Simon Gabay
email: simon.gabay@unine.ch

Lucie Rondeau Du Noyer

Mohamed Khemakhem
email: mohamed.khemakhem@inria.fr

digitising the Revue des Autographes

20 del XIX secolo. In questo contesto, mercanti specializzati vendono regolarmente cataloghi d'asta a prezzo fisso che descrivono minuziosamente ogni documento acquistabile. Tale testi hanno delle strutture ricorrenti che è possibile estrarre e strutturarle (quasi) automaticamente, creando così un database di tutti i manoscritti venduti nella Parigi del XIX secolo.

Introduction

The number of projects dealing with French related objects circulating on the private market is increasing: research is currently being carried out in art history [START_REF] Saint-Raymond | Le pari des enchères : le lancement de nouveaux marchés artistiques à Paris entre les années 1830 et[END_REF], book history [START_REF] Montoya | The MEDIATE project[END_REF], medieval manuscripts [START_REF] Wijsman | The Bibale Database at the IRHT[END_REF]. . . Following recent trends, the latter project, that is the most relevant to our own work, is now sharing its data with other teams [START_REF] Burrows | Mapping Manuscript Migrations: Digging into Data for the History and Provenance of Medieval and Renaissance Manuscripts[END_REF] in the USA1 and in the UK 2 to trace the history of manuscripts over time and places, across national and linguistic borders.

Unfortunately, no similar survey has been conducted yet on modern French autographs. If Renaissance manuscripts and their history are better known thanks to the Biblissima project [START_REF] Turcan | BIBLISSIMA : Bibliotheca bibliothecarum novissima, an observatory for written cultural heritage of the Middle Age and the Renaissance[END_REF], no systematic work has been carried out on 17th, 18th and 19th c. materials. However, sale catalogues are recognised as being useful, since they are, for instance, regularly used as sources for critical editions [START_REF] De | [END_REF], p. 158) (Voltaire, 1960, p. 18) (Lamartine, 2001, p. 348).

Such a hole in our knowledge is due to the fact that it remains extremely tedious to extract information, because this task is either performed manually or with imperfect digital solutions [START_REF] Cuadra | Publishing German Sales, A Look under the Hood of the Getty Provenance Index[END_REF][START_REF] Raphael | Aucase -auction catalog segmentation[END_REF]. In the present paper, we therefore want to propose an (almost) automated workflow for the retroconversion of catalogues to transform images into structured information and create a database of sold items.

The corpus

The manuscript market

Since the beginning 19th c., rich collectors have been selling manuscripts on the private market [START_REF] Bodin | Les grandes collections de manuscrits littéraires[END_REF]. Archives and libraries still keep sales catalogues that have been published on a regular basis (fixed price catalogues) or for special sales (auction catalogues) by dealers. Most of the manuscripts sold in these catalogues are modern and contemporary autograph manuscripts.

The information contained in these catalogues is crucial for at least four different reasons.

• It helps assessing the authenticity of autographs: it is unlikely that a document sold repeatedly on the private market, and therefore authenticated each time by an expert, is a forgery.

• It documents the reception of authors, via the history of collections (i.e. who collected what?) and prices (i.e. who costs how much?).

• It informs us on the distance between what has been sold and what is available in libraries (i.e. are there autographs we do not know about?).

• It provides us with images of documents which are still in private hands, because catalogues sometimes offer either facsimiles or pictures of autographs sold.

For a first test phase, we have concentrated our efforts on the Revue des Autographes, a journal published since 1860's in Paris by Gabriel Charavay.

The RDA collection

In the second half of the 19th c., the autograph market is mature and the first generation of dealers begins to retire. In 1865, Gabriel Charavay (1818-1879) cease the opportunity of Auguste Laverdet's (1809-1867) retirement to take over his business, following the example of his elder brother Jacques (1809-1867), who opened his own shop in 1830. At that moment, Gabriel abandons his role of editor for L'Amateur d'autographes, a journal about the autograph market in Paris created in 1862, which keeps being published by his brother Jacques.

Realising the importance of a publication attached to his activities, Gabriel creates another journal one year after his installation, in 1866: the Revue des autographes, des curiosités de l'histoire et de la biographie (RDA). Two journals for such a small market is however too much: in December 1868, after eight months of interruption, the price of the publication is divided by two and part of the content consists now of a list based on the autographs for sale in Gabriel's stock. Over time, the proportion of articles keeps diminishing and the RDA becomes first a hybrid publication mixing news and items to be sold, and eventually a fixed-price catalogue with the name of a journal published (almost) monthly until 1936. In the meantime, Gabriel's shop is taken over by Gabriel's son Eugène (1879-1892), and then by Eugène's widow and by Eugène's daughter (1918)(1919)(1920)(1921)(1922)(1923)(1924)(1925)(1926)(1927)(1928)(1929)(1930)(1931)(1932)(1933)(1934)(1935)(1936).

The transformation of an hybrid journal into a disguised fixed-price catalogue under a journal's name is confirmed by a modification of the format: Eugène Charavay opts for a two-columns layout and a smaller font (cf. figure 1), harder to read but easier to browse for readers, who are now buyers, looking for the autograph of their dreams.

Encoding

Entries

It is the images of these catalogues that we want to transform into minable data. Each of them generally contains a minimum of c. 200 entries, all of them being extremely dense in information and always following the same structure: We can clearly see the lot number (in brown), the name of the author (in red), a short biography (in orange), the material description of the autograph (in blue), the price (in pruple) and an additional description (in green).

To render the structure of the document, we propose the following encoding in XML-TEI:

<item n="55" xml:id="CAT_000037_e55"> <num>55</num> <name type="author">Scudéry (Madeleine de),</name> <trait>
célèbre romancière du XVIIe siècle, créatrice du genre précieux, surnommée la Sapho moderne, née au Havre.
 </trait> <desc>L.a.s. à Huet, (1686), 3 p. in-4. 100 »</desc> <note>Curieuse épître sur le Lutrigot de M. de Bonnecorse. L'auteur est un fort honnête homme de ses amis. « Boileau a fait six vers sur Lutrigot qui ne valent rien.» (Il s'agit de la célèbre épigramme qui commence par ces vers: Venez Pradon et Bonnecorse, etc.)</note> </item> Such an encoding allows a simple disambiguation of the entry and increases the accuracy of the search. In our example, we have the names of three major 17th c. French writers: the novelist Madeleine de Scudéry (1607-1701), the bishop Daniel Huet (1630-1721), and the satirist and poet Nicolas Boileau-Despréaux (1636-1711). The three names are enclosed in three different tags (name, desc and note) reflecting their status in the document (author, addressee, mention): we can therefore easily narrow down our query to a name depending on its role.

