Aurélien Agnèsa

Fleury Sylvaina

Rishi Vanukurub

Simon Richira

Studying the Effect of Symmetry in Team Structures on Collaborative Tasks in Virtual Reality

Keywords: Multi-user Virtual Reality, Symmetric Collaboration, Creativity Workshops

Studies involving collaboration in groups are frequently carried out under symmetrical conditions, meaning that all team members have the same role at the same time. In Virtual Reality, symmetrical collaboration often seems attractive as it allows all participants to be included in the VR environment, but it is not clear whether this provides any benefits over asymmetric forms of collaboration, such as when some team members are in VR while others are working on laptops. We conducted a study to compare the conditions of symmetric configurations (both team members use VR) and asymmetric configurations (one member in VR, and the other uses a laptop) when two participants performed a creativity task together. We found that there were differences in participant behavior and the ideas generated. We conclude by proposing guidelines for future work in the area, particularly related to the use of Virtual Reality for creativity workshops.

Introduction and State of the Art

In recent years, Virtual Environments (VE) have been used in order to better understand creativity, and find ways to improve it [START_REF] Ward | Creative Expression in Virtual Worlds: Imitation, Imagination, and Individualized Collaboration[END_REF][START_REF] Guegan | Avatar-mediated creativity: When embodying inventors makes engineers more creative[END_REF][START_REF] Fleury | Studying the Effects of Visual Movement on Creativity[END_REF]. More recently, with the democratisation of virtual reality and increased access to the required hardware, scientific research has highlighted the capacity of immersive VEs to improve creativity when compared to conventional tools and systems [START_REF] Feeman | Exploration and evaluation of CAD 19odelling in virtual reality[END_REF][START_REF] Mille | Effects of digital tools feature on creativity and communicability of ideas for upstream phase of conception[END_REF]Yang and Lee 2020), particularly in collaborative settings.

Creativity workshops, regularly used in companies [START_REF] Nutzmann | Study on learning concepts applying creativity methods in education and industry[END_REF], are often designed so that all participants have the same role, also known as symmetric collaboration [START_REF] Osborn | Applied Imagination[END_REF][START_REF] Kohn | Collaborative fixation: Effects of others' ideas on brainstorming[END_REF]. Some specific methods involve giving different roles to participants, such as the six hats method [START_REF] Alkhateeb | The Effect of the Six Hats Based on Program in the Development of the Pivotal Thinking of Islamic Concepts Students in Hussein University[END_REF][START_REF] Ekahitanond | Adopting the Six Thinking Hats to Develop Critical Thinking Abilities through LINE[END_REF] or world café [START_REF] Chang | The impact of World Café on entrepreneurial strategic planning capability[END_REF][START_REF] Fouché | An Invitation to Dialogue: 'The World Café'In Social Work Research[END_REF].

However, when collaboration formats are compared, it is usually symmetrical arrangements that are considered, such as classic brainstorming [START_REF] Coskun | Individual brainstorming performance as a function of velocity and comparison feedback[END_REF][START_REF] Dugosh | Cognitive stimulation in brainstorming[END_REF] or the use of sketching as an ideation tool [START_REF] Van Der Lugt | Brainsketching and how it differs from brainstorming[END_REF]. Similarly, many studies dealing with VR in a multi-user context focus on symmetric collaboration, where all participants have the same role (e.g. [START_REF] Guegan | Avatar-mediated creativity: When embodying inventors makes engineers more creative[END_REF][START_REF] Eynard | Impact of verbal communication on user experience in 3D immersive virtual environments[END_REF][START_REF] Slater | Small-group behavior in a virtual and real environment: A comparative study[END_REF]. There are clear benefits to collaboration provided by symmetric team structures -all participants are equally included, and have similar viewpoints and methods of interacting with the tools involved, thereby facilitating intra-group communication. Symmetrical setups also seem to perform better than asymmetrical ones at tasks such as solving puzzles [START_REF] Heldal | Immersiveness and symmetry in copresent scenarios[END_REF].

Despite these benefits, symmetric collaboration may not always be possible. Virtual and Augmented Reality technologies are increasingly being used for creative tasks because of superior spatial visualisation and manipulation abilities [START_REF] Grandi | Characterizing Asymmetric Collaborative Interactions in Virtual and Augmented Realities[END_REF]), but many teams may not be able to provide VR or AR headsets to all members due to lack of access to the required infrastructure. Teams may find themselves in situations where few members are able to use immersive media headsets, while others have screen-based video feedback of the virtual environment, and collaborate using conventional devices like tablets or laptops. Issues of access and team dynamics may be adversely affected by this asymmetry, and hierarchical asymmetry may be reinforced if team leaders or superiors are given access to these 'better' forms of technology.

That being said, the real constraints of technology access make the question of studying the effect of asymmetrical collaboration in VR an important one. Technological asymmetry can take different forms. It can regard for instance ability (users have different actions they can take), challenge (they have different obstacles), interface (different ways of interaction with the system), roles and levels of controls or information receives by the users [START_REF] Ouverson | A Composite Framework of Co-located Asymmetric Virtual Reality[END_REF]. Some recent studies show rather positive effects of technological asymmetry, for example in education when several students use technology together [START_REF] Thomsen | Asymmetric collaboration in virtual reality: A taxonomy of asymmetric interfaces for collaborative immersive learning[END_REF]. In a learning situation, the VR+tablet configuration appears to be better than the VR+VR configuration in terms of presence, immersion and reduction of cognitive load [START_REF] Drey | Towards Collaborative Learning in Virtual Reality: A Comparison of Co-Located Symmetric and Asymmetric Pair-Learning[END_REF]. In collocated asymmetric VR, the use of a tablet with functionalities of visualization and annotations improve the performance of the collaboration (less time wasted, less errors on the task) in comparison with a simple streaming of the VR content on a screen [START_REF] Thoravi Kumaravel | TransceiVR: Bridging asymmetrical communication between VR users and external collaborators[END_REF].

Collaboration with a larger team can also be efficient in asymmetric configuration based on videoconference tools and VR when use a specific software with relevant functionalities to facilitate this collaboration [START_REF] Burova | Distributed Asymmetric Virtual Reality in Industrial Context: Enhancing the Collaboration of Geographically Dispersed Teams in the Pipeline of Maintenance Method Development and Technical Documentation Creation[END_REF].

Asymmetry in collaboration is not always technological. Past work has shown that hierarchical asymmetry (involving superiors or leaders in the collaborative task) can be useful at the stage of idea selection [START_REF] Keum | The influence of hierarchy on idea generation and selection in the innovation process[END_REF], if not during idea generation.

Recent studies have looked at such asymmetrical structures [START_REF] Chan | FrontFace: facilitating communication between HMD users and outsiders using front-facing-screen HMDs[END_REF][START_REF] Clergeaud | Towards seamless interaction between physical and virtual locations for asymmetric collaboration[END_REF][START_REF] Yang | Sharespace: Facilitating shared use of the physical space by both vr head-mounted display and external users[END_REF], and have shown improvements in social engagement, interest, and interaction among team members [START_REF] Gugenheimer | Sharevr: Enabling co-located experiences for virtual reality between hmd and non-hmd users[END_REF][START_REF] Serubugo | Facilitating Asymmetric Collaborative Navigation in Room-Scale Virtual Reality for Public Spaces[END_REF][START_REF] Abadia | Salient features of an effective immersive non-collaborative virtual reality learning environment[END_REF][START_REF] Lee | RoleVR: Multi-experience in immersive virtual reality between co-located HMD and non-HMD users[END_REF]. Through our work, we hope to more clearly compare the effects of symmetry on collaboration in VR. Therefore, the aim of this study was to evaluate the effects of technological asymmetry on the performance on a creative task in VR while collaborating in pairs.

The research question of the present study was "how do technological asymmetry influence creativity and behavior during collaboration in pairs?". Two hypotheses related to this question have been formulated.

In a collaborative situation aimed at generating ideas of solutions to a specific problem, the performance of this collaborative task can be measured by the quality of the ideas generated or by their quantity. In this study, we consider asymmetry as an adaptation to a material constraint that can hinder of slow down collaboration because of the reduction of possible interactions (less immersion and different pattern of humanmachine interactions). Similar viewpoints and methods of interaction may improve the collaboration [START_REF] Heldal | Immersiveness and symmetry in copresent scenarios[END_REF]. The abovementioned researches revealing positive effects of asymmetry regard activities of communication and learning, but not cocreation of creative solution. In the case of co-creation, the ideas co-created are analysable output to assess the effectiveness of the collaboration.

