Jacobsen Bjørkås

Marja Van Den Heuvel-Panhuizen
email: m.vandenheuvel-panhuizen@uu.nl

Measuring area on the geoboard focusing on using flexible strategies

Keywords: Measurement, area, digital geoboard, strategies

In this paper we report about a small-scale classroom experiment on teaching area measurement with a digital geoboard. The two-lesson teaching sequence that was developed, was tested in two sixth-grade classes and was aimed at teaching students using measurement strategies in a flexible way. The pre-and posttest data were analysed both on task and student level. The results show that most of the students developed more flexible strategies, but many of them did not use them accurately. Our results indicate that apart from working on the ability of students to find correct answers, also the flexible use of strategies can be realized through education.

Introduction

Students often say that area "is length times width". This suggest both that the conceptual understanding of area is lacking, and that they only have one strategy available for finding the area of figures, even if they have learned about finding area by counting units earlier: "It [...] is typical of many learning processes, especially in mathematicsthat the original sources of insight have been clogged, and the way back to insight is blocked by the processes of algorithmising and automatising." (Freudenthal, 1983, p. 209). In the new curriculum for Norwegian primary school, in-depth learning is important, and among other things the students are expected do "construct a robust and flexible understanding" (Ministry of Education and Research, 2015). Therefore, it is important to develop teaching that supports the learning of flexible strategies for area measurement.

Theoretical framework and research question

Knowledge about measuring the area of polygons consists of both conceptual and procedural knowledge which are mutually supportive and should both be fostered [START_REF] Rittle-Johnson | Not a One-Way Street: Bidirectional Relations Between Procedural and Conceptual Knowledge of Mathematics[END_REF]. Having a good understanding of measuring area means that students have several strategies available to determine the area of polygons. For example, they may count square units and/or half square units, make a multiplication to find the area of rectangles, decompose polygons into smaller, simpler polygons and use addition, may embed a polygon into a larger, simpler polygon and use subtraction, and may be able to apply a formula to calculate the area. Essential is that students can apply these strategies in a flexible way fitting to the problems at hand. However, there is evidence [START_REF] Huang | Developing children's conceptual understanding of area measurement: A curriculum and teaching experiment[END_REF]) that primary school students fail often in area measurement problems and have difficulties in flexibly handling area measurement problems. According to [START_REF] Huang | Developing children's conceptual understanding of area measurement: A curriculum and teaching experiment[END_REF] this finding is considered to be linked to the emphasis that in teaching is put on formula memorization rather than on conceptual understanding and explaining your reasoning. Further, students who develop flexibility in problem solving use or adapt existing strategies to unfamiliar problems, and show a greater understanding of concepts [START_REF] Star | The role of prior knowledge in the development of strategy flexibility: The case of computational estimation[END_REF]. Therefore, it is important to investigate how students' flexible use of strategies in measuring area and their awareness of using them can be supported.

Most of the research on strategy flexibility or strategy adaptivity has been on mental calculation (e.g., [START_REF] Threlfall | Strategies and flexibility in mental calculation[END_REF][START_REF] Verschaffel | Conceptualizing, investigating, and enhancing adaptive expertise in elementary mathematics education[END_REF] or on solving nonroutine word problems [START_REF] Elia | Exploring strategy use and strategy flexibility in non-routine problem solving by primary school high achievers in mathematics[END_REF]. In the study of [START_REF] Verschaffel | Conceptualizing, investigating, and enhancing adaptive expertise in elementary mathematics education[END_REF] a distinction is made between task, subject, and context adaptivity of strategies. In the present study we built on this research and explored how task flexibility of strategies for finding the area of polygons can be taught. We will not use the term "adaptivity" in this paper because although we will discuss appropriateness of strategies with the students, the main goal is to teach multiple strategies and discuss the relationship between them. Further, we will focus on inter-task flexibility in the tests [START_REF] Elia | Exploring strategy use and strategy flexibility in non-routine problem solving by primary school high achievers in mathematics[END_REF], investigating whether students change their strategies from pre-to posttest.

One tool that is sometimes used for teaching measurement of area is the geoboard. However, there are only a limited number of empirical studies on this topic. For example, Bair and Cady (2014) observed that by means of a geoboard students could find Pick's theorem and the study of Britton and Stump (2001) showed that students could develop cut-and-paste strategies. To our knowledge, no research has been carried out to investigate whether the geoboard is suited to foster the development of flexible strategies. Therefore in this study our research question was: Does working with the geoboard help students to use flexible strategies for measuring area?

Method

To answer the research question, a small-scale classroom experiment comprising a pre-postretention-test research design was carried in two sixth-grade classes with 46 students during the period of February-July 2017. For this experiment we designed and tested a teaching sequence of two 75-minute lessons on reasoning about area measurement based on working with a digital geoboard.

The digital geoboard ("Geoboard [Computer software]," 2017) is an app for a tablet or computer that resembles the physical geoboard. Students can manipulate polygons and write computations or explanations on the digital geoboard, and they can share their work with the teacher through a sharing platform ("Showbie [Computer software]," 2018) or show it on a projector in the classroom. This makes it possible to select and present students' strategies to emphasize the connections between different strategies and give value to different strategies for the same task.

The lessons were taught by the researcher in the presence of the classroom teacher. In the lessons whole-class discussions led by the researcher were interchanged with students' individual work on the geoboard. In the discussions the use of different measurement strategies was stimulated, both strategies proposed by the students and strategies planned by the researcher, and students were made aware of the relationships between the different strategies. To support these relationships, the strategies were represented visually by drawings on the geoboard. To encourage students to think about their strategies it was emphasized that it is very important to be aware of the used strategy and that good understanding means that one can explain how the answer was found. This encouragement was given to the students both during their individual work and the whole-class discussions.

Type of task Description

Task The pre-, post-, and retention tests were administered in Week 5, 7 and 22 respectively. They contained several types of measurement tasks presented on paper-and-pencil test sheets. Table 1 shows which type of task is included in which test. In every test, for each type of task the to be measured area was varied a bit in order to avoid retest effect.

Figure 1 shows an example of how the tasks in the retention test were presented to the students. The question asked to the students was: «Can you find the area of the shape? Write how you were thinking. You may use the picture to explain.». The yellow square was given as an area unit. On a next page the students were asked to to give a justification for their result. During the pretest it turned out that after giving this prompt it was for most students easy to say what their thinking was, but writing this down was harder for them. Therefore, the teacher and the researcher made clear to the student that they just had to write down what they said orally. The data from the test sheets were entered into NVivo. Both the students' measurement results and their explanations of the used strategies (see for examples Figure 2 and 3) were coded. When the students found the correct area, the answer was coded as "1" and an incorrect answer was coded as "0". Based on these codes we divided the students per task into four classes: the "00" students (incorrect answer in pretest and posttest), the "01" students (incorrect answer in pretest, correct answer in posttest), the "10" students (correct answer in pretest, incorrect answer in posttest), and the "11" students (correct answer in pretest and posttest). For coding the strategies, we used three main categories, namely "counting", "multiplication", and "restructuring" (decomposing or embedding strategies). Furthermore, we had two additional categories, namely "other" and "no strategy information available".

Results

Table 2 shows for each type of task and for each test the percentage of correct answers. For all the tasks which were both in the pre-and posttest, the students improved their performance of which the most gain was obtained for the Blue task. In the retention test this progress continued only for the Red and the Brown task. The next step in our analysis was figuring out what strategies the students used to find the area of the polygons and how flexible they were in using these strategies. We analyse the Red task as an example, because this is in our view the most interesting task that was on both the pre-and the posttest.

The strategies used in the Red task (see Table 4) were counting (in this case the number of squares and triangles), multiplication (length × width; leading to mistakes because diagonal distances cannot be measured by counting), and restructuring. In the lessons it was made clear that this latter way of measuring the area for this task is very suitable, and we also showed why the multiplication strategy could not work easily. For this Red task, no student used the restructuring strategy before the lessons. Yet, after the lessons, nine students tried to apply this strategy, of which three got the correct answer. The work of one of these students is shown in Figure 2. The six students who got a wrong answer made various mistakes. Four of them give a clear explanation of the procedure of embedding the rectangle in a larger square and "subtracting what is in the corners which is not in the red [rectangle]", but proceeded with computing the wrong area of the square (one student got 7 × 7 as a result) or computed the wrong area that had to be subtracted (three students). Of the other two students the source of the error is unclear.

The fact that four "10" students go from correctly using a counting strategy to making mistakes with a restructuring strategy, is possibly related to the value we gave to this strategy in the plenary parts of the lessons.