Workflow

The presented encoding can be compiled semi-automatically through a simple three steps workflow: The scan is OCRised with Transkribus [START_REF] Kahle | Transkribus -A Service Platform for Transcription, Recognition and Retrieval of Historical Documents[END_REF], for which a substantive model of 125,000 words has been created (CER of 0.59%). The pdf with a text layer is then processed with GROBID-Dictionaries (Khemakhem et al., 2018b), a tool relying on text and layout features (cf. Figure 4) to perform a supervised classification of the parsed text and generate a TEI compliant encoding where the various segmentation levels are associated with an appropriate XML tessellation [START_REF] Khemakhem | Automatic Extraction of TEI Structures in Digitized Lexical Resources using Conditional Random Fields[END_REF]. Preliminary designed for the retroconversion of dictionaries [START_REF] Bohbot | Presenting the Nénufar Project: a Diachronic Digital Edition of the Petit Larousse Illustré[END_REF], it is also used to parse large bibliographical collections [START_REF] Lindemann | Retro-digitizing and Automatically Structuring a Large Bibliography Collection[END_REF] or address directories (Khemakhem et al., 2018a). GROBID-dictionaries provides an answer to important issues left open by previous attempts, that do not produce standardised data (e.g. in XML-TEI), are not open source [START_REF] Cuadra | Publishing German Sales, A Look under the Hood of the Getty Provenance Index[END_REF] or do not offer fine-grained encoding (Barman, 2019). On top of this, GROBID is a free, language agnostic, easily trainable solution compatible with other sub-projects of the GROBID galaxy, which leaves the door open to further analysis with complementary tools, such as GROBID-NERD (Named-Entity Recognition and Disambiguation).

After preliminary tests ensuring the compatibility of GROBID-Dictionaries with sale catalogues (Khemakhem et al., 2018c) models have been created for the RDA and many other catalogues [START_REF] Rondeau | Scaling up Automatic Structuring of Manuscript Sales Catalogues[END_REF] with a new bigram template for the GROBID-Dictionaries models [START_REF] Rondeau | Scaling up Automatic Structuring of Manuscript Sales Catalogues[END_REF] to reinforce the parsing of the structure of each entry.

With GROBID-Dictionaries, the document is annotated using a cascading approach: several Conditional Random Fields (CRF) models are applied one after the other, each of them corresponding to a granularity level in the final XML hierarchy: Levels Tag(s) Task 1 body separates the content from running titles, page numbers, . . . 2 entry separates the entries in the  3 num, form and sense separates the lot n°, information on the author and the MS in <entry> 4 name and desc separates the name of the author and its biography in <form> 5 subsense and note separates the MS description and the additional note in <sense> GROBID-Dictionaries being developed for lexicographic purposes, its results are encoded in a TEI compliant output, but with tags reflecting the content of dictionaries rather than catalogues. Therefore, we automatically convert the output into a second TEI document whose tags are dedicated for the described catalogue elements. The consistency of the transformation output is controlled with a specific schema, prior to its final publication via an XML database.

Future work

As for future work, four tasks will be undertaken. First, we will move towards a fully open source workflow and therefore abandon Transkribus for Kraken [START_REF] Kiessling | Kraken -an universal text recognizer for the humanities[END_REF]. Second, we will increase the size of our database by retroconverting the entire RDA collection (c. 500 catalogues), but also another important series of catalogues: the Lettres autographes et documents published by the other branch of the Charavay family up to the First World War (c. 500 catalogues). Third, we will improve our modeling and increase the granularity of our data collection to capture more informations regarding the document (size, format, length) and named entities (people, places). Fourth, we will use this additional information to reconcile entries and share our work with similar projects via an RDF export.

Ideally, we should eventually be able to go from the TEI digital edition of sales catalogues to a semantic dataset, described using controlled vocabularies where authors, places and manuscripts would be referred to using unique identifiers (ISNI, ISMI. . . ). It would allow federated search with other databases of sold manuscripts, but also with catalogues of libraries in France and abroad.

Figure 1 :

 1 Figure 1: One-column layout (1873) vs two-columns layout (1893).

Figure 2 :

 2 Figure 2: RDA, n°67 (March 1881), lot N°55.

Figure 3 :

 3 Figure 3: Workflow.

Figure

  Figure 4: Features.

Table 1 :

 1 GROBID-Dictionaries Segmentation levelsThe GROBID-Dictionaries output for our example is therefore the following:

	<entry>
	<num>55</num>
	<form type="lemma">
	<name type="author">Scudéry (Madeleine de</name>
	<pc>),</pc>
	<desc>célèbre romancière du XVIIe siècle, créatrice du genre précieux,...</desc>
	</form>
	<pc>. </pc>

1https://sdbm.library.upenn.edu 2http://mappingmanuscriptmigrations.org


Simon Gabay
email: simon.gabay@unine.ch

Giovanni Pietro Vitali
email: giovanni.vitali@ucc.ie

A Theatre of Places Mapping 17 th c. French Theatre

Keywords: Theatre, Literary genre, Geocriticism, Distant reading, GIR, NER, map

As a form of distant reading, mapping texts allows scholars to read classical works anew. Using 17 th French theatre as a test case, we describe an easily reproducible and fully open-source workflow used for extracting and mapping place names, then reach conclusions on literary influences and the strength of genre during the Grand Siècle based.

Introduction

After Ch. Schöch's stylometric [START_REF] Schöch | Fine-tuning our stylometric tools: Investigating authorship, genre, and form in French classical theater[END_REF] and thematic [START_REF] Schöch | Topic Modeling Genre: An Exploration of French Classical and Enlightenment Drama[END_REF] studies, we would like to return to French classical theatre from another perspective: spatial analysis. Places mentioned in text are indeed highly informative regarding genre, but also literary influences, and are therefore worth considering. Using a corpus of 36 plays written by prominent authors across three genres (comédie, tragédie and tragi-comédie) and three periods (first, second and third generation of authors), we clean texts and extract data to project toponyms on a map, which has been conceived as a dynamic research interface for literary analysis. The workflow used is composed solely of simple open-source softwares to maximize its reusability by other scholars who would like to pursue our work.