H1: Quality of ideas is better in symmetrical collaboration than in asymmetrical collaboration.

H2: The participants generate more content in symmetrical condition.

Method

Participants

We recruited 28 participants for our study (20 men and 8 women). The age of the youngest participant was 21, and of the oldest was 27. The mean age was 23.1 (SD = 1.65). All participants were students in the field of virtual reality, which avoids possible biases related to the first use of virtual reality which can make it difficult for participants to concentrate [START_REF] Ochs | The interplay Between Presence and Learning[END_REF].

Procedure

We conducted a within-subjects experiment, where each pair of participants performed a similar creativity task but in two different configurations: a symmetrical one with two VR headsets, and an asymmetrical one with one VR headset and a desktop. In both cases, participants were given 25 minutes to complete the task. The HTC Vive headset was used for the VR component

The virtual environment in which participants were placed was that of an underground train station. The task involved the generation of creative ideas to help solve the problem of air pollution of this type of station by sketching there ideas of solution in the virtual environment. This environment and task was chosen because it is a creative problem, in the sense that several new solutions are possible. Participants in VR are represented by their headset and controllers floating in the VE. The participant on the computer controlled a virtual drone that can be controlled using a keyboard to move around in the VE. Asymmetry here is characterized by immersive or non-immersive visualization (what Ouverson and Gilbert, 2021 name "Information"), but especially by the fact that the participant in VR can draw while the participant on desktop cannot (what Ouverson and Gilbert, 2021, name "Ability").During the experiment, one or both participants using a VR headset could draw and move around using a teleportation system. In the asymmetrical configuration, the participant on the computer could interact with their partner with a laser pointer that they could orient appropriately with a mouse. In the asymmetric configuration, participants had the liberty to exchange their devices. While we realise that the drone allows for a limited range of interaction, our goal was to develop a system that could realistically be suggested as part of a product or service offer.

During the experiment, in the two conditions, the participants were collocated in the same room and the desktop participants could see the VR participants. They could communicate directly by voice. After the completion of each configuration (symmetric or asymmetric), participants filled a questionnaire that evaluated their collaboration.

Measures

The creativity of the proposed solutions was measured using the [START_REF] Cropley | Elements of a universal aesthetic of creativity[END_REF] test to reflect creativity according to four criteria: Relevance and effectiveness (the output is fit for purpose; CC1), generation of novelty (the product is original, surprising and germinal; CC2), elegance (the output is well-executed; CC3) and genesis (which means "generalizability", the output changes how the problem is understood; CC4). All the ideas were rated by a panel of 3 judges.

To measure the level of detail, objective criteria were chosen: the number of lines drawn by the group, the number of mode changes (colour, size of the line) and the time spent drawing (for this variable, data is considered for each participant in VR and not by dyad, to make it comparable between the two experimental conditions).

Finally, the level of collaboration of the group was measured with a short version of the Team Climate Inventory (TCI) questionnaire [START_REF] Anderson | The innovation imperative: The relationships between team climate, innovation, and performance in research and development teams[END_REF]. This version contains 14 items (Kivimaki and Elovaino 1999), each on a 5-point Likert scale. The dimensions measured by the TCI are vision (focusing on clear and realistic objectives in which the team members are committed), participatory safety (interaction between team members in a participative and interpersonally non-threatening climate), task orientation (commitment to high standards of performance and preparedness for basic questions and appraisal of weakness) and support for innovation (enacted support for innovation attempts including cooperation to develop and apply new ideas). Quantifying and measuring collaboration is all the more interesting, as it has been shown that during collaborative tasks, participants behave differently despite not 'performing' the task differently depending on the medium used [START_REF] Lisiecka | Medium Moderates the Message. How Users Adjust Their Communication Trajectories to Different Media in Collaborative Task Solving[END_REF]. Screenshots of the virtual drawings were recorded to allow the panel of judges to view and rate them (see Figure 1 for examples of the ideas generated). The internal consistency of the judges' scores was found to be quite high (Cronbach's alpha of the three judges was 0.940). Some of the scores do not follow a normal distribution as measured by Shapiro-Wilk tests. In the symmetric configuration, the overall CC score (.903, p = .024), CC1 (.726, <.001), CC2 (.879, p = .008), and CC3 (.837, p = .001) are not normally distributed, and in the asymmetric configuration the measures of CC1 (.804, p = <.001), and CC4 (.871, p = .004) do not follow a normal distribution. Table 1 contains all the means and standard deviations for the creativity scores. The results reveal that relevance of ideas are significantly higher in symmetrical condition than in asymmetrical condition, which is consistent with the hypothesis 1. However, none of the other dimensions of [START_REF] Cropley | Elements of a universal aesthetic of creativity[END_REF] is significantly influenced by the experimental conditions (novelty, elegance and genesis). Some dimensions of the TCI do not follow a normal distribution: in the symmetric configuration, Vision (0.918, p = 0.031), Participatory Safety (0.880, p = 0.004) and Support for Innovation (0.919, p = 0.033); in the asymmetric configuration Participatory Safety (0.867, p = 0.002) and Support for Innovation (0.923, p = 0.042). The dimensions were calculated by averaging their items (see Table 3), and each item is on a 5-point scale. As shown in table 3, no significant difference appeared between the two experimental conditions for the dimensions of TCI. Therefore, regarding the variables measuring the perception by the users of the way in which the collaboration takes places, symmetry or asymmetry does not make a significant difference.

Results

Comparison

Correlations

We conducted a correlational analysis to interpret the results. There is a correlation between the general creativity scores between both conditions: rho=.509, p=.016.

Because there were the same participants in the two experimental conditions, this correlation mean that the participants who are the more creative in one condition tend to be also the more creative in the other condition.

Symmetry

As for the relationship between the results of the TCI and the creativity scores, there are several results. As a reminder, TCI1 corresponds to "Agreement with the objectives", TCI2 corresponds to "team's objectives clearly understood", TCI3 corresponds to "Team's objectives achievable" and TCI13 corresponds "Time taken to develop ideas".

First, Spearman's correlation showed a significant correlation between TCI1 and CCS (rho=.681, p<.001). More specifically, it showed a significant correlation between TCI1 and CCS1, CCS2 and CCS3 (respectively rho=0.633, p<.001; rho=.621, p=.001; rho=.660, p<.001). This means that in symmetry condition, agreement with the objectives is positively correlated with relevance of the ideas, their novelty and their elegance. Secondly, Spearman's correlation showed a significant correlation between TCI3 and CCS4 (rho=.651, p<.001). Finally, Spearman's correlation showed a significant negative correlation between TCI13 and CCS (rho=-.490, p=.015). More specifically, it showed a significant negative correlation between TCI13 and CCS3 and CCS4 (respectively rho=-.524, p=.009; rho=-.488, p=.016). As for the dimensions of the TCI, Spearman's correlation showed a significant relation between general creativity scores and Vision (or clarity) (rho=.501, p=.013). More specifically, it showed a significant correlation between Vision and CCS1 (rho=.434, p=.034), CCS2 (rho=.446, p=.029) and CCS3 (rho=.497, p=.014). Spearman's correlation also showed a negative link between time spent drawing and TCI2: rho=-.462, p=.018. To sum up, in symmetry condition, the main correlations with creativity scores are with the items of Vision.

Asymmetry

In the asymmetric condition, only the correlations between some TCI and CCA4 (genesis) were significant. As a reminder, TCI5 corresponds to "'We are together' attitude", TCI10 to "Critical appraisal of weaknesses", TCI12 to "Search for new ways of looking at problems" and TCI14 to "Cooperation in developing and applying ideas".

Firstly, Spearman's correlation showed a significant negative correlation between TCI5 and CCA4 (rho=-.524, p=.006). Secondly, Spearman's correlation showed a significant correlation between TCI10 and CCA4 (rho=.443, p=.023). A negative correlation also appeared between the number of mode changes and the TCI dimension "Support for Innovation" (rho=-.538, p=.003), as well as the items in it (12: rho=-.446, p=.017; 13: -.447, p=.017; 14: rho=-.448, p=.017). Finally, a Spearman's correlation showed a negative link between the time spent drawing and TCI14: rho=-.378, p=.047. To sum up, in asymmetry condition, the main correlations with creativity scores are with the items of participatory safety, task orientation and support for innovation.

