Aya Steiner
email: ayasteiner@gmail.com

The individualization of rational numbers discursive routines

Keywords: Fraction, rational numbers, discourse, individualization

The research presented deals with the process of learning rational numbers. Assuming that at the end of a successful learning process the formal routines taught at the school become useful for practical activities, we have documented the activities of children from different grade levels regarding school assignments and daily practical tasks. We document the participants' activities from their entrance to the 1st grade, where they have no school experience with fractions, all the way through the sixth grade, where they are expected to arrive at the fully satisfactory mastery of rational numbers. In analyzing the data, we examined how the formal routines for performing certain fractions-related school assignments changed over time and how (if at all) they converge with routines for daily practical tasks. In this paper, we present findings from two school tasks:

(1) locating a fraction on the number line, and (2) naming a point on the number line.

Introduction and theoretical background

In the research presented in this paper we investigate the development of students' thinking about rational numbers. The research is part of my doctoral dissertation under the supervision of Anna Sfard. In spite of the abundance of former research on the topic, this investigation may be expected to make a novel contribution because of two features that set it apart from former ones: (1) the study follows the process of development rather than just snapshots of children's performance with rational numbers; (2) the study is grounded in a conceptual framework markedly different from those that have been guiding the majority of other investigations.

In the last quarter of the twentieth century, learning rational numbers has been one of the topics most vigorously studied by mathematics education researchers. Numerous studies recognized the complexity of the concept and agreed that rational numbers should be characterised as a set of related but distinct constructs rather than as a homogenous single one [START_REF] Behr | Rational number concepts[END_REF][START_REF] Kieren | On the Mathematical, cognitive and instructional[END_REF]Rappaport, 1962). Although this model has been the point of departure for many projects, some researchers argued that the division into interpretations of the rational number is insufficient for describing children's construction of the concept [START_REF] Olive | The learning of rational number concepts using technology[END_REF]. Psychologists took the research in a different direction and focused on the mental operations involved in constructing knowledge of non-integer quantities [START_REF] Confrey | Splitting reexamined: Results from a three year longitudinal study of children in grades three to five[END_REF]. Nevertheless, only few of the studies offered a continuous picture of the learning of rational numbers. In the Second Handbook on Mathematic Teaching and Learning, [START_REF] Lamon | Rational numbers and proportional reasoning: Toward a theoretical framework for research[END_REF] writes: "Multiplicative ideas, in particular fractions, ratio and proportion, are difficult and develop over a long period of time. Brief teaching experiments have had disappointing results. There seems to be no substitute for longitudinal research" (p. 651). She also writes that such research should consider students' intuitive and experimental knowledge as well as their formal school knowledge. In this longitudinal study I interviewed 24 children, from a wide range of ages (6-13), for two years, looking at how their execution of an activity changed over time -before, during and after the formal learning of fractions.

By adopting a theoretical perspective that views individual learning as a collective endeavor [START_REF] Vygotsky | Mind in society: The development of higher psychological processes[END_REF], I take yet another approach to the topic. According to the basic tenet of the proposed conceptualization, known as commognitive [START_REF] Sfard | Thinking as communicating: Human development, the growth of discourses, and mathematizing[END_REF], mathematics is a form of communication -a discourse. The discourse on rational numbers is a mathematical discourse dealing with the mathematical object rational number. A mathematical object is introduced in order to account for the equivalence of many signifiers, which are the terms (names) and symbols used for communication. The term "three quarters" and the symbols 0.75, 3/4 and 6/8 are declared to be signifiers of the mathematical object called rational number. The leading example of mathematical signifiers featured in this article are fractions, whereas the objects signified by fractions are rational numbers. The discourse of rational numbers is identifiable by four characteristic features: its special words and their use, its visual mediators and their use, the narratives that are endorsed by the discourse community, and discursive routines, which are patterns of actions a person tends to perform in response to a task situation, a situation in which she feels obliged to act. Given a task-situation, the decision about what it is that needs to be done is made by the performer based on precedents -past task-situations she considers as similar enough to the present one to justify repeating at least some of the things that were done then. The participant is not always aware of what and why she chooses for repetition, or that she is even repeating anything, in the first place. The repetition-requiring elements constitute the task the performer feels obliged to perform. We will now say that routine performed in a given tasksituation by a given person is the task the performer saw herself performing together with the procedure she executed to perform the task.

The origins of the historical discourse of rational numbers go back to early human attempts to expand some everyday practical activities, such as those that require comparisons of continuous quantities -length, area, etc. The development of that discourse through human history involved assigning a name (signifier) to germinal routines, routines of practical activities that are likely to evoke the use of fractional words. While most of these germinal routines can also be successfully dealt with without any mathematical discourse, the use of fractions allows refinement of communication: it makes it more compact, more accurate or more widely applicable. For example the activity of sharing fairly a roll of fabric among four women can be communicated as: "each woman get one quarter of the roll". It also involves consolidations of different germinal routines into a single one. Such consolidation was due to the use of a signifier which is applicable to several hitherto unrelated types of activities. For example, the deed of sharing a roll of fabric among four women and the deed of dividing land among four heirs can both be described as "finding one fourth of the whole". Today, it is through the process of learning that children gradually become participants of this historically established discourse. The way it happens is bound to deviate from the historical trajectory because the formal discourse has already been established and the child is not required to name signifiers on her own initiative or to formulate discursive patterns. Hence, learning rational numbers is the process of individualization of the formal discourse of rational numbers. It is the process at the end of which the learner fluently participates in the discourse, according to her needs. Individualization begins with the learner's exposure to new words and symbols embedded in such everyday expressions as "half an hour" or "quarter to six", and proceeds with the formal learning of rational numbers in school. Since rational number is not a physical object, it cannot be displayed in class. Instead, the child is introduced to the discourse of rational numbers as a formal language: she is presented with the signifiers within its typical discursive context, and the formal routines. According to the commognitive approach there is no other way to begin the process of individualizing a new routine than by adopting it as a ritual. The adoption of a routine begins with an imitation of an expert's moves (e.g., father, mother, and teacher) which is motivated by the child's social needs, meaning participation is ritual. Being unaware of the practical application of the outcome, the child would not recognize the outcome of such routine as the exclusive aim of the performance. Later, the child's ritual routine undergoes de-ritualization; there is a change of focus in the performance of the routine, from the procedure being performed to the desired result. Eventually it becomes exploration, that is, a routine whose focus is on the outcome and whose success is evaluated by answering the question of whether a new endorsed narrative has been produced [START_REF] Sfard | Why cannot children see as the same what grown-ups cannot see as different? Early numerical thinking revisited[END_REF]. This paper focuses on the individualization of two formal routines for manipulating rational numbers performed by Ada and Noa, two fourth graders who are new participants in the formal discourse of rational numbers. Through presenting the girls with the same two school-like tasks repeatedly, five times over two years, we describe the de-ritualization of these formal routines. Ada and Noa were randomly chosen. The way the girls acted appeared rather standard; what they did was similar to what was done by most other participants.

Research questions and method

The purpose of the study is to describe the process of individualization of formal discourse routines of rational numbers taught in school. In the part of the study presented we will describe one case of the individualization of two school routines: "locate a fraction on the number line (as a point)" and "name a point on the number line". Hence we focus here on the questions:

1. How do these routines change throughout the school learning process? 2. Are they applicable in reproducing a point on the axis according to verbal instructions?

In order to answer these questions, I repeatedly interviewed 12 pairs of children from grades one to six (two pairs from each grade level) over two years. In Israeli schools, fractions are introduced in the third trimester of the third grade. The curriculum of grades 4, 5 and 6 was designed to gradually introduce fractions so that by the end of 6th grade, students would have satisfactory mastery of the fraction and its different interpretations. This is an ongoing study, in which interviews are conducted by introducing a predesigned assignments. The children are then engaged in the activity in order to accomplish the task. As a preliminary step to this project, we composed a battery of 28 assignments of two kinds: (i) school tasks; (ii) assignments which are meant to spur a performance of a germinal routine that occasions the use of rational numbers (such as finding parts and fair sharing of both discrete and continues quantities). This second group of assignments are meant to check the applicability of the formal routines. As we noted earlier in this paper, our aim is to characterize the way in which Ada and Noa individualized the formal routines of locating a fraction and naming a point on the number line (routines of lasting and naming, for short). We do so by looking at five consecutive interviews in which the girls repeatedly performed the respective school assignments and one practical activity. We labeled the interview as presented in Table 1. Ritual and exploration were defined by considering the task of the routine. We are dealing with a ritual if the routine is oriented exclusively at the process, that is, the task at hand -the set of elements of the precedent performances the performer regards as requiring repetition -is related to the process in its entirety. In other words, these repetition-requiring elements are the specific operations implemented in the precedent situation, not just their outcome. We are dealing with an exploration if only the outcome of precedent performances counts as important. The focal signifiers are used as a noun, a name of an independently existing object.

Was there an increase in the performer's use of the focal signifiers as nouns?

Flexibility

The performer, over time, uses the same single procedure in similar task-situations.

In a given type of tasksituation, the performer uses a range of procedures, with their choice depending on the parameters of the situation (such as the numbers that were given)

Was there an increase in the number of different procedures P's performs in reaction to what can count as the same task-situation?

