Chloé Mercier
email: chloe.mercier@inria.fr

Lisa Roux
email: lisaroux.87@gmail.com

Margarida Romero
email: margarida.romero@unice.fr

Alexandre Frédéric
email: frederic.alexandre@inria.fr

Mnemosyne

Thierry Viéville
email: thierry.vieville@inria.fr

Formalizing Problem Solving in Computational Thinking : an Ontology approach

Keywords: Cognitive Neuroscience, Ontology, Problem-Solving, Learning Sciences, Computational Thinking

We introduce the idea of a symbolic description of a complex human learning task, in order to contribute to better understand how we learn. The learner is modeled on the basis of knowledge from learning sciences with the contribution of cognitive neurosciences, including machine learning formalism, in the very precise framework of a task, named #CreaCube reviewed here, related to initiation to computational thinking presented as an open-ended problem, which involves solving a problem and appealing to creativity.

We target problem-solving tasks using tangible interfaces for computational thinking initiation, and describe in details how we model the task and the learner behavior in this task, including goal-driven versus stimulus-driven behavior and the learner knowledge construction. We show how formalizing these elements using an ontology offers a well-defined computational model and the possibility of inferences about model elements, analyzing and predicting the learner behavior.

This operationalization of a creative problem-solving task is still at a preliminary stage, but an effective proof of concept is described in this study.

I. INTRODUCTION

Understanding how we learn is a key issue for improving education worldwide. This is especially true for transversal competencies sometimes referred to as "21st-century skills", such as computational thinking ( [START_REF] Wing | Computational thinking[END_REF], see also [START_REF] Lodi | Introducing Computational Thinking in K-12 Education: Historical, Epistemological, Pedagogical, Cognitive, and Affective Aspects[END_REF] for a recent review). Other transversal skills such as creativity, problem solving and critical thinking are especially important in the present period for the development of citizenship [START_REF] Romero | Computational thinking development through creative programming in higher education[END_REF]. Such competencies are linked and must be considered in interdependence. For instance, computational thinking is intrinsically related to problem solving (i.e., requires to solve ill-defined tasks, not only solving well-defined problems using a step-bystep approach) and involves creativity in practice (e.g., creative programming), beyond specific know-what (knowledge) and know-how skills. Computational thinking requires technocreative activities to be developed and must integrate critical thinking development, especially when acculturating citizens of all ages to Artificial Intelligence [START_REF] Alexandre | Open Educational Resources and MOOC for Citizen Understanding of Artificial Intelligence[END_REF]. How to better understand such human learning ? Our contribution is a modeling proposal of such a problem-solving learning activity as a step towards better understanding these transversal competencies, thus tackling a key challenge in learning science (also referred to as educational science, from the French terminology).

Problem solving: Problem solving in everyday life takes different forms depending on the task and context. Despite the pioneering work of [START_REF] Newell | Human problem solving[END_REF] aiming to develop a general problemsolving framework, human problem solving is dependent on the task and context, requiring to consider prior knowledge but also the knowledge construction that is developed through the task process based on the different operations the subject performs on the task. Research on problem solving has focused on a diversity of tasks, most of all on well-defined problems such as the chess game or the Tower of Hanoi, allowing to define task models and problem-solving methods. In short, the idea is to represent the problem space, including the positions of the initial state and of the desired goal (possibly a region of the space if the goal is not defined as a unique position but specified through instructions) and to define a path between the positions. This can be done in both forward and backward directions, where knowledge can provide rules for moving in the space or constraints to obey.

Using tangible artifacts: Taking into account the need to consider the set of specific knowledge required to solve a task, we focus our study on a specific problem-solving task with tangible material. Using tangible artifacts allows not only to infer cognitive process but also to observe behavioral patterns [START_REF] Romero | CreaCube, a Playful Activity with Modular Robotics[END_REF]. At a more concrete level, we aim to study creative problem solving through screen-less activities, also known as unplugged activities [START_REF] Huang | A critical review of literature on "unplugged" pedagogies in K-12 computer science and computational thinking education[END_REF], for instance, using pedagogical robots or connected objects [START_REF] Romero | CreaCube, a Playful Activity with Modular Robotics[END_REF]. These activities require a non-trivial setup in order to track the learning process through the activity. They aim to consider the problem-solving process instead of considering only performance scores of the task. Methodological challenges in process-based data collection also take into account the ethical issues of collecting human data in a way that ensures data privacy. One challenge is the use of efficient measurement devices of reasonable cost and it appears that a precise model of the task allows to define specific observables that are thus more robust to estimate [START_REF] Romero | CreaCube, a Playful Activity with Modular Robotics[END_REF]. Another key point is the bias induced by the fact that the learner is observed and monitored, and one solution is to take this as a chance to involve the learner in their own learning process, but this is to be properly studied, see [START_REF] Barnabé | A low-cost tabletop set-up to collect learning analytics during computational thinking unplugged or tangible activities[END_REF] for a discussion on these points. In both cases, the construction of a precise model of the task and of the learner involved in this task is a lever to take up this challenge.

The data challenge: A step further, the challenge here is to work from a relatively small batch of data (a few dozen to compare with the thousands of data used with classic statistical methods). In order for the results to be meaningful, we need to introduce a maximum of prior information upstream to the analysis of the observables, resulting in highly structured data. An ontology is an appropriate tool when it comes to describing and structuring data. We are thus making use of ontology modeling to model the learner and the task, including the observables collected during such a learning activity. This model will then be applicable to the observed learning analytics and allow to interpret them. Phenomenological models, indeed, already exist: the ICAP framework [START_REF] Chi | The ICAP Framework: Linking Cognitive Engagement to Active Learning Outcomes[END_REF] proposes to categorize cognitive engagement behaviors into four modes (Interactive, Constructive, Active,and Passive) and links these behaviors to learning outcomes, while the activity theory [START_REF] Engestrom | Activity theory as a framework for analyzing and redesigning work[END_REF] is an appropriate framework providing the theoretical context of this study. Based on these epistemological frameworks, our present study is positioned at an operational level.

II. MATERIAL AND METHOD

A. The #CreaCube task

In the #CreaCube study1 , we aim to analyze problemsolving strategies using modular robotic cubelets, targeting children between 8 and 12 years old. The task engages the participants into building an autonomous vehicle, composed of 4 items, able to move from a point A to another point B. In this context, the participant is given a set of 4 modular robotic cubelets which differ in their appearances (e.g. different colors) and affordances (e.g. wheels, switch...). The learner is expected to understand that the 4 "items" mentioned by the guideline are the 4 cubelets on the table, and that they should assemble those cubelets into a certain configuration which will be able to move autonomously. It is worth noting that this problem does not have a unique solution as several configurations may satisfy the goal. The task is relatively easy in the sense that most participants come up with a solution in less than 15 minutes, but complex to model because of the large problem space and the lack of specification (e.g. the possible actions are not clearly stated).

The #CreaCube task engages the participant in an ill-defined problem, for which the participant has no prior experience of the modular robotic cubelets. Considering this lack of knowledge, there is a need in exploratory orientation, which is defined from a learning science perspective, as a way to gather information from the environment [START_REF] Kaplan | Achievement goal orientations and identity formation styles[END_REF] in order to complete the task. Throughout the task, the participant moves towards the goal by mobilizing prior knowledge which appears to make sense in relation to their experience with the material being manipulated and the requirements defined by the guideline. Specifically, in this particular task, exploration aims to understand the artifacts, their characteristics and technological affordances, allowing to formulate hypotheses on the material and define sub-goals to solve the problem. The exploration of the means given to solve the task should facilitate the generation of new stimuli which can then help to make progress towards the goal, by fulfilling sub-goals and validating or invalidating hypotheses.

The task is modeled based on the knowledge needed to solve it but also on the initial states of the hardware and the final state for its success, as detailed in Fig. 1. We can see the different observables taken into account, i.e. the possible configurations of the cubelets (disassembled or assembled into a certain shape), the discovery of affordances, i.e. the practical possibilities offered by an element (e.g. there is a switch, so it can be flipped), the different results obtained (e.g. at the motion level), but also elements linked to the subject, such as their emotions or their attitude (perseverance, abandonment) regarding the task. All of these observables will serve as a basis for modeling the task in the form of structured knowledge.

The file model generated from this interface has been developed in a typed hierarchical format (JSON syntax, with both the raw and computed data as well as the description of each type of information and its relationships to other types) to allow its formal manipulation and to best represent the structure of the collected information, in order to make the link with knowledge representations related to the computational notion of ontology (as used in the Semantic Web) and integrated as an interactive platform to facilitate the ergonomics of manual video analysis [START_REF] Romero | Analyse d'activités d'apprentissage médiatisées en robotique pédagogique[END_REF]. Each video for the #CreaCube task is encoded as a temporal sequence of states corresponding either to a configuration of the task or to a state of the learner in the task as explained above (Fig. 1). Here is an example of such a JSON file generated from the interface, describing for a given participant at which moment he reached which observable: Fig. 1: Interface for the identification of observables, this figure describes a part of the system states to which is added, for example, the identification of each cubelet (recognizable by its color: navy blue battery, black sensor, white motor, red inverter) and the states of the cubelets (e.g. "connected/disconnected" or "on/not on wheels"). This sub-ensemble of the possible states corresponds to the observables that have been chosen to analyse the task (see http://aide-line.inria.fr/public/doc/vid.mp4 for a video of the experiment.) {"time": "101","click": "AS01"}, {"time": "109","click": "AS02"} ],"idParticipant": "p362"}

The methodology for generating an organized set of learning analytic log from video analysis has been developed from a learning sciences perspective through the ANR CreaMaker project to study problem-solving activities. These videos serve a double purpose: (i) analysis of the observables (ii) and training of the teachers [START_REF] Albero | L'intérêt pour l'activité en sciences de l'éducation. Vers une épistémologie fédératrice ?[END_REF].

In order to allow for the analysis of the learner's activity and the context of the learning activity, the description of the learning activity is developed in the form of ontology in the computational sense that we will now explicit.

B. The ontology approach

Terminology: Ontologies, as used in the Semantic Web, allow to represent structured knowledge by defining concepts as well as relationships and hierarchies between these concepts. In the following, we use the Web Ontology Language (OWL) terminology:

(i) Individuals represent atomic, real-world objects (ii) Classes represent concepts; a class may therefore be a collection of individuals (which are called instances of the class) (iii) Individuals may be linked by relationships, which are labeled by properties. Properties may have a domain, i.e. the class that they can be applied to, and a range, i.e. the class that they can take values from. OWL also defines both class hierarchy and property hierarchy.