Discussion

The objective of this study was to determine how a difference in collaboration patterns (symmetric and asymmetric configurations) influences participant creativity and behaviour in a paired task. Our results tend to show that there are certain relationships between specific aspects of behaviour and creativity, although there is no significant difference in the overall creativity score between the two configurations.

Links between behaviour and creativity

To begin with, the correlations highlight different links between participants' behaviour and their creative performance. In the symmetrical configuration, concerning the TCI dimensions, a good Vision (or clarity) has a positive impact on creativity, especially on the criteria of relevance and effectiveness, generation of novelty, and elegance.

Creativity and these same three criteria are correlated with the item 'Agreement with Objectives' as well. Similarly, believing the team's objectives are achievable positively impacts genesis. However, the time taken to develop ideas negatively impacts global creativity, and more particularly elegance and genesis.

The more the objectives were understood, the less the participants spent time drawing.

Note that the understanding of the objectives impacted neither creativity nor the other measures of the level of details. Moreover, the time spent drawing in the symmetric configuration is significantly higher than in the asymmetric configuration, by nearly a factor of 2 (1.70). An additional observation that can be made by looking at the drawings is that some are kind of 'colored': some sides are filled with a line, as to show it is solid. It thus seems that there is a line efficiency concept: participants drew more schematic ideas when they understood the objectives well.

In the asymmetrical configuration, a 'we are together' attitude (TCI5) negatively impacts genesis, while the critical appraisal of weaknesses impacts it positively. This means that being more focused on the task than on improving group dynamics allows for an improved performance, while also having a critical debate. Negative correlations appeared between the number of mode changes and the dimension of 'Support for Innovation', as well as the 3 items composing it. Yet, participants changed modes more in the symmetrical configuration when compared to the asymmetrical configuration, with no difference in their behaviour towards innovation. Thus, participants were more efficient on their use of proposed options during asymmetrical task performance, and this has had a positive effect in terms of innovation.

There is a negative link between the time spent drawing and the cooperation in developing and applying ideas. However, participants spent less time in the asymmetrical configuration, and the item cited above was not different between the two conditions. Thus, asymmetric collaboration could indeed allow teams to be more efficient in the production of ideas.

More generally, this means that in order to produce the best ideas in symmetry, users have to have a good 'Vision', and be in agreement with the objectives. This will also impact the time spent drawing: they will be more efficient. Believing the objectives are achievable positively impacts genesis, while taking more time to develop ideas negatively impacts creativity, and notably elegance and genesis. In other words, isolation and intrinsic motivation can help improve creativity. This relationship between intrinsic motivation and creativity seems to be coherent with recent literature (Tan et al.

2019).

In asymmetrical configurations, users should focus on the task itself, while encouraging constructive criticism within the group. This state of collaboration can allow for improved creativity, especially genesis. Moreover, we saw that relevance and effectiveness, generation of novelty, and elegance are correlated with the item of 'agreement with objectives' in symmetrical configurations but not in asymmetrical configuration. In asymmetrical configurations, participants are forced to cooperate because the laptop-bound participant cannot draw, they can only participate in the task through interactions with their partner. Maybe this is a way to reduce social loafing that tend to appears in collaborative work (Stieglitz, Mirabaie, Möllmann & Rzyski, 2021).

Make the participants communicate with each other could be beneficial to favour mutual trust that can promote teamwork engagement (Zhan, Meng, de Pablos & Sun, 2019), and to favour knowledge sharing that improve group creativity [START_REF] Zhang | Exploring the effect of transformational leadership on individual creativity in e-learning: a perspective of social exchange theory[END_REF]. On the other hand, participants in the symmetrical condition could draw things without directly cooperating with their partner. The fact that they are not compelled to cooperate under the symmetrical condition makes their group performance dependent on their agreement with the objective.

Finally, concerning the results of creativity, it has been seen that there is a strong correlation between the results of the two configurations. In other words, the configurations themselves did not fundamentally impact the creativity of the group, and thus of the individuals, because there is a strong correlation between the creativity of the group and those of its individuals (Pirola-Merlo and Mann 2004). However, there is a significant difference in one of the dimensions: relevance and effectiveness, with a superior score in symmetrical collaboration than in asymmetry. Thus, the symmetric configuration leads to more routine ideas as defined by Cropley and Cropley than the asymmetric one. This can be interpreted as the result of an easier collaboration due to the similar methods of interaction and viewpoint [START_REF] Heldal | Immersiveness and symmetry in copresent scenarios[END_REF]. Regarding the hypothesis 2, participants in symmetric condition spent more time in drawing activity and done more lines. Looking at the whole picture, we may interpret that participants in symmetric condition were less focused on cooperation, but more focused on their own drawing. Therefore, they were more "productive" in their drawing activity in comparison with asymmetric condition.

General recommendations

In light of these results, in the symmetric configuration, having good clarity of the objectives and not taking too much time per idea improves creative output. This seems to fulfill the basic requirements of a divergent creativity workshop, where the aim is to have as many ideas as possible in as little time as possible. Furthermore, the more time is spent drawing, the less elegant and generic the idea becomes, which may sometimes be inefficient for the divergent phase. Finally, this would allow the generation of a better set of ideas to select from during the next phase of convergence. In the asymmetric configuration, focusing on the task at hand more than group dynamics, and being critical towards the proposed ideas leads to better ideas in our study. These seem like good qualities for a convergence phase during ideation. Moreover, drawing from the results of [START_REF] Keum | The influence of hierarchy on idea generation and selection in the innovation process[END_REF], giving the control of the drone to a superior (team leader) could also be beneficial to the selection process. R&D teams can also benefit from the results pertaining to the asymmetric condition. According to [START_REF] Anderson | The innovation imperative: The relationships between team climate, innovation, and performance in research and development teams[END_REF], the focus on ideation tasks is of crucial importance to such teams. Therefore, an asymmetrical tool could be particularly useful for this type of team.

In conclusion and in more general terms, the asymmetric configuration forces people to communicate and work together, whereas symmetry allows users to work individually on a creative level towards a broader group goal. Therefore, at the scale of a pairwise collaboration, we now have general recommendations for circumstances where it may or may not be possible to provide VR tools to both participants, and discuss the consequences on creativity and behaviour in both situations. At this stage, we cannot be sure that these results are also applicable to larger groups, neither in asynchronous collaboration. We have also detailed the behaviours in both configurations in order to improve the quality of ideas generated by participants. Through this study and the analysis that has followed, we hope to have given some insights for teams or organisations that want to make the most appropriate decision while setting up collaborative tasks in Virtual Reality, with focus on creativity and ideation.

We also have another recommendation regarding the evaluation of creativity. We found that there was no correlation between the level of details measured in each idea and creativity. As such, it seems that at least the specific levels of details we considered are not relevant to measure creativity without using questionnaires. However, there may be other measures that could be relevant in order to have another source to measure creativity, and we hope that future studies will help investigate the same.

There are a few concerns regarding the experimental setup as well. During the whole experiment, participants were under the attention of at least two experimenters. The question may arise as to whether the participants would have been as active in problem solving had it not been for the experimenters. Moreover, participants also complained about the lack of interactivity of the drone. An increase of the same could also have led to different results, e.g. move closer to the results of the symmetric condition if we had added a drawing tool to the drone. From this point of view, we can even go further and wonder what role(s) can be given to people who would not wear a headset. Should they explicitly have the role of a supervisor? If there are not enough headsets for everyone, what is the sufficient number of asymmetric roles?

The final question we have is whether encouraging the behaviours we observed in the best cases actually improves creativity. Indeed, in the design of the experiment, participants were not incentivised to behave in a certain way. Will encouragement (as so often happens during group work) lead to a perceptible difference in performance or clarity of objectives compared to a control group, and would that improvement lead to a better creativity? This is but one line of questioning that arises from our study. We hope that our work helps spark conversations on the subject of collaboration, creativity and VR, and serves as a good starting point for future work in the area.

Conclusion

Technological asymmetry corresponds to work situations that could be frequent in the future. Indeed, the disability of certain employees or the unavailability of equipment in the organization make it critical for these asymmetric situations to function satisfactorily. [START_REF] Drey | Towards Collaborative Learning in Virtual Reality: A Comparison of Co-Located Symmetric and Asymmetric Pair-Learning[END_REF] revealed some positive results with technological asymmetry, but their control group was also asymmetric, with only less elaborated functions. In the present study, we revealed some positive impacts of symmetry in comparison with asymmetric condition. In the present study, we find that this asymmetry is not necessarily negative in terms of its impact on collaborative work. A key point is that technological asymmetry can lead to participants having different possibilities of action in the digital environment. In this case, it can create situations of dependencies in which they can be forced to communicate because some participants cannot perform certain actions.

