Nicolas Bouleau
email: bouleau@enpc.fr

Mathematization of risks and economic studies in global change modelling

Keywords: risk, tails of probability distribution, finance, interpretation, VaR, modelling I. The

With respect to the climate change, and more generally to the energy problem, in the laboratories working on the subject, scientists contribute to clarify the situation and to help decision makers by yielding and updating factual physical informations, and also by modelling. This conceptual work is mainly done in the language of economics. This discipline, which appears therefore in the core of the reflecting process in action at present, is however rather peculiar is the sense that it uses mathematics in order to think social phenomena.

It is on this methodological configuration that we hold a philosophical enquiry. Our analysis focuses on risks, incertitude and on the role of mathematics to represent them. It concludes on the importance of a certain type of modelization, investigation modelling, which reveals new significations.

This study attempts to enlighten some part of the limit between mathematized knowledge, as economics, and interpretative meaning used in every day life and social and human sciences

The probability distributions are badly known in the region where the probability is low, in particular in the neighborhood of infinity. Although well known today, the awareness of this important question has been progressive during the history of sciences and explains some present difficulties.

First of all, it has been remarked that asymptotically thick distributions possess strange unusual properties. The discovery of the fact that the law dx π(x 2 +1) (today called Cauchy law) has no expectation and has the stable property, hence doesn't fulfilled the central limit theorem, goes back to Siméon Denis Poisson around 1830, twenty years before Cauchy. Henri Poincaré in his probability course at the Sorbonne at the close end of the nineteenth century notes that this type of distribution contradicts the least squares method. Nevertheless during the twentieth century, with the development of mathematical economics, operational research and decision theory, the quadratic mean reasonings expand in a whole general rationality.

It may be sketched in the following manner: if, for a decision in uncertainty, you choose the value of a parameter so that the mean of the squares of the discrepancies between this quantity and the chosen value be minimal, then this kind of choice is compatible with other subsequent choices taken with the same method and with possible sub-choices done along the same principle. It is what mechanicists know when a solid body is shared into several parts, the gravity center of the gravity centers of the parts weighted with the masses of the parts, is still the gravity center of the whole body. This rationality impregnate economical reasonings of optimal decision as soon as they involve the notion of expectation or of conditional expectation which carry in themselves the idea of mean quadratic minimization.

A remark must be done here that the use of utility functions in order to escape from the concept of mathematical expectation, although yielding true services to represent the behaviour of the agents with their subjective probabilities (and avoid St Petersburg paradox), doesn't solve at all the problem of the tails of probability distributions because the utility function also is badly known on extreme events.

On one hand the reasonings in quadratic mean are iteratively coherent and develop in a whole rationality, on the other hand, procedures and arguments using thick probability tails build a language which tends to deteriorate, what leads necessarily to methodological and philosophical questions: given a family of probability distributions P α on R, each P α possessing an expectation, if the real parameter α is random with law ν(dα), it is possible that the resulting probability distribution

P = P α [ . ] ν(dα)
have no expectation. This is not related to the existence of an expectation for α, but comes only from the fact that an everywhere finite function may be non integrable. This has been known for long time and concrete decisional consequences have been emphasized by Mandelbrot in the 1970's (Mandelbrot 1970 and 1973) and more recently by Weitzman (2007) in climate change models about variances what is similar. In other words, the intersubjective rationality based on the least squares is fragile, when you leave it a little, no return force leads you back to it.

To these remarks, we must add specific features related to the identification of models by mean of statistical experiments 1 . Of course no histogram is able to yield the tails of the distributions and this happens also, in the multivariate case, for estimating correlation in any form of this notion (copulas, etc.). When the dimension is high, as usual in physical or economical models for global change, to the preceding difficulties comes in addition the famous phenomenon of the "curse of dimensionality" which makes identification much more fuzzy [START_REF] Artzner | Coherent measures of risk[END_REF] .

Let us mention also that the use of probability distributions on R or R + to evaluate probabilities of exceeding a threshold thanks to the extreme value theory is most often a doubtful operation because the extreme value theorem yields a convergence in distribution to a limit law which depends on the scale of the measurement device. Except for very particular physical cases where the quantities to be measured are clearly additive, the obtained thresholds have no intrinsic value and depends on the choice of a linear or logarithmic scale for instance [START_REF] Bruckner | Methodological Aspects of the Tolerable Windows Approach[END_REF] .

In order to tackle the methodological difficulties related to the tails of probability distributions, we will enlarge the problem and open new angles of attack.

II. Teachings drawn from the mathematical finance.

Finance developed since the 1970's several techniques of risk management, which are interesting examples both to understand their efficiency and to attempt to draw lessons of their weakness as recently revealed by the subprime crisis. The amounts staked on the markets are finite but not a priori bounded, and the problems of tails of distributions occur for all models, either models of options pricing and hedging, for interest rate models, or for assessing the exposure to risk of a financial company.

Slightly after the development of the organized derivative markets (the 1970's in the US, the 1980's in Europe and Japan), under the impulse of the bank JP Morgan, appeared a simple principle in order to evaluate the exposure of a financial organism: the "value at risk" denoted V aR represents the maximal loss that is not exceeded with the probability p (in general p = 95% or p = 99%) at a given date. The computation of V aR for an assets portfolio or for a bank presents several difficulties due especially to the restlessness of the spot prices (mark to market) and the discounting renormalization of term products. Practically, at present, in the banks and in most of the textbooks, V aR is computed with the help of explicit probability distributions calibrated on reality by usual statistical parametrical tests, hence hypotheses on tails of distributions are assumed. However, in its philosophy, the criterion V aR doesn't involve at all the tail of the distribution and its logic may be extended further by the following principle: to have a good knowledge of the parameters when they are in some domain and to content oneself with an upperbound of the probability of the exterior of the domain. We shall come back to this idea later on.

At present, the use of V aR is going to be supplanted by more sophisticated techniques.

These new methods appeared to avoid a default of the criterion V aR. The criterion V aR is not sudadditive in general. If V aR(X 1 ) and V aR(X 2 ) are the absolute values of the maximal losses on the portfolios X 1 and X 2 at some given date and with some given probability threshold, for the union of the two portfolios, where compensations may occur, we would expect to have V aR(X 1 + X 2 ) ≤ V ar(X 1 ) + V ar(X 2 ) but this inequality is not fulfilled in general [START_REF] Burdzy | The Search for Certainty[END_REF] .