Corpus

The plays are written by nine of the most important playwrights of 17 th France

Creating the gazetteer

As all the plays were already encoded in XML-TEI 2 [3], the texts have been cleaned with XSLT to keep only the content of the <l> element -i.e. the content of the verses. It must be noted here that place names are not only present within these dialogic parts, but also in stage directions (<stage>, <set>…). Removing the latter is therefore a literary choice to map not the play itself, but the imaginary world created by the characters' speeches perceived by the audience.

As for place names not tagged with <placeName>, the challenge is to find a way to extract a maximum of them for each text. We have decided to use HER [START_REF] Alexander | Practical, Efficient, and Customizable Active Learning for Named Entity Recognition in the Digital Humanities[END_REF], an active learning-based Named Entities Recognizer (NER) toolkit, which accelerates the process of manual annotation by pre-selecting sentences that are maximally informative to the model. It operates iteratively: a first limited wave of sentences are annotated to produce a first gazetteer, from which a second bigger wave is prepared for annotation, etc. In other words, HER does not guarantee the creation of an exhaustive gazetteer, but maximizes its exhaustiveness without reading entirely large corpora.

From an initial list of 3,000 sentences (c. 59,000 tokens) manually annotated, 69 place names were identified. Based on this data, a second list of sentences is created, all ranked according to the probability that they contain a toponym. This time the 100,000 first tokens (out of 694,000, i.e. 14.4% of the corpus) were manually annotated, which was enough to identify 924 place names (162 different types) in the corpus. This process of annotation has been repeated using NLP tools to automatically detect proper nouns, cross-validate the results and minimize omissions. To select the most efficient technique, we have tested three POS taggers:

1. TreeTagger [START_REF] Schmid | Improvements in Part-of-Speech Tagging with an Application to German[END_REF] 4 Geo-referencing and categorization

All occurrences of places have as much metadata as possible, such as the author's name, the generation he belongs to, the play's name and genre, longitude and latitude, etc.

To find the coordinates, we have first used georeference, a geolocation R package developed by J. L. Losada [START_REF] Palenzuela | Georeference: Geolocation R package[END_REF] that automatically queries Genonames3 . Due to the complexity of the names mentioned in our plays, it has only been efficient with the most important ones (Madrid, Paris, etc.), and we have decided to process the data manually as it relies on two databases: Genonames for early-modern places, and Pleaides4 for ancient ones.

Disambiguation has not been done automatically for several reasons:

1. Many foreign place names are "francized", usually following French customs of the 17 th century (Gaeta is translated Gaïette and not Gaète like today). 2. There is a superposition of chronological period (ancient world & classical era). 3. Not all places actually exist: some are imaginary places such as Chicuchiquizèque or Gaula (from the novel Amadis de Gaula) in Scarron's Dom Japhet d'Arménie. Some are legendary places such as Hell or underworld rivers like the Cocytus in Racine's Phèdre.

For the latter, we have tried to rely on actual geography when possible (Mt Olympus as the home of the Greek gods, the Acheron as the entrance to the underworld), thinking that approximate locations were better than none. Regarding rivers, regions or kingdoms, the coordinates are those of the aforementioned databases despite, once again, inevitable approximations -as no polygons were available, or even feasible, for regions like Hyrcania (a historical region composed of the land south-east of the Caspian Sea) or Scythia (a region of Central Eurasia in classical antiquity).

When needed, disambiguation has been done with the following criteria:

1. Existence of coordinates. 2. Common sense: Thessalia is most probably the region in continental Greece (Pleiades n°1332) rather than the city in Peloponnese (Pleiades n°991374). 3. Context: Rome in Geonames (n°3169070) is the modern city, Rome in Pleiades (n°423025) is the ancient one. 4. Historical importance: Troia is the city at its apex (Pleiades n°550595) and not the first settlement in Kumtepe (Pleiades n°210709924).

In order to deal with these loose coordinates, but also to strengthen our final visualization, we have decided to group our results in different macro zones: France, Italy, the Iberian Peninsula (Portugal and Spain), Italy, England, Germany, Greece, Asia Minor/Anatolia, the Levant (Syria, Palestine/Israel, Lebanon), East (Irak, Iran, central Asia), Far East (China, India, Japan), Africa (Maghreb, Libya, Egypt, Ethiopia) and America.

Mapping process

In order to display these macro zones, shapefiles of contemporary countries are downloaded, and we work on the vectors to merge, divide and re-draw the boundaries with QGIS. To avoid slowdown or timeout and to lower the size of the final map, shapefiles have been resized with MapShaper 5 .

All our data is contained in 14 datasets, 12 of which are csvbased and the 2 others are shp-based. We processed it with R, which allows us to easily publish and share our code. The result is composed of 19 superimposed layers of elements, which allows the reader to select precisely the information he wants to see. Each type of information is displayed with a specific marker to ease the map's readability: It is now possible to start exploring our data. As a first observation, French 17 th c. theater is already global: it mentions Europe, but also America, Africa and Asia. This distribution is unbalanced, however, since most of the toponyms are concentrated on the Mediterranean basin, which remains the core of the period's imaginary (fig. 2). This reveals the strong impact of Spanish Golden Age theatre on comédies, and of the classical and the biblical world on tragédies. Tragi-comédies, as its name suggests, appears as a truly intermediary genre, not leaning in either direction. When comparing the use of place names mentioned by author, it also appears that some authors, like Molière or Du Ryer, do not use geography as a stylistic marker (fig. 4), while others make an intensive use of it (fig. 5).



  Genres are displayed with Google Maps pin  Authors are displayed with circles  Periods are displayed as a heat map  Macro zones are displayed with a coloured background A selector on the right allows the user to choose what he wants to display, and a map legend on the left provides basic information for understanding the results.

Figure 1 :

 1 Figure 1: the final map.

Figure 2 :

 2 Figure 2: Global distribution of toponyms.It seems that there is a very clear generic distribution of places, with Italy as a shared border. Tragédies are located east, comédies west, and tragi-comédies both east and west (fig.2).