Substantiation The performer is either unable to give any substantiation of the correctness of her performance or simply repeats the performance claiming its correctness.

The performer argues for the correctness of her previous performance by employing a different procedure and showing that she gets the same outcome

Is P now less dependent on other people's judgement in gauging the appropriateness of the execution?

Applicability

The routine is performed only if task-situation is reproduced in almost all its details.

No restrictions

Was there an increase in the use of the procedure in new contexts?

The analyst's question is how she can diagnose a degree of ritualization of a routine or a change in it by considering the records of specific performances. According to the commognitive approach, there are features of performances that the analyst can actually see and can take as indicative of a degree of ritualization (or de-ritualization). The features of Table 2 indicate how much the performance is directed toward a specific procedure and how much toward the specific outcome [START_REF] Lavie | Routines we live by: From ritual to exploration[END_REF]. Among those features are: i) vi the procedure is functional; iii) Agentivity: the degree of decision making while operating; iv) Substantiation: the performance arguments for the correction of the outcome; v) Objectification: the transition to referring to the rational number as an independent object in the mediating discourse; vi) Applicability: the performer finds the routine useful in situations where there is no significant similarity between the two task situations.

Here are some principles that the analyst must keep in mind while performing such diagnosis:

1. Most of the features appearing in Table 2 cannot be diagnosed directly on the basis of any specific performance. Instead, one needs to look at a whole series of performances for what counts for an expert as the same task-situation. 2. The properties of performances are not independent from one another. Sometimes, when you decide on one of them, the other almost automatically turns thru as well. 3. Diagnosing the features is an interpretive activity, in which any claim is tentative and subject to change; the change in interpretation may come at any time as a result from broadening the context and considering additional performances.

Because of space limitations, we will present in this article illustration only for some of the properties listed in Table 2. We reiterate that although the examples we chose are typical, in that they represented phenomena we saw in the rest of Noa and Ada's data and also in those of other participants, we will not present here finding but rather an illustration of the analytic method.

Illustrations of the analytic method

We will present representative examples showing how different features of routines summarized in Table 2 were identified in our data. The examples presented here are taken from the five consecutive interviews in which the girls repeatedly performed the following two fourth grade textbook assignments:

Flexibility

The rise in the flexibility of a routine means that there is now more than one way to perform the task. Stating that a routine preformed in a task-situation became more flexible means that there is now more than one optional procedure to perform the task. The routine become more flexible through the adaption of the procedure to typical features of the current task. Looking at Table 3, which present the procedures performed by Ada and Noa in the activity over Assignment 1, it might seems at first glance that the same procedure of dividing into twelfth and then reaching to the fifth twelfth is executed over and over again. However, despite the similarity between the procedures implemented by the girls, we argue that there is a process of streamlining -the adaptation of the procedure of the formal routine that they have learned to the specific assignment with which they are coping. Ada shifted from dividing the interval [0,1] into twelfth to another method of division that was easier for her to implement. Then she made yet another alternation when she divided only the interval [0,0.5].

would be substantiation of the outcome by showing that an alternative procedure would yield the same outcome. We found that over time the routines of the two task-situations became more substantiate. One of the signs of substantiation can be found in the change in Noa's explanation of her choice to divide the interval unto eight equal parts in performing Assignment 2 (Table 2). In ITV1 Noa is having hard time substantiate her actions. In ITV3 she is more confident and describes the calculation that led her to eighth, but she does not use words that are typical to the discourse of rational numbers as we would expect. In ITV4 she is describing the calculation again but this time she is using new words that are typical to the discourse of rational numbers that she didn't use before, like half quarter and denominator. In addition Noa is using another procedure that of division that her partner Ada was using in TLV4 to in order to substation her action.

Applicability of the two formal routines

We speak about routine's applicability while considering the range of task-situations for which its performances so far are likely to constitute precedents. In order to learn of the applicability of the two formal routines we presented the girls with the following activity that is supposed to evoke the use of the two formal routines. Each girl is given a card. Ada got the card in Figure 3 and had to instruct Noa over the phone to draw the point on her card (Figure 4) in the same location.

Using her demonstrated ability to name the point with a fractional number (23/4) would have helped Ada in this assignment, and would have been useful for Noa too, since she could translate a fraction into a number line location, as she had already shown during the execution of assignment 1. And yet, none of these routines were applied. We take this fact as an evidence of their ritualized character. Noa and Ada were asked to perform the telephone assignment over and over again in every session, but although their procedure underwent several refinements as is shown above, it did not involve point-naming and point-reading. It was only 18 months after ITR1 took place that the girls evoked that routine in that context, as shown in excerpt from ITR5:

Ada: Ok, divide between 2 and 3 into quarters (Looks at the page and marks with a finger the division of the interval between 2 and 3) Noa: Ok... three quarters? (Looks at her page) Ada:

Yes (Looks at her page and smiles)

Both: (Laugh)

Noa:

(Divides the interval between 2 and 3 into four more-or-less equal parts) Ada: Perfect. So two and three quarters, approximately (Looks at her page)

Here, it was the first time Ada was preforming the formal routine for naming the point to be copied with a fractional number. Later, Noa used the point-reading formal routine for reproducing

Figure 1 .Figure 2 .

 12 Figure 1. Text with Assignment 1

Table 1 .

 1 The five rounds of interviews (ITV) with Ada and Noa

	Labeled	ITV1	ITV2	ITV3	ITV4	ITV5
	Date	4.2015	12.2015	5.2016	11.2016	5.2017
	Grade	4 th	5 th	5 th	6 th g	6 th

Table 2 .

 2 Features of routine performances in a task-situation

	aspect	extremal types	examples of analysis-
	of routine R	Ritual	Exploration	guiding questions about the available performances
	Agentivity	The performer, in a given	The procedure	From one performance to
		type of task-situation, is	implemented by the	another, was there any
		always executing the	performer required	increase in the number of
		procedure as it was in the	making independent	independent decisions
		precedent. She is not	decisions; in some	made by the performer?
		making independent	performances, the
		decisions.	performer was task-
			setter.
	Boundedness	Different steps of the	Outcome of one step in	Does each step in the
		performance do not	the performance feeds	performance that should
		depend on one another,	into the next one.	depend on the outcome of
		even if they should.		the former does, indeed,
				utilize that outcome?
	Objectification The talk is about
		processes; The focal
		signifiers, if used, are
		either stand-alone or
		appear as adjectives or
		adverbs.

Table 4 .

 4 Signs of substantiation in Noa's explanationITV1because four times two equals eight ... do not know how to explain it, so it's eighths ITV3 And that four times two, that's eight.ITV4 Here it was divided into quarters, I deliberately emphasized the quadrants and that's half, oh and it was here, it was exactly half between quarter and two quarters. So I had to divide it, to multiply ... multiply the denominator by two to get it to the middle. Because half ... that's ... for that you have to double by two. So I multiplied the fo ... four times two, it's equal to eight, and I divided it so that each part is eighth.

Another sign of flexibility can be found in the excerpt taken from ITV3 where the equivalence of different procedures is explicitly recognized by Ada.

Ada:

Ok, I did it between one to zero because I know it is not yet a whole and then I divided it into half, and then I sevises both halves to sixth, I did not complete the division of that one' and here is the five.

Noa: I am dividing it to half to half and then divide into three.

Ada: But it is the method, like, it is less important, the how we divide it.

Objectification

For the discourse of rational numbers to be considered sufficiently developed, it is not enough that the discussant is acquainted with the different signifiers and realizations of rational numbers.

It is also important that she objectifies rational numbers, that is, becomes able to see all of the relevant signifiers as signifying full-fledged objects in their own right. Hence, the increasing tendency of using the signifiers as nouns is yet another indication for the progress of deritualization. Throughout the repeated performances of Assignment 2 we could see changes in the linguistic use of the signifier fourth in the girls mediating discourse. The transformation from referring to the location of the wolf as: "It is in the middle of the second quarter" in ITV1, to utterance like this: "It is right in the middle between forth and two forth" in ITV3 indicates that the girls have objectified the signifier fourth. In the repeated performance of assignment 1 Ada have objectified the signifier five over twelve. Along ITV1,2 and 3, Ada did not related to the fraction five over twelve as a number, but rather as a set of twelfths with five entities. As can be seen in the following excerpt taken from ITV2, Five is the number of twelfths that we need to count from zero:

Ada: I divided it into twelfth and then I checked where the five was, because it is five over twelve.

The objectification of five over twelve in the discourse of Ada is expressed in the following excerpt taken from TLV4 as Ada referred to it as a name of the location that is needed to be found.

Ada: I divided the zero to one into twelfth and then I just found the five over twelve.

Substantiability

When focused only on the procedure, the performer would substantiate her actions by simply describing the procedure she executed. Hence, a step forward in the process of de-ritualization

Concluding remarks

In the above, we have described the analytic method by which we operate in this study. As timeconsuming and laborious a task it is to conduct this kind of analysis we feel that it is worthwhile to undertake this effort. Underlying the decision to study the notion of routine is our belief that conceptualizing learning as a process of routinization may generate new insights.

