with time is monitored. Three distinct regimes are identified after ignition:

Regime I: Combustion remains stable and the burner evolves naturally towards thermal equilibrium.

Regime II: Initially the flame stabilizes on the multi-perforated burner surface but, as T w reaches a certain threshold, flashback occurs. This value is defined as T w,f b .

Regime III: Ignition itself generates a sudden flashback. In this circumstance the flame is never stabilized on the burner, which remains at ambient temperature. experiments do not belong to the stabilization maps in Figs. 6a-b. In fact, it would be impossible to gather this level of insight operating the burner in quasi-steady state mode. As expected, only the operating conditions leading to regime I correspond to the ones displayed in Figs. 6a-b. At high equivalence ratio ϕ g and large hydrogen power content P H2 , instead, the burners undergo flashback during ignition corresponding to regime III in Figs. 14a-b. For this pool of operating points, the reactivity of the mixture increases so much that it is impossible to operate the burners. In between these two regimes, there is a range of mixtures for which flashback is thermally activated by a specific burner surface temperature T w,f b . This mechanism defines regime II. Both maps in Fig. 3 show that this limit can be attained from regime I either by increasing the hydrogen power content P H2 or raising the equivalence ratio ϕ g towards stoichiometry. Moreover, results reveal that this transition depends on the burner design.

The maps presented in

Regime II identifies a strict dependence between flashback and the thermal state of the burner. As shown, increasing either the hydrogen power content P H2 or equivalence ratio ϕ g leads to a reduction of this activation temperature, presumably due to the boosted reactivity of the mixture.

Figure 15b, for instance, puts in evidence that increasing P H2 from 80% to 90% at constant ϕ g = 0.6, reduces T w,f b from 967 K to 912 K. On the other side, increasing ϕ g to 0.7 keeping P H2 constant to 80%, reduces T w,f b to 880 K. Measurements have been repeated at least three times for each condition and the variation of T f b w lies within ±5 K, which is below the instrumentation error. Furthermore, Figs. 14a-b illustrate that burner B1 undergoes flashback for a wall temperature that is lower than for burner B2. This is particularly evident for a hydrogen power content P H2 = 50% and equivalence ratio ϕ g = 0.8, which is the unique point shared by the two burners lying in regime II. In this case, T w,f b is equal to 872 K for burner B1, while it increases to 1068 K for burner B2.

Despite results highlight the crucial role of the wall temperature T w , they do not shed lights on the fundamental mechanisms that drive this process and additional work is needed to interpret the observations made.

Conclusions

Hydrogen hybridization of two commercial premixed burners, used in condensing boiler applications and designed to operate with natural gas -air mixtures, has been investigated. The burners are tested in the low-power range, analyzing blow-off and flashback limits over a wide range of equivalence ratio for methane, hydrogen and air mixtures. Stability maps demonstrate that H 2 addition improves the blow-off resistance, but it also limits the maximum achievable equivalence ratio due to occurrence of overheating or flashback, leaving the range of stable operating conditions roughly unchanged. Because of these constraints, pure hydrogen combustion is possible only at ultra-lean conditions, while the maximum hydrogen concentration for a given thermal power and equivalence ratio is found to be variable with the burner design.

The study underlines the impact of hydrogen low Le number on lean flame stabilization. To this regard, it has been highlighted a specific mechanism that contributes to blow-off resistance related to hydrogen diffusion properties, through which flames stabilize in the near-surface region among holes rather than in their wake.

For a fixed equivalence ratio, the increment of H 2 enrichment above a certain threshold leads to flashback. It has been shown that this limit is strongly related to the burner wall temperature T w .

The three regimes have been identified depending on the mixture composition and on the burner design. The first regime (I) is characterized by stable flame stabilization, without risk of flashback.

In regime II, flashback is piloted by the thermal state of the burner and occurs at a specific wall temperature T w = T w,f b . This threshold temperature T w,f b decreases with increasing the reactivity of the mixture. In regime III, flashback occurs during the ignition process, irrespective of the burner wall temperature T w . This regime is characterized by very high hydrogen content and equivalence ratio close to stoichiometry. Further studies will be focused on this specific issue to shed lights on the fundamental mechanisms that drive flashback in this kind of systems with the objective of designing a concept burner suitable for full H 2 combustion over a wide range of equivalence ratio.