Figure 3 :

 3 Figure 3: Comédies (red), tragédies (green), tragi-comédies (blue).

Figure 4 :

 4 Figure 4: Molière (pink) and Du Ryer (purple).

Figure 5 :

 5 Figure 5: Pierre Corneille (yellow) and Rotrou (red).

  7 NER modelsHER offers the possibility to train NER models (CRF or Bi-LSTM) based the previously annotated data. To keep our promise of an easily reproducible workflow, we have decided to compare this model with two other popular solutions using two other architectures: Stanford NER (CRF) and Spacy (CNN.) It has been trained on all the texts of our corpus but one, kept for testing the model: Scudéry's Le Prince déguisé (c. 15,935 tokens, 17 place names, 12 different types, 7 in the training data). false) We have also developed alternative models with additional training data taken from the Europeana newspapers project 6 to ameliorate the results (out of the 12 different types, 8 are in the training data

  a relatively old POS tagger. Though outperformed by many others, it is currently the most commonly used by literary scholars[START_REF] Noémie | Pour une cartographie romanesque de Paris au XIXe siècle. Proposition méthodologique. Spatial Analysis and GEOmatics[END_REF]. 2. MElt[START_REF] Pascal | Coupling an annotated corpus and a lexicon for state-of-the-art POS tagging[END_REF] a recent POS tagger already trained on French. 3. Talismane[START_REF] Urieli | Robust French syntax analysis: reconciling statistical methods and linguistic knowledge in the Talismane toolkit[END_REF] a recent POS tagger and lemmatizer trained on French.

					HER	HER+GROBID HER accuracy after
					100,000	tokens
					annotated
					Toponyms	162	188	86.2%
					Occurrences	924	1,032	89.5%
	Tests have been conducted on Boyer's Agamemnon, which
	contains c. 16,500 tokens and a high number of toponyms (116
	occurrences, 18 types, 2 compounds ones):
		Proper nouns Toponyms	%
	TreeTagger	983	16	1.63%
	MElt	777	18	2.32%
	Talismane	785	18	2.29%
	Only MElt and Talismane have properly tagged all 17 place
	names as proper nouns. Because MElt has identified slightly less
	proper nouns than Talismane, the former is slightly more efficient
	than the latter.
	Recent research has however shown the superiority of NER
	taggers for the extraction of anthroponyms and toponyms (Brando
	et alii 2016). We have therefore tested GROBID nerd, a new
	named entity recognizer for English and French with a knowledge
	base of 37 million entities from Wikidata [6].
			Entities Positive %
	GROBID nerd		419	18	4.30%
	GROBID nerd, location type 83	17	20.48%
	Since GROBID nerd seems to offer not only perfect precision but
	also the best recall, it has been used to control the places tagged
	with HER in each text and update the gazetteer and offer a first
	evaluation of HER:

Dates are those of the first and the last play of each author.

http://dramacode.github.io

http://www.geonames.org.

https://pleiades.stoa.org.

https://github.com/EuropeanaNewspapers/ner-corpora.

ACKNOWLEDGMENTS

Names are in alphabetical order, as the paper was equally written by the two authors. Merci à Jean-Baptiste Camps pour le corpus, Alex Erdmann pour le TAL et Patience Haggin pour la relecture.

DATA

Gabay Simon, & Giovanni Pietro Vitali. (2019). CARTO17: Data and Scripts for Mapping 17th c. French Theatre (Version 1.0) [Data set]. Zenodo. http://doi.org/10.5281/zenodo.3530791


Guillaume Castanet

William Chaze

Ophélie Caballina
email: ophelie.caballina@univ-lorraine.fr

Romain Collignon

Fabrice Lemoine

Drop impact in the regime of film boiling : transient evolution of the heat transfer and the vapor film thickness

Keywords: Film boiling, Drop impact, Vapor film thickness

When a drop impinges onto a wall heated above the Leidenfrost temperature, a very thin vapor film is formed at the interface between the liquid and the solid substrate. This vapor layer modifies the impact behavior of the drop and induces a significant decrease in heat transfer. To study this phenomenon, a model is proposed for the growth of the vapor film and the heat transfer at the impact. The main assumptions are: (i) a uniform but time varying thickness of the vapor film, (ii) a quasi-steady Poiseuille flow inside the vapor film, and (iii) a constant wall temperature. Heat energy and momentum balances are employed to obtain an ordinary differential equation describing the evolution of the vapor film thickness during the drop impact. Upon a one-dimensional analysis (nonetheless including some effects due to the complex fluid flow), the local heat flux transferred to the liquid qL can be evaluated. When the initial drop temperature is sufficiently lower than the saturation temperature, qL predominates over the heat flux used for liquid evaporation. This results in a simplified model for the vapor film thickness that we were able to validate against experiments carried out with ethanol droplets impacting an overheated sapphire surface (typically above 250°C). Two optical measurement techniques, based on laser-induced fluorescence imaging and infrared thermography, are associated to temporally and spatially characterize the heat transfer as well as the thickness of the vapor film.