In terms of how to foster collaboration, trust between individuals was known to be an important element [START_REF] Zhang | Learning analytics in collaborative learning supported by Slack: From the perspective of engagement[END_REF], as was knowledge sharing [START_REF] Zhang | Exploring the effect of transformational leadership on individual creativity in e-learning: a perspective of social exchange theory[END_REF].

Figure 1 .

 1 Figure 1. Examples of ideas proposed by participants inside the VR environment (remove electricity and add ramp to allow cycles and pedestrian replace trains, air purifying fan, showerlike device for cleaning travellers).

Table 1 .

 1 Values of the creativity scores

	Measure	Symmetrical Means (SD)	Asymmetrical Means (SD) Wilcoxon test (p value)
	CC	11.75 (4.511)	10.103 (4.874)	185.000 (.060)
	CC1	2.333 (0.933)	1.744 (.906)	95.000 (.008)
	CC2	5.306 (1.988)	4.872 (2.331)	84.000 (.422)
	CC3	2.444 (1.486)	1.949 (1.466)	157.000 (.053)
	CC4	1.667 (.667)	1.538 (.589)	95.000 (.694)

Table 2

 2

	Table 2. Cronbach's alphas of the TCI dimensions
	Dimension	Cronbach's alpha S	Cronbach's alpha A
	Vision	.704 (TCI 2)	.270
	Participatory Safety	.751	.822
	Task orientation	.639 (TCI 11)	.348
	Support for innovation	.639 (TCI 13)	.776

presents Cronbach's alpha values of the TCI dimensions. The items that were removed to adjust the dimensions are indicated in parentheses, and the items in bold indicate acceptable results

[START_REF] Taber | The use of Cronbach's alpha when developing and reporting research instruments in science education[END_REF]

).

Table 3 .

 3 Values of the TCI items and dimensionsSome dimensions of level of detail measures do not follow a normal distribution: in the symmetric configuration, time spent drawing (.914, p = .032); in the asymmetric configuration, number of mode changes (.900; p = .011), time spent drawing (.914, p = .024) and number of lines (.838, p < .001). The results presented in Table4reveal that mode changes and time spent drawing are significantly higher in symmetrical condition than in asymmetrical condition, which is consistent with the hypothesis 2.

	Measure	Symmetrical Means (SD)	Asymmetrical Means (SD) Wilcoxon test (p value)
	Vision	4.161 (.541)	4.170 (.457)	-
	TCI1	4.250 (.967)	4.393 (.737)	30.500 (.510)
	TCI2	4.714 (.535)	4.750 (.701)	10.000 (1.000)
	TCI3	3.893 (.786)	3.786 (.957)	122.000 (.512)
	TCI4	3.786 (.917)	3.750 (.844)	63.000 (.882)
	Participatory Safety	4.321 (.645)	4.438 (.530)	151.000 (.430)
	TCI5	4.464 (.693)	4.214 (.787)	82.500 (.190)
	TCI6	4.143 (1.008)	4.536 (.576)	29.500 (.081)
	TCI7	4.429 (.742)	4.357 (.678)	58.000 (.743)
	TCI8	4.250 (.928)	4.643 (.559)	18.500 (.057)
	Task orientation	3.833 (.657)	4.202 (.604)	-
	TCI9	3.929 (.940)	4.286 (.897)	44.000 (.206)
	TCI10	3.5000 (.923)	3.857 (1.044)	76.000 (.161)
	TCI11	4.071 (.858)	4.464 (.793)	42.500 (.098)
	Support for innovation	4.060 (.529)	4.024 (.660)	230.500 (.166)
	TCI12	4.143 (.705)	3.893 (.786)	101.000 (.230)
	TCI13	3.821 (.819)	3.893 (.875)	68.500 (.704)
	TCI14	4.214 (.686)	4.286 (.713)	49.000 (.835)

Table 4 .

 4 Values of the level of details

	Measure	Symmetrical Means (SD)	Asymmetrical Means (SD) Wilcoxon test (p value)
	Number of mode change	224.692 (97.463)	151.143 (53.262)	38.000 (<.001)
	Time spent drawing	457.474 (168.811)	266.420 (71.639)	<.001 (<.001)
	Number of lines	397.000 (188.959)	401.500 (244.358)	143.000 (.416)

This research shows that technological choices can also be a determining factor, providing constraints and opportunities.


Didier Lasseux
email: didier.lasseux@ensam.eu

Pascal Jolly Trefle

Yves Jannot

Emmanuel Sauger

Permeability Measurement of Graphite Compression Packings

In this work, we address the issue of the sealing performance of ring-shape valve compression packings. Our analysis is focused on the characterization of the permeability of the rings made of die-formed exfoliated graphite. Because of the tight character of the material, significant Klinkenberg effects are expected. In addition, due to the manufacturing process, permeabilities k z and k r as well as Klinkenberg coefficients b z and b r in the respective axial and radial directions are markedly different and strongly dependent upon the applied stress. A specific experimental device based on pressure pulse decay of nitrogen through the material was designed for the measurement in each direction under a controlled axial compression. Determination of k z and k r and b z and b r is performed on the basis of a nonstationary gas flow model in the radial and axial directions using an inverse procedure applied to the pressure decay signal. Our results confirm the efficiency of the method developed here. They clearly show the anisotropic character of the material (k z is roughly one order of magnitude larger than k r ) and the dependence upon axial compression. The present analysis is the key step before further quantification of the leak rate that may result from the permeation through the material as envisaged here as well as through interfaces between the housing, the packings, and the stem.

Introduction

While chemical leakage through devices such as valves present on piping is estimated to contribute up to 60% of fugitive emission from petrochemical units, sealing efficiency of stem packings is crucial. To improve sealing performance, a thorough characterization of the possible leak paths is required. In fact, while leak may result from both permeation through the porous material of the packings and flow through the stem-packings-housing interfaces, the net contribution of each of the two to the total flow rate is not clear so far. However, a relying direct and separate characterization of the two contributions represents a very difficult problem. An alternative two-stage strategy underlying the present work is first to characterize the packing material itself, i.e., the permeation through the porous material, and second to determine the total leak in a real configuration so that the contribution of the interfacial flow can be inferred. The aim of this work is hence to contribute to the diagnosis of packing tightness efficiency by addressing the first of these two stages through permeability measurement of the packing material only. Our analysis is focused on die-formed exfoliated graphite compression packings. Since these materials are selected for their low permeability, the measurement of this quantity is a tricky task that requires specific techniques. Packings under investigation are rings whose permeability k z and k r in the axial and radial directions, respectively, are expected to be significantly different as a consequence of their manufacturing process. Moreover, k z and k r are expected to depend on the stress applied to the material. A special measurement device for each direction of investigation is hence necessary for their determination. Both devices are designed for a controlled axial compression to be applied.

Because these materials are expected to have very small permeabilities, the use of a steady-state gas flow method, although appealing, is not well adapted. In fact, this can lead to excessively long experiments since the characteristic time to reach steady-state flow typically varies with the inverse of the permeability and the square of the sample length in a 1D experiment. Moreover, in this permeability range, important Klinkenberg effects are expected ͓1͔ unless the mean gas pressure within the sample is relatively high. If performed on the basis of a steady-state method, the measurement hence requires several different experiments carried out at different mean pressure levels ͓2,3͔. In addition, precise measurement of gas flow rates in this type of experiment remains a difficult task. Conversely, an unsteady-state gas flow is much faster and, as detailed below, allows simultaneous determination of k and b. The use of this method was initiated in the early 1950s ͓4͔ and has been the subject of many studies until recently ͓5-11͔. The unsteady-state experiment basically consists in recording the pressure difference between two gas chambers connected to the sample edges, the upstream one being suddenly submitted to a pressure larger than the equilibrium one featuring what is often referred to as a pulse-decay experiment.