Acknowledgements

Figure 1 :

 1 Figure 1: The solid line ( ) illustrates the relation between the percentage of the total thermal power P th provided by hydrogen oxidation P H2 and the molar fraction of hydrogen X H2 in the fuel blend. The dashed line ( ) shows the reduction percentage of the emission factor EF (kg CO2 /J) of hydrogen enriched mixtures with respect to pure methane combustion.

Figure 2 :

 2 Figure 2: Schematic representation of the test bench cross sectional area with on board instrumentation.

Figure 3 :

 3 Figure3: Example of the exponential fitting for experimental data to evaluate the fresh gas preheating due to the burner hot walls. Three independent gas temperature measurements over time are shown using the burner B1 at 4 kW for P H2 = 40 % and ϕg = 0.6.

Figure 4 :

 4 Figure 4: Illustration of external and internal designs of the burners B1 (a) and B2 (b) with qualitative description of the evolution of the gas stream temperature inside the burners. The two zoom-boxes at the bottom of each image represent a detailed view of the exit hole pattern for each burner.

Figures 4 (Figure 5 :

 45 Figures 4(a-b) display photos of the two burners with a sketch of their internal structure. They are made of stainless steel and feature similar sizes (Tab. 1). Burner B1 is characterized by both round and slit holes (Fig.4(a)). Circular holes have 0.5 mm diameter with l = 2.5 mm pitch and the slits are 4.0 mm wide by 0.5 mm long. Burner B1 features a double metallic layer with a small gap that makes a U-turn to guide the flow through the holes. For the burner B2 (Fig.4(b)), the outlets are distributed following the repetition of the same geometrical pattern: two circular holes of 0.7 mm

Figure 6 :

 6 Figure 6: Steady-state stabilization maps displaying stable operating conditions as function of equivalence ratio ϕg and H2 content P H2 . Plots refer to burner B1 operated at 4 kW (a) and 2 kW (c) and to burner B2 operated at 4 kW (b) and 2 kW (d), showing the operating limits associated to blow-off ( ),flashback ( ) overheating (). The red, green and blue curves on each plot identify the isolines at U b /S L = 1 for fresh gas temperature

Figure 7 :

 7 Figure 7: Flame shape along burner B1 longitudinal side, captured from a direction tangential to the burner surface above the slit holes as shown in (a). Operation at 4 kW with pure methane flames P H2 = 0% at equivalence ratios (b) 0.70, (c) 0.65 and (d) 0.55.

Figure 8 :

 8 Figure 8: Flame shape along burner B1 longitudinal side, captured from a direction tangential to the burner surface above the slit holes as shown in (a). Operation at 4 kW with an hydrogen content P H2 = 50% at equivalence ratios (b) 0.50, (c) 0.45 and (d) 0.40.

  Fig. 6(a) for burner B1. The windows of investigation are illustrated in Figs. 7(a) and 8(a), while

Figure 9 :

 9 Figure 9: Flame shape along burner B2 longitudinal side, captured from a direction tangential to the burner surface above round holes as shown in (a). Operation at 4 kW with an hydrogen content P H2 = 50% at equivalence ratios (b) 0.60, (c) 0.55 and (d) 0.50.

  3.1), imposing a volumetric flow rate representative of reactive operating conditions at 4 kW.

Figure 10

 10 shows U r with respect to the axial direction of the burner, taken at a distance d = 0.5 mm above the burner surface. Interestingly, despite the maximum velocity being around 3 m/s, U r drops below 0.3 m/s between two adjacent holes, reaching similar values of the H 2 diffusion velocity V d,H2 evaluated previously. Hence, the local flow velocity is comparable with the diffusion velocity of hydrogen in air and so compatible with the conjectured mechanism of preferential diffusion.