Introduction

Cooling of hot surfaces by sprays, drops or droplets is common in many applications, such as in the quenching of hot metals [START_REF] Ji | Effect of inclination angle onto heat transport characteristics of bubble-actuated circulating heat pipe covering high-temperature region[END_REF][START_REF] Kim | Effect of initial temperature of a cylindrical steel block on heat transfer characteristics of staggered array jets during water jet quenching[END_REF], or in the cooling of power electronics [START_REF] Kim | Spray cooling heat transfer: The state of the art[END_REF]. It has driven the attention of many researchers because of the ability of sprays to remove high quantities of heat using limited mass flow rates of liquid coolant. However, while spray cooling has been applied for decades, its integration remains complex due to ill understanding of the heat transfer characteristics of spray cooling. Correlations for the heat transfer coefficient are generally determined experimentally on a global scale, based on the mean parameters of the spray (for example the mean size and velocity of the impinging droplets). Such correlations can serve well for an intended application, but in general they poorly perform whenever the configuration of the spray is changed. There is therefore an important need to obtain a glimpse of the underlying physics, which implies investigating the fundamental case of the impact of a single droplet. Numerous studies have focused on the impact of a single droplet onto an isothermal surface, which does not require consideration of heat transfer and phase change. The emphasis was mainly placed on the drop behavior evaluating the diameter and the velocity of the outcoming droplets (partial/total rebound, splashing) [START_REF] Rioboo | Outcomes from a drop impact on solid surfaces[END_REF], the spreading of the liquid film [START_REF] Roisman | Inertia dominated drop collisions. i. on the universal flow in the lamella[END_REF]. These investigations mainly aimed at determining the influence of the dynamics of the impinging droplet (velocity, diameter...), the physical properties of the liquid (viscosity, surface tension...) and of the solid surface (surface roughness, wettability...) on the impact process. All these works must be reexamined carefully when the wall is heated above the boiling point of the liquid. Even when the temperature of the solid substrate exceeds the saturation temperature of the liquid by a few degrees, boiling can considerably change the droplet behavior. As for pool boiling, the heat flux is a complex non-monotonic function of the wall temperature. From the boiling curve, heat transfer during drop impingement can be categorized into boiling regimes: (i) natural convection boiling, (ii) nucleate boiling, (iii) transition boiling and (iv) film boiling. At very high substrate temperatures, in the regime of film boiling, the liquid does not remain in contact with the solid. A vapor layer is formed underneath the droplet and prevent at any time a direct contact between the liquid and the solid surface, which is hugely detrimental to heat transfer. The main subject of the present study is the introduction of a predictive theoretical model for heat transfer during drop impact in the film boiling regime. Some theoretical models are available for sessile droplets, deposited without velocity on the solid surface. Biance et al. [START_REF] Biance | Leidenfrost drops[END_REF] derived some scaling laws for the vapor film thickness. They evidenced distinctive evolution of the vapor thickness depending on whether the droplet size exceeds or not the capillary length. Sobac et al. [START_REF] Sobac | Leidenfrost effect: Accurate drop shape modeling and refined scaling laws[END_REF] proposed a model based upon lubrication theory which accounts for the curvature of the liquid/vapor interface. This approach provides good agreements with experimental works showing the presence of a vapor pocket underneath the droplet and a neck in the vapor film profile. However, these theoretical approaches are difficult to adapt to the case of a drop impact, since liquid inertia considerably modifies the force acting upon the vapor film and the spreading evolution of the droplet. Recently, Breitenbach et al. [START_REF] Breitenbach | Heat transfer in the film boiling regime: Single drop impact and spray cooling[END_REF] derived a theoretical model to predict the heat transferred from the hot substrate to the single drop during impact. They considered a uniform but time variable thickness of the vapor film and they proposed an explicit expression for the thickness of the vapor layer obtained from the energy balance at the liquid-vapor and vaporsolid interfaces. A significant difference with the present study, however, concerns the heat flux used for liquid evaporation, denoted qv. Breitenbach et al. [START_REF] Breitenbach | Heat transfer in the film boiling regime: Single drop impact and spray cooling[END_REF] postulated that the growth rate of the vapor film thickness is directly related to the rate of liquid change into vapor since they considered that qv = ρ l Lv dh/dt, where h is the thickness of the vapor film, ρ l is the liquid density and Lv the latent heat of vaporization. This approximation allows them obtaining an explicit expression for the vapor film thickness without having to involve the momentum balance and the forces acting upon the vapor film. In order to evaluate the theoretical model proposed in the present study, the results from the model are confronted with measurements obtained for ethanol droplets using recently developed optical techniques.

Measurement techniques and experimental observations

A first technique based on two-color laser-induced fluorescence (2cLIF) imaging allows to determine the temperature of the liquid drop during the impact process. Details of the method can be found in Chaze et al. [START_REF] Chaze | Spatially and temporally resolved measurements of the temperature inside droplets impinging on a hot solid surface[END_REF]. An example of temperature maps is presented in Fig. 1a. It corresponds to the case of an ethanol drop impinging onto the sapphire window at Tw=300°C and a value of the Weber number We equal to 90. A short time after the beginning of the impact, the droplet takes the shape of a thin lamella of liquid surrounded by an annular rim. In the first times, the heating seems to proceed at a faster rate in the bounding rim, where hot liquid accumulates and mixes with colder liquid already there. As time goes by, the liquid lamella becomes thinner and thinner and the temperature in its inside increases very sharply. The temperature of the lamella gets close to the boiling point of ethanol (TSat = 78°C) at t = 4 ms. Nonetheless, the lamella does not account for a significant part of the drop volume at the end of the spreading. The receding of the drop edge accelerates the internal mixing of the liquid. At t = 10 ms, the drop leaves the solid surface with a rather uniform temperature, even though the top of the vertically elongated droplet has a slightly higher temperature.

In addition to measuring the droplet temperature , Chaze et al. [START_REF] Chaze | Heat flux reconstruction by inversion of experimental infrared temperature measurements -Application to the impact of a droplet in the film boiling regime[END_REF] recently undertook a characterization of the heat flux removed from the hot solid surface. To this end, they used IR thermography. The solid substrate, made of a flat sapphire window, is coated with TiAlN, which is a material very emissive in the IR and resistant to high temperatures.

The deposited sub-micron layer of TiAlN is also totally opaque in the IR, and does not modify the thermal impedance of the solid sample. Sapphire being transparent for the wavelengths up to 5 µm-7 µm, the thermal footprint of the droplet on the rear face of the TiAlN deposit can be observed by an IR camera through the opposite face of the sapphire window which is impacted by the droplet. Figure 1b shows a typical set of temperature images recorded with the IR camera. In this example of an impinging ethanol droplet, the cooling of the sapphire surface does not exceed 10°C. An analytical model taking into account the heat conduction in the sapphire substrate was developed to estimate, by an inverse method, the heat flux extracted by the droplet from the solid surface. This model uses the axisymmetry of the polar angle-integrated problem to take advantage of the Hankel transform for a parsimonious parameterization of the variations of temperature and heat flux in the radial direction. This approximation appears to be well justified in Fig. 1b where the temperature distribution on the solid surface seems rather axisymmetric. A maximum of heat flux is reached very rapidly after the beginning of the impact in Fig. 1c, then the heat flux decreases progressively with time. A reliable estimates of the heat flux can be obtained in the film boiling regime, where heat transfer is particularly reduced.

Modeling of the evolution of the vapor film thickness

Given the small thickness of the vapor film, thermal conduction is predominant in the direction perpendicular to the solid surface [START_REF] Breitenbach | Heat transfer in the film boiling regime: Single drop impact and spray cooling[END_REF][START_REF] Guo | A non-equilibrium mechanistic heat transfer model for post-dryout dispersed flow regime[END_REF][START_REF] Biance | Leidenfrost drops[END_REF][START_REF] Sobac | Leidenfrost effect: Accurate drop shape modeling and refined scaling laws[END_REF][START_REF] Sobac | Self-propelled leidenfrost drops on a thermal gradient: A theoretical study[END_REF]. As a result, the temperature in the vapor film varies linearly between the liquid surface at the saturation temperature TSat and the solid surface at Tw,s. Knowing the value of the heat flux qw from the resolution of the IHCP, the thickness of the vapor film h can be determined by:

h = λv • ∆T /qw, (1)
where λv is the thermal conductivity of the vapor and ∆T = Tw,s -TSat. Fig. 2 shows profiles of the vapor film at different times corresponding to the same drop impact as in Figs. 1. It can be observed that the vapor film is steadily growing with time until t ≈ 5 ms, then the thickness increases very rapidly because of the formation of a hole in the middle of the liquid lamella (Fig. 1.a). At t=0.9 ms, the vapor film profile takes a characteristic shape with a bottleneck at the edge. Globally, the difference in thickness between the center and the edge of the film remains very moderate, which justifies the assumption of a uniform thickness of the vapor film made in the following. The configuration considered for modeling is shown in Fig. 3. An axial symmetry is assumed for the problem and the vapor film thickness h is supposed to be uniform but time-dependent. The parameter dc denotes the diameter of the area of effective interaction between the droplet and the solid surface. It is usually close but not exactly equal to the spreading diameter dmax, as the droplet edge is curved.

