Eduardo Redondo-Iglesias

Emmanuel Vinot

Pascal Venet

Serge Pelissier

Electric vehicle range and battery lifetime: a trade-off

Keywords: EV (electric vehicle), battery ageing, battery management, LCC (life cycle cost)

Extended range of electric vehicles is seen as a mean to accelerate the market penetration of electric vehicles. However, putting more and more battery cells in a pack may lead to an unsuitable usage regarding battery lifetime. In this paper, we assess the impact of oversizing batteries in their lifetime and their total cost of ownership. Simulation results reveal a reduction in battery lifetime of larger batteries when charge is not adapted, dramatically increasing the total cost of ownership. On the contrary, optimisation of the charging strategy preserves the battery's performances for a longer time and lead to important cost savings.

Introduction

Electric vehicles constitute a part of the solutions to stop climate change and reduce air pollution. The market penetration of electric vehicles is increasing fast in many developed countries [START_REF]Global EV Outlook 2018[END_REF]. Governments encourage purchasing this type of vehicles by using tax exemptions, subsidies and other. However, the increase of electric vehicles sales since 2010 is also due to the manufacturing cost reductions mainly performed in the battery. As the battery pack size increases, electric vehicles' range is growing. For example, Renault Zoe had a battery pack of 22kWh in 2012 and of 41kWh since 2016; other electric car models have increased their battery size (Nissan Leaf from 24 to 40kWh, BMW i3 from 22 to 42kWh) expanding ranges from about 150 to more than 300km. With ranges up to 350km, electric vehicles could replace internal combustion engine vehicles in most of cases. However, the daily required range is most often less than 50km [START_REF] Derollepot | Optimizing components size of an extended range electric vehicle according to the use specifications[END_REF]. With larger batteries, state of charge may be often higher leading to faster battery degradations. Size and charging of batteries can be optimised to a minimal range in order to prolong their lifetime.

Despite of advances in cost and performances, batteries are still the weak point of electric vehicles because they limit their range and have a significant impact on the total cost of ownership. For these reasons, battery management to reduce its degradation sparks much interest in the scientific community. For example, some authors aimed to optimise the charging protocol of lithium-ion batteries to minimise charging time and/or maximise battery's lifetime [START_REF] Monem | Lithium-ion batteries: Evaluation study of different charging methodologies based on aging process[END_REF][START_REF] Chu | Optimal charge current of lithium ion battery[END_REF][START_REF] Zhang | Charging optimization in lithium-ion batteries based on temperature rise and charge time[END_REF]. Other works, for example [START_REF] Petit | Development of an empirical aging model for Li-ion batteries and application to assess the impact of Vehicle-to-Grid strategies on battery lifetime[END_REF][START_REF] Dubarry | Durability and reliability of electric vehicle batteries under electric utility grid operations: Bidirectional charging impact analysis[END_REF], studied the influence of V2G (vehicle to grid) applications in battery degradation.

In this work, we are exploring the consequences of increasing electric vehicles' battery size on its lifetime. We are considering two distinct scenarios: first one, consists in fully charging the battery every day (worst-case scenario); second scenario consists in charging the battery to a defined state of charge ensuring a reduced range.

Battery ageing

A battery allows to reversibly store energy by transforming electricity into chemical energy. The main characteristics of a battery are energy (capacity) and maximum power (impedance). Batteries are complex systems: their performances depend on the composition of their parts (electrodes, current collectors, electrolyte, etc.) and on the reactions occurring between them.

Batteries' performances degrade over time because of ageing mechanisms. These mechanisms are parasitic physico-chemical reactions causing capacity fade and impedance rise [START_REF] Vetter | Ageing mechanisms in lithium-ion batteries[END_REF]. Battery ageing is classified in two types: calendar and cycling ageing. Calendar ageing is the degradation during rest times whereas cycling ageing is the degradation induced by charging and discharging the battery. In this work, we are considering the electric vehicle application where average current rates are very low, about C/2 in discharge (during the vehicle use) and C/5 in charging. Moreover, private owned vehicles are parked most of time (more than 95% of time). For this reasons, in this paper we will consider only calendar ageing. So cycling ageing with little solicitation will be neglected. The main calendar ageing factors are temperature (T ) and state of charge (SoC).