Figure 10 :

 10 Figure 10: The radial velocity profile U r, measured along the axial direction of burner B1 at d = 0.5 mm above the external surface ( ), is compared to an estimation of the hydrogen diffusion velocity at the burner outlets ( ) (a). The axial and radial directions are indicated in (b), as well as the distance d at which the velocity profile is determined.

Figure 11 :

 11 Figure 11: Wall temperature T w contour maps as function of the hydrogen content P H2 and equivalence ratio ϕg for burner B1 (a) and B2 (b) at power of 4 kW.

Figure 12 :Figure 13 :

 1213 Figure 12: Flame shape evolution along burner B1 longitudinal side, captured from a direction tangential to the burner surface (a). Images correspond to steady state conditions for ϕg = 0.6 and P H2 equals to 0% (b), 25% (c) and 46% (d).

Figure 14 :ggFigure 15 :

 1415 Figure 14: Description of the region of influence of regimes I, II and III with respect to the hydrogen power content P H2 and equivalence ratio ϕg for burners B1 (a) and B2 (b). Operating conditions belonging to regime II are sized and colored considering the value of T w,f b .

Figures

  Figures 15a-b display the evolution of the burner surface temperature T w , after ignition, for three different mixture compositions with burners B1 and B2. In each case the wall temperature T w increases up to a peculiar value T w,f b at which flashback occurs. It was verified that this value only depends on the mixture composition and it is not related to the time needed to attain T w,f b .

  This project has received funding from the European Research Council under the European Union's Horizon 2020 research and innovation programme Grant Agreement 832248, SCIROCCO and under the Horizon 2020, COEC (Center of Excellence in Combustion) program, Grant Agreement 952181.

Table 2 :

 2 List of operating conditions at 4 kW used to provide the average normalized axial wall temperature profile

	6 -40%	635	0.6 -70%	658
	0.7 -20%	698	0.7 -50%	695
	0.8 -0%	718	0.8 -30%	714

Table 3 :

 3 Preheated gas temperatures T u for burners B1 and B2 are compared at fixed wall temperature T w. The burners are operated at the same equivalence ratio to minimize the mass flow variations. The hydrogen content is tuned to match the targeted wall temperature T w with the different burners.

Table 4 :

 4 Five operating conditions obtained with burner B1 at 4 kW are compared in terms of wall temperature

	1.	0.6 -10%	407 ± 10	14.4	28.9	710 ± 2.1 1765
	2.	0.6 -20%	481 ± 12	17.8	54.1	813 ± 2.4 1838
	3.	0.6 -30%	512 ± 9	22.1	78.2	883 ± 2.6 1878
	4.	0.6 -40%	530 ± 8	26.8	105.0	943 ± 2.8 1909
	5.	0.5 -62%	521 ± 11	20.0	91.6	863 ± 2.6 1748

  table lists five operating points for burner B1 at 4 kW, where two stable operating points are

Table 5 :

 5 List of operating conditions used to investigate the role of the ratio between the bulk flow velocity U b and the laminar burning velocity S L with respect to flashback occurrence. This ratio is calculated for the gas temperature T u(after being heated by the combustor walls), showing the decisive role of the unburned mixture temperature. The first stable condition identified by * is not included in the stabilization maps since the wall temperature T w overcomes the overheating threshold of 1500 K, but is specifically considered in this table because of the low value of the ratio (U b /S L ) T u .


domain: scco phys

text:
Dr Pierre Uzan

Correlations: How to Interpret the No-Signaling Condition? Super-Quantum Correlations: How to Interpret the No-Signaling Condition?

Keywords: Uzan, laboratory CHArt. Super-Quantum Super-quantum correlations, no-signaling condition, Tsirelson bound

This article deals with the question of the maximal correlation degree of two intelligent machines that cannot exchange any signals. After reminding the reader of the incorrectness of the mainstream statistical interpretation of the "no-signaling" condition, its informational meaning is explored. It is emphasized that if Pawlowski et al.'s Information Causality Principle correctly expresses (and generalizes) the no-signaling condition, its application is, for now, based on a specific scenario (suggested by van Dam) and a no less specific (and simplified) relationship between mutual information and correlators. A more general informational interpretation of the no-signaling condition from which the Tsirelson bound can be derived is then formulated in terms of correlational independence.