This formalism allows to perform logical reasoning in order to validate the model, as well as logical inference to find all the assertions that can be deduced from the user-defined statements. Indeed, under certain constraints2 , OWL specifications correspond to description logics, providing computational completeness and decidability (a didactic introduction can be found in the guide [START_REF] Allemang | Semantic Web for the Working Ontologist[END_REF]).

Ontologies in education: Here we consider the ontology approach to model a very specific problem-solving task, as it has been done with well-defined tasks such as the Tower of Hanoi [START_REF] Eriksson | Task modeling with reusable problem-solving methods[END_REF]. A rather large literature is already available regarding the use of ontologies in e-learning systems, as reviewed in [START_REF] George | Review of ontology-based recommender systems in e-learning[END_REF]. These approaches attempt to model the learning contents, or the learner (rarely both), in order to perform semantic inference on the observables (e.g [START_REF] Lebis | Recherche intelligente de processus d'analyse de traces d'e-learning via des inférences sémantiques[END_REF]). They generally aim to propose adaptive learning resources in order to improve the learner's performances, while our main purpose is to better understand the learner behavior. Other related work includes the ontology specification of serious games [START_REF] Tang | Game Content Model: An Ontology for Documenting Serious Game Design[END_REF] 3which is inspiring although it does not apply to the description of unplugged activities. In our case, we want to use the OWL vocabulary to specify the #CreaCube task on two levels: (i) the material environment (cubelets, points...) and its affordances, and (ii) the learner in the context of this task, their behaviour and their cognition. A step further, we wish to query our model against the JSON data, using logical reasoning,in order to validate its relevance regarding the observable analysis. We also introduce here computational and cognitive neuroscience knowledge in the modeling process, which does not seem a current practice.

C. Borrowing from cognitive and learning sciences

Goal-driven vs stimulus-driven behavior: In order to advance in the understanding of creative problem solving, we propose to observe a specific problem-solving task -in this case, the aforementioned #CreaCube task -for which we aim to consider a functional neurocognitive model of the brain [START_REF]A global framework for a systemic view of brain modeling[END_REF] and the specific knowledge related to the task in terms of the subjects' prior knowledge, the technological affordances they might discover and their potential interpretation in the process of problem solving (how these affordances might contribute to the task). The behavior of the participant in relation to her or their concurrent goals during the task (choosing among the sub-goals to be addressed in the task, performing the task as fast as possible, preserving their ego when facing problems, etc.) is considered in relation to two main potential strategies/behaviors to reduce the problem space into the solution-space: (i) a top-down goal-driven strategy (a subgoal is considered and it is questioned which existing skills might be recruited to solve it) and (ii) a bottom-up stimulusdriven strategy (affordances generate stimuli that are selected and explored, potentially leading to revise the sub-goals). Globally, the behavior can be voluntarily goal-driven when the participant regulates the problem-solving process at the metacognitive level. However, we assume that the situation will engage the participant in stimulus-driven behavior in certain situations when the participant is lost or confronted with unknown stimuli (such as the robotic cubelets) that drive their actions to satisfy their curiosity. The stimulus-driven behavior could be considered within a larger goal-directed mechanism (see, e.g., [START_REF] Moors | The role of stimulus-driven versus goal-directed processes in fight and flight tendencies measured with motor evoked potentials induced by Transcranial Magnetic Stimulation[END_REF]). We then assume that when a participant is engaged in a problem-solving activity, the main goal accepted by the participant is to solve the problem. The sub-goals for advancing towards one of the potential configurations which will satisfy the requirements of the problem, will engage the participant in behaviors which can sometimes be stimulusdriven, alternating with analysis of the stimuli based on the main goal of the task, exploiting previous knowledge and skills and also possibly with discovery and creativity [START_REF]Creativity explained by Computational Cognitive Neuroscience[END_REF].

Related cognitive functions in the brain: In addition, also based on the same neurocognitive model of the brain [START_REF]A global framework for a systemic view of brain modeling[END_REF], we consider different ways of specifying and identifying a goal by defining (i) What is this goal (its sensory description), (ii) Why (and how) it answers current motivation or specification, (iii) Where it is located and can be accessed and (iv) How it can be manipulated and more generally addressed by a skill. These different motivational and sensori-motor aspects of information representation in the brain, as described in our cerebral architecture, represent an operational way to deal with all aspects of a goal to be considered in problem solving.

Exploration vs Exploitation: Having made the distinction between goal-driven and stimulus-driven behavior, we can now discuss the notions of exploration and exploitation. While this pair is well known in cognitive science and in machine learning at the level of reinforcement learning, they are two distinct conceptualizations that we will now make explicit. In problem-solving activities, subjects alternate between two main strategies: exploration, which aims at experimenting with new alternatives or at acting on the environment in order to generate new stimuli [START_REF] Kaplan | Achievement goal orientations and identity formation styles[END_REF], and exploitation, which is the use of existing knowledge (declarative, procedural) in a given situation. At this level of description, we can draw parallels with these notions in reinforcement-based machine learning, but in this second domain exploration is mostly random, whereas in problem solving, exploration is seen as a strategy for generating new stimuli or new ideas or behaviors by recombining pre-existing knowledge or behaviors, or by manipulating the environment. These are therefore two quite distinct notions, one corresponding to an internal process of generating new internal representations, and the other to a behavior that is only defined in relation to a direct interaction with the environment.

III. RESULTS

We implemented the ontology4 using the software Protégé 5 . First, we have formalized, on one hand the classes corresponding to the material environment of the task, on the other hand the classes referring to the learner's cognition and behavior. Then, we have extensively instantiated these classes, listing their individuals based on the task design itself and the observations collected during the experiments. Finally, we have defined some properties to express the relationships between these instances in order to describe the task process from both a goal-driven and a stimulus-driven point of view.

A. Modeling the task and the material environment

The task involves several objects (cubelets, points), which have characteristics or features (colors of the cubelets, visual and functional affordances, functions, etc.) and states (assembled/disassembled cubelets, wheels facing the ground/not facing the ground, switch on/off, complete/incomplete cubelet configuration, etc.). These concepts allow to a class hierarchy that characterizes the task material as shown on Fig. 2.

The material states represent the problem space of the task in the sense of [START_REF] Newell | Human problem solving[END_REF]: indeed, the goal states of the task are those where the cubelets are in a configuration that satisfies the guideline, i.e. in an "assembled" state, "stable" and "moving" "into the right direction", with the switch "on", and the wheels "on the ground". The learner may interact with the material and change its state through their actions, as described in the next section.

The objects have different functions (e.g the white cubelet rolls, the black cubelet detects nearby objects, etc). The learner may guess these functions, or at least some of them, by looking for clues provided by affordances. These affordances may be either perceptible (i.e. suggested by sensitive information obtained through the discovery of objects), or functional (i.e. discovered through the interaction of one object with others). In the #CreaCube task, the perceptible affordances are all visual. For example, the switch of the blue cubelet, or the wheels of the white cubelet are visual perceptible affordances. Conversely, the red cubelet has no specific visual affluence : there is nothing on its surface that shows its usefulness for the task at hand. However, by making it interact with the other cubelets, the learner may understand its functionality and its operating principle, which is inverting the sensor's behaviour,: hence disclosing a functional affordance. Affordances participate in the construction of memory traces, since they allow the learner to discover, little by little, their material environment and its logic, in order to know how to use it to solve the problem.

B. Modeling the learner's cognition and the knowledge construction

Stimuli and actions: Throughout the task, the learners discover and manipulate the material environment, from which they receive various stimuli, i.e. events that cause them organic reactions. In #CreaCube, the stimuli are sensory in nature and give the learner new information about the material environment of the task; this information can then be processed by the learner to constitute traces of stimuli that are kept active during the task, before being consolidated as knowledge (as we will described further).

In the #CreaCube task, stimuli occur when the learner proactively interacts with their environment; for instance, when they flip the switch on the blue cubelet, a diode lights up as a response to their action. The only exception is at the very beginning of the task when the material is revealed (visual stimulus) and the guideline is played (auditory stimulus), without any previous action from the learner. Some actions are only available to the learner in some particular states of the material environment; those states define the preconditions of the action. Similarly, post-conditions define changes of state that an action results in. These preand post-conditions (specified by the ontology properties, see III-C) also constitute knowledge to acquire.

Knowledge retrieval and acquisition: In order to complete the task, the learner needs to retrieve prior knowledge that has been acquired in the course of past experiences and learning. Most of the time, this prior knowledge is activated through the discovery of affordances: for instance, after finding of the wheels, the learner might retrieve the fact that wheels on a vehicle must be in contact with the ground to allow it to move. However, prior knowledge may not always be transferable to the situation at hand: for example, the fact that wheels automatically start to turn after the user has pulled the vehicle backwards is only applicable in certain situations -most often, this mechanism is found in children car toys. However, in this task, trying to press down on the white cubelet by manually rolling it backwards will not cause it to move forward autonomously. This piece of knowledge might even hinder the completion of the task, if it is too strongly anchored (i.e., if it considered to be a general truth when it is actually valid only in a limited set of situations): as long as the learner keeps on trying to make the wheels work this way, they will not go on their research any further.

Confronting their prior knowledge to stimuli perceived through their interactions with the material, allow the learner to maintain a representation of their environment. We refer to this representation as contextual knowledge in our ontology, i.e. facts that are related to the specific context of the task. Some of this contextual knowledge is acquired when the learner listens to the instruction and is thus directly related to the goal; the other part is built up during the task and relates to the resources available to complete the task.

Hypotheses and goals: In order to refine their representation of the environment, the learner needs to understand which pieces of prior knowledge are effectively applicable in the context of the task and relevant to solve it. For example, the learner might want to test the hypothesis that the wheels on the white cubelet can be activated by pull-back action, because they have previously seen such a mechanism on other toys. The outcome of the pull-back action will allow them to either discard this hypothesis or confirm it and store it as contextual knowledge. Testing hypotheses thus define subgoals to acquire knowledge that will help solve the task. As previously described in section II-C, goals are a way for the participant to organize their understanding of the task and plan their actions hierarchically. Global goals (essentially responding to the cognitive questionings What and Why, in link with high-level executive functions [START_REF]A global framework for a systemic view of brain modeling[END_REF]) may be instructed by an external direction -in this case, the guideline -or a metacognitive reflexion. These high-level goals yield new questionings, inducing (Where, How) behavior mechanisms [START_REF]A global framework for a systemic view of brain modeling[END_REF] thus resulting in sub-goals at a more local level, in link with sensori-motor behavior.