Recently, we have clearly shown that the optimal configuration from the point of view of the pressure signal sensitivity to the parameters k and b is the so-called draw-down experiment, a variant of the pulse-decay. In this configuration, the downstream chamber has an infinite volume, which means that the downstream face of the sample remains at the atmospheric pressure ͓12͔. With this method, the sample to be tested does not need to be preconditioned at a high gas-pressure level. This last option was however often considered in the literature to eliminate Klinkenberg effects but has the main drawback of requiring an equilibrium stage at this pressure level, which can be very lengthy as indicated above.

In many reported analysis on this unsteady-state technique, interpretation of the pressure decay signal is often carried out by assuming that the gas flow is insensitive to accumulation effects in the pores ͑i.e., that the mass flow rate does not explicitly depends on time͒ and/or that Klinkenberg effects are negligible. In this work, we do not retain this hypothesis and a more complete method to interpret our experiments is developed in addition to the two specific experimental draw-down setups that were designed for the measurement of transport coefficients in the radial and axial directions. The analysis allows estimating k z and k r and Klinkenberg coefficients, b z and b r ͑as well as the porosity ͒, of the material at the axial compression level under investigation. axial compression. The compression stress, , is applied to the graphite ring by means of a screw-nut system. It is transmitted from the screw to the graphite ring through the piston, the load cell ͑item 6͒, allowing the measurement and the control of the axial compressive force, and a centering ring ͑item 5͒. A displacement sensor ͑not represented in Fig. 1͒, whose finger is placed on a pin ͑item 9͒, is used to measure the axial corresponding deformation.

In addition, as in the real situation, internal and external radial displacements are limited by rigid and highly permeable rings ͑items 7 and 8͒. These two rings play the role of the stuffing box and stem and are made of sintered bronze. Their intrinsic permeability is roughly 3 ϫ 10 -14 m 2 , a value which is far larger than those expected for the graphite rings ͑see results below͒. As a consequence, their pressure drop contribution to the flow, both from the viscous and Klinkenberg effect points of view, can be safely neglected in the interpretation of the pressure signal to estimate the intrinsic properties of the graphite rings.

To validate the design of the device and control the sealing efficiency of the assembly, sealing tests were carried out. For these tests, the graphite ring was replaced by an impermeable metal ring, and the change in the initial pressure of typically 50 bars was recorded in the upstream chamber closed for a period of several days without noticing any decrease.

As detailed below, permeability and Klinkenberg coefficient estimation is extracted from the interpretation of an unsteady gas flow through the graphite ring. When the upstream chamber containing the pressurized gas ͑nitrogen͒ is opened at the inner part of the ring at t = 0, gas flows through the material in the radial direction and pressure decreases in the upstream chamber. As justified in the introduction, our experiments were carried out in the drawdown configuration that is optimal for unsteady-state experiments. Consequently, the following measurements are performed during the test:

• Compressive force ͑load cell range 0-50 kN, precision 20 N͒. This corresponds to an axial stress range from 0 MPa to 52 MPa and a precision of 0.02 MPa. • Deformation ⌬h of the graphite ring due to compression ͑displacement sensor range Ϯ5 mm, precision 0.02 mm͒ • Pressure over time p 0 ͑t͒ in the upstream chamber until t = t f ͑pressure gauge range 0-60 bars, precision 5 ϫ 10 -3 bars͒. In the draw-down experiment used here, the pressure in the downstream chamber remains constant equal to P 1 = P atm .

For parameters estimation, the volume V 0 of the upstream chamber must be determined. This was performed by replacing the graphite ring by an impermeable metal ring having the same dimensions and by relaxing the pressure of a mass of gas initially confined in a tank of known volume into V 0 . The upstream volume was found to be V 0 = 38.2ϫ 10 -6 m 3 . This volume corresponds to the height of the stress-free graphite ring and is corrected according to the deformation after compression.

The experimental procedure is as follows: Graphite rings are prepared with glued metal rings respecting coaxiality of the assembly which height is measured prior to the test. This assembly is positioned within the experimental device, and the axial compressive force is applied. Two different compression stresses were investigated ͑ ϳ 15.2 MPa and ϳ 29.3 MPa͒. The deformation is measured, and V 0 is recalculated. The last step is the gas pressure relaxation record through the ring from its initial value P 0 = p 0 ͑t =0͒. Typically, P 0 was chosen in the range 5-40 bars. Tests were carried out on different rings, and some of them were repeated several times on the same sample. Estimation is performed on the basis of a creeping, nonstationary, isothermal, and weakly compressible gas flow model along with an inverse technique. Our results indicate that the procedure and the technique employed here to estimate k z and k r and b z and b r are efficient. Estimated permeabilities are in the range 10 -20 -10 -21 m 2 , while k z is roughly one order of magnitude larger than k r . Permeabilities in both directions ͑as well as b z and b r ͒ are significantly affected by the axial compression. The perspective of this work is to compare the leak rate that can be estimated from the present analysis to that observed on a valve. In this last configuration, leak may result from the permeation through the packing material, on the one hand, and from the flow through interfaces between the housing, the packings, and the stem, on the other.

Experimental

Experimental Technique

In this section, the two specific experimental devices developed for the measurement of the permeability and Klinkenberg coefficient in the radial and axial directions are presented. Graphite rings employed in the tests have an inner radius r 0 =15 mm, an outer radius r 1 =23 mm, and a height h=8 mm.

Experimental Device for the Measurement of k r and b r .

The experimental device to measure the permeability k r and Klinkenberg coefficient b r ͑and porosity ͒ of the die-formed exfoliated graphite rings in the radial direction is schematically represented in Fig. 1. It basically consists of a rigid frame ͑item 2͒ closed by a flange ͑item 4͒. Within this assembly, high-and lowpressure gas chambers are formed, corresponding respectively to the inner and outer regions of the ring. To measure k r and b r , the graphite ring must be sealed axially to avoid flow in this direction. To do so, two metal rings ͑item 1͒ are glued on the upper and lower faces of the ring. Sealing between the upper and lower axial faces of this three-ring assembly ͑graphite and metal rings͒ is achieved using O-rings placed between the lower metal ring and the frame on the one hand and the upper metal ring and the piston ͑item 3͒ on the other. Two other O-rings placed in grooves of the piston ensure tightness between the low-pressure ͑or downstream͒ chamber and the outside of the device. The high-pressure ͑or up-stream͒ chamber and the downstream one can be connected to external tanks if required; i.e., the experiment can be run in the pulse-decay configuration.

Tests are performed in a configuration identical to that in the real situation. In particular, graphite rings undergo a controlled box of a valve. In order to avoid the presence of interfacial leakage and flow in the radial direction, the graphite ring is fitted with two sealing rings ͑item 1 in Fig. 2͒ on its inner and outer faces. The assembly is positioned between an upper and a lower porous rings made of sintered bronze ͑item 2͒ having the same characteristics as those employed in the radial configuration. These porous rings allow gas flow in the axial direction through the graphite ring and transmission of the axial compressive force. Again, their contribution to the flow of gas can be completely disregarded while interpreting the signal.

The ring assembly is placed inside a cylinder ͑item 3͒ that simulates the stuffing box, while a piston ͑item 4͒ plays the role of the stem and is used to transmit the axial compressive force. The upstream chamber--including the pore space of the sintered bronze upper ring--formed between the cylinder, the upper face of the graphite ring, and the lower face of the piston is made tight by two O-rings positioned in grooves machined in the piston. Gas is pressurized in this chamber through a connecting orifice ͑item 10͒. The downstream chamber is formed between the bottom of the stuffing box and the lower face of the graphite ring and is connected to the atmosphere through the orifice ͑item 11͒. Tightness with respect to the upstream chamber is ensured by the two sealing rings ͑item 1͒. The compressive force is transmitted to the graphite ring via the piston and the upper porous ring by means of a screw ͑item 5͒ moving through the upper flange ͑item 6͒. A system made of spring washers, a thrust ball, and a spacer ͑items 7͒ was introduced to ensure the axial alignment of the force. Compression is measured with the load cell ͑item 8͒ positioned between the cylinder ͑stuffing box͒ and the lower flange.

As in the radial configuration, sealing tests were performed, setting an initial pressure of typically 50 bars in the upstream chamber and replacing the graphite ring by an impermeable metal one while checking the pressure change over a period of several days. These conclusive tests allow the validation of the conception of the device from the sealing point of view.