Introduction

The no-signaling condition (NS) obviously is a condition asserting the impossibility of any exchange of signals or information between physical systems. However, this informational condition is currently reduced to the only statistical independence of the observables that can be respectively measured in these systems. As explained in this article, such a statistical interpretation of the no-signaling condition is far too weak to really express the informational independence of all possible events occurring, respectively, in each of these systems. Section 1 will remind the reader of the incorrectness of this mainstream statistical interpretation of the no-signaling condition. In section 2, its informational interpretation by Pawlowski et al's Information Causality Principle will be considered. It will be emphasized that the derivation of the Tsirelson bound they propose lacks of generality. Section 3 will propose a more general informational formulation of the no-signaling condition in terms of correlational independence that allows a purely informational derivation of the Tsirelson bound (section 4).

Incorrectness of the mainstream statistical interpretation of the (NS) condition

One refers to the usual bipartite (Alice and Bob) Bell game where the question of superquantum correlations is generally discussed -see, for example Popescu 1 , Barrett et al. [START_REF] Barrett | Nonlocal correlations as an informationtheoretic resource[END_REF] or Gisin [START_REF] Gisin | L'impensable Hasard -Non-localité, téléportation et autres merveilles quantiques[END_REF] . In this bipartite game, which can be seen as modeling the possible connection between two intelligent machines, the incomes chosen by Alice and Bob are noted as x and y respectively, and the corresponding outcomes, noted as a and b respectively, can take the values 0 or 1. A "box" is an abstract description of the relation between inputs and corresponding outputs, that is, it is the given of the conditional probabilities P(a,b/x,y) for all combinations of a, b, x and y. In the current literature, the no-signaling condition, called (NS) in the following, is interpreted as a condition of statistical independence. This condition asserts that the probability that Alice obtains a particular outcome "a" is independent of Bob's choice, that is, of the value of y, and vice versa [START_REF] Popescu | Non-locality beyond quantum mechanics[END_REF][START_REF] Barrett | Nonlocal correlations as an informationtheoretic resource[END_REF][START_REF] Gisin | L'impensable Hasard -Non-localité, téléportation et autres merveilles quantiques[END_REF][START_REF] Jones | Interconversion of Nonlocal Correlations[END_REF][START_REF] Bub | Why the tsirelson bound?[END_REF] . This statistical condition is formulated as follows: (NS)stat For all possible actions x, x', y, y' and for all possible outcomes a and b:

b P(a, b / x, y) = b P(a, b / x, y') and a P(a, b / x, y) = a P(a, b / x', y), which can be more simply written as: P (a / x, y) = P (a / x) and P (b / x, y) = P (b / y) for all a, b, x and y.

As shown by Popescu and Rohrlich, (NS)stat allows the "theoretical" existence of super-quantum correlations, that is, correlations whose CHSH degree is greater than the Tsirelson bound 6 : (SQ) R  > 2 2, where the Bell number R is defined as [START_REF] Clauser | Proposed experiment to test local hidden-variable theories[END_REF] : R = C00 + C01 + C10 -C11 and the correlators are defined as the expectation values of the products of the respective outcomes ai and bj of the conjunction of actions "Alice chooses x and Bob chooses y": Cxy = i,j p (ai , bj / x, y) ai bj In particular, Popescu and Rohrlich have shown that PR-boxes, defined as: a  b = x.y, where "" is the addition modulo 2, or, equivalently, in terms of probabilities of outcomes: (PR) P (a, b / x, y) = ½ if a  b = x.y is realized = 0 otherwise satisfy the condition (NS)stat and violates maximally (SQ) -that is, for PR-boxes R= 4. However, as shown by Uzan [START_REF] Uzan | Super-Quantum Correlations: a Necessary Clarification[END_REF][START_REF] Uzan | The Debate on Super-quantum Correlations: Statistical Misinterpretation and Logical Fallacy[END_REF] , counter-examples of boxes that satisfy the statistical condition (NS)stat and allow the exchange of signals informing one of the parties of the choice of the other can easily be found. For example, in a deterministic box, for which xi determines the outcome ai for Alice and yj determines the outcome bj for Bob, the fact that Bob is informed about Alice's choice does not change at all the conditional probability distribution that defines this deterministic box: (D) P (ai, bj / xi',y j') = i i' . j j'. In this situation, (NS)stat is satisfied since p (ai/xi',y) = p (ai / xi') = ii', which is independent of y, and p (bj/x,yj') = p (bj / yj') = j j', which is independent of x. However, (NS) is obviously not satisfied since Bob is informed about Alice's choice. This communication between Alice and Bob has no influence on the conditional probabilities of (D) that regard the probability of obtaining a result if they chose to draw such or such inputs (which are the values of x and y).