C. Modeling the task process

Instantiating the concepts: Having defined the previous concepts, we have instantiated them by extensively (although not exhaustively) listing all possible hypotheses, actions, stimuli, goals... of the learner, as well as all states, affordances, functions... of the material. This instantiation was based on the task design itself, e.g. the cubelet operating modes, and on the expected behaviors predicted by the theory, but also updated during the course of the experiments to account for newly observed behaviors. These instances (also known as individuals, see II.B) are linked to each other through relationships that describe the interactions between the material and the learner, the knowledge construction and the goal hierarchy, all of these contributing to better understand the task process.

Describing relationships: Relationships between individuals are represented by properties for which we specify the domain and the range, i.e. the classes they link, as shown on Fig. 2. These relationships allow us, for example, to identify preconditions and post-conditions of an action (respectively represented by the properties technicallyRequires and changesTheStateOf), describe the goal intricacy (mayBeAnUnderGoalOf) or the knowledge dependence (operationallyRequires, mayBeInducedBy). These properties have been defined with regard to the goal-driven and stimulus-driven behaviors described in section II-C, as shown on Fig 3 . It is worth noting that we are using modal verbs when the relationship may exist while not being systematic. This representation could be improved using a numerical weighting of RDF statements based on the notions of possibility and necessity (in the sense of [START_REF] Denoeux | Representations of Uncertainty in AI: Beyond Probability and Possibility[END_REF]) as discussed in [START_REF] Mercier | Ontology as neuronal-space manifold: towards symbolic and numerical artificial embedding[END_REF].

IV. APPLICATION A. Modeling as an interdisciplinary tool

This work allowed us to confirm the multiple interest of working with ontological modeling when we bring together three disciplines (educational, cognitive and computer sciences) to tackle a problem:

• terminological interest to begin with: beyond periphrases and phenomenological descriptions in each discipline, constraining us to define things through lexical choices and fully specified properties and relations between these key words, forces us to clearly posit what we are talking about ; • interest in formalization: this approach allows, without even using algorithmic reasoning skills at this stage, to take stock of what can be defined rigorously, to formalize completely. Defining a well-formulated ontology is therefore already by itself a structuring exercise, an epistemological tool in a way, before even using it. As with any modeling, it gives an exhaustive and explicit view of what is to retain compared to what is neglected. Once the specification has been set, the use of available reasoners such as Pellet or Hermit allows to check its consistency and query the data on the consequences that may or may not be drawn from it. It also allows us to evaluate the order of magnitude of such development, showing that it is quite a huge work but feasible.

This being done, we also have an immediate check of the model properties, in particular its logical coherence and soundness. It not only allows us to verify the absence of contradictory statements, but we also experimented that if our definitions are not sufficiently specified, we can easily deduce spurious or odd consequences (e.g., mixing categories of concepts). We thus have an operational tool to verify the completeness of our formalization.

B. Reasoning about the model 1) Inferring internal observable of the learner from outer observable:

Having stored these behavioral processes in an ontology allows to retrieve them to interpret new data. A query language, such as SPARQL in the case of RDF data, may fulfill that purpose. For example, we may observe a new action which changes the state of the switch, and wonder what motivated the learner to execute this action. As illustrated by Fig. 4, a SPARQL query on the model helps us retrieve that the action may have been a simple reaction to the stimulus of seeing the switch, in the case of a stimulus-driven behavior, or, in a goal-driven mode, the result of a more elaborate reasoning such as an attempt to understand the function of the blue cubelet (thus testing the hypothesis that it is indeed a switch) or, this hypothesis being verified, to activate the movement of the vehicle.

Characterizing whether the learner is rather in a goal-driven or stimulus-driven mode for a given sequence of observables might be inferred from the timestamps recorded in the data, thus helping us to decide which statements of the ontology are more appropriate to describe the observed situation. For example, we could assume that once the learner has identified a goal and the means and resources at their disposal to fulfill it, the flow of their actions might be smoother, whereas in an exploration situation, there might be more pauses corresponding to the wait between new stimuli. However, this link between timestamps and switch of strategy is not well understood yet and needs further investigation.

2) Predicting the learner's behavior:

Instead of querying the model using backward chaining as in the previous example, we can also use forward chaining in order to simulate the learner's deduction, hence predicting their behavior. For example, we might wonder how the learner would react to the visual stimulus of seeing the switch. In a stimulus-driven mode, they would likely touch it or try to flip it (assuming that the knowledge relative to switches being flipped has become a procedural knowledge, that is to say automated into an almost reflex behavior). In a goaldriven mode, we may infer that the learner will elaborate the hypothesis that the affordance is indeed a switch and can be used to activate the vehicle, which will result in planning goals (flip the switch, connect the cubelet to the figure) and subgoals (they have to grasp the cubelet before connecting it to the others).

Again, this ontology provides a subset of plausible behaviors, however we still need a way to decide which of these behaviors to choose to simulate the learner. Such mechanisms are still to be developed to explain what the learner's attention is more focused on in a given situation. Moreover, we need to take into account inter-individual differences: some participants might be more interested in achieving a performance goal, thus appealing for a predominant exploitation strategy, whereas others might pursue a mastery goal and try to understand as much as possible about the environment of the task, displaying more exploratory behaviors.

V. DISCUSSION AND CONCLUSION

This study introduces the operationalization of a creative problem-solving task. Within this type of task, we need to consider first extrinsic motivation (e.g., task performance) and intrinsic motivation (e.g., increase knowledge), i.e., the "Why". Then consider the "What", i.e., both exploration and exploitation strategies directed toward different concurrent goals (performance goal versus mastery goal in relation to the knowledge related to the task). Our ontology formalizes these behaviors involved in the creative problem-solving process to model the learner's behavior in interaction with the educational robots, with regard to a cognitive model of the brain. Through this process, we take into account the stimuli received during the interactions with the modular robots, the perception and interpretation of the different technological affordances that will support the problem-solving activity through concrete actions to be developed. The learner's prior knowledge is mobilised to understand current perceptions and transform knowledge into a way that makes sense within the context of task [START_REF] Musial | Précis d'ingénierie pédagogique[END_REF], formulating and testing in-task hypotheses.

The key point is that the description does not remain phenomenological, that is to say, guided by a theoretical framework expressed in human language to guide the analysis of the data, while we hereby reformulate these notions at a computational level to give oneself the means to process them in a more systematic way using an algorithmic approach, as discussed here, formalizing in a precise language the concepts invoked. We also consider, based on this formalism, biologically plausible implementations of internal reasoning mechanisms in a companion work [START_REF] Mercier | Ontology as neuronal-space manifold: towards symbolic and numerical artificial embedding[END_REF] in order to contribute to cognitive neuroscience modeling of related mental processes. This is only a first step, which already shows that formalizing such a learning task is definitely a rather huge, but attainable work. Next steps will be to complete this formal description, and develop reasoning mechanisms to better infer the subject internal motivation and behaviors. We will also complete the linking between this task and learner modeling and experiment observables (e.g., learner state observation, post-activity evaluation).

Though we have the feeling that it is a rather disruptive approach with respect to learning science, we are really aware that it is still a very preliminary work. Eventually, the key point is to consider a triple multi-disciplinary approach from learning science, cognitive neuroscience and computer science, something that might be called computational learning science [START_REF] Romero | LINE -Mnémosyne : Des neurosciences computationnelles aux sciences de l'éducation computationnelles pour la modélisation du cerveau de l'apprenant et du contexte de l'activité d'apprentissage[END_REF].

Fig. 2 :

 2 Fig. 2: Main classes defined in our ontology. Classes related to the task material are encompassed in the upper frame, while the lower frame includes classes that characterize the learner. Some of the properties are also represented through their domains and ranges, to show how those classes are related to each other. (Figure edited using the Protégé OntoGraph plugin; lower-level classes and individuals are not shown)

Fig. 3 :

 3 Fig.3: Goal-driven behavior on the left versus stimulus-driven behavior on the right. Accordingly relevant properties defined in our ontology are represented in blue.

( a )

 a An example of SPARQL query (b) SPARQL results returned for the query (a): we enquired about which actions change the state of the blue switch, what stimuli may have provoked them, orwhat goal they might be aiming at, and if that goal might be the sub-goal of another one, as well as what knowledge they are using and what hypothesis might be tested. We used the Protégé Snap-SPARQL plugin6 (which allows us to perform logical inference and answer queries not only based on asserted data but also inferred data).

Fig. 4 :

 4 Fig. 4: Example of inferences performed by the Hermit reasoner and retrieved by SPARQL.

See https://creamaker.wordpress.com/2019/02/06/publicationswithin-the-creamaker-project for full details

{"clicks": [ {"time": "46","click": "AS01"}, {"time": "50","click": "AS02"}, {"time": "58","click": "AS01"}, {"time": "67","click": "AS00"}, {"time": "82","click": "AS01"}, {"time": "93","click": "AS00"},

Implemented in OWL DL, a sub-language of OWL

Implemented in the Ludo Ontology: http://ns.inria.fr/ludo

This ontology is publicly available here: https://gitlab.inria.fr/line/aide-group/creaonto.

https://github.com/protegeproject

https://github.com/protegeproject/snap-sparql-query

Acknowledgments

Hugo Chateau-Laurent and Lola Denet are highly acknowledged for their contribution to scientific discussions in link with this work.


Abstract-A composite constituted of a steel sheet taken in a sandwich between two aluminum (Al) sheets is produced by cold roll bonding. A good adherence between Al and steel sheets, ensuring a good mechanical resistance, is obtained with a specific process. A previous work of the authors has shown that optimal condition between adherence and magnetic shielding effectiveness (SEH) is obtained with a 230 µm composite produced with an initial thickness of Al and steel sheets respectively of 250 and 100 µm. In this paper, the 230 µm Al/Steel/Al composite is used in three applications modelized by 2D numerical simulations. To facilitate calculations, a homogenization method is applied to the composite. Studied applications are a cylindrical box containing a coil, a square box under an external magnetic field and a high voltage cable. In each application, SEH is calculated at low frequency (1 Hz -100 kHz) and different materials (Al/Steel/Al, Al, steel, and copper) are compared. It is observed that, in each application, the composite is more advantageous at equal mass and especially at frequencies higher than around 5-10 kHz.