In order to correct this default "coherent measures of risk" were proposed [START_REF] Corfee-Morlot | Climate change: long-term targets and shortterm commitments[END_REF] . It is possible to show that any risk criterion satisfying the common sense rules that are expected when two portfolios are gathered or when a portfolios is multiplied by a deterministic constant, (subadditivity

C(X 1 + X 2 ) ≤ C(X 1 ) + C(X 2 ); homogeneity C(λX 1 ) = λC(X 1 ); monotony C(X 1 ) ≤ C(X 2 ) if X 1 ≤ X 2 ; invariance by translation) are necessarily of the form m(L) = sup{E P [L] | P ∈ P} (1)
where P is a set of probability laws, E P the expectation symbol under the probability P and L the loss (i.e. the opposite of the value of the portfolio L = -X).

An interest of this result is to emphasize the importance of scenarios. The probability distributions which govern financial assets generally depend on the observer. The coherent measures of risk suggest to consider several probabilistic hypotheses to perform the risk analysis and to take eventually the most unfavorable estimate.

The mean loss above a threshold (expected shortfall), and all the coherent measures of risk, because of the formula (1), need the knowledge of the whole probability distribution of the loss, including the tail with its very rare events. They yield therefore numerical quantification criteria satisfying more harmonious logical principles but carrying the congenital default of throwing mathematical writings on a partially or totally unknown reality.

This leads us to deepen the reflection on the over-mathematization of the finance today, because this is the source of more general teachings. The recent crisis is the convergence of several causes which are not our purpose to examine exhaustively. But it reveals a fundamental point about risks and their mathematization. The above mentionned techniques of risk quantization favored the organization of markets of credit risks, that denotes the term of titrization. When insurance companies or banks have to manage risks in their country or region and if these risks are independent from a region to another or less correlated, they have advantage to exchange their risks. In order to install a market of risks it is necessary that sellers and buyers be able to quantify their own risks and they use for this the mathematization that we described above. The idea of organizing reinsurance as a market whose participant are insurance companies and where the product to be exchanged is the risk is an old "genious" idea studied by Maurice Allais and Kenneth Arrow in the 1950's. It applies as well to banks for the management of all their risks [START_REF] Delbaen | Coherent Risk Measures[END_REF] . In Arrow's model, each participant has his own subjective probabilities. This extend to the case where everyone looks at several scenarios to evaluate his risk. Since agents do not have the same measure of risk because they dont interpret the underlying reality to each asset in the same way and because they have different risk aversion and utility functions hence different behaviors in uncertainty, exchanges of risks against payment may be profitable for buyers and sellers. Prices may be computed thanks to mathematical models.

On such bases, credit risk markets have developed especially in the US these last twenty years.

For the management of completely probabilized risks (known probability distribution and known function of damages) the interest of risk markets are clear. It amounts not to put all one's eggs in one basket. This would be the case for instance for the material damages of traffic car crashes. But most of risks possess a purely semantic and interpretative component which is based on a reading of economy and which remains in every case concerning the tails of distributions. For such risks, to place one's confidence in the common opinion -as done by putting the risks in market -is a logical mistake, since most of participants go by a ready-made mathematization drawn from notation agencies, and therefore do not bring anything to the lack of understanting of anyone. In other words, putting the risks in market belongs to a rationality only for completely calibrated and perfectly statistically known situations.

The probabilistic representation of risk (with objective probabilities or subjective probabilities from the points of view of the agents) is classically a pair of mathematical quantities, 1 o a probability distribution which governs the state of nature that may happen, 2 o a function which to each state gives a damage or a benefit. For instance the height of the river Loire at Orléans follows some probability distribution, and to any given height may be (theoretically) associated the cost of the possible flood.

However this representation by a pair of mathematical quantities is a too simple and too ideal scheme of thinking.

-Too ideal because we are almost never in a situation where this model is well filled in. We dont know the tails of the distributions because the data are lacking on rare events. We ignore the correlations in order to estimate the damages and no complete description of what may happen is available.

-But, above all, the model is simplistic because it is hiding the reasons why we are interested to these events by doing as if they could be translated automatically and objectively into costs.

The true goal of risk analysis is not to do computations but to progress with a little more clear-sightedness in social facts and uses. It may be about the threat a child be run over by a car when crossing the street, about the risk that the air of London poisons the inhabitants, or that the bankruptcy of an firm induces other bankruptcies, etc.

The intellectual operation consisting in probabilizing a situation is fundamentally an ousting, an erasing of the meaning. In the particular case of physics, it is the result of the construction process of the knowledge on the laws of matter. But it is widely problematic for all what concerns the human behavior. The risk analysis is necessarily understanding of interpretations.

Computations, even with methodological precautions, have the innate defect to hide ignorance. As we dont know precisely to quantify neither the return risk of a loan, nor the market or liquidity risk, nor the risks due to human mistakes or due to a regulation change, very accurate computations are mixed with rough estimates hopping they will have no sensitive consequences on the result. During the manufacturing of packages of several credits, a standardization of the description occurs (reduced to the notations of the agencies) which moves away from reality. At the limit, a mortgage loan in Iowa or Kansas is thought in the same manner as a credit at New-York on Madison avenue if they are both well assessed.

As the mathematical theory if far in advance before the available data, sophisticated methods are applied to rough hypotheses whose coarseness is forgotten along the computation. The bankruptcies of the banks or firms are represented by Poisson processes whose parameter, the mean flow, is in fact unknown. It is a characteristic of coherent measures of risk to involve the whole tail of the probability distribution. But the tails are never known. Hence their use implies necessarily tacking a mathematical model on an ignored reality.

It is the meaning of an event that makes the risk. Let us take an example, suppose a particular type of cancer be enumerated within the population. This subset of the suffering people is the concrete event. The ratio to the whole population will be taken for an estimate of the risk. If it would happen that observation and inquiry reveal that a significant part of the concerned persons have taken -say cannabis -twenty years before, then all the cannabis consumers become potential patient. If another common property of the sick persons is pointed out -e.g. they have used mobile phones as soon as these were available -then almost the whole population is now threatened. The risk changes when the interpretation modifies the meaning of the event.