Task type Strategy

Number of students belonging to a class of students Total "00" student "01" student "10" student "11" student Table 5 summarizes the findings of the correctness of the answers per type of task and whether there was a change in strategy between the pre-and posttest. In the Blue and the Brown task more students retained the same strategy than in the other tasks. Moreover, they even did not change the strategy when they got a wrong result in the pretest. In the Red task more changes happened. Here 60% of the students changed their strategy. We see in this table that many students use different strategies in the pre-and posttest, for instance 18 students changed strategies in the red task, but 7 (3+4) did not come up with the correct answer. Taking again the perspective of the students, we could identify for 22 students their strategy flexibility in the pre-and posttest over the Blue, Red, and Brown task. As it is revealed by Table 6, a few students only used one strategy, but most students changed their strategy from the pretest to the posttest. Note, that in this analysis the students which applied a strategy that could not be classified or was not available in the pre-and/or the posttest are not taken into account.

Student's inter-task flexibility Pretest Posttest

Using the same strategy on all tasks 7 5

Using two different strategies 15 14

Using three different strategies 0 3 We also analysed whether students used the same strategies or different kinds of strategies on the three tasks on the pretest, and the same question on the posttest, to investigate the inter-task flexibility of the students. We see that the inter-task flexibility of the students have increased on the posttest relatively to the pretest (table 7).

Discussion

In this study, we investigated whether working with the geoboard could help students use flexible strategies for measuring area of polygons.

The problems on the pretest and posttest were routine in the sense that the polygons were shapes that the students could have worked with before. However, the problems were given on a grid and without rulers and calculators that the students might be use to have when measuring areas of rectangles. The students showed some inter-task strategy flexibility already at the pretest, choosing Not changing strategy between pre-and posttest but changing it over tasks 2

Changing strategy between pre-and posttest in one task 11

Changed strategy between pre-and posttest in two tasks between multiplication and counting strategies. This is in contrast to the situation where students are given unfamiliar, complex problems, where the unfamiliarity and complexity of the problems may hinder the flexible change of strategies [START_REF] Elia | Exploring strategy use and strategy flexibility in non-routine problem solving by primary school high achievers in mathematics[END_REF]. However, the pretest also showed that some students chose inappropriate strategies, e.g. multiplication on tasks where this could not lead to a correct answer, even though they showed that they knew how to use a counting strategy that would have been appropriate.

Working with the geoboard, the students tried different strategies and shared with each other through whole-class discussion. They also tried the strategies that the researcher emphasized or introduced, e.g. counting or different rearranging strategies. The pre-and posttest results taken together (table 6 and7) show that many students have developed their inter-task flexibility. Heinze et al. [START_REF] Heinze | Addition and subtraction of three-digit numbers: Adaptive strategy use and the influence of instruction in German third grade[END_REF] raise the question whether the "investigative approach" that was used in the teaching, where students discuss and practice selected strategies, "prevent students from choosing strategies adaptively because they tend to ignore problem characteristics". Because the posttest items are related to tasks which were discussed in the teaching, we can't address this question with certainty; however, the results show that many students on the posttest used more appropriate strategies e.g. on the Red task, even if they didn't use them accurately.

Conclusion

Considering that this was only a small-scale classroom experiment with an intervention consisting of only two lessons, it was hard to draw conclusions. Nevertheless, although the students did not substantial improve their performance in measuring area, we can say that we have experienced that most of the students were active in thinking and trying out strategies for measuring area which were new to them, so working with the geoboard and discussing strategies in whole class helped many students develop their inter-task flexibility for measuring area. This made us more aware that apart from working on the ability of students to find correct answers, also the development of the flexible use of strategies is an important goal in mathematics education to work on.

In further work, we will do a controlled design experiment where the teaching period is longer, to be able to compare working with the geoboard with the ordinary curriculum, and to give the students the opportunity to develop more accuracy and strategy adaptivity.

Figure 1 :

 1 Figure 1: Test sheet with the polygons and the questions the students have to answer

Figure 2 :

 2 Figure 2: Test sheets of students using restructuring as a strategy applied in the Red task in the posttest resulting in a correct answer, and in the Brown task in the pretest making an error

Student

Table 1 : Overview of the tasks in the tests which are used in the paper

 1

		included in
	Pretest	Posttest	Retention test

Table 2 : Percentage of students who gave correct answers to the different types of tasks

 2 pretest but right on the posttest, but 8 students got the answer right on the pretest but wrong on the posttest. Complete results are in table 3.

			Number of students belonging to a class of students
	Type of task
		"00" student	"01" student	"10" student	"11" student
	Blue task	0	8	2	26
	Red task	8	11	8	9
	Brown task	3	5	3	25

Even if the overall performance on each task was increasing, individual students showed both progress and deteriorating performance on each individual task. Using the red task as an example: Of the 36 students that participated on both the pre-and posttests, 11 students got the answer wrong on the

Table 3 : Number of students (n = 36) belonging to a particular class of students split out for the three types of tasks included in the pre-and posttest

 3

Table 4 : Students' (n = 36) strategies in the pre-and posttest applied in the Red task

 4

	Used strategy in the Red task in	Number of students belonging to a class of students
	Pretest  Posttest	"00" student	"01" student	"10" student	"11" student	Total
	Counting  Counting	0	3	3	5	11
	Counting  Restructuring	0	0	4	3	7
	Multiplication  Counting	1	7	0	1	9
	Multiplication  Restructuring	2	0	0	0	2
	Multiplication  Multiplication	1	0	0	0	1
	Other	2	1	1	0	4
	No strategy in pre-and/or posttest	2	0	0	0	2

Table 5 : Number of students using the same or a different strategy in the pre-and posttest split out per task type

 5

Table 6 : Number of students who showed a particular strategy flexibility

 6

Table 7 : Student's inter-task flexibility on the blue, red and brown tasks (N=22)

 7


Caroline Bulf
email: caroline.bulf@u-bordeaux.fr

Professional actions of novice teachers in the context of teaching and learning geometry

Keywords: Geometry education, teacher professional actions, beginning teachers

This paper aims to describe the professional actions of beginning teachers in the context of teaching and learning geometry. Using different theoretical tools, our research is based on the observation of lessons (with students aged between 6 and 11) delivered by beginning teachers in France. The paper seeks to show how the specific context of geometry impacts on professional actions especially among beginning teachers.

The first part of the article describes the theoretical background and research questions. The second part is dedicated to the methodology and results from several observations of novice teachers in the context of teaching geometry. The final part of the article is an opportunity to discuss with new perspectives in the context of pre and in-service teacher training.

Theoretical framework in terms of teacher professional actions

This research takes part of a bigger collaborative research which deals with different fields (mathematics, literature, sciences…); the goal is to study conditions of development of teacher professional actions through the follow-up of a cohort of dozen of beginning teachers. In order to compare analyses we need a common theoretical framework, we describe it in next part. [START_REF] Jorro | Professionnaliser le métier d'ensei nant[END_REF] distinguishes generic teaching actions from professional actions 1 . The former are attached to a collective culture shared by a community such as stereotyped actions (writing on the blackboard, calling a student up to the black-board, putting a finger to the lips meaning quiet, etc.) while the latter are more a singular and contextual conception. We refer mainly to [START_REF] Bucheton | Les gestes professionnels et le jeu de postures de l'enseignant dans la classe : un multi-agenda de préoccupations enchâssées[END_REF] to define more precisely teacher professional actions. These authors define professional actions as teacher's acts (physical or verbal) and the permanent "adjustments" in relation to student activity and knowledge at stake. These authors describe their organization and dynamics through the "multi-agenda of entrenched concerns [our traduction]" (Ibid., p.32; schema 1 for an adapted version). The five concerns are:

leading the lesson (organization and coherence of the lesson, through its material, spatial and functional dimensions); atmosphere (maintaining dialogical spaces); linking (making connections with what pupils already know or connections between other knowledges);

1 Our traduction from Jorro's original notions of « geste du métier » and « geste professionnel ».

scaffolding according to Bruner's meaning i.e. helping pupils do and say things without doing it for them [START_REF] Bruner | Le développement de l'enfant, savoir-faire[END_REF]; the targeted knowledge (central component).

For us, professional actions are specific, situated, on a given context and moment, and led through didactical intention(s) (that means these "intentions" are more or less with direct and conscious links with the teaching project). It is important to notice now this theoretical framework's goal is not about to give a binary classification generic/specific we recuse. What seems rightly interesting in the use of this general theoretical framework is that it can be mobilized in order to analyze any classroom session. Therefore, this is interesting in the cross of didactical analysis from different fields, which is the goal of the bigger research in which our collection of data fits. This paper deals with professional actions of beginning teachers in the context of teaching geometry.