Index Terms-Cold Roll Bonding, Finite Element Method, Homogenization, Shielding Effectiveness

I. INTRODUCTION

Electromagnetic pollution is a source of disturbance to sensitive electrical and electronic devices [START_REF] Smolenski | Timedomain-based assessment of data transmission error probability in smart grids with electromagnetic interference[END_REF] and can be potentially harmful to human beings [START_REF] Dawson | Pacemaker interference by magnetic fields at power line frequencies[END_REF]. Living and critical components must be then protected from electric and magnetic fields, and electronic devices must confine them thanks to electromagnetic shielding [START_REF] Abdelli | Impact of composite materials on the shielding effectiveness of enclosures[END_REF][START_REF] Ahn | Low frequency electromagnetic field reduction techniques for the On-Line Electric Vehicle (OLEV)[END_REF] Common materials used for electromagnetic shields are metals [START_REF] Liu | Electromagnetic interference shielding effectiveness of titanium carbide sheets[END_REF][START_REF] Xu | Electromagnetic interference shielding effectiveness of aluminum foams with different porosity[END_REF] and carbons [START_REF] Chung | Electromagnetic interference shielding effectiveness of carbon materials[END_REF][START_REF] Li | Robust carbon nanotube foam for efficient electromagnetic interference shielding and microwave absorption[END_REF]. Lightweight materials, required in the transportation industry, composed of metal or carbon reinforced polymer have been greatly studied [START_REF] Al-Ghamdi | Novel polyvinyl alcohol/silver hybrid nanocomposites for high performance electromagnetic wave shielding effectiveness[END_REF][START_REF] Ray | An investigation of the electromagnetic shielding effectiveness of ethylene vinyl acetate elastomer reinforced with carbon nanofillers[END_REF]. However, polymers are insulating, and reinforced polymer composites show lower electrical conductivity than metal bulk. Multilayer composite is also an effective means to shield against electromagnetic fields [START_REF] Ma | A novel structure of Ferro-Aluminum based sandwich composite for magnetic and electromagnetic interference shielding[END_REF][START_REF] Kim | Transparent and flexible film for shielding electromagnetic interference[END_REF].

Low frequency magnetic fields can cause noise and disturbance to sensitive devices and are emitted, for example, by motors, power supplies, and switches. In this paper, the shielding effectiveness (SEH) of a 230 µm Al/Steel/Al composite with homogenized properties is studied and compared to Al, steel, and copper ones in various applications modelized by the finite element method (FEM). The idea is to propose a solution that gives an efficient shielding effectiveness against both static and low frequency electromagnetic fields. Furthermore, the thin composite thickness allows ensuring its lightness and flexibility. Another main interesting advantage of this solution is that Al layers in the composite protect the steel layer from potential oxidation. Thus, the composite presents the advantage of being adapted to oxidizing atmospheres. Consequently, SEH is studied in this paper for a magnetic field at low frequencies from 1 Hz to 100 kHz. Both experimental and numerical approaches, including homogenization process, are used to present the global study, from the elaboration of the composite material to the shielding application.

II. MATERIALS AND APPLICATIONS

A. Al/Steel/Al composite

The studied composite is composed of one layer of steel sandwiched between two layers of aluminum (Al). Commercial low carbon steel DC01 and 8011-aluminum alloy are chosen as raw materials with a respectively initial thickness of 100 µm and 250 µm.

DC01 steel is mainly used for drawing and forming applications. 8011-Al is a Fe and Si based aluminum alloy which is widely used in the industry. The chemical compositions of these two materials are listed in Table 1.

The Al/Steel/Al composite is produced by cold roll bonding (CRB) with different reduction rate (hf / hi). Figure 1 presents a schematic diagram of the composite CRB following rolling-normal cross-section (RD-ND). In a previous study [START_REF] Clérico | Compromise between magnetic shielding and mechanical strength of thin Al/Steel/Al sandwiches produced by cold roll bonding: experimental and numerical approaches[END_REF], the trade-off between shielding effectiveness and Al/Steel interfaces adherence has been studied. The following part proposes a summary that justifies the later thickness of the composite used for our applications.

Study of Low

Figure 2 shows the electronic assembly used to measure the experimental shielding effectiveness of the composite. A sinusoidal current is produced by a lowfrequency generator associated with a linear amplifier. The effective intensity of this current is 2 A. A coil with a height of 18.5 mm, an inner and outer diameter of 15 and 30 mm generates a magnetic field. This magnetic field is then measured by a Hall effect sensor with a sensibility of 5.0 ± 0.1 mV/G. A cooling fan is employed to limit the temperature influence. Figure 4 shows that the quality of Al/Steel interface adherence increases with the reduction rate. Adherence has been determined by tensile bond strength test (TBST) and the mean value has been calculated for each reduction rate. This method is detailed in a previous paper [START_REF] Clérico | Compromise between magnetic shielding and mechanical strength of thin Al/Steel/Al sandwiches produced by cold roll bonding: experimental and numerical approaches[END_REF]. The optimal condition between shielding effectiveness and adherence of Al/Steel interfaces is then obtained with a 230 µm thickness composite. Indeed, the 230 µm composite shows very little steel fractures, that limit the negative effect on shielding effectiveness, and attain a good quality of adherence with a tensile bond strength around 19-20 MPa. In this case, aluminum and steel layer thicknesses of the composite considered in the numerical approach are then around 191.6 (2 × 95.8) and 38.4 µm.

B. Applications

In this paper, three applications are modelized in Comsol Multiphysics with AC/DC module and studied in the range of [1 Hz -100 kHz]: a cylindrical box containing a coil to simulate the confinement of a magnetic field, a squared box placed in an external magnetic field to simulate the protection of a sensitive component and finally a simplified model of a high voltage cable that is closer to actual industrial applications. Figures 5, 6 and7 show the modelized geometry of the three applications. The cylindrical box is modelized with a 2D axisymmetric model. The box has a diameter and a height of 50 mm. In this first approach, the coil is considered centered inside the cylindrical box. To simulate the magnetic field, an effective current of 2 A, corresponding to a current density of around 1.44e6 A/m², is considered. The shielding effectiveness of this box is studied at two points, one horizontal and one vertical, both situated at 15 mm of the shield. This first application allows simulating the magnetic shielding with a near-field condition. The squared box is modelized with a 2D plane model. The box has a side of 50 mm and is placed in an external magnetic field obtained with magnetic vector boundaries condition (n × A = n × A0). The shielding effectiveness is studied at the center of the box. This second application allows simulating the magnetic shielding with a far-field condition.

The high voltage cable is also modelized with a 2D plane model. The simplified modelized geometry (Figure 7) has been inspired by the work of [START_REF] Duc | Développement d'un outil d'investigation pour le diagnostic des phénomènes hautes fréquences dans les câbles électriques[END_REF]. Figure 8 shows the actual geometry of the high voltage cable studied by [START_REF] Duc | Développement d'un outil d'investigation pour le diagnostic des phénomènes hautes fréquences dans les câbles électriques[END_REF]. In a high voltage cable, perturbations can take place affecting cable integrity and leading to malfunctions. A correct shielding can prevent these perturbations.

The total diameter of the cable is 78 mm. The cable has four copper wires with a diameter of 16.4 mm. The current density running through the copper wires is equal to 1e6 A/m² and is defined positively in two wires and negatively in the two others. A shield surrounds each wire to limit proximity effects, and another one surrounds the four wires for a global shielding of the cable. The area, other than copper and shield, is considered as PET (polyethylene terephthalate). The shielding effectiveness is studied at 1 cm from the high voltage cable. In each application, the following equations are resolved:

(1)

(2) (3) (4)
with H the magnetic field, B the magnetic induction, A the magnetic vector potential, E the electrical field, J the total current density, Je the external current density, σ the electrical conductivity, and ω the angular frequency.

For the near-field application and the high voltage cable, the modelized system is bounded by a layer of infinite elements.

The magnetic shielding effectiveness (SEH) is defined by: [START_REF] Liu | Electromagnetic interference shielding effectiveness of titanium carbide sheets[END_REF] with H0 and Hshield respectively the magnetic field without and with the shield at the different measurement points defined previously. SEH is studied in each application at low frequencies from 1 Hz to 100 kHz.

The use of the 230 µm Al/Steel/Al composite in these applications is compared to aluminum (Al), steel, and copper (Cu) in two cases: first with equal thickness for applications with low space available, and then with equal mass especially adapted for embedded applications. The following densities are used for calculations: 2.71 for Al, 8.96 for Cu, 7.85 for steel and 3.57 g/cm 3 for the composite. Thus, the 230 µm Al/Steel/Al composite is 2.5 and 2.2 times less dense than Cu and steel, but 1.3 times denser than Al.

Materials properties considered into the numerical model for the magnetic shielding are summed up in Table 2.

TABLE 2 RELATIVE PERMEABILITY AND CONDUCTIVITY OF CU, AL AND STEEL

The numerical modeling of actual devices with the consideration of the heterogeneous layers of the composite is a real challenge regarding the scale differences from the material to the application. As a consequence, the composite material properties are here homogenized to give reasonable computation time for the simulations. The principle of homogenization is shown in Figure 9. The properties of an equivalent layer are determined in order to obtain the same shielding effectiveness of the tri-layer composite.

In the case of the 230 µm composite, the determined equivalent properties are resumed in Table 3. A slight difference is observed between RD and TD properties. The presence of a few steel fractures considered in the homogenization process explains this difference. More details of the method to determine these properties by an energy approach can be found in our previous paper [START_REF] Clérico | Compromise between magnetic shielding and mechanical strength of thin Al/Steel/Al sandwiches produced by cold roll bonding: experimental and numerical approaches[END_REF].

A. Homogenization validation

Before comparing the shielding effectiveness of the composite with Al, Cu and steel, homogenization method has to be validated by comparing the shielding effectiveness of the equivalent layer with the tri-layer one.

The comparison between the tri-layer and the equivalent layer is shown in Figure 10 for the cylindrical box with the measurement of the magnetic field at the vertical point. It is observed that the equivalent layer, with its properties determined by homogenization, has an excellent concordance with the tri-layer.

However, a slight difference can be seen for frequencies higher than 5 kHz. Indeed, the equivalent layer shows shielding effectiveness lower than the trilayer with a maximal difference around 4 dB.

One parameter linked to the shielding effectiveness at high frequencies is the skin depth [START_REF] Schelkunoff | Electromagnetic Waves, Bell Telephone Laboratories series[END_REF]. Table 4 gives the skin depth of Al, steel and the equivalent layer at different frequencies. As a reminder, the Al layer and steel layer thicknesses are respectively around 95.8 and 38.4 µm. Figure 11 shows the layer thickness divided by the skin depth in function of frequency for Al, steel and the equivalent layer.