Reducing the risk to a probability distribution and a money amount rubs out the main part of the difficulties. It is to attach confidence to the mathematization as approximation, as if the physical reality was concerned, but the question deals with meaning whose subjectivity impregnates all the social relations of the agents. We are not in an approximation process, as usual in applied mathematics with discretization or finite elements methods. It is interpretation, hence meaning which is transformed into numbers.

We said that significant improvements have recently been brought to these procedures thanks to the coherent risk measures. But all these methods have the inborn inadequacy of considering that the interpretating process is closed. Now on the contrary, instead of being closed, it is in permanent innovation. As soon as a new reading appears, it generates new risks which are only perceived by those ones who understand it. If in 2006, nobody sees the increase of the real estate prices and the decrease of the households savings in the US as a phenomenon allowing several readings, the corresponding risk is not detected. The mathematization sweeps up these difficulties thanks to hypotheses on the tails of the distributions. It is not enough to say that they are badly known. They are, by nature, temporary and fluctuating depending on the interpretative knowledge that the agents acquire by their understanding of economical phenomena.

This limitation of the relevant mathematization holds as well for the other risks due to scientific and technical innovations on which focused Ulrich Beck as soon as the 1980's.

In fact any advance in the knowledge makes us discover features of the world on which we were not aware before and, by this, generate new risks. Whatever way is taken for mathematizing the risks, they are congealed, canned in a box which hides the interpretative. It is compulsory, the weakness of mathematics is that they are formal ! III. Can we transpose these arguments to the economical models of global change ?

Uncertainties are so numerous about CO 2 future concentration that IPCC, in his methodological recommendations to contributers, looks for a clear and well calibrated language to speak on vague notions. Is it the right way? As things are taken, it is doubtful [START_REF]Guidance Notes for Lead Authors on the IPCC Fourth Assessment Report on Adressing Uncertainties[END_REF] .

First of all, we can think that it is possible to avoid any assumption on the tails of the distributions, as with the criterion V aR, by reasonning inside a domain with explicit boundary.

In a modelling framework, the frontiers are defined by specifying a domain to all the quantities: to the data, to the parameters, and, thanks to the model to the computed outputs. It is possible to write down a methodology BaR (Boundary at Risk) which extends V aR, and consists for each quantity, say X, to define a domain D and a threshold [START_REF]Guidance Notes for Lead Authors on the IPCC Fourth Assessment Report on Adressing Uncertainties[END_REF] It is suggested 15 steps in order to take in account the desirable precautions from the beginning of the work until its final expression, evaluating the doubts, the ignorances, the uncertainties and the degrees of confidence which may be attached to each part of the discourse. Three categories of uncertainty are distinguished: unpredictability, structural uncertainty, uncertainty on the numerical values. This philosophy adopts the rule of claiming as few as possible and using mood auxiliaries watering down any remarkable insight in order all what is interpretative desappears. For instance the degrees of confidence are splitted in five levels: Quantitatively calibrated levels of confidence

Terminology

Degree of confidence in being correct Very High confidence At least 9 about 10 chance of being correct High confidence About 8 out of 10 chance Medium confidence About 5 out of 10 chance Low confidence About 2 out of 10 chance Very low confidence Less than 1 out of 10 chance This rather curious table shows how the care of well doing may leads to obscure nomenclatures: how is it possible to have a low degree of confidence in a model which has 2 out of 10 chances of being correct? With 5 chances on 10 already, the model doesn't yield information any longer, lower it brings a thesis about what will not happen. We see that the two discourses of the expert and of the model are confused. The calibration proposed by the IPCC is a false rigour. It simply reduces to carefully evade any formal criticism. This point has been objected in the CIRED: "have we not to consider that a model correct one times over ten yields an information? If it rains 1 day over 2, and if some model, correct one times on 10 predicts the rain, we can say that the probability of the rain is greater than 1/2." This objection shows the difficulty to keep clear ideas even among scientists about the meaning of words. For myself I argue in the following way: if the model is correct 1 times on 10, it is wrong 9 times on 10, then if it says today "rain", there are 9 chances on 10 that the sun shines. In the objection the expression "to be correct one times over ten" is taken as meaning "the model has a correlation greater than 1/10 with the reality".

of probability p such that P {X / ∈ D} ≤ p. This way of thinking possesses the natural following coherence: if several situations are considered depending on a parameter α and if

P α {X / ∈ D 1 } ≤ p
for α in the domain D 2 , if α becomes random and if the probability that α be outside D 2 is less than q, then for the resulting probability one has evidently:

P {α / ∈ D 2 , X / ∈ D 1 } ≤ p + q.
The language BaR allows to improve a model by taking in account new aleas provided that the new randomized parameters be managed in the same way. then we know the joint law of the pair (X, Y ) given the event {X ∈ D 1 } and {Y ∈ D 2 }. Despite several interesting works related to this way of reasonning (cf. Petschel-Held et al. 1999, Bruckner et al. 2003, Corfee-Morlot et al. 2003), this approach presents necessarily the counterpart of its advantages : nothing is said in case of overcrossing the boundary. In fact, with a suitable mathematical formalism, BaR may be seen as a special case of VaR, so that it may be subject to the same citicism as VaR in particular to give no information on what happens when the boundary is crossed. But the ultimate goal of the models is to favor the emergence of an intersubjective truth in order that the parties, inside their interest conflict, may begin a constructive dialog. The philosophy of Boundary at Risk can only be relevant if everyone hopes to remain in the studied domain and acts for this at any cost. But the action of economic forces makes instead that everyone is oriented toward a progress of his energy consumption what leads straitforward to the boundary. In other words communication rules are of the first importance, as is well known in matter of climate change, and this imposes that extreme events occuring behind the boundary be described and discussed.

The logic of scenarios suggested by the "coherent measures of risk" is much more promising. We find here an approach which was already in practice among civil engineers. Since it is actually impossible to explore all the branches of the tree of consequences leading to the ruin of the structure, some a priori damagings are considered and the probability of the scenarios leading to such damages is estimated. This logic of scenarios penetrates more and more in every field of the futurology, and especialy in the studies of the relations between economic decisions and climate modifications.