Flow in the vapor layer-Considering that dc is, at any time, much larger than h, the problem falls within the scope of the lubrication theory. Taking into account the axisymmetry of the spreading drop, the momentum balance equation in the radial direction r can be expressed as:

∂vr ∂t = - 1 ρv ∂p ∂r + νv ∂ 2 vr ∂z 2 , ( 2
)
where vr is the radial component of the velocity inside the vapor film and ρv and νv the density and the kinematic viscosity of the vapor respectively. In this expression, the unsteady term ∂vr/∂t is negligible in comparison to the viscous term. A rough estimate for these terms can be made assuming that ∂vr/∂t ∼ vr/ti and νv vr h 2 , where the characteristic time of an impact ti is typically a few of ms, h ∼ 1 µm and νv ≈ 10 -4 m 2 /s. Although the thickness of the film h varies with time, the problem can be treated as a quasi-steady one. The mass conservation applied to the vapor film yields:

πr 2 ṁ = 2πr h ρv vr (r) , (3)
where

vr (r) = 1 h h ∫ 0
vr dz and ṁ is the vapor mass flow rate per unit area. According to Eq.( 3), the radial velocity vr increases linearly with r. This leads to the following velocity field within the vapor layer:

vr = bz + cz 2 r and vz = - 1 r ∂rvr ∂r dz = -b z 2 - 2 3 c z 3 , (4)
where b and c are determined in regard to the boundary conditions. When the droplet spreads, a thin liquid lamella bounded by a rim is formed. The flow within this lamella can be approximated by the asymptotic flow solution proposed by Yarin and Weiss. [START_REF] Yarin | Impact of drops on solid surfaces: self-similar capillary waves, and splashing as a new type of kinematic discontinuity[END_REF] v

r,l = r t , v z,l = -2ξ t . (5)
where ξ = z -h. The subscript l refers to the liquid phase. Eq. ( 5) corresponds to a non-viscous flow solution.

The slip velocity of the liquid/vapor interface is commonly assumed to be not affected by the vapor flow. The liquid drop moves almost freely over the vapor film. Numerous studies have emphasized a significant reduction of friction associated with the presence of a vapor film. [START_REF] Berry | Navier slip model of drag reduction by leidenfrost vapor layers[END_REF] In the case of a drop impact, the vapor film is squeezed and the radial velocity vr of the vapor is usually much larger than the liquid velocity v r,l . Therefore, it has been suspected that the liquid can be entrained radially by the vapor flow. In this scenario, according to Castanet et al. [START_REF] Castanet | Drop spreading at the impact in the leidenfrost boiling[END_REF] the radial velocity at ξ = 0 can be determined by:

v r,l = α r t at ξ = 0, (6)
where α is a parameter that can be slightly larger than 1. The following conditions are considered at the liquid/vapor interface:

vr = v r,l at z = h, (7)
ρv,s • vz - dh dt = -ṁ at z = h. (8)
In this expression, ρv,s is the vapor density at the liquid/vapor interface where T = TSat. Using Eqs. ( 4)-( 8), the parameters b and c can be determined by:

c = α h 2 t - b h , (9)
b = - 3 h 2 dh dt - ṁ ρv,s + 2 3 α h t . ( 10
)
Energy balance-The heat flux taken from the wall qw is used first to heat up the liquid in the drop, to vaporize the liquid and finally to increase the temperature of the released vapor up to the average temperature in the vapor film which is denoted T film . Hence, the balance of energy flux can be summarized as followed:

qW = qL + Lve • ṁ , (11)
where qL corresponds to the heat flux entering into the liquid drop. This heat flux is responsible for the temperature increase of the droplet during the impact. The parameter Lve is introduced with Lve = Lv+(Cpv • T f ilm -Cpv,s • Tsat) with Cpv,s and Cpv respectively the heat capacity of the vapor at Tsat and T film . Since the variation in temperature in the vapor film is linear along the z-axis, the average temperature of the vapor T film can be well approximated by T film = (TSat + Tw,s)/2. Using Eqs. ( 1) and ( 11), the vapor mass flowrate ṁ can be determined by:

Lve • ṁ = λv∆T h -qL, (12)
Momentum balance-The reaction force F of the vapor film on the impinging droplet is due to pressure inside the vapor film. Using Eq. ( 4), the integration of Eq.( 2) allows obtaining the following expression of the pressure:

p (r) -p (r = 0) = µv c r 2 , (13)
On the outer edge of the vapor film at r = dc/2, the pressure returns to the external pressure, meaning that p = 0. This yields:

p (r) = µv c • r 2 - d 2 c 4 . (14)
The reaction force F can be determined by:

F = 2π dc/2 r=0 p (r) r dr = - π 32 µv c d 4 c . (15)
Using Eqs.( 9) and ( 10), the following expression of the force F can be obtained:

F = 3π 32 µv d 4 c h 3 ṁ ρv,s - α h t - dh dt . ( 16
)
This expression of F is equivalent to the Stefan-Reynolds force opposed to the approach of two parallel disks [START_REF] Stefan | Versuch über die scheinbere adhäsion, Sitzungsberichte der Mathematisch-naturwissenschaften Klasse der Kaiserlichen Akademie der Wissenschaften[END_REF] when neglecting the liquid vaporization ṁ and the sliding of the liquid/vapor interface (ie. α = 0 and ṁ = 0). Based on Eqs.( 12) and ( 16), the evolution of the vapor film thickness can be determined by solving:

dh dt = - α h t + 1 ρv,s Lve • λv∆T h -qL - 32 3π h 3 F µvd 4 c . ( 17
)

Modeling of the droplet heating

A one-dimensional model for the droplet heating can be associated with the previous description of the vapor film [START_REF] Breitenbach | Heat transfer in the film boiling regime: Single drop impact and spray cooling[END_REF].