Modelling

We are focusing in battery degradation as a consequence of the electric vehicle use. As explained above, T and SoC are the main factors for calendar ageing. Calendar ageing will be computed by using a degradation model based on Eyring relationship [START_REF] Redondo-Iglesias | Eyring acceleration model for predicting calendar ageing of lithium-ion batteries[END_REF].The following expression is used to calculate the capacity fade as a function of T and SoC:

Q L = A • e (-Ea/kT +B•SoC) (1)
To obtain the SoC profile, we need to define a use scenario; that is a daily speed profile and a charging strategy.

The chosen mission profile is shown in Figure 1. From Monday to Friday, two WLTC (World-wide harmonized Light duty Test Cycle) class 3 [START_REF] Tutuianu | Development of the World-wide harmonized Light duty Test Cycle (WLTC) and a possible pathway for its introduction in the European legislation[END_REF] speed profiles are used to represent a daily trip hometo-work (at 8 a.m.) and work-to-home (at 5 p.m.). The WLTC class 3 is 23km long and it includes four phases (low, medium, high and extra-high speed) corresponding to different road types (urban, rural, motorway). The cycle length corresponds approximately to the average daily travelled distance in the French case (25km) [START_REF]Enquête nationale transports et déplacements[END_REF]. On Saturday, we supposed that a longer trip is made mainly for leisure activities. For this reason, WLTC class 3 speed profile is used twice at each trip (at 9 a.m. and 9 p.m.), making the travelled distance (46km) two times higher compared to the preceding case. Finally, on Sunday, a WLTC class 1 is performed at 10 a.m. and 5 p.m. as we supposed less leisure activities. The WLTC class 1 is 8km long and it includes two phases (low and medium speed). These choices are arbitrary, but sufficiently realistic for the purpose of the work.

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The VEHLIB library [START_REF] Jeanneret | New Hybrid concept simulation tools, evaluation on the Toyota Prius car[END_REF] is used to assess the electric power going upstream from vehicle's speed to battery power (P bat ). Vehicle chassis is modelled considering its total weight (including battery) and the drag forces due to aerodynamics and tires. Electrical machines are modelled considering efficiency maps and battery behaviour by an electric equivalent circuit.

Note that P bat is the output battery power. In order to properly quantify the amount of available energy in the battery, the net power P bat,0 must be calculated by adding the power losses inside the battery. This is achieved by using the relations given in Appendix (Useful equations).

Finally, the charging strategy consists in defining some rules to make the decision of charging the battery. For example, some users may decide to completely charge their batteries every day, but other may adapt the battery charge to the expected distance to travel the day after. The charge time is fixed to 9 p.m. from Sunday to Friday and 11 p.m. on Saturday.

In this paper, a small city car is considered (47kW, 945kg excluding batteries). Battery mass is calculated for each battery size from 20 to 60kWh. Temperature is constant and equal to 25 °C.

Simulation results: effects of oversizing the battery depending of the charging strategy

Depending on the charging strategy, battery lifetime and total cost of ownership may change according to the size of the battery. Indeed, according to equation 1, ageing is amplified by high temperature and high SoC. In this work, we consider constant temperature all along the year but scenarios differ on the value of the daily average SoC. Consequently, different capacity fade rates occur.

We will consider two charging scenarios. In the first one, there is no charging strategy depending on the next day's travelled distance: the battery is fully charged each night independently the needed range for the day after. On the contrary, in the second scenario, the charging strategy will take into account the expected next day's travelled distance.

For each scenario, the mission profile and the charging strategy will be repeated until end of life (EoL) of the battery (i.e. capacity fade = 20%). Different battery sizes from 20 to 60kWh are simulated to measure the impact of the battery size on its lifetime (in number of days) and cost (in C/ day) for each charging scenario.

Scenario 1: charging the battery to a maximum SoC level

In this scenario there is no charging strategy: the battery is fully charged every day regardless of the travelled distance. Real world use shows that this situation of charging "as soon as possible" occurs frequently [START_REF] Kreczanik | Construction of database on real world uses of electric vehicles -a French case[END_REF].

In figure 2, the SoC level for a Monday is shown for three battery sizes: 20, 40 and 60kWh. The battery is charged to 100% every day. Depending on the battery size, the SoC range is different. For example, at beginning of life (BoL), with 60kWh the SoC level varies from 100 to 90% whereas it varies from 100 to 69% with a 20kWh battery. At end of life (EoL), as capacity decreases, the SoC range is larger: minimum SoC is 86 and 61% for 60 and 20 kWh respectively.

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These differences in the SoC profile have a direct consequence in battery ageing according to equation 1: with higher SoC levels, battery degradation is faster. So, in big batteries, as their SoC remains higher than in small ones, degradation will be faster if no charging strategy is used.