Conceptualizing the learning of rational number as de-ritualization of the formal school routines, which is reflected primarily in the shift of focus from the implementation of a certain procedure to achieving a desired outcome, allows us to look at familiar phenomena that are considered challenging in the world of education in a new and insightful way. We know from previous research that the process of de-ritualization is gradual and slow and only too often will not be completed in school. Such analytic method may provide answers to outstanding questions in the research of learning rational numbers. Questions like the ones listed in the Second Handbook on Mathematics Teaching and Learning [START_REF] Lamon | Rational numbers and proportional reasoning: Toward a theoretical framework for research[END_REF] may not get holistic and comprehensive responses. They are questions like: How does one measure rational number sense? How does a child come to understand a rational number as a single quantity as oppose to regarding it as a pair of numbers? What are the benchmarks by which to judge that children's knowledge is moving in a desirable direction? And how can we assess depth of understanding of the rational numbers?


B Franzelli

A Vié

N Darabiha

A three-equation model for the prediction of soot emissions in LES of gas turbines

Keywords: Soot modelling, three-equation model, Particle Size Distribution, gas turbine

The design of new low-emission systems requires the development of models providing an accurate prediction of soot production for a small computational cost.

In this work, a three-equation model is developed based on mono-disperse closure of the source terms from a sectional method. In addition, a post-processing technique to estimate the particles size distribution (PSD) from global quantities is proposed by combining Pareto and log-normal distributions. After validation, the developed strategy is used to perform a large eddy simulation of soot production in a model combustor representative of gas turbine combustion chambers. It is shown that the three-equation model is able to provide a good estimation of soot volume fraction and information on PSD in complex geometries for a low computational time.

Introduction

Numerical simulations of soot production are essential to design low-emission burners, but require the development of reliable models. This is a challenging task because of the complexity of soot physics, which leads to a poly-disperse population of particles with complex morphologies. Moreover, models should require a minimal CPU cost to perform parametric studies of realistic configurations using Large Eddy Simulation (LES).

Different methods have been applied to LES of sooting turbulent flames with increasing accuracy and CPU cost. Semi-empirical models [1] transport two quantities, generally soot mass fraction and total number density, whose source terms are chosen 'ad hoc' and fitted to experimental data, so that their validity is limited.

They rely on a spherical assumption for soot particles and they do not give access to the number density function (NDF). However, their low CPU cost and ease of implementation into CFD solvers have made them preferred candidates for the investigation of complex configurations [2,[START_REF] Franzelli | Proceedings of ASME Turbo Expo GT2015-43630[END_REF]. Alternatively, methods of moments (MOM) solve the N first moments of the NDF, the main issue being to compute their source terms, which requires untransported moments. In the Hybrid MOM (HMOM) [START_REF] Mueller | [END_REF], this is done using an algebraic relationship between moments [5].

This method has been used to simulate turbulent complex configurations [6][START_REF] Koo | AIAA Aerospace Sciences Meeting[END_REF][START_REF] Wick | Proceedings of ASME Turbo Expo GT2017-63293[END_REF].

However, even if HMOM accounts for soot fractality, the NDF is not accessible.

Finally, the sectional method (SM) discretizes the NDF into a discrete number of sections. The soot mass fraction of each section is transported. This method gives access to the NDF and has already been used for LES of model combustors [START_REF] Grader | Proceedings of ASME Turbo Expo GT2018-75254[END_REF]. However, it is highly CPU-time demanding since at least 25 sections are required for a reasonable accuracy. In addition, extension to fractality requires even more transport equations.

Here, the objective is to develop a three-equation (3-eq) model that accounts for soot fractality and gives an accurate prediction of soot global quantities and an estimation of the NDF for a small CPU cost. The governing equations are presented in Sec. 2, accounting for soot fractality. A strategy to estimate the NDF based on an analysis of NDF shapes obtained with SM simulations is proposed in Sec. 3. In Sec. [START_REF] Mueller | [END_REF], the 3-eq model is validated in terms of global quantities and NDF reconstruction. Finally, in Sec. 5, the 3-eq model is used in a LES of the model combustor FIRST [START_REF] Geigle | Proceedings of ASME Turbo Expo paper GT2013-95316[END_REF]. This configuration is considered today as a reference for the investigation of soot production under conditions representative of gas turbines. Results are validated against experiments and the convenience of the NDF reconstruction method is discussed.

The three-equation model

The soot particle population is described by the NDF, f (t, x, , s), giving the particles number density at volume and surface s.

Total number density N s = f , soot volume fraction f = f and total soot surface S s = s f are moments of the NDF, where Φ = Φd ds. The mean particle volume and surface are s = f N -1 s and s s = S s N -1 s , respectively. Here, a model is developed to describe the global characteristics of soot population, such as f and N s , for a small CPU cost. This model is based on the transport of three global variables: N s , S s and the soot mass fraction Y s = ρ s ρ -1 f (ρ and ρ s are the gas and soot density, respectively). Each of these three quantities is solved using the following equation:

∂Ψ ∂t + ∇ • (uΨ) = ∇ • C th ν ∇T T Ψ + ωΨ (1)
where u, ν and T are the gas velocity, kinetimatic viscosity and temperature,ρ s = 1800 kg m -3 , and C th = 0.55 in the limit of small particles with respect to the mean free path of the gas phase [START_REF] Derjaguin | [END_REF]. The source terms ωΨ are derived from the sectional model in [12] by assuming a mono-disperse distribution f ( , s) = N s δ s δ s s (δ is the Dirac's delta function): Spherical dimers are assumed for nucleation, whose number density N dim is computed using a quasi-steady state approximation from the collision of polycyclic aromatic hydrocarbons (PAHs) [13]. For condensation and surface growth, a particle volume variation δ leads to a particle surface variation δs frac δ following a fractal behavior

ωY s ρ s = dim β fm dim N 2 dim nucleation + dim β fm dim , s N dim N s condensation (2)
+ C 2 λk sg s s N s surface growth -C 2 λk ox s s N s surface oxidation ωN s = β fm dim N 2 dim 2 nucleation -(1 -H [ s -C 2 ])λk ox s s N s surface oxidation - β s N 2 s 2 coagulation ωS s = (18π) 1/3 ( dim )
δs frac δ s s = 2 3 δ s n χ p [4]
, where n p = (36π) -1 -2 s s 3 s is the number of primary particles and χ = -0.2043. As in [START_REF] Mueller | [END_REF], oxidation modifies the surface of a factor δs spher δ defined as

δs spher δ s s = 2 3 δ s .
The surface reactions constants k sg and k ox are obtained from the HACA-RC mechanism as in [12]. λ = 1/s C 2 is the number of active sites per unit surface, where s C 2 is the surface of two carbon atoms. The collisional rates are modelled as:

β fm dim = nu πk b T 2ρ s 1 dim + 1 dim 1/2 d dim p + d dim p 2 β fm dim , s = cd πk b T 2ρ s 1 dim + 1 s 1/2 d dim p + d c 2 (3)
β s = max        cg πk b T 2ρ s 2 s 1/2 (2d c ) 2 , 8k b T 3ρν Cu        with nu = 2.5 , cd = 1.3, cg = 2.2. d c = d p n 1/D f p
is the collisional diameter, d p = 6 s s -1 s is the primary particle diameter, D f = 1.8 is the fractal dimension, Cu = 1 + 2.154 λ g d -1

c is the Cunningham corrective coefficient and λ g is the mean free path of the gas phase [12]. k b is the Boltzmann constant. Subscript or superscript dim stand for dimer properties.

N s , Y s and S s correspond to the 0 th -and 1 st -order moments used in the bivariate HMOM [START_REF] Mueller | [END_REF]. However, the source terms in Eqs. (2) are derived using a monodisperse NDF where HMOM uses assumptions on the moments themselves. This simplifies its formulation, physical understanding and implementation into CFD solvers. Compared to semi-empirical models, our method is a mono-disperse transcription based on a sectional model. Therefore, the source terms do not follow an 'ad-hoc' law fitted to experiments but they account for the physics included into the detailed SM.

Reconstruction of the number density function

The 3-eq model is expected to reproduce the global behavior of the soot population, but no information is available on the particles size distribution (PSD).

In order to characterize the PSD, a simple strategy is proposed to estimate the marginal NDF, n( ) = f ( , s)ds, from the transported quantities. For simplicity, in the following we will refer to the marginal-NDF as NDF.

On the one side, fine particles such as metal-oxides are known to be characterized by a self-similar distribution [14,15], approximately log-normal (LN), significantly simplifying the prediction of the NDF. The same approach has been used to investigate the soot particles distribution in fires [16]. On the other side, as observed in laminar [12,17] and, only recently, in turbulent flames with experiments [18,19] and numerical simulations [13], the PSD considerably varies along the flame, starting in a one-peak shape and evolving into a two-peaks distribution along the flame. To verify the validity of the self-similarities assumption for sooting flames, results obtained with a SM [12] are presented in Fig. 1(a). They correspond to the burner-stabilized stagnation laminar premixed C2H4/Ar/O2 flame experimentally investigated in [17] for a distance between the burner and the stagnation surface equal to H = 10 mm. These numerical results were already proven to be in good agreement with the experimental data [12]. The discussed transition from one-to two-peaks shape is recognized.