Focus on beginners

In this framework, the dynamical organization of professional actions determines teachers' "postures of scaffolding2 " some of which can be exaggerated among beginning teachers. The "posture of control", for example, is frequently observed because they try above all to organize their work in relation to time and space, without taking into account the students' potential activity. "Linking is rare. Pupils are usually addressed collectively and the atmosphere is relatively tense [our traduction]" [START_REF] Bucheton | Les gestes professionnels et le jeu de postures de l'enseignant dans la classe : un multi-agenda de préoccupations enchâssées[END_REF]Soulé, 2009, p. 40). Furthermore, Robert andRogalski (in Vandebrouck, 2013) define teacher's practices as complex, stable and coherent. According to these considerations, it is therefore relevant to study the practices of teacher since their beginning to better understand the conditions of genesis and development of professional teacher actions. Especially since the generic teaching actions have been identified as being the ones that beginning teachers will try to incorporate into their practices [START_REF] Jorro | Professionnaliser le métier d'ensei nant[END_REF]. How do the teacher actions become professional and relevant? What impact do the specificities of teaching geometry have on the implementation and articulation of these actions? What impact on potential students' learning?

The specific context of teaching and learning geometry A lot of works relate the difficulties of teaching and learning geometry. Indeed, geometry is unique and complex in terms of the role of material activity (using instrument, manipulation, modeling…), visualization and language for the construction of mathematical concepts, as described in previous CERME [START_REF] Jones | Introduction to the papers of WG4: Geometry Education[END_REF]. We assume in our work there are mutual influences (and not subordinated one to another) between visualization (according to Duval's meaning, 2005), instrumental action (graphical register) and language (Bulf, Mathé & Miltalal, 2011). In doing so we seek to describe teacher's actions more precisely in our analysis, taking into account the different manifestations of geometrical concept (graphical register, verbal formulation, coding, physical gesture, etc.). According to Duval's framework (2005), we refer to the "dimensional deconstruction" to analyze geometrical object at stake in terms of figural units (2D for surface, 1D for line and 0D for point as intersection of lines), and relations between these different units. In other words, this is our additional and specific theoretical background to take into account the context of teaching and learning geometry.

The goal of this paper is to highlight the fact that professional actions are formed early (as mentioned at the beginning of this paper) but, depending on the task assigned to students in the context of teaching geometry, that they may hinder understanding of the geometrical concepts under consideration. We try to describe the reasons of these misunderstanding because, precisely, same professional actions could be, on the contrary, relevant in other context of teaching. That's why our theoretical framework seems particularly interesting because it contributes to show clearly the limits of dichotomy genericity / specificity of professional actions in pre and in-service teachers training. We summarize our theoretical framework in the schema 1. The central component (from Bucheton and Soulé' framework) is here the geometrical knowledge at stake, and therefore according to our theoretical framework, it is observed in activity through three dimensions: visual, oral and instrumental. The specific context of geometry teaching with beginning teachers allow us to highlight obstacles or potential levers to better understanding professional development of teacher. This paper tries to answer to the research questions : How do the specificities of teaching geometry (in terms of visualization, instrumental action and language) have an impact on the implementation and articulation of professional actions (schema 1) ? What impact on potential students' learning?

Methodology and results

Our corpus is composed of several videos and transcripts of lessons, visit reports, professional writings, interviews, etc. collected during pre-service teachers and during the first years of practice of a dozen of beginning teachers since 2013. In the present contribution we chose to refer to the professional actions of three teachers, Maya, Emilie and Celine observed in 2016 and 2017. We think these case studies are representative of novice teachers' practices in the context of teaching and learning geometry3 .

Our analysis methodology is qualitative and based on classical models in the French field of didactics of mathematics, known as a priori or a posteriori analysis according to the Theory of didactical situations in mathematics [START_REF] Brousseau | Theory of didactical situations in mathematics[END_REF]. A priori analysis is an epistemological reasoning which does not have a predictive sense, but instead a causal one; it involves describing various possible (and therefore potentially reproductible) phenomena in teaching session. Our a posteriori analysis places contingent facts in the context created by the a priori analysis. Furthermore, our data are analyzed in accordance with our theoretical framework, which means we describe geometrical activity (a priori and a posteriori) through its differents manifestations: visual, instrumental or verbal (as already described in previous paragraph). We infer from these analyzes hypotheses in terms of professional actions as we have described in our general theoretical framework (schema 1). We pay attention on the different type of teacher's scaffolding, leading, linking and atmosphere own to these teachers in relation with the geometrical knowledge at stake.

Over-scaffolding through different dimensions (verbal / visual / instrumental) of geometrical activity

This episode is frequently encountered in classes ; it's about talking collectively about a figure on the blackboard. We refer here to a observation occurred in 2017 in Celine's class (with students aged 8 to 9). Students have to reproduce the figure given on the left extremity of the figures 1. The different ways of seeing one side of the square as also the diameter of the half-circle4 is the key to solve this problem. It is necessary to go beyond a first vision as a simple border line between two juxtaposed figures:

-this side is perpendicular to 2 other sides (relationship between 2 objects of dimension 1D);

-this side is a segment, which is also a diameter of the half-circle. The middle of this segment is the center of the circle (different objects 1D and 0D, and their relationships).

These different ways of seeing this "simple" figure are not self-evident because it involves different figural units (lines, points) and different kind of relationships (belonging, middle, perpendicular, equal distance...). If we focus on the center of the circle, there is a priori a lot of different ways to understand it (without being exhaustive):

-a point belonging to the side of the square (which requires to see a line as a set ofinfinitepoints, but it is far from being acquired for students of this age);

-the middle of a segment of which the ends are the vertices of the square: here coexist two ways to see points (vertex / end of segment);

-the center of the circle is where we will put one branch of compass ;

-the center is the point located to equidistant from the line (or points) defining the half-circle;

-the center is the junction point of the union of two rays in the same direction (here); In less than two minutes, there are 9 different oral designations (square, side, line, diameter, radius, middle of the line, center of the circle, middle of the diameter, this-there) and 3 different physical actions that all rely on the same graphical sign on the blackboard. The same sweeping movement of the hand sometimes refers to the diameter of the circle or the side of the square or the radius (if the amplitude is shorter). The pointing finger indicates at the same time where to put the branch of the compass, the center of the circle, the middle of the diameter, the middle of the side of the square, the junction between two rays. This kind of scalfodding, both verbal and physical, characterizes the teacher's will to help student through multiple designations, reformulations,… and as the beginners think they should provide. But this exemple is, in our opinion, representative of potential misunderstandings on the construction of meaning through these teaching actions. Indeed, different ways of thinking of the same objects are consciously mobilized, but it is as if going from one vision to another is self-evident, without linking: we move from the designation of objects to their relationships and their operationalization through instruments. At this student's age all these connections are not yet built. Here we are one of classical teacher interventions in geometry teaching, not especially with novice... The big question is : how to manage the complex specificity of geometrical activity : the way of seeing a figure, the way of acting with geometrical tools or not, the way of speaking ? We refer here to researches already exposed in previous CERME [START_REF] Bulf | Language in geometry classroom[END_REF]) (Barrier, Hache, Mathé, 2013) ; they proposed different theoretical tools to analyse geometrical activity in this way.

Different impacts on students' learning from scaffolding on the necessary and sufficient conditions for building a square

Two novice teachers, Maya and Emilie, in 2016 prepared together a lesson (for students aged 8 to 9 years old). The students have to reproduce a square of 10 cm of side. At the end of the collective phase, both teachers chose to proceed by "dictation to the adult" that means students gives oral instructions and the teacher executes them to build the figure. An interesting moment happens when three sides of 10 cm and 2 right angles are correctly drawn at blackboard with expected geomerical tools. Both teachers ask to students « do we need set square to close the square ? » which is, in both cases, a spontaneous move. The question is : is it a relevant professional action ?

In Maya's class, she retains the one who talk about the set square to finish the construction. But Maya asks to students about the validity of another student's proposal who suggests to draw the line directly without the set square. This student, Nicolas, is also a student usually in difficulty.

Maya: Would Nicolas's technique have worked? Nicolas can you explain why it would have worked? […] So you tell me we did not need the set square to draw the last line we could do directly with the rule to draw the last line.

The students do not validate Nicolas's technic because precisely it does not use the set square. On one hand they may reply this because it corresponds to what is worked precisely in this lesson, this is the didactical contract [START_REF] Brousseau | Theory of didactical situations in mathematics[END_REF] and on the other hand probably because Nicolas is a student usually in difficulty… But here teacher's question is about the necessary and sufficient conditions for building a square, which is out of didactical contract for these students at this age.