Thus, the experiment consists of the following measurements:

• Compressive force ͑load cell range 0-50 kN, precision 20 N͒. The corresponding axial stress range is 0-52 MPa with a precision of 0.02 MPa. • Deformation ⌬h of the graphite ring due to compression: This is performed using the displacement sensor ͑range Ϯ5 mm, item 9͒. Precision of this sensor is given by the maximum between 2.5 m and 0.5% of the measured displacement value. • Pressure over time p 0 ͑t͒ in the upstream chamber ͑pressure gauge range 0-60 bars, precision 5 ϫ 10 -3 bars͒. As in the radial configuration, the downstream chamber is connected to the atmosphere ͑P 1 = P atm ͒ in the context of a draw-down experiment. The change is recorded until t = t f .

Estimation of the permeability, k z ͑and Klinkenberg coefficient b z , as well as porosity , see below͒ is made by interpreting the upstream pressure signal decay using a physical model that is developed in the following sections and requiring the knowledge of V 0 , the volume of the upstream chamber. Using the same technique as in the radial case, we found V 0 = 35.71ϫ 10 -6 m 3 , a value that is not subjected to variations after compression provided all parts transmitting the stress are assumed to be infinitely rigid.

The first stage of the experiment is to glue the sealing system onto the graphite ring. Deformation of the graphite ring is then measured after the compressive force is applied. Next is the recording of the upstream pressure variation p 0 ͑t͒ due to axial gas flow ͑nitrogen͒ through the ring. Two axial compression stresses ͑ ϳ 14.3 MPa and ϳ 28.3 MPa͒ were considered in this analysis. The initial upstream pressure, P 0 = p 0 ͑t =0͒, was roughly 45 bars in all our tests. Experiments were repeated on several different rings and, for some of them, several times on the same ring.

All the experiments were performed at room temperature with a fluctuation from 2 °C to 3 °C.

Physical Models

In this section, the physical models used to interpret pressure decay signals in the radial and axial configurations are detailed. These models are presented considering the general case of pulsedecay when the upstream and downstream volumes are of finite extent, keeping in mind that the draw-down case is only a particular situation where the downstream volume tends to infinity. The porous sample is assumed to be homogeneous under the form of a ring of inner radius r 0 , outer radius r 1 , and thickness h. Its porosity is denoted by . We assume an isothermal gas flow at very low Reynolds number ͑this is usually the case in practice͒ so that no significant inertial ͑or Forchheimer͒ effects are present. In addition, gas is considered as ideal, which is a valid approximation for gases such as N 2 at experimental operating pressures used here. Under these circumstances, the general form of the mass and momentum conservation equations can be written as

ץ ץt + ٌ • ͑v͒ = 0 ͑1͒
Here, is the gas density and v the filtration velocity through the porous medium given by Darcy's law including Klinkenberg effects:

v = K • ͩI+ B p ͪ•ٌp ͑2͒
In this last relationship, K, B, and I are the intrinsic permeability, Klinkenberg correction, and unit tensors, respectively. The ideal gas law is given by

= p R G T ͑3͒
where p and T are the pressure and temperature, respectively, of the gas at a given time and position in the porous medium and R G is the constant for the gas under consideration ͑R G = R / M with R = 8.314 J mol -1 K -1 , M being the molar mass of the gas, R G = 296.93 J kg -1 K -1 for nitrogen͒. These equations are now used to derive the complete initial boundary value problem in each direction in order to interpret our experiments.

Radial Direction.

We first consider a pressure gradient in the radial direction ͑see Fig. 3͒, i.e., ٌp = ͑ץp / ץr͒e r . As in the sequel of this paper, we assume that the principal axes of K and B are e r and e z , respectively. Along with the fact that the material is homogeneous, this leads to a pure radial flow so that p = p͑r , t͒. Under these circumstances, the combination of Eqs. ͑1͒-͑3͒ leads to the following time and space distribution of pressure:

k r ץp ץt = 1 r ץ ץr ͫ r ץp ץr ͑p + b r ͒ ͬ ͑4͒
to which, for completeness, one must associate boundary and initial conditions as specified below.

Boundary Conditions.

The upstream boundary condition can be simply derived by writing that the mass flow rate through the upstream or inlet face of the porous sample is exactly that out of the upstream chamber. The mass flow rate through the inlet face of section S 0 at r = r 0 can be written as

q m = 0 v 0 S 0 ͑5͒
where v 0 denotes the filtration velocity in the radial direction at the inlet face of the ring as given by Eq. ͑2͒. If p 0 denotes the pressure on this face of the ring ͑p 0 = p͑r = r 0 , t͒͒, which is equal to that in the upstream chamber, and when ideal gas and Darcy's laws are taken into account, this flow rate can be expressed as

q m = -2r 0 h p 0 R G T k r ͩ1+ b r p 0 ͪͯ ץp ץr ͯ r=r 0

͑6͒

If the experiment is performed at constant temperature, the mass flow rate out of the chamber is simply

q m = dm dt = - V 0 R G T ץp 0 ץt ͑7͒
m being the mass of gas at t in this chamber. Matching the two above expressions leads to the boundary condition at the upstream face given by

ץp 0 ץt = 2k r r 0 h V 0 ͑p 0 + b r ͒ͯ ץp ץr ͯ r=r 0 at r = r 0 ͑8͒
Following the same lines, the boundary condition at the outer ͑downstream͒ face of the sample can be readily written as

ץp 1 ץt = - 2k r r 1 h V 1 ͑p 1 + b r ͒ͯ ץp ץr ͯ r=r 1 at r = r 1 ͑9͒
where p 1 = p͑r = r 1 , t͒. Note that in the case of the draw-down ͑V 1 → ϱ͒, this last condition is simply replaced by a classical Dirichlet condition on the pressure,

p 1 ͑t͒ = Cst = P 1 at r = r 1 ͑10͒
3.1.2 Initial Condition. At the initial stage of the experiment, the system is at equilibrium at a pressure equal to the downstream pressure leading to p͑r,t͒ = P 1 , t = 0, r 0 Ͻ r Յ r 1 ͑11͒

Gas flow through the material starts at t = 0 when the upstream chamber at pressure P 0 Ͼ P 1 is connected to the inner face of the ring, yielding the following condition: p 0 = P 0 , t = 0, r = r 0 ͑12͒

As will be detailed below, this initial boundary value problem has no simple analytical solution and requires a numerical treatment.

Axial Direction.

In this section, we consider that the pressure gradient is along e z only, leading to a flow along this axis, i.e., p = p͑z , t͒. Combining the mass and momentum equations and the ideal gas law yields the following pressure equation ͓12͔:

k z ץp ץt = ץ ץz ͫ ץp ץz ͑p + b z ͒ ͬ ͑13͒ 3.
2.1 Boundary Conditions. Boundary conditions at the upstream and downstream faces are obtained using the same arguments as in the radial configuration. The upstream face of the sample is taken at z = 0 and the downstream one is taken at z = h, thus leading to

ץp 0 ץt = k z ͑r 1 2 -r 0 2 ͒ V 0 ͑p 0 + b z ͒ͯ ץp ץz ͯ z=0 at z = 0 ͑14͒ and ץp 1 ץt = - k z ͑r 1 2 -r 0 2 ͒ V 1 ͑p 0 + b z ͒ͯ ץp ץz ͯ z=h at z = h ͑15͒
In these equations, p 0 and p 1 are, respectively, p͑z =0,t͒ and p͑z = h , t͒. The latter condition reduces to

p 1 ͑t͒ = Cst = P 1 at z = h ͑16͒
in the draw-down condition.

Initial Conditions.

Just as in the radial direction, the initial condition is simply given by p͑z,t͒ = P 1 , t = 0, 0 Ͻ z Յ h ͑17͒ the flow being initiated at t = 0 by the pressure P 0 at the upstream face, i.e., p 0 = P 0 , t = 0, z = 0 ͑18͒

For subsequent interpretation of experimental data, this problem is solved numerically.

For each configuration ͑radial and axial͒, the initial and boundary value problems presented above is discretized using a finite difference scheme that is second order in space and explicit first order in time. A stability condition of the scheme must be ensured by a time step condition given by ⌬t Ͻ ␣ x ͑⌬x͒ 2 / 2͓p 0 ͑t =0͒ + b x ͔, with ␣ x = / k x , x = r or z, depending on the radial or axial configuration. The numerical solution resulting from the direct simulation yields p 0 ͑t͒, which quadratic deviation with respect to the measured signal is minimized using an inverse algorithm to estimate the desired parameters.