Let us emphasize that this exchange of information between Alice and Bob does not define a distinct communication channel beside the one defined by this deterministic box (that is, by the given of the conditional probabilities P (a, b/ x, y) above, in (D)): the definition (D) correctly characterizes such a deterministic box where Alice and Bob exchange information about their inputs. This exchange of information can be regarded as an integral part of this deterministic box since it is taken into account in its very definition -nothing more needs to be added to this definition to say that Alice and Bob can communicate about their choice of input. Also note that even in the case Bob explicitly uses the information sent by Alice about her input, for example by drawing y = x, this protocol still remains compatible with the definition (D). This deterministic box satisfies (NS)stat even though Bob is informed about Alice's choice of input and can use this information for selecting his input. The definition (D) does not impose that an exchange of information between Alice and Bob is impossible.

Another paradigmatic example of such a signalling box that nevertheless satisfies the statistical condition (NS)stat is nothing but… the PR-box defined above. The definition of this box, which satisfies (NS)stat (see above), does not at all prevent Bob to be explicitly informed about Alice's choice and even to choose his income according to the latter, since this exchange of information is quite compatible with the conditional probability distribution (PR) that defines the PR-box. Indeed, as was the case for the deterministic box above, this exchange of information between Alice and Bob and the fact that Bob's choice is made according to Alice's choice is independent of the definition of this PR-box, which is the given of the conditional probabilities (PR). A PRbox does not prevent Alice and Bob to exchange information about their input.

Thus, it can be concluded that the statistical condition (NS)stat is too weak to characterize (NS), which denotes the absence of any exchange of information between the parties, and, in particular, any exchange of "significant" information that could be used by the parties to choose their input. Consequently, the mainstream results about super-quantum correlations is skewed by this statistical misinterpretation: if (NS)stat allows the existence of super-quantum correlations that satisfy (SQ), it could be quite wrong that the stronger no-signaling condition (NS) allows their existence -which is actually the case (see below)! We then have to reject this statistical interpretation of (NS) -unless we use (NS)stat only as a necessary (but not sufficient) condition of (NS).

The no-signaling condition as an informational constraint

(NS) obviously is a condition of informational independence asserting that what occurs in Alice's subsystem cannot be known by Bob, whatever he does -and symmetrically, that what occurs in Bob's subsystem cannot be known by Alice, whatever she does. Along this line of thought, Pawlowski et al. [START_REF] Pawlowski | Information Causality as a physical principle[END_REF] provided such a purely informational formulation of the no-signaling condition. These authors have suggested that (NS) could be reasonably interpreted by the Information Causality Principle according to which "Bob can gain no more than m (classical) bits of information about Alice's data if she sends him only m bits", with the additive constraint that m = 0 (no information is sent by Alice). This informational condition is then applied by considering the amount of Shannon information that can be exchanged between the two parties in a specific scenario designed by van Dam 11 : Alice is endowed with a list of N random and independent bits {a0, …, ak…, aN-1} and sends m classical bits to Bob, with the help of which Bob tries to guess the value of the bth bit in Alice's list. If Bob can perfectly determine this bth bit when Alice sends him the sequence {a0, …, am-1} and 0  b  m-1, it is not the case for b > m-1. In order to evaluate his chance of success, these authors then compute the Shannon mutual information between Alice's data and Bob's output, denoted as β, when Alice sends him this message of m classical bits: I  ∑ k = 0 to N I (ak : β / b = k), and they assert, in agreement with their Information Causality Principle, that this mutual information must be smaller than m: I ≤ m. In this scenario, they compute Bob's probabilities of success for guessing specific bits in Alice's data, probabilities which are involved in the CHSH correlation factor and in the mutual information I. They can show that, always in this scenario, the Information Causality principle is violated as soon as the Tsirelson bound is.