An alternative variant of this scenario is to modify the value of the target SoC (SoC max ) in the daily charge (90, 80 or 70% instead of 100% for instance). By decreasing the average SoC, this is expected to limit the capacity fade.

Scenario 2: charging the battery to the required SoC level

In this scenario the charging strategy consists in dynamically calculate SoC max as a function of the expected next day's travelled distance.

First we need to calculate the required energy (E required ) in the battery as the product of the average energy consumption (Ec, in Wh/m) and tomorrow's trip length. A security margin should be included to avoid excessive depth of discharge and to cope with unexpected trip overlength. The auxiliary devices consumption (P aux ) can also be considered. This energy consumption is very dependent of ambient temperature:

E required = Ec • (distance + security margin) + P aux • duration (2)
Then, the target state of energy (SoE max ) is computed as :

SoE max (p.u.) = SoE end of trip + E required E bat (3)
Finally, SoC max can be calculated with relations included in Appendix (Useful equations). Figure 3 illustrates the SoC profile during a week following the mission profile given in the preceding section with this charging strategy for a 40kWh battery. In contrast to scenario 1, maximum SoC is not the same every day, but it depends of the subsequent daily trip. At beginning of life, the initial SoC (fixed by the preceding day's SoC max ) is 26% from Monday to Friday, 43% on Saturday and 14% on Sunday. These levels ensure that the minimum SoC is about 10%. The minimum SoC directly depends of the security margin. In these simulations security margin was fixed to 20 km, although it could be adjustable by the user. SoC levels change progressively with battery ageing. For example, the initial SoC from Monday to Friday raised from 26% at BoL to about 31% at EoL.

Comparison between scenarios 4.3.1 Daily state of charge levels

Notice that, as illustrated by figures 2 and 3, SoC levels change with battery's capacity fade because weekly mission profile is the same but stored energy in the battery decrease. For example, in scenario 1 (figure 2), the maximum SoC is constant but average and minium SoC levels decrease.

Table 1 shows the values of the daily SoC levels (maximum, average, minimum) for each scenario at beginning of life for a 40kWh battery. From these values, one could predict that the ageing would be lower in scenario with lower SoC.

Lifetime

Figure 4 shows the battery lifetime (days to EoL) depending on battery size for scenarios 1 and 2.

For scenario 1, when SoC max is 100% the battery lifetime is between 2380 and 2100 days. By reducing SoC max , the battery lifetime is increased for every battery size. For example for 40kWh, battery lifetime is 2170, 2630, 3210 and 3910 days with SoC max = 100, 90, 80 and 70% respectively. In other words, battery lifetime is multiplied by a factor of 1.2, 1.5 or 1.8 when SoC max is 90, 80 and 70% respectively compared to SoC max = 100.

When the charging strategy of scenario 2 is used, the battery lifetime varies from 5320 to 9950 days when battery size varies from 20 to 60 kWh. Compared to scenario 1 with SoC max = 100%, lifetime is 2.2 to 4.7 times greater (for 20 and 60kWh respectively). This demonstrates that this strategy can be used to preserve the battery's life.

Another interesting result appears in figure 4. In fact, with this charging strategy, in contrast to scenario 1, bigger batteries lead to longer longevities, as the average SoC becomes lower when the size increases.

Battery cost

Figure 5 shows the daily battery cost, i.e. the purchase cost divided by the lifetime, assuming that the battery's purchase cost is proportional to its size and equal to 500C/kWh. We also neglected in the cost estimation an eventual resale of the batteries for second life applications.

For every case of scenario 1, the battery's daily cost grows rapidly with battery size. For example, for SoC max = 100%, a 20 kWh battery costs about 4 C/day while a 60 kWh one costs about 14 C/day (3.5 times more).

By limiting SoC max , the battery cost is reduced: for example, in the case of a 60kWh battery,limiting the SoC to 70%, instead of 100%, reduce the cost from 14 to 8C/day.

But the charge optimisation leads to considerable higher cost reductions. In fact, when a charge optimisation strategy is applied, the battery cost is 3C/day for 60kWh batteries.

Figure 5 shows that increasing the battery size could drastically increase the cost per day in case of scenario 1 (full charging every day). On the contrary, when considering a charging optimisation like in case scenario 2, the influence of the size on the battery cost is minimised. It could be deduced that increasing the battery size could be a not so good idea if no care to SoC is taken.

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Finally, the following rules could be concluded from the results in figures 4 and 5:

• Increasing the battery size with no adapted strategy accelerates the battery ageing.