The teacher is stuck because Nicolas is right, it is mathematically correct. Therefore Maya chose to validate it and she concludes with theoretical argument: « As long as I have 3 sides of the same length and 2 right angles », which is, actually out of contract but in the same time it is also an opportunity to value this student (atmosphere concern).

Maya: Nicolas' proposal it would have worked / I connected well / I took my rule and my set square to do it but if I had connected this first line that I had done with it, it would have made me too the 2 right angles in the case ok where I already had the 2 right angles and the 3 sides which made the same length. Alright ... With my set square I did the same thing as Nicolas proposes, I connected the 2 parts of the line, I did it with the square but we could do it directly with the rule .

Whereas in Emilie's class, Emilie also relies on a student's proposal but the student stands on a graphical argument that Emilie resumes directly : « yes you have already drawn both lines you need to connect» « I just have to connect». We stay in a primary school contract strongly linked to graphical register.

Conclusion, discussion and perspectives about pre and in-service teacher training

From the previous paragraph, we pointed out the necessary and sufficient conditions (here to recognize a square) are usually not anticipated by novice teachers and this may has a direct impact on teacher's scaffolding and therefore on student's learning. In one case, students may are confused at the end of the session on the contract at stake about the use of set square whereas in the other case, we stay in a contract in a paradigm of Geometry I [START_REF] Kuzniak | Thinking About the Teaching of Geometry Through the Lens of the Theory of Geometric Working Spaces[END_REF]) that means validation is doing within a graphical register.

This paper is also based on other case studies. We described at the beginning of the paper how Celine was overwhelmed by the managing of different dimensions own to geometrical activity (way of speaking, way of seeing and way of acting -with geometrical tools or not) about graphical figure. This is a typical novice's teacher action : novice teachers want to help students (when they give instructions at the beginning of the lesson) with multiple designations, reformulations,… but in the geometrical context, this may involves over-scaffolfing through all registers in the same time (graphical, visual, discursif) and brings more confusion than help.

In previous work [START_REF] Bulf | Etude didactique des gestes professionnels d'un enseignant débutant en formation[END_REF], we described how novice teachers try to play school teacher through the embodiment of « caricatural » teaching actions with an over-leading about rigor on semiotic representation of geometrical objects (vocabulary, coding, or quality of drawing5 ).

We believe these teacher's actions with novice teachers are actually not reserved for novices… That's why our research gives perspectives for pre-service and in-service teacher training to better understand the conditions of professional development in geometry teaching but not only. Our research in progress crosses results with other fields. We think this kind of research gives the opportunity to discuss the limit about a theoretical framework based on genericity of teaching actions. Indeed, in our different examples we may recognize teachers' actions shared by teacher community (have students talk about the validity and rigor of their production; to explain in different ways the same thing ; to compare opinions between students etc.) but it seems important to highlight the fact that these actions may be rich in potential misunderstandings in the context of teaching geometry. Even more, our study, in the context of teaching and learning geometry, suggests that it makes no sense to consider teacher's actions from a generic point of view because

  Schema 1. "multi-agenda of entrenched concerns" (Bucheton and soulé, 2009, p.33) adapted to the context of teaching geometry.

Figures 1 .

 1 Figures 1. Different ways of seeing one side of the square

  of the session, Celine tries to explain the task by making the students talk about the instrumented actions. It is kind of over-scaffolding without linking the different possible ways of seeing the side of the square and the center of the circle: Celine: How do I put my compass on my square? Student: on the board there is a line in the middle to put the starting mine in the middle of the line Celine: then in the middle of the line you speak about side [She sweeps the side with a hand gesture and points the middle of the side, figure 2] okay what is it in relation with our circle /// you remember where he is the center of the circle? // How is it called? [Sweeps the side of a hand gesture like the moment before from top to bottom] /// all of this / yes it is the diameter / there [gap between the auriculare and the thumb, figure 3] it is the radius / you remember the center of circle [pointing finger, figure 2] it is the middle of the diameter/ it shares the diameter in 2 radius.

Figure

  Figure 2. physical gesture of pointing finger Figure 3. physical gesture of finger spacing

"Accompaniment, control, letting go, teaching-conceptualization, magician"(Bucheton & soulé, 2009, p. 41)

Given what we have observed and analyzed so far in our previous researches and in the corpus for this research (corpus made up of a dozen novice teachers).

« double designation »(Duval, 2014, p.5) of a figural unit.

This is quite typical for beginning teacher in France. We suppose that may comes from the fact the competive exam that these young teachers had to pass requires a theoretical and deductive hypothetico reasoning of the Euclidean geometry.

each teaching actions are always specific and connected to a specific knowledge which requires its own and specific actions. This opens the more theoretical discussion of theoretical frameworks that seek to model teaching activities.


Martin Kerler
email: martin.kerler@tum.de

Christian Schäffer

Wolfgang Erhard

Volker Gümmer

Experimental Investigation of an Engine Quick-Start System with Compact Air Supply for Rotorcraft Application

Keywords: Turboshaft, Engine, Quick-Start, Helicopter, Air Supply

A quick-start system for turboshaft engines is an appropriate method to enhance safety and the flight operation range of a helicopter during intended single engine operation. Such a system was designed and successfully tested at the institute. However, the quick-start system used shop air of 13 bar(a) pressure for proof of concept. The shop air supply has to be replaced by an airworthy, compact and lightweight air supply system for feasible rotorcraft application. This comprises in first case general identification of adjustable system parameters by numerical studies and experimental tests in consideration of certain system constraints. Therewith, the ideal nozzle count is determined to 3 and the required nozzle entry pressure has to be about 19 bar(a). Based on these findings the final air supply system for rotorcraft application is designed and simulated with ESPSS. This system is built up at the engine testbed with some modifications for experimental testing. Gas generator acceleration tests as well as engine starting tests to idle speed are performed with the new system. A final engine start-up time of 2.8 s was achieved. Finally, further advice is given for feasibility.

INTRODUCTION

Future challenges in rotorcraft and propulsion design are diverse. Governmental regulations are tending to achieve a reduction in carbon emissions of engines and lowering the noise level of rotorcrafts. More strict regulations will apply in the future [START_REF] Warwick | Collective Approach[END_REF]. Then, rotorcraft operators claimed the development of efficient engines with significant lower fuel consumption a few years ago, due to high fuel prices. In addition, lowering other direct operational costs as well as maintenance and repair costs is crucial for the success of future rotorcrafts [START_REF] Norris | Little Bigger[END_REF]. An entire new development of a complex system like an engine or rotorcraft is often usually risky and costly in terms of time and money. Some objectives can be partially achieved due to evolution of available and reliable technologies.

Potential of Intended Single Engine Operation

On account of this, a flying technology demonstrator was introduced by Airbus Helicopters in 2015. This rotorcraft called Bluecopter [START_REF] Bebesel | Bluecopter Demonstrator -An Approach to Eco-Efficient Helicopter Design[END_REF] (see Fig. 1) should comprise besides aerodynamic improvements also a modified operational usage strategy of the installed turboshaft engines for fuel saving. The helicopters original powertrain design is mainly driven by safety reasons. Thus, two engines are installed to cover an engine failure during flight and avoiding autorotation mode. Due to this requirement, the installed maximum engine power exceeds by far the usual required power during ordinary helicopter flight missions. The engines are loafing most of the mission time at part load at worse specific fuel consumption (SFC) [START_REF] Walsh | Gas Turbine Performance[END_REF]. To resolve this predicament, a modified operational usage strategy can be an intended shutdown of one engine at appropriate helicopter flight states. Then, the shut off engine does not burn any fuel and the remaining running engine has a better SFC by means of higher power loads. This can be observed in Fig. 2 in the power required range in which ISEO mode is possible. Here, a sample engine with max. 500 kW output power is used. Overall, this results in significant fuel savings up to 20% depending on the flight mission [START_REF] Kerler | Evaluation of Helicopter Flight Missions with Intended Single Engine Operation[END_REF]. The savings can be used either for extending mission range and mission time or overall fuel burn. However, if this intended single engine operation (ISEO) strategy is applied on current powertrain designs, neither low and high speed flights nor steep climb flights are possible during ISEO. The required power of these flight states of exceeds the available maximum continuous power (MCP) of one engine. The fuel advantage of ISEO is also a drawback for flight safety.