In the case of the damaging of a civil engineering structure, for security reasons the engineer will choose the unfavorable envelope of all the considered scenarios. It is also what bankers do for estimating the exposure of their company.

But in the case of economical models for the global change, it is rather the specific appearence and the qualitative description of the consequences of each scenario which is liable to interest the decision makers.

Of course these approaches have the defect which is the counterpart of their advantages: they use tails of distributions which are badly known and hence may be criticized. In addition, we must keep in mind that the main defect of the methods using tails of distribution is not so much to propose an imperfect representation of the risks but rather to let believe that the interpretative process is closed. But continuously the agent think to improve their knowledge and forecastings, and for this re-interpret all the information they capture. This is well illustrated by the language used by the IPCC. If we read today the first report of 1990, the supplement of 1992 and the second report approved in December 1995, we are necessarily amazed by the fact that the meaning of the tails of distribution changed. They reflected a high ignorance on physical phenomena so that a lot of accelerating effects were unsuspected. They reflect today much more the doubts about the behavior and the political decisions.

Sometimes a distinction is done between the alea related to the (natural and social) reality and that one related to our own ignorance which may be more or less "probabilizable". The point is that generally the statistical methods do not allow to share clearly between these two categories which in practice overlap. But is that really the problem? In any way it is an ignorance about the plausibility of a phenomenon, the main point is that we are interested by this phenomenon because it means something.

If we consider a very elaborate model of the type of IMAGE for instance and that a lighter simulation is proposed which reveals new phenomena like the permafrost fermentation, or the lubrification of the rock by the water under the glaciers, etc., all the preceding formalization of the risk becomes lapsed because new significations appear. A modelization is always an interpretation [START_REF] Mandelbrot | Long run dependence in price records and other economic time series[END_REF] . Already for natural phenomena it is impossible to be exhaustive and new effects are always possible, but this is yet much more relevant in the case of modelization of economic phenomena. The more physical phenomena of climate change are better known the more we are closer to a similar situation to finance where it is the interpretations and anticipations of the actors that we have to put into the risks. The hope of tending to a "reality" thanks to a finer and finer mathematization is an illusion because in mathematizing the economic risks one attempt truly to modelize all the possible interpretations of a situation.

The investigation modelling has the ambition to reveal new significations.

In some sense, it is with respect to heavy mathematization what sociological enquiry is with respect to statistical poll. Let us explicit this comparison. Econometricians often criticize sociologists that they content themselves of polling about fifty persons, an unsufficient sample of the population to deduce anything. On the contrary they construct polls, as for them, statistically exploiting thousand or more questionnaires. But the sociologist doesn't proceed with questionnaires, he lets people talk. What he is looking for, when he investigates on daily transportation or on the use of water by the inhabitants for instance, it is explanations to which he wouldn't have thought a priori, significant reasons which enlighten some behaviors and which he will use later on to deepen the subject. The questionnaire encapsulates thought categories. That yields numbers, ratios, but it is closed. It is not with its help that what is not yet imagined will be discovered.

Therefore investigation modelling possesses, from this point of view, a fundamental justification. It is not a clumsy sketch of truly scientific models which run gigantic black boxes. It has an essential epistemological role, that of regenerating the risks signification. For that it can avoid the heavy, accurate and opaque quantitative which has the innate flaw of a hight intellectual inertia, it can remain light, movable following the aim of making significant phenomena appear.

  Therefore this language doesn't have the defects explicited by Weitzman 2007. More generally, it is easy to see that if two random quantities are considered X and Y and if N (y, dx) is the conditional law of X given Y = y. If we know i) the measure N (y, dx)1 D 1 (x) when y is in D 2 ii) the law of Y given the event Y ∈ D 2 i.e. the probability measure

	1 D 2 (y)P Y (dy)
	P Y (D 2 )

Cf. K. Burdzy

for different philosophies of probability theory and the question of the use of statistical estimate for predictions.

Actually, from a pragmatic point of view, in concrete cases where the tail is unknown, even the total mass of the distribution may be indeterminate. Our measurement devices are not always designed to distinguish accurately between very large and infinite quantities. In other words, it may happen that we hesitate about the possible presence of a mass at infinity. This might be written as usual in the theory of Markov processes when dealing with sub-Markov kernels.

For other pecularities of this theorem see N. Bouleau "Splendeurs et misères des lois de valeurs extrêmes." RISQUES 3, p 85-92 (1991).

Cf. e.g. R. Portait and P. Poncet, Finance de marché, Dalloz 2008, p. 901 et seq.

Cf.[START_REF] Artzner | Thinking coherently[END_REF] 1999;[START_REF] Delbaen | Coherent Risk Measures[END_REF]

Cf. K. Borch "Equilibrium in a Reinsurance Market" Econometrica vol 30, n3, July 1962; and R. Portait and P. Poncet op. cit. p. 931 et seq.

Cf. N. Bouleau Philosophies des mathématiques et de la modélisation, L'Harmattan 1999.


Gustav Nyrenstedt

Romain Grosseuvres

Andrea Comandini

Alexandre Bleyer

Ahmed Bentaib

Nabiha Chaumeix
email: chaumeix@cnrs-orleans.fr

COMBUSTION REGIMES OF HYDROGEN-AIR-STEAM MIXTURES

In the case of a severe nuclear power plant accident, hydrogen gas formation may occur, from the core degradation, and cooling water evaporation and subsequent oxidation of zircaloy. These phenomena increase the risk of hazardous combustion events in the reactor, especially when combined with an ignition source. If not handled carefully, these types of accidents can cause severe damage to the reactor building with potential radioactive effects on the environment. Although hydrogen-air combustion has been investigated before, hydrogen-air-steam mixtures remain unstudied under reactor-like conditions. Thus, this study investigated such mixtures' combustion regimes. A closed tube of 318 liters (7.65m tall and 0.23m inner diameter) measures the flame speed, flame propagation and shock wave behaviors for 11-15 %vol hydrogen mixtures combined with 0, 20 or 30 %vol steam, and air. Thus, both the effect of steam and hydrogen content was investigated and compared. The experimental setup combined photomultiplier tubes, pressure sensors, and shock detectors to give a full view of the different combustion regimes. A number of obstacles changed the in-chamber turbulence, during flame propagation, to provide further reactor-like environments. This changed turbulence affected the combustion regimes and enhanced the flame speed for some cases. The results showed varying combustion behaviors depending on the water vapor concentration, where a higher concentration meant a lower flame speed, reduced pressure load, and sometimes combustion extinction. At 0 %vol steam dilution, the flame speed remained supersonic for all H2 concentrations, while at 30 %vol steam dilution, the flame speed remained subsonic for all H2 concentrations. Thus, with high levels of steam dilution, the risk for shock waves, leading to potential reactor building destruction, decreases.