The droplet is assimilated to a semi-infinite liquid medium (Fig. 3). The thickness of the thermal boundary layer, which develops inside the drop at the interface with the vapor film, is small in comparison to the radial dimension of the spreading drop. Therefore, the thermal gradient in the z direction ∂T /∂z is dominant relative to that in the radial direction ∂T /∂r . The heat equation can be written:

∂T ∂t -v z,l (ξ) • ∂T ∂ξ = a l ∂ 2 T ∂ξ 2 , ( 18
)
where ξ = z -h, a l is the thermal diffusivity of the liquid and v z,l is the z component of the liquid velocity, which can be evalued using Eq.( 5). Hence, the convection induced by the liquid flow within the lamella, is taken into account in the equation above. At the liquid/vapor interface, the temperature is imposed by the liquid-vapor saturation:

T = TSat, at ξ = 0. ( 19
)
Assuming that the thickness of the thermal layer is small compared to the drop thickness, one can expect:

T = T d0 , at ξ → ∞. ( 20
)
where T d0 is the drop temperature before impinging onto the hot solid surface. As presented by Breitenbach et al. [START_REF] Breitenbach | Heat transfer in the film boiling regime: Single drop impact and spray cooling[END_REF], the above problem can be solved analytically. The temperature field inside the spreading drop is determined by:

T (ξ, t) = TSat + (T d0 -TSat) • erf √ 5 ξ 2 √ a l t . ( 21
)
The heat flux entering the liquid phase can be calculated from the following expression:

qL = λ l dT dξ ξ=0 + = Cq L t -1/2 , ( 22
)
Cq L = √ 5e l (TSat -T d0 ) √ π . ( 23
)
Here e l = ρ l Cp l λ l is the thermal effusivity of the liquid. Thus, the increase ∆T d in the mean temperature of the drop Tm can be evaluated by: where Se is the effective surface of heat exchange between the droplet and the solid substrate, i.e. Se = πd 2 c /4. The diameter dc is evaluated from the sideview images taken by laser-induced fluorescence, which allows evaluating the length of the "apparent contact" line between the solid surface and the drop (Fig. 1a). On the hand, it is also possible to estimate the area Se of most efficient heat transfers directly from the reconstructed distributions of the wall heat flux qw (Fig. 1c), and then to determine its diameter. These two approaches give roughly equivalent results. The above-described model of drop heating can be validated against the average temperature in the maps obtained experimentally by laser-induced fluorescence (Fig. 1a). The results of this comparison are presented in Fig. 5 in the case of three different values of the Weber number We. A good agreement between the experimental data and the results of the theoretical model can be obtained applying Eq. ( 24). It should be noted that the accuracy of the temperature measurements is of the order of ±1°C. On the other hand, the temperature calculated from the experimental maps is not rigorously the volume-average temperature (see Ref. [START_REF] Chaze | Spatially and temporally resolved measurements of the temperature inside droplets impinging on a hot solid surface[END_REF] for an overview of the measurement biases).

∆T d = Tm -T d0 = 1 m Cp l t 0 qL (t) • Se (t) dt, (24)

Evolution of the vapor film thickness

To determine the evolution of the vapor film, Eq. ( 17) has to be solved which implies having a description of the reaction force F and of the "contact" diameter dc. In the vast majority of applications, the term in F in Eq. ( 17) can be neglected as it scales in h 3 . In addition, F decreases exponentially with time when the drop spreads [START_REF] Roisman | Inertia dominated drop collisions. i. on the universal flow in the lamella[END_REF]. Replacing qL by its expression in Eq. ( 22) and assuming F = 0, the following differential equation can be obtained:

dh dt = - α h t + 1 ρv,s Lve • λv∆T h - Cq L √ t . (25)
Leaving aside the term αh/t , this equation is very similar to that proposed by Breitenbach et al. [START_REF] Breitenbach | Heat transfer in the film boiling regime: Single drop impact and spray cooling[END_REF] to describe the evolution of the vapor film thickness. However, an important difference is that the density of the vapor ρv,s is replaced by the density of the liquid ρ l (see Eq. 1 p. 37 in Ref. [START_REF] Breitenbach | Heat transfer in the film boiling regime: Single drop impact and spray cooling[END_REF]), which introduces an overwhelming influence of the evaporation heat flux qv in the energy balance. This difference comes from the fact that Breitenbach et al. [START_REF] Breitenbach | Heat transfer in the film boiling regime: Single drop impact and spray cooling[END_REF] did not consider exactly the same problem. They assume the situation to be the analog of a vapor/liquid front moving in a semi-infinite liquid medium under the effect of phase change without any regard to pressure forces. An analytical solution of Eq. ( 25) can be found in the form:

h(t) = C √ t. (26)
Given that C > 0, the only possible value of C is determined by:

C = -Cq L + C 2 q L + (4α + 2) ρv,sLve λv∆T ρv,sLve • (2α + 1) . ( 27
)
When C 2 q L (4α + 2) ρv,sLve λv∆T , a Taylor expansion of Eq. ( 27) to the 1 st order with respect to C -1 q L yields:

C = λv ∆T /Cq L . (28)
This solution implies that the two predominant terms of Eq. ( 25) are the heat flux taken to the solid surface qw = λv∆T /h and the heat flux entering the liquid phase qL = Cq L √ t . Indeed, neglecting the other terms, Eq. ( 25) is equivalent to:

λv∆T h = Cq L √ t , (29)
which immediately yields Eq. (28). The above condition C 2 q L (4α + 2) ρv,sLve λv∆T can be rewritten as:

TSat -T d0 Tw,s -TSat A • ev e l , (30)
where ev = ρvCpvλv is the thermal effusivity of vapor and the parameter A is determined by:

A = π 5 (4α + 2) ρv,s ρv Lve Cp v ∆T 1/2 . ( 31
)
To satisfy the condition in Eq. ( 30), the initial temperature T d0 must be sufficiently low in comparison to TSat. For example, in the case of an ethanol droplet impinging onto a sapphire substrate at 300°C, parameter A in Eq. ( 30) is about equal to 3.4, while e l ≈ 565