Flight Safety and Capability Enhancement with

Engine Quick-Start During ISEO flights, the twin-powered helicopter is just like a single engine powered helicopter. In case of an engine failure, flight cannot be maintained and autorotation has to be initiated. A solution to ease this emergency situation is a re-start of the engine which was shut off before due to switching in ISEO mode. Regular starts to idle of turboshaft engines of 500 to 1000 kW class are up to 30 s. Further time is elapsing until sufficient shaft power can be provided. A helicopter has sink rates of about 8 to 15 m/s which means in worst case the helicopter has lost about 700 m of altitude until engine power is available again. Thus, sufficient altitude margin must be available during active ISEO. This cuts the flight envelope of the helicopter and the reasonable usage of ISEO. A solution to overcome this drawback is a reduction of the turboshaft engine's start-up time.

Evaluation of Engine Quick-Start Methods

An engine quick-start (QS) can be achieved by additional acceleration torque for the gas generator GG provided by a technical device. Pascoe [START_REF] Pascoe | Start systems for aero gas turbines[END_REF] gives a general overview of common gas turbine start devices. Hull and Santo [START_REF] Hull | Development of a Rapid-Start System for the Boeing Model 502-2E Gas Turbine Engine[END_REF] made an evaluation of suitable devices for high torque demands, too. One system comprises a hydraulic motor with an appropriate hydraulic system of high pressure. Experimental tests of the system were promising but the prerequisite of a hydraulic system for proper operation may not apply for a helicopter due to excessive additional weight. Another method is Hydrazine impingement or cartridge impingement. This is realized by directing a hot gas stream of high velocity onto the blades of a GG turbine stage. However, the hydrazine solution has many drawbacks like costs, complexity and health hazards and is thus, not really a practical solution. The cartridge impingement requires for operational usage too much installation space and has to be replaced after each use [START_REF] Hull | Development of a Rapid-Start System for the Boeing Model 502-2E Gas Turbine Engine[END_REF]. Since the required system for quick starts should be an emergency system, the replacement after use is not an issue. But installation space is restricted at helicopters. Thus, this variant seems not really applicable for a compact and lightweight quick-start system (QSS). Rodgers [START_REF] Rodgers | Impingement Starting and Power Boosting of Small Gas Turbines[END_REF] picks up the basic idea of impingement nozzles but utilizes high-pressure air or hot-gas as propellant at the turbine. The hot-gas was of approximately 533 K and the gas generator was accelerated just to 4770 rpm. Due to radial turbine design and its operational principle, a less nozzle expansion efficiency [START_REF] Rodgers | Impingement Starting and Power Boosting of Small Gas Turbines[END_REF] and therewith lower acceleration rates were achieved. Another problem which cannot be satisfactorily resolved is the required proper long term storage of pressurized air at temperatures above 500 K. A gas impingement at the compressor section was also investigated by Rodgers [START_REF] Rodgers | Impingement Starting and Power Boosting of Small Gas Turbines[END_REF]. Utilization of pressurized gas seems feasible and compressor shroud impingement was the most convenient method for GG acceleration with impingement nozzles [START_REF] Rodgers | Impingement Starting and Power Boosting of Small Gas Turbines[END_REF]. QS tests of a gas turbine with 260 kW power output were conducted with compressed nitrogen at several operating conditions. Start times could be reduced by 55%. More results of this investigation can be found in [START_REF] Rodgers | Impingement Starting and Power Boosting of Small Gas Turbines[END_REF] and [START_REF] Rodgers | Fast Start System for a 200-KW Gas Turbine Generator Set[END_REF].

Design of a Quick-Start System for the Allison 250

Based on these findings, a QSS comprising impingement nozzles directing a gas jet onto the trailing edge of a radial compressor rotor seems to be a proper solution. However, the use of compressed nitrogen seems not feasible, since it cannot be used for combustion during QSS operation. This would not be an issue if the compressor is regularly operating during QS. But the compressor of Rodger's experiments did not. Compressor surge and backflows out of the compressor occurred. This means, the compressor does not deliver any air for combustion during QS. Thus, regular air injection instead of nitrogen would be the better case, since this air can be used for combustion. For proof of concept with a common helicopter turboshaft engine, the institute's Allison 250-C20B was chosen. The engine is fully functional as well as instrumented for further performance analysis. The modular design of the engine enables an easy replacement of engine modules. MCP is about 300 kW and the compressor has a radial last stage. Thus, this engine is similar to the engine modified by Rodgers [START_REF] Rodgers | Impingement Starting and Power Boosting of Small Gas Turbines[END_REF]. Hence, the Allison QSS has similar design parameters as the QSS of Rodgers [START_REF] Rodgers | Fast Start System for a 200-KW Gas Turbine Generator Set[END_REF]. The new compressor casing is equipped with de Laval nozzles which are asymmetrical positioned in circumferential direction due to available design space and to avoid harmonic excitation (see Fig. 3).

Figure 3. New radial compressor casing with 5 nozzles, supply pipes and its integration into the compressor section.

With these restrictions a maximum nozzle count of 5 is possible. The angle of the injected air is defined to provide best impulse propagation to the impeller's trailing edge and to allow proper connection to an air supply. The available shop air for nozzle supply has a static pressure of about 13 bar(a). This QSS with shop air and 5 nozzles is called Testbed Quick-Start System (TQSS). Due to total pressure losses caused by valves, pipes and hoses the nozzle entry working total pressure is about 12.3 bar(a). Based on these specifications and with a required start performance of less than 3 s from off-state to idle, the calculations revealed the following nozzle parameters. The nozzle's throat diameter is 6.6 mm which means an air mass flow of 0.1 kg/s resulting in 0.5 kg/s overall air mass flow. The design Mach number of the nozzle is 2.3. Operation time is 2.2 s. During GG acceleration, the starter motor is also working. Approximately 0.85 s from off-state, ignition of the injected fuel takes place and the high pressure turbine supports the GG acceleration. Finally, start-up time was reduced by over 90% from 26s to 2.4 s for ambient conditions close to the International Standard Atmosphere at sea level at a Standard Day (see Fig. 4). For sufficient power delivery, the GG has to be further accelerated. Initial conservative tests were performed with 100 Nm brake torque preset at the testbed. As Fig. 5 shows, 60 kW (equals 20% of MCP) is available within approximately 8 s from off-state. Further details on design and initial testing of the Allison QSS supplied with 13 bar(a) shop air can be found in Hönle et al. [START_REF] Hönle | Engine Quick Start in Case of Emergency -A Requirement for Saving Fuel by Means of Engine Shutdown[END_REF] and Hönle [START_REF] Hönle | Ökonomische Optimierung von Triebwerksbetriebsstrategien von zweimotorigen Hubschraubern[END_REF]. These results show the successful proof of concept of a TQSS for a turboshaft engine of 300 kW power class. The TQSS is supplied at this development stage with shop air which is not available in a rotorcraft. Hence, the next step is design and test of a pressurized air supply system which can be integrated into a helicopter airframe. If an existing airframe is used, extensive structural airframe modifications have to be avoided and the system must be compact as well as of lightweight design.

DESIGN PARAMETERS IDENTIFACTION

Prior to final airworthy design, adjustable design parameters of the QSS have to be identified. Due to cost-containment and development time optimization, the components of the air impingement system (AIS) are predefined and thus the modified compressor casing as well as the nozzles cannot be modified or exchanged. Consequently, changeable parameters are on the air supply system (ASS) side. The ideal acceleration torque M QSS,id is basically the result of the relative nozzle forces F noz to the surface of the trailing edge of the radial compressor and the distance r from the origin of force to the shaft axis. Since the acting force of the nozzle's jet is not perpendicular to the trailing edge, the relative deflection angles α and β have to be considered (see Eq. 1). These angles are based on design space restrictions. Further information about this can be found in Hönle [START_REF] Hönle | Ökonomische Optimierung von Triebwerksbetriebsstrategien von zweimotorigen Hubschraubern[END_REF]. Hönle sets the transfer efficiency 𝜂 𝑛𝑜𝑧 of the force between nozzle and trailing edge for an initial TQSS design to a fixed value of 0.7 based on literature data. Later on (see section 4.1.) this value is redefined.

𝑀 𝑄𝑆𝑆,𝑖𝑑 = ∑ 𝜂 𝑛𝑜𝑧 ⋅ 𝑟 ⋅ 𝐹 𝑛𝑜𝑧,𝑖 ⋅ sin(𝛼) ⋅ sin (𝛽) 𝑛 𝑖=1 [START_REF] Warwick | Collective Approach[END_REF] Each nozzle has the same distance to the axis as well as the same deflection angles. Thus, 𝑀 𝑄𝑆𝑆 varies with varying nozzle force 𝐹 𝑛𝑜𝑧 . This force can be adjusted by changing nozzle mass flow ṁ, nozzle exit velocity v noz,exit or the nozzle count n (see Eq. 2).