INTRODUCTION

Nuclear reactor electricity generation presents one of the cleanest alternatives when it comes to lifecycle CO2 emissions [START_REF] Van Der Zwaan | The role of nuclear power in mitigating emissions from electricity generation[END_REF][START_REF] Kharecha | Prevented mortality and greenhouse gas emissions from historical and projected nuclear power[END_REF][START_REF] Siddiqui | Comparative assessment of the environmental impacts of nuclear, wind and hydro-electric power plants in Ontario: a life cycle assessment[END_REF]. However, the safety concerns for potential accidents remain, especially after the Fukushima accident [START_REF] Visschers | How a nuclear power plant accident influences acceptance of nuclear power: Results of a longitudinal study before and after the Fukushima disaster[END_REF]. This accident involved hydrogen (H2) explosions, causing structural damage, leading to the release of radioactive material to the surroundings [5].

Light water nuclear reactors can in general form H2 through for example reactions between the zircaloy fuel-coating and the surrounding steam [START_REF] Suman | Hydrogen in Zircaloy: Mechanism and its impacts[END_REF][START_REF] Courty | Modeling and simulation of hydrogen behavior in Zircaloy-4 fuel cladding[END_REF]. This H2-generation occurs during reactor overheating since the steam generation and reaction initiation require certain temperature levels. To avoid this overheating, occurring during cooling system failure, several nuclear power plants (NPP) utilize pressure release valves allowing H2 -and steam flow to the reactor containment building.

To avoid NPP H2-ignition, with risk for detonation [START_REF] Yanez | An analysis of the hydrogen explosion in the Fukushima-Daiichi accident[END_REF][START_REF] Nishimura | Current research and development activities on fission products and hydrogen risk after the accident at Fukushima Daiichi nuclear power station[END_REF], a thorough understanding of combustion behavior at NPP-like conditions is needed. Fundamental studies [START_REF] Liu | Influence of initial pressure and temperature on flammability limits of hydrogen-air[END_REF][START_REF] Grosseuvres | Combustion properties of H2/N2/O2/steam mixtures[END_REF][START_REF] Kuznetsov | Flammability limits and laminar flame speed of hydrogen-air mixtures at sub-atmospheric pressures[END_REF] concluded a H2 lean flammability limit at 4 %vol when mixed with air at atmospheric pressure and temperature. However, the laminar flame speed remains low at low equivalence ratios (ER) [START_REF] Grosseuvres | Combustion properties of H2/N2/O2/steam mixtures[END_REF][START_REF] Ciccarelli | Fast-flame limit for hydrogen/methane-air mixtures[END_REF] suggesting a low detonation probability. For the NPP, the transition limit from non-hazardous to hazardous conditions must be known.

To understand this transition limit, different CFD-simulations [START_REF] Bentaib | Overview on hydrogen risk research and development activities: methodology and open issues[END_REF][START_REF] Nicolás-Pérez | Evaluation of different models for turbulent combustion of hydrogen-air mixtures. Large Eddy Simulation of a LOVA sequence with hydrogen deflagration in ITER Vacuum Vessel[END_REF] predict the combustion regimes and flame propagation inside the NPP containment building. The mere building size represents a computational challenge leading to large mesh sizes [START_REF] Bocanegra | Fukushima Unit 2 Accident Simulation With MELCOR 2.1[END_REF]. Thus, the mesh sizes needed for accurately predicting detonation waves cannot be reached. Instead, different assumptions are made which need to be validated against experimental data.

Previous studies aimed to provide this validation data [START_REF] Grosseuvres | Effect of initial temperature and temperature gradient on H2/Air flame propagation in confined area[END_REF] for varying H2 concentrations and temperatures. These studies included turbulence created from different obstacle configurations to further mimic the NPP environment. A specific vertical combustion vessel, aiming to mimic the NPP, performed the experiments showing little or no risk for detonation at H2 concentrations below 11 %vol. However, this behavior changes with diluent addition [START_REF] Sathiah | The role of CFD combustion modeling in hydrogen safety management-III: Validation based on homogeneous hydrogen-air-diluent experiments[END_REF].

The present study investigates (in the framework of the MITHYGENE project) how steam, potentially released together with the H2 in a NPP accident, affects the combustion regimes with respect to diluting the mixtures. A vertical, bottom-ignited vessel [START_REF] Grosseuvres | Effect of initial temperature and temperature gradient on H2/Air flame propagation in confined area[END_REF] evaluated H2-concentrations, ranging from 10 to 15 %vol, mixed with air and steam (0, 20 or 30 %vol). This work provides new insights on how steam improves or deteriorates the detonation risk during a nuclear reactor accident.

METHODOLOGY

This section summarizes and describes the different experimental and post-processing methods used for this article.

Experimental Setup

A 7.65m tall vertical combustion chamber (ENACCEF2) [START_REF] Grosseuvres | Effect of initial temperature and temperature gradient on H2/Air flame propagation in confined area[END_REF][START_REF] Bentaïb | ETSON-MITHYGENE benchmark on simulations of upward flame propagation experiment in the ENACCEF2 experimental facility[END_REF] performed the H2/air/steam experiments. ENACCEF2 consists of nine different sections forming a cylindrical tube (230 mm internal diameter), where each section includes three photomultiplier tubes (PMs) placed with a 150 mm height difference. These PMs detect light around a 306 nm wavelength via a 5 mm diameter optical passage through the ENACEFF2 wall. The small optical passage diameter allows flame detection via a thin horizontal sheet (using a plano-convex lens of 100 mm focal length and a slit of 1 mm thickness). Each section also contains a pressure sensor. The two lowest sections, usually experiencing subsonic combustion, wield PCB 113B03 sensors suitable for that regime. The other sections use Kistler 601CA sensors to better account for the possible supersonic combustion here. The top section also includes a Kistler sensor at the end wall. A silicon coating protects all pressure sensors from hot gases ensuring measurement accuracy. To further follow the shock waves, the top sections used five shock detectors.