In Fig. 1(b), following [16], the dimensionless NDF, ψ(η) = n f /N 2 s , is evaluated at different heights above the burner, where η = / s is the dimensionless volume. Figure 1 shows that a single shape cannot reproduce the NDF along the flame. However, we suggest that the PSD can be approximated as the sum of two distributions n 1 ( ) and n 2 ( ) considering the following normalized NDF,

v[nm 3 ] ndf [nm 6 ]
n( ) = n( )/N s : n( ) ≈ na ( ) = α n1 ( ) + (1 -α) n2 ( ), (4)
where na ( ) is an approximation of n( ). By integrating Eq. ( 4) over the volume, we get:

s = α mean 1 + (1 -α) mean 2 , (5)
where mean

n1 ( ) = k k nucl k+1 with ≥ nucl , ( 6
)
where k is the Pareto index. In this case, mean k k-1 nucl . Assuming that the contribution of n2 is negligible at nucl , Eq. ( 6) gives the first-peak value as n0 = α n1 ( nucl ) = αk -1 nucl . Then, given α and n0 , the Pareto index k can be obtained as k = max n0 α -1 nucl , (1 -α nucl / s ) -1 . This latter criterion guarantees α mean 1 ≤ s (here = 1.01).

To reproduce the second peak, which is usually observed downstream the flame, due to condensation, coagulation and surface reactions on the biggest particles, an LN function is used for n2 with parameters µ and σ:

n2 ( ) = 1 σ √ 2π exp - (ln( ) -µ) 2 2σ 2 . ( 7
)
Once α and σ are provided, we get µ = ln( mean 2 ) -0.5σ 2 , where mean 2

= (1 - α) -1 ( s -α mean 1
). An example of NDF reconstruction is shown in Fig. 2 for the results corresponding to two axial positions of Fig. 1. As it is observed in Fig. 2, to correctly reproduce the NDF, α, n0 and σ have to vary along the flame height.

Although NDF evolves as a function of f , N s and s , here we choose s as the best parameter to trace its shape along the flame and to derive empirical expressions:

α = max       0, 1.0 -0.18 s nucl 0.12      , ( 8
) n0 = 8(1 -α) 2 , σ = 1 + 0.65(1 -α).
Even if constants have been fitted, their evolution is physically justified. α represents the balance between the Pareto (α = 1) and the LN distributions (α = 0). As observed experimentally and numerically [12,14,15], for small s (and small f ), a one-peak shape is predominant whereas two peaks are retrieved for large mean 8) may depend on the targeted fuel and/or operating conditions. However, the decomposition into Pareto and LN distributions seems a promising candidate to represent the general transition from one-to two-peaks shapes observed in sooting flames. (a) Nucleation and surface growth only.

N s [cm 3 ] f v [ppb] x[mm] x[mm]
N s [cm 3 ] f v [ppb] x[mm] x[mm]

Validation cases

In this session, the 3-eq model is validated against detailed Monte-Carlo or SM simulations. Comparisons with experiments are here avoided since any discrepancy (or agreement) of the 3-eqn model with experiments may potentially depend on the choice of more or less accurate physical and collisional description (in particular to the modelâ Ȃ Źs constants), whose evaluation is out of the scope of this work. However, the validation of the 3-eqn model on experimental laminar premixed flames [17] is presented in the supplementary material for completeness.

Global quantities

The accuracy of the 3-eq model is assessed in a laminar freely-propagating premixed C2H4/Ar/O2 flame, simulated with the REGATH solver [20] using the KM2 detailed gas phase reaction mechanism [21] for equivalence ratio φ = 2.1, initial temperature T = 300 K and atmospheric pressure. Soot production has been calculated with the 3-eq model as well as a reference Monte-Carlo (MC) approach using the Sweep2 solver [START_REF]Sweep2: Cambridge soot simulator, cambridge[END_REF]. In addition, a two-equation model (2eq) is also considered by solving Eqs. (1) for N s and Y s and imposing s 2eq s = (36π) 1/3 2/3 s . In order to guarantee consistency among MC, 3-eq and 2-eq models, spherical particles are considered with MC and δs frac = δs spher for 3-eq model.

First, f and N s profiles are presented in Fig. 3(a) by taking into account only nucleation and surface growth.

The 2-eq model overestimates f , whereas the 3-eq model is able to correctly reproduce the soot production, showing that the equation for S s is essential to reproduce the evolution of the particle surface and, consequently, of the growth and destruction source terms, even for spherical particles. Then, results obtained while considering also oxidation and coagulation are presented in Fig. 3(b). Discrepancies with the MC profiles are observed since the mono-disperse closure assumption is less valid since the resultant MC soot population is more poly-disperse in this case (not shown). However, the quality of the 3-eq model is better than the 2-eq approach and is quite satisfactory with regards to its simplicity and low CPU cost. This model is then a good candidate to estimate soot production using LES of complex industrial applications and it can be used to perform parametric studies necessary for engineering design whose CPU cost is today prohibitive with other models.

NDF reconstruction 4.2.1. Laminar flames

The NDF reconstruction is validated a priori on two laminar flame simulations using the SM [12]: a premixed flame stabilized by a stagnation plate and a counterflow diffusion flame. The spatial evolutions of f and N s are presented in Fig. 4. The two flame profiles greatly differ, leading to a large variability of the NDF. f and N s have been extracted at six different positions along both flames, identified by the vertical lines in Fig. 4, to calculate s . The NDF is then re-constructed at six locations as N s na ( ) and compared to the NDF provided by the SM in Fig. 5. The reconstructed NDF (R-NDF) reproduces the expected behavior, the Pareto and the LN functions mainly contributing to small and large particles, respectively. The R-NDF also correctly guarantees the first two moments, i.e. N s and f , with a maximum error of about 10%, and captures very well the transition between one-peak to two-peak shapes for the two different flame structures.

Turbulent flames

As a final validation, the R-NDF is tested on a turbulent ethylene/air jet flame (Fig. 6, [START_REF] Shaddix | Understanding and Predicting Soot Generation in Turbulent Non-Premixed Jet Flames[END_REF]). The temporal and spatial evolutions of the NDF have been characterized using SM [13]. Here, an instantaneous solution of the SM is used to To conclude, it should be reminded that the reconstruction procedure depends not only on the global transported quantities but also on some parameters (α, σ and n0 1 ) fitted in Sec. 3 on a specific flame from the results obtained with a specific SM. Even if this reconstruction has been validated a priori on different flame archetypes, a different evolution of the NDF with s may be observed for different soot modes, fuels or operating conditions. However, the decomposition of the NDF into a Pareto and a LN distribution seems to be a general strategy to represent the NDF without the transport of additional information, allowing parametric studies on gas turbines at low CPU cost.

LES of the FIRST combustor

The 3-eq model is now used to perform an LES of soot production in the model combustor FIRST, investigated experimentally at DLR [START_REF] Geigle | Proceedings of ASME Turbo Expo paper GT2013-95316[END_REF]. Once the simulation is validated against the experimental data, results on the PSD validity are discussed.

Numerical setup

The FIRST configuration of DLR [START_REF] Geigle | Proceedings of ASME Turbo Expo paper GT2013-95316[END_REF] is a classical benchmark for validation of soot modelling [START_REF] Franzelli | Proceedings of ASME Turbo Expo GT2015-43630[END_REF][START_REF] Koo | AIAA Aerospace Sciences Meeting[END_REF][START_REF] Wick | Proceedings of ASME Turbo Expo GT2017-63293[END_REF][START_REF] Eberle | [END_REF]25]. The burner consists in a plenum, a swirler and a combustion chamber. Ethylene and air are injected separately and mixed directly in the chamber where a swirled flame is stabilized.

The case studied here corresponds to a pressure of 3 atm, an injection equivalence ratio of φ inj = 1.2 and a secondary air injection reducing the global equivalence ratio down to φ global = 0.86. Simulation is performed with the AVBP solver [26]. The flow is simulated on the 40 million tetrahedra unstructured mesh presented in [27]. Adiabatic no-slip walls are considered except for the chamber walls, whose temperature is imposed based on experiments [28]. A look-up table based on the RFPV model [29] is built from the KM2 [21] detailed mechanism, to obtain information on gaseous species, as well as on precursors and on gaseous quantities required for the solid phase. Turbulent subgrid stresses are modeled using the WALE approach [30], with constant turbulent Prandtl and Schmidt numbers (Pr t = Sc t,k = 0.6). The subgrid chemical source terms of gaseous species are described using a β-pdf model [31]. The intermittency model for subgrid soot source terms developed in [32] for HMOM is used here. An optically-thin model is used for radiation. Time-averaged quantities have been obtained over 30 ms, corresponding to approximately six convective times. The gas phase simulation has already been validated in [START_REF] Rodrigues | Multi-physics modelling of turbulent sooting flames including thermal radiation and wall heat transfer[END_REF], showing a negligible impact of soot radiation on the gas phase. Therefore, the validation of the gas phase is not presented here and only results on soot are discussed.