∑ 𝑑𝐼 𝑎𝑖𝑟,𝑖 𝑑𝑡

𝑛 𝑖=1 = ∑ 𝐹 𝑛𝑜𝑧,𝑖 𝑛 𝑖=1 = ∑(𝑚̇⋅ 𝑣 𝑛𝑜𝑧,𝑒𝑥𝑖𝑡 ) 𝑖 𝑛 𝑖=1 (2)
Since weight saving is important for helicopters, QSS nozzle count reduction saves weight due to fewer parts. To keep F noz at the same level, mass flow or exit velocity has to be increased. The nozzles of the QSS are basically operating at choking conditions and consequently the air mass flow 𝑚̇ does not depend on the ambient pressure at nozzle outlet anymore. A feasible exit velocity increase is not possible. The geometry of the nozzle and thus the critical throat cross section area 𝐴 * is fixed. Since variation of gas constant 𝑅 and isentropic coefficient 𝜅 are negligible, an air mass flow 𝑚̇ change can only be achieved by change in total temperature 𝑇 𝑡 or total pressure 𝑝 𝑡 . This relationship is depicted in Eq. 3.

𝑚̇= 𝐴

* ⋅ √ 𝜅 ⋅ (1 + 𝜅) 2 ⋅ 𝑅 ⋅ 𝑇 𝑡 ⋅ 𝑝 𝑡 ⋅ ( 1 + 𝜅 2 ) -𝜅 𝜅-1 (3)
A lower gas temperature 𝑇 𝑡 results in higher mass flow.

A temperature drop by approximately 30 K would increase the air mass flow in best case by only 6%. A device to reduce the temperature in front of the nozzles seems not reasonable because of high air mass flow rate and low cooling effectiveness. Finally, if nozzle count is reduced, the nozzle entry total pressure has to be increased. For a final compact QSS (CQSS) design this appropriate pressure level has to be identified. Similar test were performed by Rodgers [START_REF] Rodgers | Fast Start System for a 200-KW Gas Turbine Generator Set[END_REF]. He measured the acceleration time until 40% of the GG speed is reached, depending on nozzle count and nozzle entry pressure (see Fig. 6). These tests were declared as cold-crank test, which means no additional acceleration torque, like electric starter or turbine operation, was used. With a nozzle pressure of about 160 psig (~11 bar(a)) and five active nozzles, the start-up time is about 2.45 s. This performance can be also achieved by three nozzles and an increased pressure of 275 psig (~19 bar(a)). Further pressure increase reduces the nozzle count to two or one. But this means very high nozzle entry pressures which are difficult to deal with in an helicopter environment. Finally, increasing nozzle pressure for better start-up performance is only reasonable up to a certain pressure level. Figure 6. Start-up time to 40% design gas generator speed depending on nozzle count and nozzle entry pressure [START_REF] Rodgers | Fast Start System for a 200-KW Gas Turbine Generator Set[END_REF].

It was decided to realize the CQSS with three active nozzles. One reason was to have them almost evenly distributed in circumferential direction to reduce excessive radial loads on the shaft bearings. Based on analytical start-up performance calculations, each nozzle must provide a thrust of approximately 83.5 N. For verification reasons, simulations and experiments were conducted to find the appropriate pressure level. The simulations were done with the simulation tool ESPSS. Further information about this tool can be found in Isselhorst [START_REF] Isselhorst | HM7B Simulation with ESPSS Tool on Ariane 5 ESC-A Upper Stage[END_REF], Pérez-Vara et al. [START_REF] Pérez-Vara | Overview of European Applications of EcosimPro to ECLSS, CELSS and ATCS[END_REF] and Moral et al. [START_REF] Moral | 1-D Simulation of Solid and Hybrid Combustors with EcosimPro/ESPSS[END_REF]. A compressed air cylinder bundle, a pressure regulator and a coaxial valve are main components of the experimental set-up (see Fig. 9). Further information about this setup is given in Kerler et al. [START_REF] Kerler | Design Parameter Identification of the Air Supply for a Turboshaft Engine Quick-Start System[END_REF]. Evaluation of the simulation results revealed a nozzle entry total pressure of about 18 bar(a) (see Fig. 7). The total pressure measured inside the measuring pipe was 17.9 bar(a). A measured thrust of about 80 N is achieved with this pressure. The simulated thrust is 83.5 N (Fig. 8). The differences between both values rely on measurement accuracy as well as manufacturing tolerances. The load cell which measures the thrust has an uncertainty of measurement of about 1% of its whole measurement range. This means, the maximum error can be up to +/-5 N. In addition, further flow effects inside the measuring tube may influence the result, too. The slower increase in pressure of the measured curve depends mainly on the response time of the pressure transducers. The results are very satisfying and define objectives for the CQSS design.

AIRWORTHY QUICK-START SYSTEM DESIGN

The design of the airworthy CQSS design is realized in two different ways:

 The first one is to define a system with parameters of the airworthy one but realized with respect to ground testbed application. This means, the used parts of the hydraulic sector are mainly not optimized for aerospace application. Therewith, a cost effective realization at the test bed is possible. Overall, this system is not optimized regarding weight and used space as well as regarding length of pipes and hoses. Furthermore, additional adapters are incorporated in the flow paths to offer measurement possibilities for temperature and pressure of the flow. These adapters are usually not required for helicopter application.  The second design is based on the parameters of the first design and accounts especially for helicopter application with respect to lightweight design, compactness, safety, reliability and robustness. However, this system is modeled just virtually and gives a prospect of how such an airworthy CQSS can look like.

Basic considerations for experimental CQSS design for testbed application

The CQSS for testbed application is designed in a way which comes close to the CQSS for helicopter application. But instead of expensive aerospace certified components, the testbed setup is realized with widely used parts of the hydraulic sector. In general, the CQSS has a mass flow of 0.5 kg/s and an operation time of approximately 2.2 s, resulting in an air mass demand of 1.1 kg at a pressure level of about 18 bar(a). The equivalent air volume is 50 l at 18 bar(a) pressure. However, for any kind of reasonable airframe integration, this volume is far too big. Feasible cost effective high pressure sources are pressurized bottles for diving applications. These are available with pressure levels up to 300 bar(a). Since the institute is owner of a 200 bar(a) air pressure pump, a pressure bottle of maximum 230 bar(a) working pressure is a reasonable choice. These bottles have standardized connecting screw diameters of 25 mm offering sufficient flow path diameter. Based on the requirement of having the capability for two consecutive QS, the bottle volume was determined to at least 15 l. A minimum pressure level of about 80 bar(a) inside the bottle ensures at all operation times the required overall mass flow of 0.5 kg/s. An adapter for pressure and temperature measurement as well as refilling the bottle is connected to the bottle. Further downstream is a ball valve which is closed for refilling process and opened when the CQSS is armed for operation. The bottle pressure has to be reduced to the nozzle working pressure of 18 bar(a). This is realized with a pressure regulator which can handle 300 bar(a) input pressure and provides an output pressure range up to 100 bar(a). The required output pressure of the pressure regulator is higher than 18 bar(a) because further downstream is a pressure releasing valve and other devices causing total pressure loss. This pressure regulator output pressure is found by a parameter study with ESPSS. Between ball valve of the bottle adapter and pressure regulator is a measuring adapter offering static pressure measurement and recovery temperature measurement. This device is also available after the pressure regulator. Then, between pressure source and AIS is a coaxial valve to release the pressurized air to the nozzles. Such a valve is the best choice regarding weight and high flow rates as well as safety and compactness. Another aspect is the fast response time. Opening and closing times are about 0.1 s. Besides these main components, some other components are required like flow distribution manifolds, adapters and valves as well as sensors for system health monitoring. The diameter of the flow paths cross section areas are defined to realize slow flow velocities inside the manifolds, adapters, pipes and hoses. A slower velocity causes less total pressure losses. Thus, a trade-off between pressure loss, cross section area and weight has to be found. In this case, flow velocities with a maximum of Mach 0.3 are acceptable. Significant further compressibility effects can be avoided.

Fig. 9 shows the scheme of the system's final design for integration in a twin engine powered helicopter. In general, the ASS comprises one bottle and one pressure regulator. After the pressure regulator, the manifold block (1/2) splits the one flow path into two feeding lines for both engines. This junction is not needed and thus not realized for the CQSS at the testbed. Further downstream is the coaxial valve and another manifold block (1/3) where the single feeding line for the engine is divided into three supply lines for each nozzle of the AIS. The control of the CQSS is integrated in the engine controller.