In ENACCEF2, ignition occurs at the bottom flange, by a spark plug, inducing an upwards flame propagation. This study also included nine annular obstacles at the bottom sections (from a height of 0.638m up to 2.478m). All obstacles used a blockage ratio of 0.63, defined as follows:

𝐵𝑅 = 1 -(𝑑/𝐷) 2 (1)
Where d is the inner obstacle diameter [m] and D the ENACCEF2 interior tube diameter [m]. The obstacles aim to mimic the possible turbulence created from real obstacles in a nuclear reactor containment building. To further mimic real conditions, an electrical resistance heating system accompanies each section. Several temperature sensors regulate this heating system to ensure a close to constant temperature along the tube. H2 is a light species and the constant temperature ensures minimum in-vessel mixture stratification.

All experiments used H2-fuel with air as oxidizer, while steam diluted some experiments. The gas injection system included species-specific Bronkhorst, F-201CV mass flowmeters to ensure a correct species concentration. The water is first supplied to a heated mixing tank where the liquid transforms to steam. After the mixing tank, three injection lines (at different heights) supplied the ENACCEF2 with the final gas mix. This approach minimized concentration gradients. Furthermore, a micro-GC (SRA Instruments) analysis verified the mixture compositions and homogeneity by sampling the gas composition at different heights.

Experimental Procedures and Conditions

All experiments followed a number of steps as: chamber filling up to 1 bar by supplying the gases as described above; gas sampling at four different heights coupled with micro-GC analyses to exclude concentration gradients and verify the mixture composition; spark plug triggering with following combustion initiation; binary data collection via a Hioki MR8827 and a Hioki MR8847-03; and data conversion (from binary to text format) with following post-processing. This data collection -and conversion approach allowed large file sizes and small time-steps at a roughly ten times faster rate compared to previous studies. Every experimental condition was repeated at least three times to ensure consistency.

This article focuses on the effect of steam dilution for H2/air mixtures at NPP-relevant conditions. Tab. 1 summarizes these conditions where three H2 concentrations (ER = 0.294, 0.356 and 0.42) and three steam concentrations (0, 20 and 30 %) alternated to create a total of nine experimental conditions. All cases utilized an initial temperature of 140 °C, and an initial pressure of 1 bar inside the ENACCEF2 tube. The initial temperature aimed to mimic NPP conditions during an accident and ensured that the water be present as vapor.

Data Analysis and Calculations

Several analyses -and calculation steps followed after the binary data conversion. A pre-flame detection PM signal analysis set the initial signal level at zero. Fig. 1 shows a typical PM voltage signal from flame detection, and highlights the maximum slope of the first peak. This slope calculates the flame speed by indicating the flame detection time. This article defines the flame detection time as the time when the maximum slope line cuts the horizontal zero-axis (around 0.6 s from spark initiation in Fig. 1). Such a definition ensures that the flame detection time reflects the actual flame passing and avoids light-pollution from a nearby flame (that did not yet reach the PM) [START_REF] Mathieu | Experimental study of the effect of CF3I addition on the ignition delay time and laminar flame speed of methane, ethylene, and propane[END_REF]. The velocity calculations then follow from knowing both the flame detection time and the PM placement height as follows:

𝒗 𝒊 = (𝒉) 𝒊+𝟏 -(𝒉) 𝒊-𝟏 (𝒕) 𝒊+𝟏 -(𝒕) 𝒊-𝟏 (2)
Where v denotes the flame speed [m/s], i the PM number, h the PM placement height [m from the spark plug], and t the flame detection time [seconds from spark initiation].

The pressure sensor signals rendered knowledge both on pressure rise (from combustion) and shock wave behavior. All pressure signals were converted to bar via earlier performed calibrations at relevant conditions. Fig. 2 shows the pressure traces for an example case, from five out of ten sensors. The leading shock (red dashed line) gives a rapid pressure rise when passing a sensor and can thus be easily detected. The shock wave then reflects at the ENACCEF2 top flange and travels back downwards. This reflected shock (blue dashed line in Fig. 2) follows the leading shock behavior with a following pressure jump. The shock wave time calculation differs from the PM signal flame detection.

Instead of the maximum slope method, the shock wave time follows from pinpointing the start of rapid pressure rise. The shock wave speed follows from the shock wave trajectory, using a polynomial fit.

Fundamental Flame Parameters and Simulations

To classify the experimental results, this study investigates several flame parameters. Dorofeev et al. [START_REF] Dorofeev | Evaluation of limits for effective flame acceleration in hydrogen mixtures[END_REF] indicated a relationship for determining the experimental geometry scale dependence. Eq. 3 highlights the ratio between the integral length scale of turbulence (LT) and the flame thickness (δ). When this ratio exceeds 100, the experiment is considered independent of the geometry scale. Here, LT is equivalent to the inner diameter of the ENACCEF2 facility, which equals 230 mm. The Zeldovich number (β) correlates to a mixture's reactivity. Eq. 4 shows one way to calculate β, from the activation energy (Ea), the universal gas constant (R), the flame temperature (Tb), and the initial temperature (Tu). The Ea follows from the correlation described in Eq. 5, where SL denotes the laminar flame speed.