Analysis of results

Results on time-averaged f using the 3-eq and SM [START_REF] Rodrigues | Multi-physics modelling of turbulent sooting flames including thermal radiation and wall heat transfer[END_REF] models are shown in Figs. 7(a)-7(b) and compared to the experimental data (Fig. 7(c)). To ease the comparison, the experimental and numerical results have been normalized by their respective maximum values ( f max,exp = 30±9 ppb, f max,num = 10 ppb). The quality of the prediction of the 3-eq model (Fig. 7(a)) is very satisfactory compared to the SM (Fig. 7(b)), for a CPU cost reduced by a factor 3. It should be noticed that in the SM an empirical relation was imposed to account for fractility [13], whereas the total particle surface is transported in the 3-eq model. Thus, the surface reactions may differ between the two calculations, modifying the production of f , so that any deeper comparison between the two models would not be pertinent.

Soot localization is correctly detected, but there is a factor 3 between experiments and simulations. Experimentally, the soot yield assumes the same level all along the flame and decreases for y > 60 mm. On the contrary, the numerical results predict a high value only for y < 40 mm and rapidly decreases downstream. The quality of these results is overall satisfactory since the prediction of f v is extremely challenging and the experimental data is characterized by an error of 30%.

The temporal evolution of global values of N s , f and s are represented in Fig. 8(a) at the point of maximum f fluctuations (x = 12 mm, y = 24 mm) showing that s highly fluctuates with time. In Fig. 8(b), the PSD obtained through the reconstruction procedure is represented for the three times indicated in Fig. 8(a).

As expected, it can be observed that the PSD shape greatly varies in time. However, the time-averaged distribution has almost everywhere a one-peak shape (not shown), as observed in the SM simulation [START_REF] Rodrigues | Multi-physics modelling of turbulent sooting flames including thermal radiation and wall heat transfer[END_REF]. This differentiates soot produc- tion in this configuration from what has been observed in the turbulent jet flames [13,18,19] and can be explained by the strong correlation evidenced between small (high) soot levels and one-(two)-peak distributions [13]. By looking at experimental results, a one-peak PSD is expected in this configuration since soot yield is of the order of a particle per billion, compared to ppm observed in jet flames.

Concerning alternative reconstruction techniques to obtain time-averaged PSD, Rittler et al. [START_REF] Rittler | [END_REF] suggested to use temporal fluctuations of s from a monodisperse model. This strategy was applied to the investigation of the titanium oxide population in a turbulent flame. However, there is no proof that the temporal evolution of s is representative of the time-averaged PSD. To evidence it, the time-averaged PSD for the data presented in Fig. 8 is shown in Fig. 9 (solid line), along with the temporal fluctuations of s (histogram). It shows that the number of particles is underestimated for the biggest particles because the mean volume s will obviously never be able to reach the volumes encountered in the tail of the PSD. Therefore, tracing the temporal evolution of s (or other moments) is not a pertinent representation of time-averaged PSD: a R-NDF or transported NDF are necessary. Another strategy may be to recalculate the PSD from the time-averaged s instead of simply reporting s fluctuations. This is represented in Fig. 9 (dashed line), showing that it is not identical to the time-averaged PSD. This is again not surprising because of the high non-linearity of the reconstruction with respect to moments. It can therefore be conclude that the only way to get the time-averaged PSD is to store the temporal evolution of the PSD (reconstructed or transported).

In conclusion, the 3-eq model provides good prediction of soot global quantities with no fitting procedure for parameters governing soot production processes. [START_REF] Rittler | [END_REF]) for the data presented in Fig. 8. The PSD reconstructed from the time-averaged mean particle volume is also represented in dashed blue line.

Concerning the PSD reconstruction, it should be noticed here that no validation has been provided on turbulent flames by comparison with experimental data, since they are today not available for this configuration. However, it represents a low-cost procedure to get information on the NDF, which cannot be represented only by looking to the fluctuations of s .

Conclusion

A simple 3-eq model for the description of soot production has been proposed based on a mono-disperse closure of the source terms of a sectional method. It presents the following advantages: the addition of a transport equation for the total soot surface guarantees a better description of surface reactions compared to the semi-empirical method; its theoretical development guarantees a larger validity of the model; it can easily be implemented into CFD solvers and provides a reasonable estimation of the PSD for a low CPU cost (the cost of a LES of a turbulent sooting flame is approximately reduced by a factor 3 compared to the sectional method using 25 sections when using a tabulation technique for gas description).

However, the NDF reconstruction still requires more validation and the accu-racy of the 3-eq model is expected to be reduced when the NDF is highly polydisperse. Nevertheless, since a correlation seems to exist between small values of f (less than ppm) and one-peak NDF, the mono-disperse model proposed here is a good candidate for LES of sooting turbulent flames of gas turbines and aeronautical burners, where the produced instantaneous f is expected to be small (i.e. less than ppm).

Figure 1 :

 1 Figure1: NDF obtained with the SM[12] in a burner-stabilized stagnation laminar premixed C2H4/Ar/O2 flame at three axial positions: x = 3 mm (black line), x = 6 mm (red dashed line) and x = 9 mm (blue dot-dashed line).

α mean 1 ≤ 2 ≤

 12 volumes over n1 and n2 , respectively. This implies s and (1 -α) mean s . To reproduce the first peak which is mainly due to the growth of the nuclei particles of volume nucl , we consider a Pareto distribution for n1 ( ):

Figure 2 :

 2 Figure 2: Reconstruction (red continuous line) of the NDF as the sum of a Pareto (blue cross) and a log-normal (green plus) distribution. Results correspond to two axial positions of Fig. 1: x = 3 mm (left), x = 9 mm (right). The reference NDF obtained from a SM is presented as black circles.

  volumes (and big f values). The evolution of α captures this trend: the NDF is Pareto-like at the beginning of the flame, evolving towards an LN distribution downstream. The first-peak n0 governs the stiffness of the Pareto distribution. As observed in Fig.2, the role of the Pareto function is initially to reproduce almost the whole soot population, which is the result of many simultaneous processes, whereas downstream it mainly reproduces only the nuclei production. As a consequence, n0 , meaning n1 stiffness, increases with s . Similarly, the parameter σ is proportional to s . When increasing the volume, the population gathers into the n2 ( ) distribution, mainly governed by the coagulation process, leading to a larger value of σ. The proposed reconstruction strategy has been developed by analyzing NDF results from a laminar premixed flame but will be validated in the following on different flame archetypes (laminar and turbulent, premixed and non-premixed flames). Even if the empirical relationships of Eq. (

  Nucleation, surface growth, oxidation and coagulation.

Figure 3 :

 3 Figure 3: Soot number density (left) and volume fraction (right) in a laminar freely-propagating flame. Results with the 2-and 3-eq models are compared to MC solution (continous red line, dashed blue line and symbols, respectively).

Figure 4 :

 4 Figure 4: Spatial evolution of f (black) and N s (blue) obtained with the sectional method [12] for a premixed (top) and a diffusion (bottom) flame. Vertical lines indicate the axial position of the NDF represented in Fig. 5.

Figure 5 :

 5 Figure 5: Validation of the NDF reconstruction model on laminar flames: the reconstructed NDF (lines) is compared to the sectional NDF (symbols) at 6 different positions indicated in Fig. 4.

Figure 6 :

 6 Figure 6: Validation of the NDF-reconstruction model on a turbulent jet flame: R-NDF (lines) is compared to the sectional NDF (symbols) from an instantaneous solution along the central axis [23] at 3 axial positions H/D = 50, 85, 125 for which s = 40, 250 and 2000 nm 3 .

Figure 7 : 3 ]

 73 Figure7: Time-averaged f normalized by its maximum value: f = f / f max . Results from the 3-eq (a) and the sectional (b,[START_REF] Rodrigues | Multi-physics modelling of turbulent sooting flames including thermal radiation and wall heat transfer[END_REF]) methods ( f max,num = 10 ppb) are compared to experiments (c, f max,exp = 30 ± 9 ppb,[START_REF] Geigle | Proceedings of ASME Turbo Expo paper GT2013-95316[END_REF]).

Figure 8 :

 8 Figure 8: Temporal evolution of (a) N s , f , s and (b) PSD at three instants for the 3-eq model at x = 12 mm and y = 24 mm.

Figure 9 :

 9 Figure9: Time-averaged R-PSD with the 3-eq model (continuous black line) is compared to temporal fluctuations of the mean particle volume (histogram,[START_REF] Rittler | [END_REF]) for the data presented in Fig.8. The PSD reconstructed from the time-averaged mean particle volume is also represented in dashed blue line.

Acknowledgments

This work was granted access to the HPC resources of CINES under the allocation x20162b6172 of GENCI (Grand Equipement National de Calcul Intensif).

CERFACS is also gratefully acknowledged for providing the AVBP code and the mesh of the FIRST configuration.