Basic considerations for an airworthy CQSS design for helicopter application

Since lightweight is crucial, regular bottles made of steel are too heavy. Thus, a high pressurized air tank made of composite material is a good choice. Here, already aerospace certified pressure bottles made of carbon and glass fiber with aluminum liner and 300 bar(a) working pressure are available. Since the maximum pressure is greater than the 200 bar(a) used in the testbed solution, the bottle volume may be reduced for further weight saving. A possible integration of the system in a BO 105 helicopter is illustrated in Fig. 10. This airworthy CQSS has a weight of 46.4 kg using common components [START_REF] Kerler | Design of an Airworthy Turboshaft Engine Quick-Start System with Compact Pressurized Air Supply for Rotorcraft Application[END_REF]. If special optimized components are used, the weight can be reduced to approximately 36 kg. All main components are located in the helicopter main transmission compartment due to safety reasons. None important component should be integrated into the fire risky engine bays. The CQSS fits almost into the transmission compartment. For this design, some small modifications of the engine cowlings have to be done. An advantage of this CQSS location is its closeness to the helicopter's center of gravity since helicopters are sensitive to center of gravity changes. Further details of the design can be found in Kerler et al. [START_REF] Kerler | Design of an Airworthy Turboshaft Engine Quick-Start System with Compact Pressurized Air Supply for Rotorcraft Application[END_REF]. Between bottle and pressure regulator is an adapter and a ball valve. The adapter is used for bottle condition monitoring as well as for bottle pressurization. The ball valve is required for safe bottle replacement. For weight saving, only one bottle is used to supply both AIS of the engines with pressurized air. This means, a manifold block with two output lines (1/2) is used between the pressure regulator and the coaxial valves. This manifold block is missing in the testbed CQSS.

EXPERIMENTAL RESULTS

The testbed QSS was set up as described in section 3.1. An overview of the installation gives the image in Fig. 11. The ASS and the (1/3) manifold block are mounted on a moveable rack. The supply hoses of the AIS can be quickly disconnected from the supply pipes. This arrangement allows an easy removal of the CQSS from the engine itself. The compressed air bottle is mounted in an own rack which is located at the top of the ASS rack. Thus, the bottle rack can be detached easily for transportation to the refilling station of the institute. In the background of the image in Fig. 11 the compressed air bundle is visible. For first GG acceleration tests (described in section 4.1) the CQSS's compressed air source was the compressed air bundle. The output port of the bundle was directly connected to the pressure regulator. Since the 600 l volume of the bundle is 40 times greater than the bottle volume, the pressure regulator entry conditions of the air are nearly constant during the short test runs. The experiments were conducted in four main test campaigns. The first two campaigns used the compressed air bundle as pressure source. The other two used the bottle. Each campaign is split into a cold quick-start (CQS) and a hot quick-start (HQS). The CQS is without any fuel injection and thus no combustion takes place. This setup is used to determine general CQSS behavior and to investigate important CQSS parameters, for instance air temperatures in the supply pipes and at the pressure regulator as well as static pressures at these stations. The HQS is important to determine the overall QS performance and proper gas turbine operation during QS.

For temperature measurements thermocouples of type K and with a diameter of 1 mm are used as compromise for fast response and good durability. Signal conditioning was achieved with appropriate 7B modules of manufacturer Analog Devices. Final measurement accuracy is about +/-2 K. The recovery factor for calculation of static and total temperature is set to 0.85. Pressure is measured with pressure sensors type A-10 of manufacturer WIKA. The sensors were chosen in a way, that the maximum design pressure of the sensor is two times the expected pressure at the measurement position. The sensors have an accuracy of +/-1.5% of their range and a response time less than 4 ms. Total pressure measurements were realized with a pitot tube. The data acquisition system from manufacturer United Electronic Industries has a sample rate of 2500 samples per second for each measurement channel.

Compressed air bundlecold quick-start

For future rotorcraft application it was decided to reduce the nozzle count from five to three to save weight. This means, that the nozzle mass flow has to be increased to ensure the required overall 500 g/s air mass flow through the nozzles. Thus, the nozzle entry total pressure must be raised until each nozzle provides a mass flow of 167 g/s. How the engine behaves during a quick GG acceleration with just three nozzles and different nozzle entry pressures is investigated in the following.

The static pressure short after the pressure regulator was chosen as variable, since it can be easily adjusted with the pressure regulator. Downstream the pressure regulator is defined as low pressure side. These static absolute pressure values are on the x-axis of Fig. 12. On the y-axis is the resulting final GG speed after an acceleration time of 4 s. After this short time period the coax valve was closed. Longer valve operation was not required since the final QS should take less than 3 s. There was no combustion and the tests were conducted with and without the starter motor to identify the influence of the starter motor on the acceleration performance during acceleration as well. The test series was started with a similar pressure level like the proof-of-concept design. But due to pressure losses over pressure regulator and coax valve and losses caused by pipes and manifolds, the final nozzle entry total pressure drops to 5.2 bar(a). However, to finish the acceleration test series, the pressure level after the pressure regulator was further increased until the required nozzle entry total pressure of about 18 bar(a) was reached with setting the low pressure side to 33 bar(a). In Fig. 12 it can be seen as well that the starter motor's influence gets less distinctive for greater GG speeds and therewith nozzle entry pressures. However, the additional rotational speed gain is between 5% and 10% and thus significant. This means, during QS the starter motor should operate to ensure greater GG accelerations. Both curves in Fig. 12 are leveling for greater nozzle entry pressures. An extrapolation of the red curve with starter motor support indicates a reasonable static pressure level after the pressure regulator of 33 to 35 bar(a). Pressure levels beyond these values would lead to insignificant additional GG speed gaining.

Figure 12. Allison 250 gas generator acceleration tests with compressed air bundle and with/without starter motor support

Similar tests were made by Hönle [START_REF] Hönle | Ökonomische Optimierung von Triebwerksbetriebsstrategien von zweimotorigen Hubschraubern[END_REF] with the TQSS. Here, the effective shaft torque was compared to the nozzle entry pressure. In addition some aspects are discussed about flow interaction of the air jet with the radial compressor. One of these is the definition of the real transfer efficiency 𝜂 𝑛𝑜𝑧 . Hönle [START_REF] Hönle | Ökonomische Optimierung von Triebwerksbetriebsstrategien von zweimotorigen Hubschraubern[END_REF] sets it to 0.85 based on his findings.

Besides the requirement of maximum GG speed gaining, the overall air mass flow provided by the three nozzles should be close to the 0.5 kg/s of the TQSS which is using shop air. This impinged air mass flow is also close to the air mass flow the engine is sucking in during idle mode. For nozzle mass flow calculation a supply pipe with measurement equipment is located upstream to one of the nozzles. Measured data are static and total pressure as well as the recovery temperature of the air flow. The mass flow can be then calculated with known geometry and Mach number. For this investigation, the CQS with compressed air bundle and a static pressure of 33 bar(a) at the low pressure side is used. This test is represented by the right most point of the red curve in Fig. 12. Due to aerodynamic effects and the measurement setup, valid mass flow results are available for a time period from 0.5 s after the open command for the coax valve until the close command. The calculation results are shown in Fig. 13. The slight descent of the mass flow curve is mainly caused by decrease of total pressure at the nozzle entry over this short time period. Since the pressure level in front of the pressure regulator is decreasing, the flow through the pressure regulator is also decreasing causing this decrease in total pressure inside the nozzle supply pipe. Calculations of the mean air mass flow during this time period of 3.5 s show a value of 0.163 kg/s. Since the first 0.5 s are not considered, the real averaged air mass flow may be slightly higher. However, summing up all nozzles, an air mass flow of 0.489 kg/s is entering the Allison 250 engine, which is quite close to the required 0.5 kg/s. For further QS tests, the pressure right after the pressure regulator can be slightly increased.