𝐿 𝑇 𝛿 > 100 (3)
𝛽 = 𝐸 𝑎 𝑅𝑇 𝑏 2 (𝑇 𝑏 -𝑇 𝑢 ) (4)
2 * ln(𝑆 𝐿 ) ∝ 𝐸 𝑎 𝑅𝑇 𝑏 (5) The expansion ratio (σ) follows from Eq. 6, where ρu and ρb denote the unburnt and burnt gas densities. Dorofeev et al. [START_REF] Dorofeev | Evaluation of limits for effective flame acceleration in hydrogen mixtures[END_REF] also showed how a critical expansion ratio [σ * ] can be used to determine the fast flame regimes. Ciccarelli et al. [START_REF] Ciccarelli | Fast-flame limit for hydrogen/methane-air mixtures[END_REF] later established a polynomial fit to known experimental compositions (see Eq. 7). If the σ-value remains above σ * , the flame is considered fast. Eq. 8 reveals another fast-flame definition. A ratio, between the maximum flame speed [vmax] and the speed of sound in the burnt gases (cBG), above 0.5 indicates a fast flame [START_REF] Dorofeev | Evaluation of limits for effective flame acceleration in hydrogen mixtures[END_REF].

𝜎 = 𝜌 𝑢 𝜌 𝑏 (6) 𝜎 * = -0.00285𝛽 2 + 0.3823𝛽 -2.2078 (7)
𝑣 𝑚𝑎𝑥 𝑐 𝐵𝐺 > 0.5

Equilibrium and flame simulations in COSILAB [START_REF] Cosilab | [END_REF] calculated the flame parameters used here. The Mével mechanism [START_REF] Mével | Hydrogennitrous oxide delay times: Shock tube experimental study and kinetic modelling[END_REF] was used for all conditions to ensure consistency. For these simulations, the laminar flame speed indicated no flame propagation with 20 -or 30 % steam-addition. Thus, an exponential fit (based on the data points without steam) estimated the laminar flame speed for the steamconditions here. The β-determination followed a procedure where the unmeasured data points were decided via a polynomial fit. In general, all flame parameter calculations followed a previous study [START_REF] Grosseuvres | Combustion properties of H2/N2/O2/steam mixtures[END_REF]

RESULTS

This section evaluates and describes the different phenomena experienced and includes an analysis for case-to-case difference, flame speed calculations and shock wave analyses.

Data Reproducibility

Every experimental condition includes at least three separate runs to ensure accuracy. Fig. 3 compares the three runs at 20 %vol steam and ER0.356 to highlight the run-to-run differences. The detection height indicates the spark plug-to-flame distance as detected by the PMs. Fig. 3a shows clear trends, confirmed by the different runs, although small local differences exist. The first speed-peak (just before 3m) and the second speed-peak (around 4.5m) occur for all three runs while the amplitude differs slightly at high flame speeds. Due to the fast flame, Fig. 3b shows only minor time differences between the three runs. This behaviour suggests good experimental accuracy, especially for high flame speeds.

Some experimental conditions suffer from larger case-to-case differences. Fig. 4 highlights the 30 %vol steam and ER0.356 case where the flame quenches for all runs inside the obstacle-region. However, reignition occurs only for two of three runs (see Fig. 4). These two runs (362 and 364) experience the reignition after the obstacle-region where the flame travels in several directions. Fig. 4a indicates a negative flame speed following from the flame traveling downwards. However, Fig. 4a also indicates an upwards traveling flame (positive speed) meaning a three-dimensional re-ignited flame propagation. The later data averaging procedure ignores the outlier runs (with largely differing results).

When occurring, GC analyses (by post-experimental gas sampling) confirmed the flame extinction by confirming no H2 consumption above the extinction height. This procedure ensured that extinction actually occurred and ruled out non-detected re-ignitions, occurring after the signal collection end time.

H2/Air/Steam Combustion Regimes

This section describes the effects of ER and steam dilution on H2/air mixtures. The results presented were obtained by averaging over experiments conducted at similar initial conditions. For any given ER, an increased steam dilution decreases the flame speed (see Fig. 5a). At the low ER = 0.294, this phenomenon becomes even more apparent. Fig. 5a depicts how the maximum flame speed falls by around 400 m/s when moving from 0 to 20 %vol steam dilution. When increasing the steam dilution further, to 30 %vol, the maximum flame speed again falls slightly and occurs later.

The higher ER cases (0.356 and 0.42) require more steam dilution to obtain the low flame speed encountered for the ER=0.294 case. Fig. 5a shows how the maximum flame speed remains high at 20 %vol steam dilution, for the ER=0.356 and ER=0.42 cases. The maximum flame speed, around 400 m/s, remains close to supersonic conditions (speed of sound = 448 m/s in the fresh gases at ER=0.42 and 20 %vol steam dilution) with the following risk of shock wave creation. This risk disappears at 30 %vol steam dilution, independent of ER. Fig. 5b concludes that the maximum flame speed remains below 200 m/s for all H2 concentrations. In the NPP, a 30 %vol steam dilution thus corresponds to a lower risk of containment building damage.

The H2 concentration matters only to a small extent at 0 or 30 %vol steam dilution. Fig. 5b shows how the flame speed remains high, independent of H2 concentration, at 0 %vol steam dilution. Simultaneously, the flame speed remains low for 30 %vol steam dilution. However, the H2 concentration changes the flame speed drastically at 20 %vol steam dilution. As described above, the maximum flame speeds remain close to the supersonic region for the ER0.356 and ER0.42 cases. Thus, in the case of an NPP accident, the experienced H2 concentration becomes increasingly important if a 20 %vol steam dilution is present.

As discussed above, shock waves can occur at flame speeds above the speed of sound. This study found a borderline between the ER=0.356 and ER=0.42 cases at 20 %vol steam dilution. Fig. 6a indicates no shock wave behaviors at ER=0.356 while a clear shock wave follows from the ER=0.42 case (see Fig. 6b). This borderline corresponds to the earlier observed flame speeds, where the ER=0.356 maximum flame speed falls below 400 m/s for a 20 %vol steam dilution.

Fig. 6b further shows how the shock wave intensity (in terms of maximum pressure) reduces as it passes the obstacle region (for PCB2 and PCB5). This is probably caused by the turbulence created here. In general, the leading shock wave speed (see Fig. 7) decreased as a function of time due to fluid mechanical effects, such as friction. Fig. 7 further correlates how an increased ER elevates the leading shock wave speed. An ER increase from 0.294 to 0.42 gives a roughly 100 m/s higher shock wave speed here.