• Limiting the maximum SoC of the battery decreases the capacity fade.

• Management of the charging target SoC according to the expected actual need decreases the capacity fade.

It must be noticed that the present work only dealt with one specific type of lithium-ion battery (LFP/graphite) for which the ageing parameters were identified [START_REF] Redondo-Iglesias | Efficiency Degradation Model of Lithium-ion Batteries for Electric Vehicles[END_REF]. But the tendency for ageing laws of any lithium-ion is similar to Equation 1. This allows to generalise the results by keeping the general rules: lower SoCs allow to prolong the battery lifetime.

Depending of the technology, batteries can be more or less sensitive to charge optimisation. For example, from results in [START_REF] Redondo-Iglesias | Efficiency Degradation Model of Lithium-ion Batteries for Electric Vehicles[END_REF], SoC sensitivity to calendar ageing is higher in NMC cells compared to LFP, then charging optimisation could be more interesting in NMC cells. In another work LFP cells were more sensitive to different charging strategies than NCA cells [START_REF] Petit | Development of an empirical aging model for Li-ion batteries and application to assess the impact of Vehicle-to-Grid strategies on battery lifetime[END_REF].

Conclusions

Battery size is growing as a direct consequence of manufacturing costs diminutions allowing higher range of electric vehicles. However, if charging strategies are not considered, this oversizing leads to a reduction of lifetime of batteries and relatively higher total ownership costs. Simulation results show that, if charging strategies are not adapted to higher battery sizes, lifetime of batteries decreases. This dramatically affects the total cost of ownership.

With an optimised charging strategy, the ownership costs still moderately rise with battery size but the battery lifetime greatly increases: having a bigger battery allows higher electric vehicle range for a little ownership costs increase. From a practical point of view, BMS (battery management systems) must provide a mean to charge the battery to a desired distance and to permit to fully charge the battery when the user needs to use the full range (e.g. holidays trips). However, the strategy proposed here supposes that the user knows exactly the driven distance the day after and accepts to limit the "operational" range of his/her vehicle. Studies about practical feasibility and user acceptability of this solution have to be conducted from a multidisciplinary point of view (engineering and human sciences together).

Further works will consider variable climatic conditions. A sensitivity analysis of battery lifetime to climate, trip conditions and user charging choices will be conducted. Global environmental impact of batteries (and the entire vehicle) will also be considered.

Figure 1 :

 1 Figure 1: Speed profile (red = Monday to Friday, green = Saturday, blue = Sunday).

Figure 2 :

 2 Figure 2: Monday's SoC profile in scenario 1 for different battery sizes at beginning of life (continuous lines) and at end of life (dashed lines).

Figure 3 :

 3 Figure 3: SoC profile of the 40kWh battery during a week with charge optimisation at beginning of life (blue) and end of life (red).

Figure 4 :Figure 5 :

 45 Figure 4: Battery lifetime for different sizes from 20 to 60kWh.

Table 1 :

 1 Comparison of SoC levels of 40kWh battery depending of the chosen scenario at beginning of life.

		(a) Monday to Friday
		Scenario 1 (SoCmax = 100)	Scenario 1 (SoCmax = 70) Scenario 2
	Maximum SoC	100	70	26
	Average SoC	95	65	20
	Minimum SoC	85	54	9
		(b) Saturday
		Scenario 1 (SoCmax = 100)	Scenario 1 (SoCmax = 70) Scenario 2
	Maximum SoC	100	70	43
	Average SoC	90	60	32
	Minimum SoC	70	38	14
		(c) Sunday
		Scenario 1 (SoCmax = 100)	Scenario 1 (SoCmax = 70) Scenario 2
	Maximum SoC	100	70	26
	Average SoC	99	69	14
	Minimum SoC	97	67	12

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Useful equations

State of charge (in p.u.):

Maximum energy of the battery (with SoC in p.u.):

Net power in the battery: P bat,0 (t) = I bat (t) • OCV (t)

Output power of the P bat (t) = P bat,0 (t) -P bat,losses (t)


Thomas Alves

Jérémy Rivière

Vincent Rodin

Thierry Duval
email: thierry.duval@imt-atlantique.fr

Immersive and interactive visualisation of a virtual honey bee colony

Keywords: Complexity, Agent-based simulation, Tangible interfaces, Immersive environment, Training and education