The skin depth in Al is thicker than the layer thickness for all frequencies, then the magnetic field is very little absorbed by Al. The skin depth in steel attains the layer thickness at around 77 kHz. By definition [START_REF] Schelkunoff | Electromagnetic Waves, Bell Telephone Laboratories series[END_REF], around 63 % of the magnetic field is then absorbed by steel at 77 kHz. The skin depth attains the equivalent layer thickness at around 4 kHz. Thus, the equivalent layer absorbs better the magnetic field than the tri-layer composite. This difference in skin depth could explain the slight difference in shielding effectiveness at a frequency higher than 5 kHz.

B. Applications

The shielding effectiveness of the cylindrical box measured at the vertical and horizontal points are respectively presented in Figures 12 and13. SEH is calculated with shields at equal thickness and at equal mass. The results at these different points are sensibly similar. The shielding effectiveness is, nevertheless, a little higher at the vertical point than at the horizontal one.

As expected, copper and aluminum do not shield low frequencies under 500 Hz due to their low relative permeability (close to 1). At higher frequencies, Cu shows better shielding effectiveness than Al with equal thickness due to its higher conductivity.

Steel can shield low frequencies thanks to its high relative permeability. Furthermore, its increase in higher frequencies is steeper than Cu and Al. This steeper slope could be explained by a thinner skin depth in steel than in Cu and Al for a given frequency. Moreover, it can be noticed that the increase of shielding effectiveness of steel begins at a higher frequency (≈ 2 kHz) than for Cu or Al (≈ 500 Hz) due to its smaller conductivity.

The homogenized composite can also shield low frequencies with a shielding effectiveness a little lower than the steel one due to its lower relative permeability. In the equal thickness study, the shielding effectiveness of the composite is lower than the steel one on the entire frequency range. The use of the composite is then more advantageous at equal mass, its shielding effectiveness is greater than other ones at frequencies higher than 2 kHz. The curves obtained are very similar to the cylindrical box ones. As previously, it is observed that the Al/Steel/Al composite is more advantageous at equal mass and at frequencies higher than 2 -10 kHz.

The composite could be still interesting at equal thickness. Indeed, its shielding effectiveness at high frequency is near to the steel one and can be higher than the Cu one with a lower density.

To give some quantitative comparisons, Figure 16 resumes the results of the equal mass study for the three applications. The shielding effectiveness at 1 Hz, 1, 10 and 100 kHz are drawn for the four studied materials. Cu has a shielding effectiveness a little bit lower than Al due to its higher density. Indeed, SEH of Cu and Al at 100 kHz is respectively around 37.8 and 43.2 dB in the far-field application and 34.2 and 39.4 dB in the nearfield application. Steel is able to shield low frequencies (< 1 kHz) but has a lower shielding effectiveness at high frequency than Cu and Al: at 1 Hz and 100 kHz, respectively 2.7 and 32 dB in the far-field application, and 4.7 and 30.7 dB in the near-field application.

The shielding effectiveness attained with the high voltage cable is greater than with the two previous applications: at 100 kHz, SEH of Cu, Al, steel, and the composite is respectively 54.5, 65.4, 55.2, and 104.5 dB. The presence of several shields, around each wire, and around the four wires, explains this difference.

It is observed that for each application, the homogenized composite is an interesting compromise at equal mass to shield low frequency (< 1 kHz) and high frequency (> 10 kHz). In the case of the cylindrical box (near-field application), its shielding effectiveness at 1 Hz is around 2.1 dB, lower than the steel one, but better than the Cu and Al ones that do not shield very low frequencies. At 100 kHz, its shielding effectiveness is around 57.7 dB, better than Cu, Al and steel ones. A composite constituted of a steel sheet sandwiched between two aluminum (Al) sheets has been successfully elaborated by cold roll bonding. The optimal condition between magnetic shielding and Al/Steel interface adherence is attained with a tri-layer composite of 230 µm. The homogenization method used to determinate the material properties of an equivalent layer of this composite has been validated by the great concordance between the shielding effectiveness of the equivalent layer and the tri-layer one.

The shielding effectiveness of the 230 µm Al/Steel/Al composite has then been studied in three applications with numerical simulation and compared to individual Cu, Al, and steel layers. It has been shown that the Al/Steel/Al composite is more advantageous at equal mass in each application. Indeed, the tri-layer composite is able to shield very low frequency (< 1 kHz) and has shielding effectiveness at high frequency (> 10 kHz) greater than Cu, Al, and steel. Its shielding effectiveness at 1 Hz varies in function of the modelized application between 0.8 and 2.1 dB compared to the steel one that varies between and 4.7 dB. At 100 kHz, it varies between 57.7 and 104.5 dB compared to the Cu, Al, and steel ones that vary respectively between 34.2 and 54.5, 39.4 and 65.4, and 30.7 and 55.2 dB.

Thereby, the study of the shielding effectiveness of the composite with more complex and realist geometry could be interesting. The equivalent layer with homogenized properties can be used in 3D models to reduce calculation and CPU time. This work can be extended to other materials in order to increase the shielding effectiveness, especially at low frequency with a higher relative permeability material.

Figure 1 :

 1 Figure 1: Cold roll bonding of Al/Steel/Al composite

Figure 2 :

 2 Figure 2: Electronic assembly scheme

Figure 3 :Figure 4 :

 34 Figure 3: Shielding effectiveness of 3 x 3 cm² tri-layer samples atdifferent reduction rate measured experimentally[START_REF] Clérico | Compromise between magnetic shielding and mechanical strength of thin Al/Steel/Al sandwiches produced by cold roll bonding: experimental and numerical approaches[END_REF]

Figure 5 :

 5 Figure 5: Cylindrical shielding box (50 x 50 mm) containing a coil (2D-axi). SEH measurement points are indicated

Figure 6 :

 6 Figure 6: Square shielding box (50 x 50 mm) under an external H-field (2D). SEH measurement point is located at the box center

Figure 7 :Figure 8 :

 78 Figure 7: Simplified model of a high voltage cable (2D). SEH measurement line is at 1 cm away from the cable

Figure 9 :

 9 Figure 9: Homogenization principle of the tri-layer composite

Figure 10 :

 10 Figure 10: Shielding effectiveness of the tri-layer Al/Steel/Al and its equivalent layer in the cylindrical box application (vertical point)

Figure 11 :

 11 Figure 11: Layer thickness divided by the skin depth for Al, steel and the equivalent layer

Figure 12 :Figure 13 :

 1213 Figure 12: Shielding effectiveness of the cylindrical box with equal thickness (a) and equal mass (b) measured at the vertical point

Figure 14 :Figure 15 :

 1415 Figure 14: Shielding effectiveness of the squared box with equal thickness (a) and equal mass (b) measured at the central point

Figure 16 :

 16 Figure 16: Shielding effectiveness of the cylindrical box, the squared box and the high voltage cable with different materials and frequencies at equal mass

  Frequency Magnetic Shielding of a Thin Al/Steel/Al Composite in Various Applications P. Clérico 1,2,3 , X. Mininger 2 , L. Prévond 3 , T. Baudin 1 , A-L. Helbert1

					TABLE 1
			CHEMICAL COMPOSITIONS (%WT.) OF DC01 STEEL AND 8011-AL ALLOY
	Al8011
	%Al	%Fe	%Si	%Mn	%Zn	%Cu	%Ti	%Cr	%Mg
	Bal.	0.6 -1	0.5 -0.9	≤ 0.2	≤ 0.1	≤ 0.1	≤ 0.08	≤ 0.05	≤ 0.05
	DC01
		%Fe	%C	%Mn	%P	%S	%Si	%Al
		Bal.	≤ 0.12	≤ 0.6	≤ 0.045	≤ 0.045	≤ 0.03	≤ 0.02

TABLE 3 EQUIVALENT

 3 RELATIVE PERMEABILITY AND EQUIVALENT CONDUCTIVITY OF A 230 µM AL/STEEL/AL COMPOSITE III. RESULTS AND DISCUSSION

TABLE 4 SKIN

 4 DEPTH AT DIFFERENT FREQUENCIES IN AL, STEEL AND THE

		EQUIVALENT LAYER
	Frequency (Hz)	Al	Skin depth (µm) Steel	Equivalent layer
	1	86 813	10 598	14 296
	1 000	2 745	335	452
	10 000	868	106	143
	100 000	274	33.5	45.2

ICMMO, SP2M, Univ. Paris-Sud, Université Paris-Saclay, UMR CNRS 8182, Orsay, France

GeePs, CentraleSupélec, Univ. Paris-Sud, Université Paris-Saclay, Sorbonne Université, UMR CNRS 8507, Gif-sur-Yvette, France

SATIE-Cnam, ENS Paris Saclay, Univ. Paris-Sud, UMR CNRS 8029, Cachan, France E-Mail: paul.clerico@u-psud.fr


Modeling of a Fragmented Composite for Shielding Applications P. Clérico 1,2,3 , X. Mininger 2 , L. Prévond 3 , T. Baudin 1 , A-L. Helbert 1 1 ICMMO, SP2M, Univ. Paris-Sud, Université Paris-Saclay, UMR CNRS 8182, Orsay, France 2 GeePs, CNRS, CentraleSupélec, Univ. Paris-Sud, Université Paris-Saclay, Sorbonne Université, Gif-sur-Yvette, France 3 SATIE-Cnam, ENS Paris Saclay, Univ. Paris-Sud, UMR CNRS 8029, Cachan, France xavier.mininger@geeps.centralesupelec.fr, anne-laure.helbert@u-psud.fr

Abstract-The paper deals with the numerical modeling of a fragmented composite applied to a low-frequency shielding application. The considered composite is made of a steel layer placed between two aluminum layers, and is obtained by cold roll bonding. A first numerical homogenization process is proposed to obtain a homogeneous material with effective properties. As this material has still to be discretized with a fine mesh to be considered in the numerical model, a second step based on the AMSL approach is then used to work with a onelayer mesh for the composite screen.

Index Terms-Cold Roll Bonding, Finite Element Method, Homogenization, Shielding Effectiveness.