W•K -1 • m -2 • s 1/2 and ev ≈ 9.5 W•K -1 • m -2 • s 1/2 (evaluated at T f ilm ). This yields T d0 64°C.
Experimental validation of the model in the case of F = 0 and a low initial drop temperature T d0 As discussed previously, different cases emerge from the analysis of the evolution of the vapor film thickness. When the liquid drop is sufficiently cold prior to the impact, the condition ( 30) is satisfied and it is possible to greatly simplify the problem given that qL ≈ qw. In this paper, the focus is on the experimental validation of the model in this limit case. Several experiments were conducted on ethanol droplets injected at the initial temperature of 20°C. The measurements techniques are used to determine both the heat transferred to the droplet QL and the heat taken to the solid surface Qw. The heat energy is evaluated from the measurements of the liquid temperature using the following expression:

QL = m Cp l (Tm -T d0 ) , ( 32
)
where Tm is the average temperature determined from the measurements by the 2cLIF technique. The heat Qw is evaluated from the measurements of the solid surface temperature by IR thermography. After determining the heat flux qw by solving the IHCP, Qw is evaluated by:

Qw = t 0 Se(t) qw dS dt, (33)
where Se is the cooled surface. We, which can be explained by the increase in the spreading surface of the droplet. It is noticeable that Qw and QL are almost equal at any moment of the impact regardless of the Weber number. This confirms that an initial temperature T d0 = 20°C is sufficiently low to neglect the contribution of the vaporisation heat flux to the energy balance, as expected from the theory (see the remark after Eq. 31). Of course, measurement errors can affect these experimental data, but it shoud be stressed that they are too weak to change the previous conclusion. Indeed, the drop temperature is determined with an uncertainty of about ± 1°C, which corresponds to an energy of ± 7.2 mJ. The read noise of the infrared camera is about ±0.2°C. Given the axisymmetry assumed for solving the IHCP, pixels at the same distance from the center are averaged altogether to reconstruct the local heat flux, which reduces this error. Furthermore, the measurement noise induced by the parasitic radiations from the sapphire holder is drastically reduced thanks to the deposition of a very emissive coating and the spatial filtering of off-field radiations by the optical system. Figure 7 shows the time evolution of h for the same impact conditions as in Fig. 6. The thickness of the vapor film seems to be independent on We. Estimated values of h are only slightly smaller than the prediction of the theoretical model. For these drops impinging the solid surface at T d0 = 20°C, the evolution of h is well described by Eqs. ( 26) and (28), which is consistent with the results presented in Fig. 6 suggesting that qw = qL.

Conclusions

The evolution of the vapor film thickness has been described in the framework of a quasi-steady approximation, assuming a uniform thickness of the film and one-dimensional heat transfer. A general model for the growth of the vapor film can be obtained from the application of the balance of heat fluxes and the balance of forces applied to the vapor film.

A limit case of the model is obtained when the temperature of the drop before impact is low compared to the saturation temperature of the liquid. In the case of this "cold" drop, heating of the drop outweighs the heat used for the evaporation of the liquid. The growth of the film is governed solely by the rate at which the drop heats up.

The impact velocity has no effect on the vapor film. Very simple expressions can be obtained for the evolution of the vapor film thickness and the decrease in temperature of the solid surface. They are validated by experiments conducted on ethanol drops at several Weber numbers.

Figure 1 .

 1 Figure 1. Characterization of heat transfer in the case of an ethanol droplet impinging onto a sapphire window at We = 90 (T d0 = 20°C, Tw = 300°C, d 0 = 1.95 mm). a) Temperature distribution in the liquid drop obtained by 2cLIF imaging. The diameter of the apparent "contact" line dc can be deduced from the sideview images. b) Temperature distribution at the solid surface measured by infrared thermography. c) Heat flux from the solid surface reconstructed by solving the IHCP. The region of most efficient heat exchange Se is indicated by the dotted lines.

Figure 2 .

 2 Figure 2. Evolution of the vapor film thickness h for We = 90 in the case of an ethanol droplet (Tw = 300°C, T d0 = 20°C and d 0 =1.95 mm).

Figure 3 .Figure 4 .

 34 Figure 3. Main features of the model configuration.

Figure 5 .

 5 Figure 5. Transient drop heating ∆T d in the case of ethanol drops impinging onto the sapphire substrate at Tw = 300°C (T d0 = 20°C, d 0 =1.95 mm). Comparison between the model of drop heating and the measurements of the drop temperature by 2cLIF thermometry.

Figure 6 .Figure 7 .

 67 Figure 6. Time evolution of the heat taken to the solid wall Q W and the heat transferred to the liquid Q L in the case of an ethanol drop (Tw = 300°C and T d0 = 20°C).

Figure 6

 6 Figure6displays the time evolution of QL and Qw for different values of the Weber number in the case of an ethanol drop initially at T d0 = 20°C and a solid substrate at Tw=300°C. As expected heat transfer increases with We, which can be explained by the increase in the spreading surface of the droplet. It is noticeable that Qw and QL are almost equal at any moment of the impact regardless of the Weber number. This confirms that an initial temperature T d0 = 20°C is sufficiently low to neglect the contribution of the vaporisation heat flux to the energy balance, as expected from the theory (see the remark after Eq. 31). Of course, measurement errors can affect these experimental data, but it shoud be stressed that they are too weak to change the previous conclusion. Indeed, the drop temperature is determined with an uncertainty of about ± 1°C, which corresponds to an energy of ± 7.2 mJ. The read noise of the infrared camera is about ±0.2°C. Given the axisymmetry assumed for solving the IHCP, pixels at the same distance from the center are averaged altogether to reconstruct the local heat flux, which reduces this error. Furthermore, the measurement noise induced by the parasitic radiations from the sapphire holder is

  • ∂ 2 vr ∂z 2 ∼νv •

		z
		𝑑 𝑥 /2	Spreading drop
		𝑑 𝑐 /2	𝑇 𝑑
			𝐿 𝑣 𝑚 ′′	𝑞 𝐿	𝑇 𝑆𝑎𝑡
	ℎ	𝑣 𝑟		𝑞 𝑤	r
			Hot wall	𝑇 𝑤

This work is licensed under a Creative Commons 4.0 International License (CC BY-NC-ND 4.0).

This work is licensed under a Creative Commons 4.0 International License (CC BY-NC-ND 4.0). ILASS -Europe 2019, 2-4 Sep. 2019, Paris, France

Acknowledgements

The authors gratefully acknowledge the financial support of the ENERBATIN CPER program and the European FEDER program.