Social insects and more specifically honey bees have very complex, powerful and interesting task allocation abilities. They are able to distribute their workforce effectively without any central control, using simple mechanisms based on stimuli (dances, pheromones), interactions, thresholds and feedback loops. Self-organisational concepts and some of those mechanisms, like pheromones, are invisible to the human eye. In order to help the user grasp the complexity of the task allocation, we propose to make them visible in a virtual, immersive and interactive environment. First, this implies that the system must be able to simulate and display in real-time around 30 000 bees interacting with each other, emitting clouds of pheromones. Secondly, this implies also that the user should be able to alter in real-time the environment of the bees (e.g. by manipulating the frame of the hive) and visualise the effects on the organisation, potentially days later. Finally, we would like to give to biologists and beekeepers some domainrelated, intuitive and natural ways of interacting with the hive. We describe in this article these issues in more details, and how we plan to tackle them. This is a "work in progress", therefore a lot of work has still to be done, mostly surveying and modelling the interactions.

Introduction

Social insects are well known for their robust and effective decentralised organisation. They are able to distribute their workforce amongst all the different tasks to be done, without any global knowledge of the state of the colony. This task allocation ability [START_REF] Bonabeau | From Natural to Artificial Swarm Intelligence[END_REF] lies on stimuli (dances, pheromones), interactions between individuals, internal thresholds and positive and negative feedback loops.

The honey bee Apis Melifera, the most common honey bee in Europe, uses these mechanisms in a variety of ways, such as dances to recruit their nest-mates to foraging activities or many different pheromones to inhibit or favour certain behaviour [START_REF] Seeley | The Wisdom of the Hive: the social physiology of honey bee colonies[END_REF][START_REF] Schmickl | Integral feedback control is at the core of task allocation and resilience of insect societies[END_REF][START_REF] Conte | Primer effects of a brood pheromone on honeybee behavioural development[END_REF]. The first part of our project is to model these mechanisms, within an agent-based simulation [START_REF] Schmickl | Analysing honeybees' division of labour in broodcare by a multi-agent model p[END_REF][START_REF] Bonabeau | From Natural to Artificial Swarm Intelligence[END_REF]. Agent-based modelling is a bottom-up approach that focuses on individuals, their behaviour and interactions (at the micro-level), to deduce the overall behaviour of the system (macro-level) by emergence. The agent-based approach is thus particularly suited to model and simulate complex systems. Following this approach, each honeybee is an agent, described by its biological cycle, behaviour, and its interactions with its environment, including other agents. We work with biologists and beekeepers to help us build and validate the model, and make it as relevant as possible.

We intend to model most of the in-hive tasks (feeding the brood, cleaning the combs, foraging etc.), triggered (or inhibited) by different stimuli such as pheromones. Selforganisational concepts are driven by invisible mechanisms, and thus quite challenging to understand. There comes the second part of the project, a virtual, immersive and interactive simulation. With virtual reality we want to allow the user to comprehend those concepts, by showing the mechanisms behind them, hiding some parts of the hive, abstracting the bees to focus only on pheromones, i.e. enhancing the view.

Related works on honey bee colony visualisation aim mainly at help beekeepers make decisions, by providing them information about honey bee populations [START_REF] Engelke | A visual analytics framework to study honey bee behaviour[END_REF][START_REF] Engelke | MelissaR: Towards augmented visual analytics of honey bee behaviour[END_REF][START_REF] Nguyen | Augmented reality based bee drift analysis: A user study[END_REF]. For example, the authors of [START_REF] Engelke | A visual analytics framework to study honey bee behaviour[END_REF] use Augmented Reality to display data from several sensors in a range of real hives and allow the user to go through all the data in an intuitive and immersive manner. These data are collected at the hive (macro) level, such as temperature and weight. They are thus able to know in which hive certain bees are and study "bee drift": when bees from a colony leave and join another colony. We, on the other hand, work on a simulated honey bee colony and focus on the "micro" world, at the bee level. Our goal is not to (directly) help the decision making of beekeepers, but to allow them to visualize and understand their actions' effects on the organisation of the (virtual) colony, inside the hive. Section 3.3 provides some examples of such beekeeping actions.

Scientific challenges

We highlight in this section three of the main issues that we face. First, we want to visualise a very large system in 3D, with tens of thousands individuals, interacting in various ways with each other, and allow the user to interact in real-time with the system, changing the state of the simulation. The performance of such a system needs to be considered.

Secondly, the user should be able to interact with the hive, and watch the colony adapt to those actions in real-time, or fast forward to a later stage of the world to see how the colony handled (or not) the action's effects in the long term. It is necessary to consider ways to handle those multiple time interactions, both on the simulator and the visualisation.