These authors also claim that this result remains true for any no-signaling box. To show this, they use the fact that, as shown by Masanes et al. [START_REF] Ll | General properties of nonsignaling theories[END_REF] , the local outcomes of any no-signaling box are uniformly random and that their conditional probabilities can be expressed as the following function of the Bell number (and then in terms of the correlators)-with the above notations: P (a  b = x.y / x, y) = ½ (1 + R / 4) This relationship between the conditional probabilities, which are involved in the computation of the mutual information I, and the correlators Cxy, which are involved in the Bell number R, is then used to derive the Tsirelson bound for any no-signaling box. However, this more general derivation (for any no-signaling box) still lacks generality since the mutual information I is still computed for the specific scenario suggested by van Dam, the consequence of which being that the mutual information I can be written as a specific function of the correlators Cxy which can be minored by an expression that directly involves (the square of) the Bell number R (see the supplementary materials of this reference 10 , page 8). However, it is not at all certain that this specific minoration of the mutual information I, which is essential to derive the Tsirelson bound, generally holds. In the general case the relationship between mutual information and correlators looks more complex and might even reverse this order [START_REF] Dong | Correlation function and mutual information[END_REF] . Pawlowski et al.'s derivation is then based on a specific scenario and a specific relationship between correlators and mutual information.

Could this derivation be made more general, that is, without appealing to a specific scenario and to a particular relationship between mutual information and correlators? As will be shown in the following, a more general derivation of the Tsirelson bound from the no-signaling condition can be proposed if the latter informational condition is expressed by the choiceindependence of the correlators between one party's choice of action and all the possible activities of the other. This derivation, which is made in terms of correlators, is not based on a specific scenario of information exchange.

A more general, informational interpretation of (NS)

What we want to express is the informational independence of the events occurring in the two subsystems: by her activity of choosing an input and observing the outcome, Alice cannot obtain any information about Bob's activity, and vice-versa. However, this independence cannot be reduced (as is the case in (NS)stat) to the only independence of Alice's conditional probabilities of outcomes and Bob's choice of input (the value of y), and vice versa. As reminded in section 1, this condition of statistical independence is not sufficient: it does not forbid the two parties to communicate.

In order to forbid any communication capable of informing Alice about Bob's choice of action, we have to set a more constraining condition bearing on the independence of Alice's all possible activities and Bob's choice of input -and symmetrically, between Bob's all possible activities and Alice's choice of input. This condition can be expressed in terms of the relevant correlators: In order to impose the condition according to which Alice's choice A0 or A1 (for the two possible inputs, noted as 0 or 1) does not have any influence on Bob's activity -and vice versa-we have to assert that the correlator of Alice's activities and Bob's choice of input does not depend on the latter and that the correlator of Bob's activities and Alice's choice of input does not depend on the latter. This condition can simply be expressed by the equality of the correlators of Bob's activity and Alice's all possible choices -and symmetrically, by the equality of Alice's activity and Bob's all possible choices.