B Franzelli

M Roussillo

P Scouflaire

J Bonnety

R Jalain

T Dormieux

S Candel

G Legros

Multi-diagnostic soot measurements in a laminar diffusion flame to assess the ISF database consistency

Keywords: soot, MAE, time-resolved LII, primary particle size distribution, polydispersion

Control of soot emission raises fundamental issues and has important practical implications requiring a full understanding of soot production and oxidation processes. The research reported in the present paper intends to contribute to the studies carried out within the frame of the International Sooting Flame workshop (ISF) on laminar sooting flames. The objective is to identify and quantify sources of experimental errors and to extend the existing database for the Yale laminar diffusion burner flame. This will especially enable more comprehensive comparisons among different experimental techniques and numerical simulations. To this end, a combined use of Modulated Absorption/Emission (MAE) and Laser Induced Incandescence (LII) techniques is presented in this work. Results are compared with already existing experimental data in terms of soot volume fraction, soot temperature and primary particle size distribution, highlighting the high variability of the experimental data depending on the measurement techniques as

Introduction

Soot is a major issue due to its adverse effects on human health and climate [1,2]. A better understanding of soot formation and oxidation, defining the total net production, is needed in order to control these emissions. This question is examined in the framework of the ISF workshop [START_REF]International Sooting Flame (ISF)[END_REF] by combining experimental and numerical investigations on a wide range of sooting flame configurations.

Among them, the Yale Diffusion Burner (YDB [START_REF]Sooting Yale Coflow Diffusion Flames[END_REF]) allows fundamental investigations of soot production in laminar axisymmetric diffusion flames at atmospheric pressure for different dilutions of the inner fuel stream, i.e. 32%, 40%, 60%, and 80% of ethylene diluted with nitrogen (by volume). The experimental setup simplicity enables the combination of different measurements to better understand the links between flame's temperature, gaseous quantities, flow field, and soot particles. A comprehensive experimental database already exists for this burner, including temperature measurements via pyrometry [START_REF] Kuhn | [END_REF] and two-line atomic fluorescence [6], laser induced incandescence (LII) [7] and pyrometry [START_REF] Kuhn | [END_REF] for soot volume fraction f , Particle Imaging Velocimetry (PIV) [8] for measurements of the velocity field; transmission electron microscope (TEM) measurements [9] to determine the primary particle size distribution (PPSD) and Time-Resolved LII (TR-LII [6]) to measure mean primary particle diameter. Unfortunately, these techniques have not been systematically applied to the four dilution cases so that the experimental database is still incomplete. This configuration is also widely used to validate soot models for laminar configurations [7,10]. However, the validation procedure requires a consistent database and estimated uncertainties, reduced to the lowest possible level by comparing different diagnostics.

Following this line of thought, the present investigation exploits for the first time a combination of LII [11] and Modulated Absorption/Emission (MAE) [12,13] techniques, and proposes a comparison with existing results to enhance the database on soot volume fraction, temperature and primary particle size distribution. New temperature measurements obtained from pyrometry with a digital camera [START_REF] Kuhn | [END_REF] complete this database.

The present paper begins with a brief description of the experimental setup (Sec.

2). Optical diagnostics used to measure f and soot temperature are introduced in Sec. [START_REF]International Sooting Flame (ISF)[END_REF] and results for the 80% case are presented. Finally, the TR-LII technique is employed in Sec. 4 to access the PPSD and its response to dilution.

Experimental setup

The YDB, originally designed at Yale University [START_REF]Sooting Yale Coflow Diffusion Flames[END_REF], consists of an inner fuel injection tube surrounded by a co-flow of air [7]. Fuel and oxidizer flows are set to guarantee an average cold-flow velocity of f =35 cm/s. Bronkhorst El-Flow electronic flow controllers are used with a calculated uncertainty of ±0.3% FS (Full Scale) and regulation loop accuracy of ±0.1% FS.

Using a high grade caliper, it was found that the central fuel injection tube has an inner diameter of d f =3.9 mm, that slightly differs from the nominal 4.0 mm value of the reference burner. This induces a difference between the nominal and the real flow rates (calculated from f and d f values) that has a measurable impact on the flame length. As an example, a variation of 4% of the flow rate induces an increase of nearly 6 mm in flame length for the 80% case. Such a sensitivity may explain discrepancies in the flame length observed in some previous measurements [START_REF] Kuhn | [END_REF][6][7]. The outer diameter of the air co-flow has been reduced to 50 mm to enhance the stability of steady flames via an additional outer aluminum ring on top of the burner, thus alienating the flickering of the flame. It has been verified that such modifications do not affect the mean flame length and its structure. However, a small fluctuation of the flame luminosity of approximately 4% is still observed for the longest flame (80%).

To improve results accuracy, MAE and LII setups have been optimized so that different laser and filters wavelengths are used for the two techniques. The LII measurements of the 2-D soot volume fraction field are carried out with a Nd:YAG laser (1064 nm) focused in the form of a sheet on the burner axis (convergent lens f=+1000, cylindric lens f=-50). The sheet is 10 cm wide and 0.35 mm thick at the focal point. Only its inner part (7 cm) is considered to get a nearly uniform laser fluence approximately equal to 0.45 J/cm 2 (shot duration ≈ 9 ns), which corresponds to the plateau region of the LII signal [14]. This value guarantees to minimize the effect of laser energy variations even if it can lead to sublimation of the smallest particles, which is here assumed as negligible compared to the total f . At a wavelength of 1064 nm, fluorescence from precursors is avoided and there is no need to discriminate between fluorescence and incandescence signals. Therefore, the camera gate delay t 0 is set to zero [15] with an exposure time of 25 ns to minimize the overestimation of the contribution of the biggest soot particles to the total f [14]. In addition, the signal is collected at six delay times (t i = 50, 75, 100, 200, 400, 600 ns) to retrieve PPSD information based on TR-LII measurements [16]. The LII signal is recorded with an intensified CCD camera (Princeton, PI-MAX 3, 1024 × 1024 pixels) equipped with a lens (Nikkor 50 mm f/2) and a 50 nm bandpass filter centered at 425 nm (ASAHI SPECTRA YBPA425). This filter reduces parasitic C 2 Swan band emission and flame luminosity [17].

The two-color MAE (2C-MAE) experiments use two different 100 mW (±0.5 mW) Spectra-Physics Excelsior CW Lasers operating at 645 nm (-5/+7 nm) and 785 nm (-5/+5 nm). MAE signals are acquired with a CMOS camera (1312×1312 pixels) equipped with filters centered at 645 nm (± 2 nm) and 785 nm (± 3 nm).

Working at 645nm for MAE measurements of f allows to optimize the absorption signal compared to the scattering by soot particles.

A digital Nikon D7000 camera, placed at 3 m from the flame to guarantee a difference of the maximum angle between the light rays of less than 2 degrees, is used to capture flame luminosity using the lowest possible value for the ISO number and the exposure time (120 and 1/8000 s, respectively). To enhance the signal-to-noise ratio, all data are averaged over 150 images.

Measurements of soot volume fraction and soot temperature combining MAE and LII

In this work, a combined use of MAE and LII techniques is proposed for the first time. On the one hand, the MAE technique is used to calibrate the LII signal, to measure temperature fields and to obtain information on the local spectral absorption field, which allows for the correction of the pyrometry signals for soot self-absorption along the line-of-sight. On the other hand, LII measurements provide information on both f and PPSD.

Soot volume fraction

The soot volume fraction field is here retrieved using two different techniques, MAE and LII. The MAE technique [12] gives access to the local spectral absorption field κ λ by comparing the flame luminosity with and without a collimated laser crossing the flame. The f field is extracted from information at one wavelength (λ 1 = 645 nm). The local spectral absorption field κ λ is then used to obtain the soot volume fraction field using the Mie theory: f = λ κ λ /(6 π E(m λ )), where E(m λ ) is a function of the complex refractive index m λ of soot, varying with the detection wavelength λ. Since f is proportional to 1/E(m λ ), the errors in characterizing E(m λ ) directly induce uncertainties in the soot volume fraction value.

The experimental measurements of E(m λ ) present a degree of variability, as they depend on the fuel considered, on soot maturity [9,18] and on λ. In this work, the value E(m λ ) is adapted from that reported by Kempena et al. [9] and Coderre et al. [19] by applying the MAE to a Santoro burner [20] and comparing with previous data. The value E(m λ=645 nm ) = 0.38 gives consistent f results in this reference flame but one has to keep in mind that the variability of E(m λ ) may induce an error in f of up to a factor two. In addition, results near the flame centerline may be affected by the in-house deconvolution process employed to reconstruct 2D-fields from the information integrated on the line-of-sight [12]. The error, estimated at 2.5% for the Santoro flame, may be reduced by using the Tikhonov regularization procedure [START_REF] Liu | CEUR Workshop Proceedings[END_REF] but it is considered as negligible compared to the other detected uncertainties.