Inversion and Results

Our inverse technique is a Levenberg-Marquardt algorithm developed to simultaneously estimate k x and b x , x = r or z ͑as well as , see below͒. This method combines a Gauss-Newton method with a gradient descent, one taking benefits of the rapidity of the former and of the stability of the latter. Details of the algorithm can be found, for instance, in Refs. ͓13,14͔. Basically, it consists in minimizing the square of the function given by 2 ͑k x , b x ͒ = ͚ i=1 N ͕͓p 0m ͑t i ͒p 0 ͑t i ͔͒ / i ͖ 2 , x = r or z. In this formula, N is the number of time records of the measured signal p 0m , p 0 is the value of the upstream pressure estimated from the model at a given set of parameters ͑k x , b x , and ͒, and i is the standard deviation on p 0m at the ith time record. Convergence is achieved once 2 falls below a user-defined criterion chosen here equal to 0.05.

To start the inversion process, it is necessary to provide initial values of k x , b x , x = r or z, and . These values were extracted from a simplified interpretation of the flow based on a hypothesis often proposed in the literature ͓7͔. It consists in assuming that gas density is not explicitly time dependent, meaning that gas accumulation in the pore space is negligible. This hypothesis is asymptotically valid when p 0 ͑t͒ reaches P 1 . For this reason, we used the late part of the recorded values of p 0 ͑t͒, i.e., the last 10% of the experimental signal. In the case of the draw-down experiment, this allows to pre-estimate k r and k z , according to the following expressions:

k r = - V 0 ln r 1 r 0 2h͓p 0 ͑t͒ -P 1 ͔ͩb r + p 0 ͑t͒ + P 1 2 ͪ dp 0 dt ͑19͒
in the radial case and

k z = - V 0 h ͑r 1 2 -r 0 2 ͓͒p 0 ͑t͒ -P 1 ͔ͩb z + p 0 ͑t͒ + P 1 2 ͪ dp 0 dt ͑20͒ 4.
1 Results in the Radial Configuration. Experiments in this configuration were performed on six different rings for in the range 14.2-17.0 MPa and in the range 29.1-29.9 MPa on the five first ones only.

To illustrate the inversion procedure, we have reported in Fig. 4 an example of a recorded experimental pressure signal ͑open sym-bols͒ along with the simulated evolution of p 0 obtained with estimated values of the parameters k r , b r , and resulting from inversion.

Although the comparison in this figure is not sufficient to check the validity of the inversion, our physical model seems to correctly reproduce the evolution of the pressure observed at the upstream face of the ring. A better quantitative verification of the estimation of the parameters is made by analyzing the residuals after inversion, i.e., the difference between the measured signal and the simulated one. This is represented in Fig. 5, showing that these residuals remain less than 80 mbars that is on the order of 1% of the smaller pressure measured during the experiment ͑i.e., at t = t f ͒.

A quick comparison between these residuals and the evolution in the axial case. These two relationships can be completed with pre-estimated values of b r and b z using an empirical correlation proposed in ͓7͔: b x =1,89ϫ10 -6 k x -0,36

, x=r or z where units for k x and b x are m 2 and bar, respectively. For , an "intermediate" value of around 20% can be used.

Prior to the inversion of experimental signals, we have analyzed the impact of time sampling of p 0 ͑t͒ as well the influence of errors on experimental parameters such as V 0 , r 0 , h, and on the estimation of k x and b x , x=r or z, resulting from inversion. To do so, we have used a synthetic signal p 0 ͑t͒ generated with a direct numerical simulation. To be more realistic, a random Gaussian noise given by ␦ p=0,01adpP 0 /3 was superimposed on p 0 ͑t͒. In this relationship, a is a random number sequence of unit standard deviation and dp is the error on p 0 ͑t͒͑in percent of the measure-ment͒. Coefficient 3 was chosen so that p 0 ͑t͒Ϯ␦ p represents 99.7% of the values of p 0 ͑t͒ if they would have been measured. Here, we took dp=0.1%. This noisy synthetic signal, generated with the nominal values of the different parameters, was then used as an input of our inversion procedure for which parameters were those embedding an error. The resulting estimation of k x and b x , x=r or z, was compared with the initial input values. In accordance with a result already reported in Ref. ͓15͔ in the axial case, this analysis leads to the following conclusions for both the radial and axial configurations: ͑i͒ Time sampling is not a crucial parameter to correctly estimate k x and b x , x=r or z. This is an indication that no significant precision can be gained from increasing the number of points to sample p 0 ͑t͒ over a given period of time; ͑ii͒ an error on V 0 , r 0 ,or h does not yield significant errors on the estimated values of the permeability or Klinkenberg coefficient; ͑iii͒ an error on has a very important impact on the estimation of these parameters. From a practical point of view, and its evolution with the compression stress are not known precisely a priori. For this reason, k x , b x , x=r or z, and were estimated simultaneously. Without detailing all the features of the inverse technique, it must be noted however that the sensitivity of the pressure signal to the porosity is much less than the sensitivity to the permeability and Klinkenberg coefficient. Interpretation of our experimental data is presented in the next sections. of the gas temperature in the upstream chamber as represented in Fig. 6 indicates that the residuals have a strong signature of the temperature fluctuations. Since this thermal effect is not accounted for in the physical model, its effect remains a perturbation source in the inversion process. All the results obtained in the lower range of stress are gathered in Table 1.

Except on ring 4, the deformation is roughly from 0.9 mm to 1 mm, which represents 11% to 12% of the ring thickness when it is free of stress. Estimated values of the radial permeability, Klinkenberg coefficient and porosity ͑see Table 1͒ show that except on ring 6, k r is in the range 4.9ϫ 10 -21 -6.9ϫ 10 -21 m 2 ͑average 6.2ϫ 10 -21 m 2 ͒, while b r is in the range 8.4-10 bars ͑average 9.2 bars͒, which remains relatively narrow while taking into account the order of magnitude of this permeability as well the variability on the material inherent to the manufacturing process.

The variability is evidenced with ring 6, having a smaller permeability in the range 1.5ϫ 10 -21 -3.5ϫ 10 -21 m 2 over the three different tests, whereas b r , except for the second test, is around 16-19 bars.

Estimated values of the porosity on the six samples are more spread, in the range 21-46% ͑average 33.6%͒. A plausible explanation of such a distribution lies again in the variability of the material as a result of the manufacturing process. This spread might also include the error on the estimation of this parameter since the sensitivity of the pressure decay to remains small compared with the permeability and Klinkenberg coefficient. This feature is even more pronounced in this low range of permeability.

When the axial compression stress is increased by roughly a factor of 2 ͑see results gathered in Table 2͒, the deformation increases ͑⌬h ϳ 1.4 mm except again for ring 4͒ while the permeability decreases by roughly a factor of 2 ͑average 3.6 ϫ 10 -21 m 2 ͒ and, accordingly, b r increases ͑average 12.3 bars͒.The porosity remains roughly in the same range ͑average 33.5%͒. This would indicate that larger flow-path constrictions were reduced in the radial direction while increasing the axial compression although no significant global compaction of the bulk material occurred.

Results in the Axial Configuration.

Experiments were carried out on three different rings and for the two ranges of axial compression stress ͑13.6 MPaՅ Յ 14.7 MPa and 27.2 MPa Յ Յ 29.3 MPa͒. Again, to check the result of the inversion process, we have represented in Fig. 7 the evolution of p 0 measured on ring 7 for = 29.3 MPa along with the simulated pressure signal obtained with the estimated values of the parameters in this case.

As in the radial case, the overall approach is validated by the residuals between the measured pressure decay and that predicted by the direct model as represented in Fig. 8. These residuals remain smaller than 150 mbars ͑and even 80 mbars if the very early stage of the experiment is ignored͒ and are strongly correlated with the temperature fluctuations in the upstream gas chamber represented in Fig. 9. Results obtained in this configuration are gathered in Table 3.

For the lower stress range, the deformation is again about 1 mm 3͒. Values of the porosity are much less spread than in the radial case. This might result from the fact that the sensitivity to this parameter is larger in this case. In addition, it must be noted that as observed on experiments in the radial direction, porosity very slightly decreases while increasing .

More important is the contrast between k r and k z for each stress range. In fact, for both ranges of the compression stress, k r is an order of magnitude smaller than k z . This clearly highlights the strong anisotropic character of the material and the contrast that was expected, considering the manufacturing process of the rings. Indeed, these rings are obtained by rolling up a sheet of exfoliated graphite around an axis corresponding to the ring axis. As a consequence, the axial permeability is expected to be much larger than the radial one, a behavior that is evidenced and quantified by these experiments.