In order to formulate this constraint, let's consider the correlators between Alice's choice A0 and Bob's all possible activities, that is, when Bob chooses B0 and obtains the outcome b0 j or when Bob chooses B1 and obtains the outcome b1 k . The respective correlators between these two exclusive activities of Bob and Alice's choice A0 are, respectively, C00 and C01. As will be shown hereafter, the correlation degree between Alice's choice A0 and Bob's all possible activities (that is, whatever Bob's choice and whatever Bob's results) can be defined by the product of these correlators: C (A0, B) = C00.C01. This definition merely extends that of the correlators for two random variables: C (A0, B) is computed as the expected value of the product of possible outcomes of the event relating the (sequential) realization of the two following independent events "Alice choose A0 and Bob chooses B0" and "Alice choose A0 and Bob chooses B1". These two joint events, which are incompatible (since B0 and B1 cannot be chosen at the same time), can be understood as occurring sequentially, say at times t1 and t2 > t1 respectively, and, as mentioned above, they are independent (because Ax and Ay are, like Bx and By, randomly chosen). If the outcomes of these sequential joint events are respectively noted as (a0 i , b0 j )t1 and (a0 i' , b1 k )t2, their expectation value, which provides an evaluation of the correlation degree between Alice's choice A0 and Bob's all possible activities, can thus be computed as: C (A0, B) = i,i',j,k p ((a0 i , b0 j )t1, (a0 i' , b1 k )t2 /(x0, y0)t1, (x0, y1)t2) a0 i b0 j a0 i' b1 k where the joint sequential probability p ((a0 i , b0 j )t1, (a0 i' , b1 k )t2 /(x0, y0)t1, (x0, y1)t2) denotes the probability of the conjunction of events "Alice draws x0 at time t1 and obtains a0 i while Bob draws y0 and obtains b0 j " and "Alice draws x0 at time t2 and obtains a0 i' while Bob draws y1 and obtains b0 k ". Since these two joint events are independent, the probability of their conjunction can be factorized: p ((a0 i , b0 j )t1, (a0 i' , b1 k )t2 /(x0, y0)t1, (x0, y1)t2) = p (a0 i , b0 j /x0, y0) . p (a0 i' , b1 k /x0, y1). (Note that in the above expression the mention of time has been deleted since it has no other significance than that of remembering that these events cannot be realized at the same time). The associated "composed" correlator C (A0, B) can thus be written as: C (A0, B) = (i,j p(a0 i , b0 j /x0, y0) a0 i b0 j ) (i',k p(a0 i' ,b1 k /x0, y1) a0 i' b1 k ) = C00 C01. A similar reasoning shows that the correlator of A1 and Bob's all possible activities is: C (A1, B) = C10.C11.

The condition that Bob's all possible activities are performed independently of Alice's choice of action (x = 0 or 1), which is expressed as C (A0, B) = C (A1, B), can thus be written as: C00 . C01 = C10 . C11 Similarly, the condition that Alice's all possible activities are performed independently of Bob's choice of action (y = 0 or 1) is: C00 . C10 = C01 . C11 Consequently, the no-signaling condition (NS) can be express as follows: (NS) C00 . C01 -C10 . C11 = C00 . C10 -C01 . C11 = 0. Note that this condition means that the absolute values of the correlators are equal two by two: C00 = C11  and C01  = C10. However, the expression of (NS) written above will be more useful below for deriving the Tsirelson bound.

4. The Tsirelson bound derived from the no-signaling condition Let's consider the square of the CHSH correlation factor R defined above: R 2 = (C00 + C01) 2 + (C10 -C11) 2 + 2 (C00 + C01) (C10 -C11).

Since the absolute values of the correlators Cxy are all bounded by 1, their squares are also bounded by 1 and then: R 2  4 + 2 (C00 C01 -C10 C11 + C00 C10 -C01 C11 + C01 C10 -C00 C11) Using now the no-signaling condition (NS) expressed above, the combination of the four first products between brackets is equal to zero: C00 C01 -C10 C11 + C00 C10 -C01 C11 = 0. Consequently, we have: R 2  4 + 2 (C01 C10 -C00 C11). Taking now into account that Cxy 1, one has C01 C10 -C00 C11   2 and then R 2  8. Consequently the Tsirelson bound is satisfied: R 2 2.

The no-signaling condition, if correctly understood as an informational constraint asserting the informational independence of the events occurring in the two subsystems is thus sufficient to derive the Tsirelson bound. Of course, this result corroborates Pawlowski's derivation, which is based on a specific (van Dam's) scenario and then on a particular relationship between mutual information and correlators (see section 2), but it has been shown here more generally. The non-signaling condition has been expressed in terms of correlators, as the independence of the activities of each of the parties and the choice of action of the other. This very general result shows that, unless non-local communication (conflicting with quantum theory) would be possible, the maximal correlation degree of two intelligent machines that cannot exchange any signal is the Tsirelson bound.