I. INTRODUCTION

Composite materials are nowadays an interesting solution for shielding applications. In this paper, a composite trilayer made of Al/Steel/Al is considered. Previous studies have shown that an interesting trade-off between shielding effectiveness and Al/Steel interfaces adherence can be obtained when the steel layer presents a small quantity of fragmentations, with the welding of the 2 Al layers [START_REF] Clérico | Compromise between magnetic shielding and mechanical strength of thin Al/Steel/Al sandwiches produced by cold roll bonding: Experimental and numerical approaches[END_REF]. Nevertheless, the simulation of shielding applications with the consideration of such heterogeneous material requires huge computation resources. Artificial Material Single-Layer (AMSL) approach gives the possibility to work with a onelayer mesh for a multilayer barrier [START_REF] Cruciani | Conductive Layer Modeling by Improved Second-Order Artificial Material Single-Layer Method[END_REF], but the layers have to be plane, and with isotropic material properties, conditions that are not present for the considered trilayer.

II. MODELING APPROACH AND RESULTS

The composite is obtained by cold roll bonding, as shown in Fig. 1. Due to the fragmented non-plane steel layer, here with a 4% rate, the AMSL method cannot be directly used to get the global material properties. As a first step, a numerical homogenization is then added, see Fig. 2. For this homogenization, a representative part of the steel layer obtained experimentally after the rolling process, including the fragmentations, has been scanned and integrated into finite element simulations with the "actual" material. Magnetostatic and electrostatic simulations have then been performed with imposed boundary conditions, so as to get the effective material properties of the equivalent layer from energy equivalence. Due to the initial trilayer structure, the obtained fictive material presents anisotropic properties. In order to apply the AMSL approach, only the influential material parameter is retained, i.e. RD component of the conductivity and permeability, leading to a new isotropic fictive material. Considering this last fictive material, Table I gives some results of simulated shielding effectiveness (2D axi FEM model, shielding barrier with one-layer mesh AMSL2), compared to the ones obtained with a 6-layer mesh with the first anisotropic fictive material and with our experimental approach, showing good agreement.

Fig. 1 .

 1 Fig. 1. Cold roll bonding of Al/Steel/Al composite

Fig. 2 .

 2 Fig. 2. Modeling approach

Table I

 I

	Result comparison for shielding effectiveness (dB)
		10 Hz	1 kHz	5 kHz	10 kHz
	Experimental	2.2	2.3	3.4	5.7
	6-layer mesh & anisotropic	2.18	2.28	4.24	7.70
	1-layer mesh & isotropic	2.08	2.19	4.18	7.67


Numerical Study of Cooling by Ferrofluids in an Electrical

Transformer using an Axisymmetric Model Raphaël Zanella 1,2 , Xavier Mininger 2 , Caroline Nore 1 , Frédéric Bouillault 2 , and Jean-Luc Guermond 3 1 LIMSI, CNRS, Univ. Paris-Sud, Université Paris-Saclay, F-91405 Orsay, France 2 GeePs, CNRS, CentraleSupélec, Univ. Paris-Sud, Université Paris-Saclay, Sorbonne Université,

3 & 11 rue Joliot-Curie, Plateau de Moulon 91192 Gif-sur-Yvette CEDEX, France 3 Department of Mathematics, Texas A&M University, College Station, TX 77843 USA Xavier.Mininger@geeps.centralesupelec.fr, Caroline.Nore@limsi.fr Abstract-We present a numerical study of the cooling of a 40 kVA electrical transformer by using an oil-based ferrofluid. The temperature and velocity fields are compared to that obtained with regular transformer oil to assess the benefits of the ferrofluid. The interest of ferrofluids is in the thermomagnetic convection phenomenon generated by the leakage magnetic field. An axisymmetric model of the transformer shows that the ferrofluid reduces the maximum temperature.

Index

Terms-Electrical transformer, ferrofluid, thermomagnetic convection, finite element method.

I. PROBLEM AND MODELING

Several research groups have proposed to enhance heat transfer in power transformers by taking advantage of the leakage magnetic flux in the insulating liquid [START_REF] Segal | An investigation of power transformer cooling with magnetic fluids[END_REF]- [START_REF] Patel | Prevention of hot spot temperature in a distribution transformer using magnetic fluid as a coolant[END_REF]. Their approach relies on the use of oil-based ferrofluid as cooling agent. For this type of fluid a phenomenon called thermomagnetic convection can be strong enough to enhance the mixing effects of thermogravitational convection.

To assess this claim, we perform computations on an electromagnetic system resembling a 40 kVA (20 kV/400 V) transformer. We use an axisymmetric model, see Fig. 1.We do not model the full magnetic circuit since we are only interested in the leakage magnetic field, the Joule effect generated by the coils and the induced natural convection of the liquid. The volume fraction of magnetic nanoparticles in the ferrofluid is of only 1%. The governing equations consist of the Navier-Stokes equations augmented with the Boussinesq approximation and a magnetic body force (Kelvin force model here), the temperature equation and Maxwell's equations. A similar set of equations is described in [START_REF] Zanella | Study of Magnetoconvection Impact on a Coil Cooling by Ferrofluid with a Spectral/Finite-Element Method[END_REF]. On the tank walls, we use Robin boundary conditions for the temperature with a high convection coefficient to model the presence of heat transfer fins. The equations are solved with the SFEMaNS finite element code [START_REF] Zanella | Study of Magnetoconvection Impact on a Coil Cooling by Ferrofluid with a Spectral/Finite-Element Method[END_REF].

II. RESULTS

Fig. 1 :

 1 Fig. 1: Transformer model (H t = 46 cm, R t = 21.1 cm).

Fig. 2

 2 Fig. 2 compares the temperature fields and velocity field streamlines at quasi-steady regime in a meridian plane (revolution axis on the left). Results with regular transformer oil are shown on the left and results with the ferrofluid are shown on the right. Using ferrofluid reduces the maximum temperature in the primary coil by 2.2 • C and in the secondary coil by 5 • C. Near the coil bottom, the streamlines show that the magnetic force reinforces the convection pattern entailing a temperature homogenization and a decrease of the maximum temperature. The main limit of this setup is that the coils are superimposed: the magnetic fields generated by the coils, carrying opposite currents, almost cancel each other everywhere except close to the coils. Other transformer geometries should be tested to improve the thermomagnetic convection effect. More details on the modeling and results are available in [4, pp. 137-154].

Fig. 2 :

 2 Fig. 2: Temperature field ( • C) at t = 20000 s and velocity field streamlines. The arrow represents the flow direction.


David • Marchal
email: david.marchal@centralesupelec.fr

Alexandre • Fougnie
email: alexandre.fougnie@centralesupelec.fr

Thomas • Schmitt

Sébastien Ducruix

Study of the acoustic damping in a nonreactive pressurized chamber combining experiments, simulations and modeling

Keywords: Thermoacoustic, High-frequency, Instability, Liquid Rocket Engine, Large Eddy Simulation Reduced Order Modeling, Damping

Acoustic damping, a key element for the accurate prediction of combustion instabilities, is studied in this work through the combined analysis of the NPCC test bench. This gaseous nonreactive test bench, developed for the study of acoustic damping in a coupled-cavity configuration, is composed of a dome, a chamber and 3 injectors. It mimics the geometry of a LRE in a simplistic way. An acoustic modulation at three different frequencies -corresponding to acoustic modes of the system -is achieved using a perforated wheel. Large Eddy Simulation (LES) properly reproduces the experimental pressure signal for the 3 cases. Then, a Reduced Order Modeling (ROM) approach is presented with a modal expansion, which retrieves numerical results in a linear framework.

Introduction

Thermoacoustic instabilities within liquid rocket engines (LRE) result from a coupling between the dynamics of the injection, the combustion process and the propulsive system's acoustics. This longstanding issue [START_REF] Crocco | Theory of combustion instability in liquid propellant rocket motors[END_REF] affects engine performance and may lead to its destruction. Several approaches exist to counter the appearance and growth of instabilities. Experimental tests on real-size engines make it possible to iteratively modify the combustion chamber's geometry to shift the modes' frequency or modularize their structures, as during the development of the F-1 engine [START_REF] Oefelein | Comprehensive review of liquid-propellant combustion instabilities in F-1 engines[END_REF]. Large Eddy Simulation (LES) allows an analysis of the phenomena at stake to understand the flame response under acoustic perturbation, the coupling between the combustion chamber and the injection system [START_REF] Hakim | Large eddy simulations of multiple transcritical coaxial flames submitted to a high-frequency transverse acoustic modulation[END_REF]. A first try to characterize the acoustic damping in a 2D LES has been presented in [START_REF] Nez | Experimental and numerical characterizations of acoustic damping rates in a coupled-cavity configuration[END_REF]. Nevertheless, although having made significant progress, LES still requires time and resources, ill-suited for engine design loops. Thus, reduced order modeling (ROM) strategies have emerged in recent years, aiming to reduce the need for long and costly hot-fire tests or LES [START_REF] Laurent | A novel modal expansion method for low-order modeling of thermoacoustic instabilities in complex geometries[END_REF]. However, mechanisms and coupling involved in combustion instabilities being still not well understood, experimental lab-scale tests and LES are still required to study the engine dynamics as a prerequisite and in order to feed the ROM solvers.

Although combustion models exist accounting for the unsteady rate of heat release under transverse acoustic solicitations, few studies offer a precise analysis of the mechanisms underlying the damping process. Within the New Pressurized Coupled Cavities (NPCC) experimental setup developed by Laboratoire EM2C in the framework of a collaboration with CNES and ArianeGroup, a three-injector cold-flow test rig mimicking the geometry of a LRE [START_REF] Gonzalez-Flesca | Reduced order modeling approach to combustion instabilities of liquid rocket engines[END_REF] presented in Sec. 2, the competition between damping and excitation is first reproduced using LES computations (see Sec. 3). Then a reduced-order representation of this test bench using a Galerkin modal expansion methodology proposed in the StaHF code (Stability High Frequency) [START_REF] Méry | Mécanismes d'instabilités de combustion haute-fréquence et application aux moteurs-fusées[END_REF] is derived (see Sec. 4).