In the LII technique [START_REF] Michelsen | [END_REF], the soot volume fraction is proportional to the measured signal:

I LII = C cal f v ,
which is here calibrated with the MAE results f MAE . The LII signal is subject to various uncertainties. First, it can be affected by laser light absorption. In the present configuration the laser sheet traverses the flame from left to right, so that in this axisymmetric flame, this issue can be easily overcome by only considering data gathered on the left side of the flame. Second, the LII signal can be affected by the calibration procedure. Here, the calibration constant C cal is calculated by averaging the first p higher pixel values of MAE and LII signals, located in the left flame wing, where results are less affected by the MAE deconvolution procedure and soot absorption. It has been observed that the calibration constant varies by up to 10% depending on the numbers p of maximum values considered (here p = 100). Third, the LII signal collected by the camera is affected by absorption due to the presence of soot in the detection pathway [START_REF] Liu | CEUR Workshop Proceedings[END_REF]23,24]. The effect of self-absorption on LII results in this flame is quantified by retrieving the emitted LII signal with an iterative procedure, described in the supplementary materials, making use of the detected signal and the postprocessing approach developed for the MAE. By comparing emitted and detected signals, it is found that the self-absorption is highest in the central region of the flame. A maximum attenuation of 5% on f is observed for the most sooting case (80%).

To conclude, error bars for results presented in Sec. 3.3 can be deduced as the result of all the uncertainties discussed in this section.

Soot temperature

New measurements of soot temperature are also provided. On the one hand, the 2C-MAE technique [12] allows to map soot temperature from information on measured fields of local spectral emission rate and local absorption coefficient κ λ at two wavelengths (λ 1 = 645 nm and λ 2 = 785 nm). For the temperature measurement itself, this technique needs no calibration or modeling to correlate f and k λ , avoiding the issue of the large discrepancies in the wavelength-dependent soot refractive index discussed in the previous section.

On the other hand, the frames required by the 2C-MAE procedure are processed by following the methodology associated with the 2C-pyrometry tech-a.

b.

Figure 1: Soot volume fraction (a) and temperature (b) at three heights above the burner for the 80% case. Comparison of different measurement techniques for f (LII from Yale [7], pyrometry from Yale [START_REF] Kuhn | [END_REF], new LII and MAE results) and temperature (pyrometry from Yale with and without constant properties [7], new pyrometry and MAE results).

nique [START_REF] Kuhn | [END_REF]. In contrast with the regular 2C-pyrometry technique, this procedure (described in the supplementary materials) inherently accounts for the soot selfabsorption along the line-of-sight [25][26][27][28]. Compared to the MAE temperature, pyrometry results depend on the model retained for the spectral dependence of m λ . Here, the model assessed by Chang and Charalampopulos [29] and usually prescribed for ethylene flames [26] is retained.

Additional temperature fields are obtained from pyrometry using a digital camera [START_REF] Kuhn | [END_REF]. Results from the three color channels (.NEF format) are post-processed using a look-up table built using the camera spectral sensitivity presented in [30] and assuming that the soot emissivity varies with λ -1. 38 . It has been observed that a 5% variation of the exponent's value leads to a temperature variation of about 60 K.

Comparison with data from literature

The objective is now to validate the present measurements and to compare results with other techniques. The most sooting case (80%) is considered in this section. Profiles of f from MAE and LII are presented in Fig. 1a Data from previous studies performed at Yale with LII [7] are also plotted in Fig. 1a for comparison. The new MAE and LII results are greater by about 20% than LII data from the literature but they present the same general aspect with a high level of soot in the lateral region of the flame. The observed differences between the present data and those from the literature are within the error uncertainties discussed at the previous section, but they may also be due to additional differences in the LII experimental setup: the choice of the laser wavelength (λ=532 nm) and the resulting choice for E(m), which in the previous studies was higher (E(m λ=532nm )=0.45), the filter used for LII and the experimental technique to obtain the calibration.

Literature f data from pyrometry [START_REF] Kuhn | [END_REF] are also plotted in Fig. 1a. They show a good agreement with the present database in the flame wings, but are smaller in the centerline compared to the LII results. This is possibly due to the deconvolution procedure as discussed for the MAE technique.

Temperature fields obtained with MAE and pyrometry techniques are compared in Fig. 1b to the pyrometry measurements with and without constant properties (R-PYRO and PYRO [START_REF] Kuhn | [END_REF], respectively) obtained at Yale. Similar results are deduced with the four pyrometry techniques in the flame wings, with a maximum temperature discrepancy of less than 100K. It can be noticed that pyrometry results obtained here with a digital camera (PYRO-DIG) are detected for a wider radial region. However, it should be noticed that also the new LII results extends to a wider radial position, possibly explaining the difference with previous pyrometry data. On the contrary, pyrometry from MAE fields needs a sufficiently high soot concentration to obtain information on κ λ , reducing the region where the signal is detectable.

Temperature deduced from MAE in the flame wing is much higher than that obtained from pyrometry, whereas results are closer in the flame middle. This may highlight the effect of the model used to represent the spectral dependence of m λ in the pyrometry procedure. However, a simulation of a 1D counterflow diffusion flame representative of the investigated flame (ambient temperature, atmospheric pressure, strain rate of 1 s -1 , 80% ethylene diluted with N 2 ) has been performed using the models presented in [31] and shows that the maximum temperature does not exceed 2150 K. Therefore, numerical results seem to indicate that the high values detected in the wings with the MAE may be un-physical. In this regard, it is worth reminding that the accuracy of the MAE temperature measurements depends on the optical thickness along the line-of-sight because inferring the temperature first requires the measurement of the absorption coefficient field [12]. In the YDB flame, the optical thickness is significantly lower than that of the Santoro burner, the MAE technique having been validated with this burner configuration [12]. The lower optical thickness of the YDB flame and the sharp temperature gradients in the flame wings possibly alter the κ λ resolution and, consequently, results accuracy.

In conclusion, measurements of f and temperature feature a significant variability that needs to be quantified by comparing different experimental techniques and to be accounted for during the validation process of numerical models.

Measurements of primary particle sizes

Time-resolved LII technique

The evolution of the LII signal decay contains information on the primary particle size distribution (PPSD) [32] so that size data may be extracted from timeresolved 2D LII signals, based on a regression method relying on modeled LII signal. An example of TR-LII images I i = I(t i ), normalized by the signal I 0 at t 0 =0 ns, is displayed in Fig. 2 for the 80% case. The decay time being longer for larger primary particles, it can be deduced that the PPSD is spatially variable. In particular, big particles are expected to be found in the flame outer region.

Among the different techniques, the one proposed by Dankers and Leipertz [33] and extensively validated by Chen et al. [16] on a Santoro flame is here retained. The LII signal is approximated by an exponential decay function:

I LII ∝ exp(-t/τ),
where τ is the exponential decay time, which can be estimated as: τ i j = (t it j )/(ln I jln I i ).

Information on PPSD is obtained for each spatial position by comparing experimental decay times τ exp i j to a look-up table containing modeled decay times τ mod i j . Thus, results on PPSD, such as the mean primary particle diameter d p , will depend on the models used to build the look-up table and to perform the non-linear regression. By assuming a monodisperse population, d p can be obtained by minimizing: 2 , where τ mod (d p ) is here calculated by performing simulations of LII decay signal for a single particle of diameter d p using the LIISim-Web tools [34] developed by Hofmann et al. [START_REF] Hofmann | Proceedings of the European Combustion Meeting[END_REF]. Eighteen diameters from d p = 4 nm to d p = 100 nm are considered to build the look-up table, imposing a gaseous temperature T g = 1700 K. Details on the modeling are described in [START_REF] Hofmann | Proceedings of the European Combustion Meeting[END_REF] and simulation parameters are provided in the supplementary material. Results in terms of mean primary particle diameter for the 80% case are illustrated in Fig. 3 for six τ i j . They confirm the qualitative trend deduced from Fig. 2, since the largest primary particles, with a diameter of about 90 nm, are located in the flame wings whereas the presence of smaller primary particles (with d p ≈ 40 nm) characterizes the central region. The quality of the results strongly depends on the decay time used to perform the minimization, i.e. on the retained gate delays. In particular, for small gating delays results are quite noisy. In [16], it is indicated that gating delays should be sufficient to avoid the effects of the vaporization period. For the fluence considered in the present experiments (0.45 J/cm 2 ), vaporization is not negligible at the beginning of the simulated LII signal and up to t 3 = 100 ns, explaining the tendencies observed in Fig. 3. However, it has been verified by performing LIISim-Web simulations with a lower 0.1 J/cm 2 that even in case of negligible vaporization, results are highly dependent on the choice of τ i j when small gating delays are considered (not shown). In contrast, when the gating delays are sufficient, results on d p become consistent. It is then important to choose detection gates that are sufficiently delayed while preserving a good signal-to-noise ratio. This implies that information will be lost in the region characterized by the smallest particles, that appears as black zones in Figs. 2 and3. The proper choice depends on the flame under investigation and on its particle size distribution.