The steam addition affects the shock wave speed only to a small extent (see Fig. 7). The initial shock wave speed decreases slightly while the speed reduction during propagation becomes smaller. Thus, the top of ENACCEF2 experiences a higher shock wave speed for 20 %vol steam dilution compared to 0 %vol (at ER=0.42). However, the speed of sound increases with steam dilution which affects the Mach number. In the NPP, the shock wave speed influences the severity of a potential ignition. Here, if the shock wave occurs, the steam addition seams to have only a small effect.

Fundamental Flame Parameters

This section describes and determines fast flame regimes for the evaluated conditions. The ratio between LT and δ falls well above 1000 for all experimental cases here. Thus, the evaluated conditions remain independent of scale, according to Eq. 2. A third fast-flame criterion will be considered here. Dorofeev et al. [START_REF] Dorofeev | Evaluation of limits for effective flame acceleration in hydrogen mixtures[END_REF] provided a sigma-criterion similar to the one later established by Cicarelli et al. Figure 10 compares the fast -and slow flames, as established by Eq. 6, with Dorofeev's criterion. Here, only three experimental points experience fast flames according to the Dorofeev-criterion. This opposes the five fast flame conditions by Eq. 6, and the seven fast flame conditions by the Cicarelli-criterion. In addition, one of the data points lies on the limitline in Figure 10, indicating difficulties to determine if it is a fast flame or not with the Dorofeevcriterion. This section concludes that neither the Cicarelli -nor the Dorofeev criteria suit these experimental conditions for determining fast-flame regimes. Thus, more experiments with H2/air/steam mixtures are needed to establish relevant sigma-criteria.

Figure 10. The present study's expansion ratios normalized by the critical expansion ratios as decided by the polynomial fits from Dorofeev et al. [START_REF] Dorofeev | Evaluation of limits for effective flame acceleration in hydrogen mixtures[END_REF], plotted against the Zeldovich number. The blue dashed line indicates the fast flame limit as decided by Dorofeev's sigma-criterion

CONCLUSIONS

This study investigated combustion behaviors in a vertical vessel, designed to mimic nuclear power plant conditions. This facility enabled detailed flame acceleration studies of fuel/air/diluent mixtures in a confined environment, with a variety of diagnostics to characterize the flame propagation history and the subsequent pressure loads. The detailed data output from this study validate future CFD codes for nuclear safety applications. In this work, H2/air mixtures diluted with steam were investigated in terms of flame speed and shock waves to evaluate the potential effects of a combustion event in a nuclear power plant. Equivalence ratios of 0.294, 0.356 and 0.42 investigated the steam addition effects at 0, 20 -and 30 %vol. The main conclusions follow as:

• Steam dilution affects the combustion behavior largely where a 30 %vol steam dilution completely prevents the creation of shock waves at the tested equivalence ratios. • The H2 concentration gives little or no effect on the combustion behavior at 0 or 30 %vol steam dilution. At 0 %vol, the flame speed remains supersonic while at 30 %vol, the flame speed remains subsonic. • Steam dilution prevents shock wave creation to a large extent due to the reduced flame speed.

• If shock waves occur, the steam dilution has little or no effect on the leading shock wave speed.

• Further experiments are needed to establish expansion ratio correlations for fast-flame characterization

Figure 1 .

 1 Figure 1. Example PM signal, as a function of time from spark initiation, illustrating the maximum slope (red line) used for flame speed calculations

Figure 2 .

 2 Figure 2. Pressure sensor data [bar] for an example case to illustrate the leading shock (red dashed line) and reflected shock (blue dashed line). This shock wave detection later calculates the shock speed. K indicates the pressure sensor type (Kistler) while the numbers 17, 20, 23 etc. indicate an ascending sensor placement height. KEW denotes the end-wall Kistler sensor.

Figure 3 .Figure 4 .

 34 Figure 3. Flame speed [m/s], vs PM flame detection height (a) and PM flame detection time (b), comparisons between different experimental runs for 20 %vol steam at ER0.356

Figure 5 .Figure 6 .

 56 Figure 5. Flame speed [m/s] as a function of PM flame detection distance, from the spark plug, for all cases. The left-hand side graphs (a) compare different H2O concentrations at every ER, while the righthand side graphs (b) compare different H2 concentrations.

Figure 7 .

 7 Figure 7. Leading shock wave speed [m/s] and Mach number [-] for all cases with shock waves

Figure 8

 8 sorts out fast flame conditions from slow flame conditions by utilizing Eq. 6, as described earlier. All data points containing 30 % steam experience slow-flame behaviour. The ER=0.294 case with 20 % steam also falls among the slow flames. All other conditions sustain fast flames according to the criterion used here. From now on, red dots indicate fast flames while black dots indicate slow flames, in the graphs.

Figure 9

 9 Figure9compares the fast-flame criterion discussed above with the fast-flame criterion (labelled the sigma-criterion) described by Cicarelli et al.[START_REF] Ciccarelli | Fast-flame limit for hydrogen/methane-air mixtures[END_REF]. All the fast flames are also considered fast with respect to the sigma-criterion. However, two of the slow flame conditions (the one at ER=0.294 with 20 % steam, and the one at ER=0.42 with 30 % steam) fall just above the critical expansion ratio, indicating that these conditions render fast flames. Thus, the two fast-flame criteria do not fully agree under the present experimental conditions.

Figure 8 .

 8 Figure 8. The maximum measured flame speed [m/s] as a function of the speed of sound in the burnt gases [m/s]. The dashed line denotes the fast-flame criterion as vmax > 0.5*cBG

Figure 9 .

 9 Figure9. Expansion ratio as a function of the Zeldovich number plotted together with the critical expansion ratio (blue dashed line) as decided by Cicarelli et al.[START_REF] Ciccarelli | Fast-flame limit for hydrogen/methane-air mixtures[END_REF]

Table 1 .

 1 Experimental conditions

	Case	Steam Conc. [%vol] H2 in the dry mixture[%vol]	ER
	1-3 4-6	0 % 20 %	11%, 13%, 15%	0.294, 0.356, 0.42
	7-9	30 %

ACKNOWLEDGEMENTS

The authors acknowledge the financial support of the French government within the program "Investissement d'Avenir -MITHYGENE grant agreement n°ANR 11-RSNR-0015"

ABBREVIATIONS