Experimental setup

Geometry The NPCC test rig is the latest evolution of the setups developed at EM2C to investigate acoustics in cavities and injector dynamics, conceived and operated during the PhD thesis of Manuel Gonzalez-Flesca [START_REF] Gonzalez-Flesca | Reduced order modeling approach to combustion instabilities of liquid rocket engines[END_REF]. The rig was designed to mimic the geometry of a LRE in a simplistic but wellcontrolled manner. It is composed of a dome and a chamber linked by three injectors, 6 mm in diameter. The chamber features three sections that can be assembled or removed to modulate the length and therefore the acoustic eigenfrequencies of the rig. The dimensions were set to allow for a coupling between the dome and the chamber. The air flow enters the system by a pipe located on the side of the dome, perpendicular to the injectors. Two nozzles at the end of the chambers allow the flow to exit the rig. It is brought to an operating pressure of 3.5 bar. The total mass flow rate is ṁtot = 3.1g/s. Fig. 1 presents a transverse cut of the rig, with the location of the pressure sensors. The probes are located to be either at a node or an antinode of pressure, depending on the excited mode. The acoustic modulation is achieved using the Very High Amplitude Modulator (VHAM) [START_REF] Méry | Experimental investigation of cryogenic flame dynamics under transverse acoustic modulations[END_REF]. A fully perforated wheel is used for continuous modulation of the eigenmodes of the rig. The VHAM is placed at the system's exit, for the wheel to block alternatively one nozzle or the other.

Cases studied Three experimental cases are reproduced here with LES: continuous modulation of mode 1T (fully transverse), 1T1L (transverse and longitudinal components with coupled dome and chamber) and 1T2L (transverse and longitudinal components, uncoupled). It is critical to pinpoint precisely the eigenfrequencies (Table 1). The experimental results were obtained by Gonzalez-Flesca [START_REF] Gonzalez-Flesca | Reduced order modeling approach to combustion instabilities of liquid rocket engines[END_REF] using a sweep over the excitation frequency. The frequencies retained for the numerical study were obtained with the Helmholtz solver AVSP from Cerfacs [START_REF] Nicoud | Acoustic modes in combustors with complex impedances and multidimensional active flames[END_REF]. They are close to the one found in experiments.

Large Eddy Simulation

Setup The AVBP code from Cerfacs and IFPEN is used [START_REF] Schönfeld | Steady and unsteady flows simulations using the hybrid flow solver AVBP[END_REF]. A dissipative Two-Step Taylor-Galerkin (TTGC) numerical scheme, third order in time and space, is used [START_REF] Rudgyard | Development of high-order Taylor-Galerkin schemes for unsteady calculations[END_REF]. Subgrid-scale momentum fluxes are modeled by the Wall Adapting Large Eddy (WALE) model [START_REF] Nicoud | Subgrid-scale stress modelling based on the square of the velocity gradient tensor[END_REF], well suited for shear flows, with eddy-viscosity vanishing in purely strained region of the flow. Compared with the experimental geometry (Fig. 1), the exit nozzles are smoothened to avoid sharp corners in zones of high velocity. Two meshes (labeled coarse and fine) are used for modulated cases which require a long physical time (at least 150ms) to reach the limit-cycle (Fig. 2). Table 2 summarizes the mesh details. At the inlet, speed and temperature are imposed in a soft way with a relaxation method. At the outlet, pressure is imposed in a soft way, with a superimposed sinusoidal acoustic wave to represent Table 2: Mesh information the excitation by the VHAM for modulated cases. An analytic study (not detailed here to limit the paper length) determines with a reasonable precision the amplitude of the acoustic modulation as a function of the mean chamber pressure : p max = 0.45 bar. The mean pressure at the outlet, 2.95 bar, represents the situation when both nozzles are half-opened. The nozzles are equipped with slippery walls to limit the headloss in the vicinity of the outlet where the excitation takes place. Two boundary conditions are tested for the injectors: adiabatic no-slip walls with zero velocity and walls equipped with an adiabatic wall function for imposing the wall shear stress. Influence of the grid resolution and injector's boundary condition is tested by computing a case without modulation on the coarse and fine meshes. In addition, the mesh adaptation strategy described by Davillier et al. [START_REF] Davillier | A mesh adaptation strategy to predict pressure losses in LES of swirled flows[END_REF] is used to obtain local mesh refinement in critical places such as the inlet and outlet of each injectors. It is found that the coarse grid equipped with law of the wall boundary conditions was able to retrieve jet topology and injector head loss close to the ones found on finer grids.

Results Mode shapes are correctly retrieved by the LES compared to the AVSP results (Fig. 4).

The LES signals are very close to the experiments (Fig. 3), highlighting the capacity of the LES to reproduce the competition between excitation and damping in this configuration, without any adjustments from the user. Very similar results are found when computing the case on the fine mesh, ensuring grid independence. The implementation of disturbance energy budgets [START_REF] Brear | Disturbance energy transport and sound production in gaseous combustion[END_REF] in the AVBP code gives insight on the contribution of each phenomena responsible for the acoustic damping in this configuration. The final goal is to develop damping models for the reduced-order approach presented below. Galerkin modal expansion StaHF (Stability High Frequency) [START_REF] Méry | Mécanismes d'instabilités de combustion haute-fréquence et application aux moteurs-fusées[END_REF], the reduced order modeling tool developed at EM2C, is based on a Galerkin modal expansion, first introduced by [START_REF] Zinn | Application of the Galerkin method in the solution of non-linear axial combustion instability problems in liquid rockets[END_REF] and extensively discussed in [START_REF] Culick | Unsteady motions in combustion chambers for propulsion systems[END_REF]. The acoustic pressure variations p (x, t) are projected onto the acoustic eigenmodes orthogonal base (Ψ n ) n≥1 such that, considering N modes, p (x, t)

= N n=1 η n (t)Ψ n (x).
The projection allows to handle the geometry complexity and offers modularity. Using this decomposition in the acoustic equation, Eq. ( 1) is obtained for each mode. Once solved, this set of ordinary differential equations provides access to pressure variations by recomposition. Driving, damping and combustion source terms are expressed in the right hand side. The VHAM is modeled through a driving term (see [START_REF] Gonzalez-Flesca | Reduced order modeling approach to combustion instabilities of liquid rocket engines[END_REF]),

ηn (t) + ω 2 n η n (t) = -2σ damp n ηn (t) + ṁtot 2Λ n ω e c 2 (Ψ n (x 1 ) -Ψ n (x 2 )) sin(ω e t) (1)
with ω n the n th mode pulsation, Λ n = S Ψ 2 n (x)dV the mode norm, m tot the mass flow rate modulated by the VHAM, c the mean sound velocity and ω e the VHAM modulation pulsation. The damping coefficient σ damp n has to be evaluated based on experimental or numerical results.

Experimental and numerical comparison

To compare to experimental results, a ramp modulation of the VHAM is made, varying ω e linearly from 0Hz to 3000Hz over a 100s period. The modal expansion is performed over the first four transverse modes, see Fig. 4. Damping coefficients and modulated mass flow rate are obtain from [START_REF] Gonzalez-Flesca | Reduced order modeling approach to combustion instabilities of liquid rocket engines[END_REF]. The StaHF code is able to retrieve the experimental pressure signals from the ramp test, see Fig. 5, while noting some differences coming from the uncertainty on the damping coefficients and mass flow rate value.

A comparison with LES is performed in the linear regime, obtained by lowering the modulation amplitude to 10% of its nominal value. This means only the excitated eigenmode responds and no energy transfer is made to the harmonics. LES provides m tot = 5.1 * 10 -5 kg/s and σ damp n = 3s -1 , and result is shown Fig. 6. Due to mass flow rate variations during the initial transient phase in LES, differences are noted in the first part of the signal. Anyway, StaHF remarkably retrieves the limit cycle LES pressure level, clearly showing the potential of this kind of approach.

Conclusion and perspective

First, a LES of a gaseous non-reactive coupled cavity test bench is performed on the 1T, 1T1L and 1T2L eigenmodes on the NPCC testrig. LES reproduces the experimental data without any adjustment of parameters and allows a damping rate characterization. Then, a ROM strategy is proposed based on the modal projection of the pressure variations. This low-order representation, while using LES input parameters as the damping rate, retrieves the limit cycle pressure amplitude in the chamber, I.e. the competition between the damping phenomena and the excitation module (VHAM). Future work will focus on studying the damping mechanisms involved, such as viscous damping and acoustic/flow coupling at the injection units' outlets, and deriving models to account for these phenomena in StaHF.

Fig. 1 :

 1 Fig. 1: Schematic cut of the NPCC test rig equipped with the VHAM

Fig. 2 :

 2 Fig. 2: Meshes used, transverse cut, zoomed-in around the central injector. Left: Injector inlet; right: injector outlet. Top: coarse mesh, bottom: fine mesh

Fig. 3 :Fig. 4 :

 34 Fig. 3: p at HFc1 (chamber, top) and HFd1 (dome, bottom), LES on coarse mesh (high-pass filtered); Experiment (high-pass filtered).Left: 1T, Mid: 1T1L, Right: 1T2L

Fig. 5 :

 5 Fig. 5: HFc1 pressure sensor under the ramp excitation from experiment and computed with StaHF

Fig. 6 :

 6 Fig. 6: Pressure variation in the chamber (HFc1) for a 1T1L modulation, from LES and StaHF

Table 1 :

 1 Modal

frequencies found in the experiment and using the Helmholtz solver AVSP from Cerfacs

Acknowledgements Support provided by ArianeGroup, the prime contractor of the Ariane launcher cryogenic propulsion system and CNES, the French National Space Agency, is gratefully acknowledged. This work was granted access to the HPC resources made available by GENCI (Grand Equipement National de Calcul Intensif) under the allocation A0042B06176. A part of this work was performed using HPC resources from the mesocentre computing center of Ecole CentraleSupélec and Ecole Normale Supérieure Paris-Saclay supported by CNRS and Région Ile-de-France. We finally thank CERFACS for sharing AVSP and CERFACS and IFPEN for sharing AVBP.


Experimental Results from a Pilot Scale Latent Heat Thermal Energy Storage for DSG Power Plants -Advanced Operating Strategies

Pierre Garcia 1, a) , Grégory Largiller 1 , Gilles Matringe a) Corresponding author: pierre.garcia@cea.fr Abstract. Since 2013, CEA has been operating a pilot scale high pressure water-steam facility called LHASSA designed to test latent thermal energy storage modules under operating conditions similar to commercial Direct Steam Generation CSP plants. A Phase Change Material (PCM) storage module connected to this facility is composed of aluminum finned steel tubes immerged into sodium nitrate and surrounded by aluminum inserts for heat transfer enhancement. This paper presents the results obtained from the third test campaign on this storage module, consisting in 25 charge-discharge cycles under a wide range of operating conditions (fixed of sliding pressure, complete and partial charge levels…). Thermal performances of the storage test section show a very good repeatability, without any performance degradation compared to the previous test campaigns. Some new operating strategies were successfully tested (charge interruption simulating a cloud transient in the solar field, discharge with fixed pressure and varying mass flows, charge-discharge transition management).