F (d p ) = [τ exp -τ mod (d p )]
To take into account polydispersion, one often presumes a log-normal probability density function featuring two parameters d cmd and σ:

f (d) = 1 √ 2πd ln σ exp - (ln d -ln d cmd ) 2 2(ln σ) 2 . ( 1
)
The mean primary particle diameter d p = exp(ln d cmd + (ln σ) 2 /2) and the variance σ 2 p = (exp((ln σ) 2 ) -1) exp(2 ln d cmd + (ln σ) 2 ) can be obtained by minimizing:

F (d cmd , σ) = [τ exp -τ mod (d cmd , σ)] 2 .
A second look-up table is then build with the LIISim-Web tool using the same parameters as in the monodisperse simulation but considering polydisperse soot particles. Six values of d cmd are considered between 10 and 60 nm and σ varies from 1.1 and 1.7 with an interval of 0.1. Results based on the polydispersed population are plotted in the right side of Fig. 3. They qualitatively reproduce the same spatial distribution of d p found in the monodisperse case with the smallest and biggest particles located on flame centerline and wings, respectively. However, compared to the monodisperse calculation, the diameter for the biggest particles is about 60 nm, a value smaller than that of the monodisperse results. The centerline primary particle diameter is approximately of 30 nm, close to the monodisperse value. This seems to indicate that a quasi-monodisperse population of primary particles characterizes the centerline, whereas a polydispersed population is observed in the flame wings. This is confirmed by standard deviation results, that are quite small on the centerline and increase in the flame wings. The present results show trends that are similar to those of Chen et al. [16] obtained for the Santoro burner, confirming the present interpretation. However, it is worth noting that the present results depend on the model used to create the look-up table. For example, a constant gaseous temperature is here assumed for simplicity, even if temperature results have been presented at the previous section. The impact of this assumption has been estimated in [16] to be about 4% in decay times. In addition, the role of the post-processing procedure and other model assumptions, such as considering single particles instead of aggregates, i.e. the shielding effect is neglected, is estimated in [16] to introduce errors of 30% in d p and 5% in σ p . More generally, modeling of LII signal is known to be characterized by a large variability, especially at high fluence [START_REF] Michelsen | [END_REF]. It is therefore important to provide access not only to the final results

Comparison with TEM measurements

TEM measurements performed at Yale are available at different locations for the 80% case [9], which are here compared to the TR-LII data. Results from TEM (vertical bars) are presented in Fig. 4 for four positions together with the lognormal fittings to the sampled TEM (continous) and to the LII (dashed) data obtained in the previous section using a polydisperse population. The values of d p using a monodisperse assumption are also added to Fig. 4. The exact radial position for TEM measurement in the flame wings not being provided, the LII data has been [6]. Middle: d p under monodisperse assumption (τ 13 for 32%, τ 34 for 40%, τ 35 for 60%, τ 36 for 80%). Right: d p and σ for a log-normal polydisperse population ( (τ 13 ,τ 23 ) for 32%, (τ 24 ,τ 34 ) for 40%, (τ 34 ,τ 35 ) for 60% and (τ 34 ,τ 36 ) for 80%) .

extracted at the location of maximum d p . In general, for all spatial locations, the LII particle size population is shifted towards larger diameters compared to TEM distributions. Concerning the centerline (top), LII results slightly overestimate d p and σ p compared to TEM measurements when using a polydisperse assumption.

With the monodisperse model, d p is strongly overestimated. In the TEM results, σ p is nearly constant for HABs of 5 to 6 cm. This tendency is correctly reproduced by the LII measurements. As expected, TEM results in the flame wings show higher values of d p and σ p compared to the centerline. Even if LII results retrieve the same qualitative behavior, d p values are strongly overestimated in the wings when using a polydisperse assumption and even higher discrepancies are found for the monodisperse case. Uncertainties on the radial position of TEM measurements may partially explain such discrepancies, since it has been observed in the LII results that a variation of 0.5 mm in the radial position decreases the d p value of about 30%. In addition, TEM measurements present in the flame wings a population that cannot be represented with a log-normal distribution, so that the assumption underlying the polydisperse LII model might not be valid. As already discussed, neglecting the shielding effect can also partially explain the observed differences. However, it should be noticed that these results present the same trend for TEM-LII comparison observed for state-of-the-art measurements on the Santoro burner [9,16]. In general, such a comparison may be critical since LII and TEM techniques do not strictly measure the same quantity (effective and measured diameters, respectively).

Dilution effects on PPSD

The TR-LII technique is now applied to investigate flames at four dilutions with the monodisperse and log-normal assumptions. Results are displayed in Fig. 5. To our knowledge, this is the first time that the effect of dilution on primary particle size distribution is investigated on the YDB flame. First, one observes that the spatial distribution of d p , which is qualitatively the same for monodisperse and polydisperse assumptions, is correlated with the f fields (shown in the supplementary materials). A nearly homogeneous value of d p is identified for 32% and 40% cases, whereas big primary particles are found in the flame wings for 60% and 80% cases. Small values of σ p are found for the 30% and 40% cases so that the primary particle distributions can be considered as quasi-monodisperse. Therefore, d p results are similar when assuming monodisperse or polydisperse populations in these two flames. In contrast, high values of σ p are observed in the wings of 60% and 80% flames, where a polydisperse population is expected and confirmed by the differences in d p corresponding to monodisperse and polydisperse results. One may generally conclude that by increasing the presence of fuel in the diluted mixture, higher values of f are observed in the flame wings, where the soot population of primary particles is also more polydispersed and shifted toward larger diameters. This is probably the results of two effects affecting chemical and collisional soot processes: the increase of precursors concentrations, since reactive species concentrations increase, and the change in the flame temperature as a consequence of the streams' physical properties variation [37,38]. It should be noticed that dilution may also affect particle optical properties, introducing an additional uncertainties to soot volume fraction measurements. Their quantification is out of the scope of this paper and they are here considered small compared to the global uncertainty on E(m).

Finally results for the 32% case (Fig. 5) are compared with the d p field measured with a TR-LII technique at Adelaide using a mono-disperse assumption [6]. While a nearly homogeneous spatial distribution of d p is also observed in the Adelaide measurement, it should be noticed that d p has been estimated to be close to 50 nm, whereas in the present measurements d p is less than 30 nm with both monodisperse and polydisperse distributions. At this point, it is not possible to identify the source of such discrepancies, which may be due to differences in measurements, processing tools or models used to calculate the look-up table associated with the LII signal. However, on the basis of TEM results for the 80% case (showing that only a small number of particles exceeds 50 nm), a mean d p value of 50 nm is quite unlikely for the 32% flame. In general, it can be concluded that PPSD results present an high variability and that it is essential to include data from as many measurement techniques as possible to create a reliable experimental database for modeling validation.

Conclusion

The objective of the present investigation is to provide new measurements for the YDB flame in order to obtain a reliable and comprehensive experimental database that can be used to validate numerical models for soot production. This is done by combining for the first time LII and MAE techniques. In particular, MAE information is used to calibrate the LII signal for f , to provide new temperature measurements and to account for soot self-absorption on pyrometry signals.

Time-resolved LII provides information on PPSD in terms of mean and standard deviation of the primary particle diameter. The study indicates that: (1) A qualitative agreement is observed among the different techniques for f and temperature measurements but quantitative differences are detected; (2) The primary particle size d p correlates with the f field and that by increasing the presence of fuel in the diluted mixture the dispersion in PPSD increases; (3) When using a monodisperse assumption, the d p distribution is qualitatively retrieved but its value is largely overestimated; (4) A high variability in PPSD is observed depending on the diagnostics and post-processing methods. In general, the need of a cross comparison between multiple data has been evidenced to define a consistent database for sooting flames. For this, access to the measured fields should be provided together with the post-processed results and the modeled signals.

Acknowledgments

This work is supported by the Air Liquide, Ecole Centrale Paris and CNRS OXYTEC Chair and by the OXYTEC project (ANR-12-CHIN-0001) from l'Agence Nationale de la Recherche.

  at the three heights above burner (HAB) presented in Fig. 2. The correspondent 2D fields are provided as supplementary material. To our knowledge, this is the first time that a comparison between MAE and LII techniques is provided. It shows a good agreement between MAE and LII in the flame wings since this region corresponds to the zone used to perform the calibration. Close to the flame centerline, differences between MAE and LII signals appear where the deconvolution process used by the MAE technique is less accurate. LII signals corrected by auto-absorption (LIIabsorp) are also presented. Only small discrepancies are detected among LII and LIIabsorp signals, so that auto-absorption contribution can be considered as negligible for this configuration as expected due to the low optical thickness of the flame.

Figure 2 :

 2 Figure 2: TR-LII images captured at six delay times t i and normalized by the signal I 0 at t 0 =0 ns.

Figure 3 :

 3 Figure 3: Fields of primary particle size distribution for 80% case. Left: mean diameter under monodisperse assumption for six decay times. Right: mean diameter and standard deviation for a log-normal polydisperse population based on delay times τ 34 and τ 36 .

Figure 4 :

 4 Figure 4: Bar graphs of d p along the centerline (top) and wing (bottom) at two HAB from TEM measurements for 80% case [9]. The solid lines are lognormal fittings to the sampled TEM data and the dashed lines are from the present TR-LII measurement.

Figure 5 :

 5 Figure 5: 2D fields of PPSD for four dilutions. Left: d p from Adelaide TR-LII measurements[6]. Middle: d p under monodisperse assumption (τ 13 for 32%, τ 34 for 40%, τ 35 for 60%, τ 36 for 80%). Right: d p and σ for a log-normal polydisperse population ( (τ 13 ,τ 23 ) for 32%, (τ 24 ,τ 34 ) for 40%, (τ 34 ,τ 35 ) for 60% and (τ 34 ,τ 36 ) for 80%) .