Conclusions

Based on a previous analysis on the unsteady-state method to measure permeability of tight materials reported in Ref. ͓12͔, an experimental procedure and two distinct experimental devices were designed to measure intrinsic permeability ͑as well as Klinkenberg coefficient and porosity͒ of die-formed exfoliated graphite compression packings in the radial and axial directions, respectively. The experimental technique consists in recording the pressure decay of a gas confined in a chamber initially pressurized above the equilibrium pressure of the whole system and connected at t = 0 to the upstream face of the ring, the downstream one remaining at the atmosphere. On the basis of a creeping, nonstationary, isothermal, and weakly compressible flow in a 1D porous medium, an inversion technique was proposed and used on measured pressure decay signals to determine the flow properties of the rings. The whole approach was illustrated on experiments performed on several different rings with two distinct axial compression stresses. Results indicate that ͑i͒ the procedure and the technique employed here to estimate permeabilities and Klinkenberg coefficients ͑as well as porosities͒ are efficient. Estimated permeabilities are in the range 10 -20 -10 -21 m 2 ; ͑ii͒ on this material, axial permeability is roughly one order of magnitude larger than the radial one; ͑iii͒ permeabilities in both directions roughly decrease by a factor of 2, while the compression stress is increased from about 14.7 MPa to 29.3 MPa. Klinkenberg coefficients decrease with permeability and are well predicted in the axial direction by an empirical correlation proposed in the literature; ͑iv͒ porosities, although more dispersed over all the experiments, seem to very slightly decrease with the axial compression. More work is necessary to improve the estimation of this parameter. Improvements of the experimental procedure can also be achieved with a better thermal regulation of the system. This work opens wide perspectives for a thorough characterization of the packing materials in order to better qualify and improve the sealing efficiency of industrial valves. In particular, this should allow a further comparison between the leak-rate fraction due to permeation through the material of the rings and the total leak-rate of the valve that includes leakage through interfaces between the housing, the packings, and the stem.

Nomenclature

B ϭ Klinkenberg correction tensor ͑Pa͒ b r ϭ Klinkenberg coefficient in the radial direction ͑Pa͒ b z ϭ Klinkenberg coefficient in the axial direction ͑Pa͒ h ϭ ring thickness ͑m͒ K ϭ intrinsic permeability tensor ͑m 2 ͒ k r ϭ permeability in the radial direction ͑m 2 ͒ k z ϭ permeability in the axial direction ͑m 2 ͒ p ϭ gas pressure ͑Pa͒ p 0 ϭ gas pressure at the inlet face of the sample ͑Pa͒ P 0 ϭ initial gas pressure in the upstream chamber ͑Pa͒ p 1 ϭ gas pressure at the outlet face of the sample ͑Pa͒ P 1 ϭ initial gas pressure in the downstream chamber ͑Pa͒ q m ϭ gas mass flow rate ͑kg s -1 ͒ r 0 ϭ inner radius of the ring ͑m͒ r 1 ϭ outer radius of the ring ͑m͒ R G ϭ gas constant ͑J kg -1 K -1 ͒ t ϭ time ͑s͒ t f ϭ final time of the experiment ͑s͒ T ϭ gas temperature ͑K͒ v ϭ filtration velocity ͑m s -1 ͒ V 0 ϭ volume of the upstream chamber ͑m 3 ͒ V 1 ϭ volume of the downstream chamber ͑m 3 ͒ ϭ porosity ϭ gas density ͑kg m -3 ͒ ϭ gas dynamic viscosity ͑Pa s͒ ϭ axial gland stress ͑MPa͒

  Fig. 1 Experimental device designed for the measurement of the radial permeability

Fig. 2

 2 Fig. 2 Experimental device designed for the measurement of the axial permeability

Fig. 3

 3 Fig. 3 Annular section of the porous ring

Fig. 4

 4 Fig. 4 Experimental pressure signal "open symbols… and p 0 "t… simulated with the estimated values of the parameters. Radial case. Ring 6, test 2. Compression stress 14.2 MPa and P 0 = 20.4 bars.

  Fig. 6 Gas temperature in the upstream reservoir. Radial case. Ring 6, test 2. Compression stress 14.2 MPa and P 0 = 20.4 bars.

  and increases to roughly 1.2 mm when this stress is increased by a factor of 2, confirming the observation already reported in the radial case. Estimated parameters k z , b z , and on the three different rings remain in a narrow range centered on the average values, respectively, given by 6.4ϫ 10 -20 m 2 , 17.3 bars, and 19.7% for 13.6 MPaՅ Յ 14.7 MPa and 2.8ϫ 10 -20 m 2 , 20.0 bars, and 18.8% for 27.2 MPaՅ Յ 29.3 MPa. As expected, k z decreases ͑again by a factor close to 2 when the stress is doubled as in the radial case͒ while b z increases. It shall be noted that all the values of b z are very well predicted by the correlation proposed in Ref. ͓7͔ mentioned above ͑see right column of Table

Fig. 7 Fig. 8

 78 Fig. 7 Experimental pressure signal "open symbols… and p 0 "t… simulated with the estimated values of the parameters. Axial case. Ring 7 and compression stress 29.3 MPa and P 0 = 44.4 bars.

Table 1 Results on six different rings obtained in a range of compression stress 14.2-17.0 MPa. Radial case.

 1

	Ring No.	͑MPa͒	P 0 ͑bar͒	P 0 ͑t f ͒ ͑bar͒	⌬h ͑mm͒	k r ͑m 2 ͒	b r ͑bar͒	͑%͒
	1	14.7	9.40	8.58	0.98	6.7ϫ 10 -21	9.6	28.0
	2	14.7	9.36	8.51	0.98	4.9ϫ 10 -21	9.2	35.0
	3	15.7	4.98	3.87	0.92	6.6ϫ 10 -21	8.4	36.1
	4	15.9	7.69	5.51	1.4	6.9ϫ 10 -21	8.6	45.7
	5	16.9	11.12	7.93	0.92	5.7ϫ 10 -21	9.9	33.9
	6_1	14.8	39.77	2.73	0.92	2.7ϫ 10 -21	18.9	21.2
	6_2	14.3	20.42	7.60	0.92	1.5ϫ 10 -21	47.1	33.1
	6_3	14.2	19.61	9.58	0.90	3.5ϫ 10 -21	16.1	35.5

Table 2 Results on five different rings obtained in a range of compression stress 29.1-29.9 MPa. Radial case.

 2

	Ring No.	͑MPa͒	P 0 ͑bar͒	P 0 ͑t f ͒ ͑bar͒	⌬h ͑mm͒	k r ͑m 2 ͒	b r ͑bar͒	͑%͒
	1	29.9	10.4	9.8	1.52	3.6ϫ 10 -21	11.0	20.8
	2	29.2	10.4	10.0	1.4	2.4ϫ 10 -21	13.6	31.6
	3	29.3	6.6	5.5	1.32	4.1ϫ 10 -21	8.2	35.7
	4	29.5	7.4	6.0	1.8	4.3ϫ 10 -21	8.8	43.7
	5_1	29.1	11.3	9.0	1.46	4.4ϫ 10 -21	7.0	37.0
	5_2	29.1	23.3	20.6	1.46	2.5ϫ 10 -21	12.8	32.1

Table 3 Results on three different rings obtained in two ranges of compression stress 13.6- 14.7 MPa and 27.2-29.3 MPa. Axial case.

 3

	Ring No.	͑MPa͒	P 0 ͑bar͒	p 0 ͑t f ͒ ͑bar͒	⌬h ͑mm͒	k z ͑m 2 ͒	b z ͑bar͒	͑%͒	b Jones ͑bar͒
	7	13.6	43.94	7.61	1.04	6.1ϫ 10 -20	17.8	20.1	15.6
	8	14.7	44.45	27.68	0.90	5.2ϫ 10 -20	17.3	19.8	16.6
	9	14.7	42.85	5.48	1.06	7.9ϫ 10 -20	16.9	19.3	14.2
	7	29.3	44.37	12.55	1.24	2.2ϫ 10 -20	21.6	17.1	22.4
	8	28.3	44.46	11.14	1.13	2.8ϫ 10 -20	18.4	21.3	20.6
	9	27.2	44.29	7.92	1.33	3.5ϫ 10 -20	20.1	18.0	19.1