CONTEXT AND OBJECTIVES

To date, there is no commercial solution for large capacity storage systems adapted to Direct Steam Generation solar (DSG) plants. Steam accumulators are widely used but this technology has rather low density and becomes expensive when large capacities are needed [1,2]. Storage technologies using Phase-Change Materials (PCM) are an interesting option as thermal storage systems for DSG plants, as they allow to minimize the steam pressure drop between charge and discharge. The most mature PCM storage technology to date is the shell-and-tube concept, in which the Heat Transfer Fluid (HTF) flows in tubes that go through a tank filled with PCM. Several prototype shelland-tube modules for steam storage have been built and studied by various groups [3][4][5]. At CEA Grenoble a shelland-tubes storage with sodium nitrate as Phase Change Material was connected to the LHASSA testing facility in 2013. Sodium nitrate shows good thermal properties as a PCM but it has a poor thermal conductivity, that is why we had to use fins and inserts to enhance heat transfer on PCM side. This new test campaign, performed in the framework of the InPower H2020 research project, has a double objective. The first objective, which is also one of the main goals of the InPower project, is to test the durability of PCM storage systems in order to demonstrate that their lifetime can be over 25 years. Most of the latent heat storage prototypes are operated during a few months and then abandoned or dismantled, but with this one we have several years of experience feedback. It is a great opportunity to study the durability of such systems and to improve their operating strategies. The second objective is to investigate advanced operating strategies, such as fixed pressure or modified-sliding pressure operation, or also cloud transients simulation, with the generic purpose of automating the control of the facility. Most of the PCM steam storage prototypes were tested in sliding pressure mode. However, fixed pressure operation or modified-sliding operation can ease the storage integration, by allowing acceptable conditions at turbine inlet all along the discharge process [6]. The present work reports and discusses PCM storage tests with advanced operating strategies, in particular steam discharge at constant power in fixed pressure mode.

DESCRIPTION OF THE TEST FACILITY AND TEST SECTION

CEA has built a pilot scale high pressure water-steam facility called LHASSA (145bar, 350°C), in order to validate at pilot scale the thermo-hydraulic behavior of thermal energy storage modules under operating conditions similar to commercial DSG CSP plants. The PCM storage module connected to this facility is composed of aluminum finned steel tubes immerged into NaNO3 salt (300 kWhth, 6.3 tons, melting temperature of 306°C) and surrounded by aluminum inserts for heat transfer enhancement (see Fig. 1). This test section is highly instrumented: the PCM temperature is measured around 25 tubes at seven locations along the module height, and at three radial positions around the tubes. Water-steam pressure, temperature, mass flow and steam quality are also measured at the inlet and outlet of the test section.

During the charging process of the storage module, steam slightly above saturation is sent from the top to the exchanger tubes where it condenses causing the melting of the PCM. Liquid water level in the storage module is either kept constant and low in sliding pressure mode so that steam could condense all over the tubes height, or moves downwards in fixed pressure mode. Inversely, during the discharging process, liquid PCM solidifies and causes the evaporation of the liquid water coming from the bottom of the test section within the exchanger tubes. In this case, the liquid water level in the storage tubes is either kept constant and high in sliding pressure mode to maintain nearly uniform temperatures all over the module height, or moves upwards in fixed pressure mode.

RESULTS FROM THE 2019 TEST CAMPAIGN Durability Analysis

Thermal performances of the storage test section show a very good repeatability, without any performance degradation compared to the previous test campaigns. To check it, the same test conditions were repeated several times during the three campaigns. Figure 2 shows some of the main storage variables (steam pressure, mean PCM pressure, and liquid water level in the tubes) during a complete charge and discharge cycle in nominal conditions tested in the first campaign in 2013 (C1) and repeated during the third campaign in 2019 (C3). These tests last 17 hours and involve two shifts of two operators each. The facility is operated at design power (50 kWth) in charge and in discharge.

FIGURE 2.

Steam pressure (in purple), liquid water level in the tubes (in orange), and average PCM temperature (in green) during a complete charge and discharge cycle in nominal conditions: results from the first campaign (C1) in continuous lines, results from the latest campaign (C3) in dotted lines Very similar performances were obtained, as it can be observed in Fig. 2 representing the pressure (in purple), the liquid water level in the tubes (in orange), and the average PCM temperature (in green): curves in dotted lines, from the latest campaign (C3), are close to the curves in continuous lines from the first campaign (C1), specially pressure variations. Some slight differences can be seen in the mean PCM temperature variations, they can be mainly attributed to the fact that the initial temperature of the storage module was lower in C3 and that the steam mass flow was not constant in charge in C1. The charged and discharged energy can be estimated from enthalpy balances on HTF flows at the inlet and outlet of the storage module, the end of discharge being defined by a steam pressure minimum threshold (68 bars in the first case and 75 bars in the second case). Steam outlet enthalpy in discharge is assessed thanks to an energy balance at the boundaries of the condenser. From these data storage efficiencies can be calculated, leading to very similar results for both test campaigns, as shown in Table 1 below. Semi-automated control was also considered, by operating various consecutive charge-discharge cycles: the storage was charged during night (without any human presence) at fixed pressure with a reduced thermal power

Advanced Operating Strategies

Some new operating strategies were successfully tested (charge interruption simulating a cloud transient in the solar field, discharge with fixed pressure and varying mass flows, charge-discharge transition management). Some experimental test results from two charging-discharging cycles with steam discharge at fixed pressure are shown in Fig. 4. In such tests, the objective is to produce a stable steam mass flow (and thus constant thermal power) at constant pressure. This is achieved by gradually increasing the liquid water level in the storage tubes to increase the heat exchange surface where evaporation occurs. Figure 4a presents a partial storage cycle with sliding pressure in charge (from 11 AM to 2:15 PM) and an outlet steam pressure controlled at 80 bars in discharge (from 2:30 PM to 4:30PM). Liquid water level in the storage tubes is controlled at 40% in charge and increases from about 30% to 90% during discharge. Figure 4b shows a complete storage cycle, that is to say that all the PCM active volume is melted in charge and solidified in discharge. An advanced pressure control was tested in charge: in sliding mode from 11:30 AM to 3:30 PM, and then fixed at 102 bars from 3:30 PM to 6:45 PM. Outlet steam pressure is controlled at 80 bars in discharge with a reduced mass flow. Liquid water level in the storage tubes increases from about 15% to 100% during discharge.

Such steam discharges at constant pressure and constant mass flow could be beneficial for the steam user, in particular for steam turbines or steam networks designed to operate in stable conditions. Other hybrid control strategies like sliding-then-fixed pressure were also successfully tested. Another control objective was to test the storage system behavior during a cloud transient: therefore, it was simulated by stopping temporarily the steam flow during the charge process. Similarly, the steam production was stopped during discharge to see how the storage behaves when the steam consumption is temporarily off. Such flow stops last about 30 minutes each; they can be observed in Fig. 5 two hours after the beginning of the test in charge, and about five hours and 30 minutes after the beginning of the test in discharge. Figure 5a shows energy balances of the storage system, and Fig. 5b shows the PCM temperatures at different altitudes. In both charge and discharge, no control difficulties were encountered and the thermal performances were not altered.

CONCLUSIONS AND PERSPECTIVES

To conclude, this new test campaign gave encouraging results on the durability of such PCM storage systems, as no signs of degradation of the storage performances were observed after 6 years and 80 charge-discharge cycles. Nevertheless, more data is still needed to better estimate the lifetime of such technologies. That is why a dedicated new facility called DURASSEL has been built and commissioned to study the corrosion of pressurized finned tubed submerged in sodium nitrate, with the objective to analyze tube samples after hundreds of melting and solidification cycles. This facility will allow to better understand the corrosion mechanisms, thanks to microstructural analyses of tubes and fins, PCM analyses by Inductively Coupled Plasma (ICP) and calorimetry measurement, and a monitoring of the gas composition above the PCM under air or nitrogen.

Moreover, this test campaign allowed us to investigate advanced control strategies, to improve the procedures towards automated tests, and to gather new experimental data to validate our dynamic models. Finally, repeated tests proved that steam can be discharged at constant pressure and at constant mass flow during more than five hours.

FIGURE 1 .

 1 FIGURE 1. Picture (left) and schematic (right) of the PCM module tested during the IN-POWER project LHASSA facility was operated during 225 days from 2013 to 2015 [5, 7] and then started again in 2019 for the IN-POWER H2020 project with the same test section and the same PCM, after passing a regulatory periodic inspection of the test section and pressurizer consisting in endoscopic visits and wall thickness measurements. Instruments (mass flowmeters, pressure sensors) were calibrated and leakage tests were done. The test campaign started in September 2019, 25 charge-discharge cycles have been performed, exploring a wide range of operating conditions (fixed of sliding pressure, complete and partial charge levels…).

FIGURE 3 .

 3 FIGURE 3. (a) Mean PCM and water temperatures at different locations, and (b) level-averaged PCM temperatures (T1 indicating the bottom of the tank and T7 indicating the top of the tank) during four consecutive complete charge and discharge cycles

FIGURE 4 .

 4 FIGURE 4. Pressure, mass flow, liquid level, and temperatures on water-steam side in the PCM storage section a) During a partial storage cycle (on September 16 th ); b) During a complete storage cycle (on December 3 rd )

FIGURE 5 .

 5 FIGURE 5. (a) Stored energy, PCM melted mass fraction and water pressure, and (b) level-averaged PCM temperatures (T1 indicating the bottom of the tank and T7 indicating the top of the tank) during a partial storage cycle with 30-minute flow stops in charge and in discharge. Charging periods are represented between dashed orange lines, and discharging period are represented between dashed blue lines.

  1 , Lucas Champelovier 1 , and Sylvie Rougé1

1 Univ. Grenoble Alpes, CEA, LITEN, DTBH, Thermal Storage Laboratory 17 rue des Martyrs, F-38054 Grenoble Cedex 9, France.

TABLE 1 .

 1 Comparison of the thermal performances of the storage module from two reference tests from 2013 and 2019

	(End of discharge criterion)	2013 test campaign	2019 test campaign
	Charged Energy	325 kWhth	374 kWhth
	(Final steam pressure: 68.5 bars)
	Discharged energy	306 kWhth	342 kWhth
	Storage efficiency	94%	92%
	(Final steam pressure: 75 bars)
	Discharged energy	249 kWhth	284 kWhth
	Storage efficiency	76%	76%

ACKNOWLEDGMENTS

This project has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement No 720749.