Figure 13. Calculated mass flow (based on measured data) of one nozzle during CQSS operation

One crucial issue during QS to keep in mind is the air temperature of the expanded injected air. Since it is cooling down during expansion, it can cause thermal stress in the radial compressor. Then, it can be too cold for reliable fuel ignition inside the combustion chamber. The ESPSS simulations prior the experimental tests revealed air temperatures close to 200 K after the nozzle. In Fig. 14 the air temperature inside the supply pipe as well as the air temperature short after the compressor section is shown. Furthermore, the engine air mass flow at the engine inlet and the coax valve position is shown. When the coax valve is opened, the air between the valve and the nozzles is being compressed, indicated by the small temperature increase at time 1.1 s. After that, the expanded air from the air bundle is approximately 10 K colder than its storage temperature. After the valve was closed, the air inside the supply pipes gets colder due to further expansion to ambient pressure. Since the impinged air is further expanding, the temperature drops in the section after the compressor, which shows the red curve. The temperature drops until 251 K. At time 3.5 s the compressor has reached a certain rotational speed, where the compressor starts to act as it should be. This can be seen in the increasing curve of the engine air mass flow. Due to the compression, the compressed air's temperature is increasing and is mixing with the cold impinged air mass flow. Thus, the air temperature after the compressor is also slightly increasing. When the coax valve is closed, the mass flow blockage due to the injected air is gone and the compressor air mass flow makes a sudden jump to greater values until it is run out. This also accounts for the temperature after the compressor. Further investigation of other air temperatures and pressures at different engine station revealed no critical values for future HQS.

Compressed air bundle -hot quick-start

Based on the findings of the CQS tests with the compressed air bundle, HQS are performed. Main parameters of FADEC adaption for a QS are fuel injection timing, amount of fuel and deactivation timing of the coax valve. For first tests, the known parameters of the TQSS were taken and adapted to have kind of conservative adjustments. An interesting point is the QSS deactivation and in the following the resumption of the compressor's work, as seen before in Fig. 14. The conservative settings are fuel injection at 20% GG rotational speed and deactivation of the QSS at 52% GG rotational speed. The result is shown as blue curve in Fig. 15. On the x-axis is the elapsed time and on the y-axis is the absolute pressure at the end of the compressor section. The coax valve opens and the pressure at the end of the compressor is slightly increasing. At time 2.2 s fuel is injected into the combustion chamber and ignition takes place. This increases the overall pressure level at the compressor end. Since the QSS is deactivated at 52% GG speed, the GG has not reached the required 60% rotational speed at idle mode. The compressor should operate in a regular manner but surge occurred for nearly a second until real regular compressor operation is achieved at 4.5 s and the GG is operating at stable 60% speed. An explanation can be that the compressor has not reached an operation point where sufficient pressure increase is possible to cope with the back pressure from the combustion chamber. Nevertheless, this surge behavior is not tolerable for a safe QS. Within a second test the deactivation limit was pushed to 58% GG rotational speed. Here the compressor works fine as the red curve in Fig. 15 shows around time 4 s. The transition from nozzle operation mode to regular compressor operation may be further smoothening by QSS deactivation at 60% GG rotational speed. This will be investigated in the future. To shorten the start-up time the beginning of fuel injection starts earlier at 14% GG speed. However, on compressor end pressure this does not have any significant influence. The fuel schedule was not changed. Further HQS tests with some changes in FADEC parameters were performed for finding a better QS performance. But in summary, the red curve in Fig. 15 represents one of the best HQS with the compressed air bundle. The result is a start-up time of 3 s from off-state until 60% GG rotational speed is reached.

Compressed air bottle -cold quick-start

The compressed air bundle tests paved the way for final tests with a bottle as pressure source. Thus, the compressed air bundle was replaced by the compressed air bottle of 15 l volume and maximum 200 bar(a) pressure level. First, CQSs were performed to identify critical engine parameters during this short phase. Since the air mass inside the bottle and the bottle pressure is decreasing during QS, the entry pressure of the pressure regulator is accordingly changing and causing less mass flow through the pressure regulator. This can be seen in Fig. 16 where the total pressure inside the supply pipe is dropping with decreasing bottle pressure. The QSS operation time was set to 3 s. For the HQS with the bottle this time should not exceed. The bottle pressure drops to 67 bar(a) within these 3 s. The lowest measured air temperature at the bottle exit was 228 K. Air temperatures at the compressor end were about 250 K. Further downstream, the air temperature was increasing, what is a good prerequisite for proper initial fuel ignition. The final GG rotational speed was 45% and is a good starting basis for the following HQS with the bottle as pressurized air source.

Compressed air bottle -hot quick-start

Based on the findings of the HQS with the cylinder bundle and the CQS with the bottle, the FADEC was modified and tests were conducted. The recovery temperatures at certain measurement points of the CQSS are shown in Fig. 17. The QS starts at 1 s and the command for coax valve opening is given. After 0.2 s the valve signals, that it is fully open. Due to compression of the stored air volume at ambient pressure inside the hoses, the temperature in the supply pipe is slightly increasing until it drops due to the following expanded cold air downstream the pressure regulator. The pressure regulator is at the beginning closed. Since it cannot deliver the air mass at adjusted pressure the stored air between pressure regulator and coax valve expands faster and the temperature drops faster as well. This can be seen during 1.2 s and 1.7 s. Like pressure vs. time curve of Fig. 16, the air temperature inside the bottle as well as the air temperature at the pressure regulator entry drops almost linearly until a minimum at -40 °C (233.15 K) is reached. The temperature short after the compressor section drops to -21 °C (252.15 K) until the compressor is working properly and a temperature increase is realized due to compression. The measured temperature between both turbines of the egnine drops to almost 273 K until combustion takes place and temperature is rising. Thus, conditions inside the combustion chamber during the QS are at all time sufficient for reliable ignition of the injected air.

The final GG acceleration result is shown in Fig. 18. Here, the blue curve is the QS with the 200 bar compressed air bottle supply and the red curve is a QS with 13 bar shop air supply of the proof of concept design. This TQSS has a start-up time advance of 0.3 sec. Nevertheless, with bottle supply, start-up times of 2.8 sec can be reliably achieved. This was proven several times at the testbed. This startup time can be improved, if combustion is starting much earlier. But a certain GG speed is required for sufficient fuel pressure to achieve reasonable spray generation of the fuel nozzle inside the combustion chamber. However, the cold air temperature in the combustion chamber has a non-negligible influence on proper combustion.

CONCLUSION AND OUTLOOK

This paper shortly describes the development process of an airworthy and compact compressed air supply for an engine QSS for rotorcraft application. Based on a given proof-ofconcept design with low pressure shop air, an analysis of adjustable system parameters was performed and a new air supply with high pressure source was designed. Tests of this system were conducted in first case with a cylinder bundle to ensure constant inlet conditions of the system, followed by final tests with a bottle as compressed air source. The experimental test campaign was successful and the engine can be quick-started within 2.8 s using the compact compressed air supply. Further work has to be done on control parameters of the system and improvements regarding combustion. In addition, the overall system has to lose weight for real rotorcraft application. This can be achieved for instance with a pressure bottle based on carbon and shorter pipes and hoses. Another investigation has to be done on systems failures. This means, how does the engine behave if the system is somehow unrequested activated during regular engine operation.

NOMENCLATURE Abbreviations

AIS

Figure 1 .

 1 Figure 1. Bluecopter Demonstrator of Airbus Helicopters. (Image: Airbus Helicopters, Charles Abarr)

Figure 2 .

 2 Figure 2. Fuel saving potential due to use of ISEO at appropriate flight operation. (similar to [3])

Figure 4 .

 4 Figure 4. Allison 250 ground idle start times of a regular engine start and a quick-start [10] [11].

Figure 5 .

 5 Figure 5. Power delivery curve after performing a quick-start to FADEC mode "flight" [10] [11].

Figure 7 .

 7 Figure 7. Total pressure in measuring pipe short before nozzle entry -experimental and simulation results [15].

Figure 8 .

 8 Figure 8. Nozzle thrust -Experimental and simulation results [15] [16].

Figure 9 .

 9 Figure 9. Scheme of an airworthy QSS for testbed application and helicopter integration.

Figure 10 .

 10 Figure 10. CAD model of a possible QSS arrangement within the BO 105 fuselage.

Figure 11 .

 11 Figure 11. CQSS test setup with compressed air bundle and compressed air bottle at the Allison 250 testbed.

Figure 14 .

 14 Figure 14. Static air temperatures of supply pipe and section after the compressor as well as engine air mass flow at the compressor inlet during QS.

Figure 15 .

 15 Figure 15. Pressure at the end of the compressor for two HQSs with cylinder bundle and different FADEC settings.

Figure 16 .

 16 Figure 16. Total pressure in the supply pipe and in the bottle for a 3 s CQS.

Figure 17 .

 17 Figure 17. Recovery temperatures of the CQSS powered by the bottle at certain measurement positions.

Figure 18 .

 18 Figure 18. GG spool speeds of QS with 13 bar and with 200 bar air supply.

Symbols𝑚̇

  Mass flow, [kg/s]𝑝Pressure,[bar, Pa]