Finally, this simulator should be used by biologists and beekeepers in different uses: a scientific use, aimed at biologists, to evaluate the role of stimuli and other mechanisms on self-organisation; and educational use, to allow new and experienced beekeepers to learn good beekeeping practices through predefined scenarios and realistic means of interaction. To do so, we would like to implement domain-related, intuitive and natural ways of interacting with the hive.

3 Preliminary Work

Visualisation and Interactions Challenges

We describe here our work on the first two challenges described previously. We propose to separate the simulation and its visualisation, following the advice of [START_REF] Louloudi | A new framework for coupling agent-based simulation and immersive visualisation[END_REF], by using a Java server calculator to run the model, and a Unity3D client for visualisation and interactions (see Figure 1). The first major effect of decoupling the calculator that holds the simulation, and the interacting app, is that we can model each part without the constraint of the other part. The simulator won't have to compute any complex visualisation nor to draw it onscreen in real-time, and the interactive system won't have to bother with the calculation of the agent-based system. We may also gain some performance, as we can easily run those two parts on separate computers: the authors of [START_REF] Louloudi | A new framework for coupling agent-based simulation and immersive visualisation[END_REF] have shown that their decoupling does not create any performance bottleneck. Moreover, by creating a "Rest API-like" server, we set the server free from the graphical client's updates, and even allow ourselves to change the interactive app later on if needed, without changing the inner behaviour of the simulation.

Another effect of this separation is the decoupling of the time between the simulator (virtual time) and the interacting app (in real-time to allow the user to interact with the frames and the colony). This is a disadvantage, as we will have to ensure that the simulator and the interacting app stay synchronised: an action performed at time t by the user must occur in the simulation at the same time. On the other hand, we need this decoupling because we intend to give the possibility to the user to fast forward in a later time to observe the effects of an action, so the simulator must be able to compute the simulation very fast. Moreover, those two highly different speeds will require us to implement a way for the user to understand the time flow and interact with it.

The Virtual Environment

The environment consists of the Dadant Hive, the standard beehive for honey harvesting. The hive is mainly a wooden box, with rails on the sides to guide a few frames inside. We modelled it in Unity3D with the hexagonal cells in the frames, which are where the larvae are raised and the resources packed. Figure 2 shows an example of such a visualisation, in which the server randomly filled all the cells, then sent the data to the client. Inside Unity3D, we can interact with these filled frames, making them rise so we can see their content. This will be the basis for our work. Our virtual world, with the hive, will be filled with honey bees. Typically, in Unity3D, we would instantiate each individual as a "GameObject", and let Unity3D handle the movements and display. But as honey bee colonies can count up to 50 000 individuals, that many GameObjects would cause severe frame rate drop and ruin the immersive experience.

Unity3D is way more effective iterating inside a GameObject as it is to iterate through them. So, instead of having a few thousand GameObject, we should have a single one, but containing a few thousands of shapes scattered around the world. We will, therefore, use a point cloud inside Unity3D, and Unity3D's implementation of a point cloud is a very good start. We managed to display a few hundred thousands points with a very decent frame rate, around 60fps. We can now use the same idea to represent different layers of the world, having, for example, a Point-Cloud object representing all the bees, and another one for the pheromones (or different objects for different types of pheromones). In section , we discuss how the visualisation of the organisation could look like.

Natural Intuitive Interactions

We also worked with beekeepers and bee biologists to know how they interact with their real hives. We recorded a dozen of interactions. This will be the basis of our model, making it as relevant as possible. We also recorded their expected effects on the organisation of the colony and will use these data to validate our task allocation model and the interaction model. For example, a beekeeper sometimes has to remove the queen of a colony, often to replace it with a younger one. This change usually increases the number of eggs laid, which will require much more nurses, food and space. Beekeepers can also change the order of the frames by performing a circular permutation of all the frames. That movement changes drastically the environment of the colony and requires a self-adaptation. The last example is the more extreme and consists of the beekeeper removing all the brood of the colony, to prevent a disease to spread. This is done by renewing all the frames of the hive and has a huge impact on the colony.

In a nutshell, each beekeeper move has a significant impact on multiple feedback loops, and, in extent, on the task allocation of the whole colony. We discuss in the next section the interaction means that could implement these actions.

Future Work and Perspectives

Different ways of interacting could help us answer to the issues listed in section 2. Natural and intuitive interactions could be implemented thanks to a virtual hive that can be manipulated with tangible interactors. For example, a common beekeeper action is to remove a frame from the hive and then hit it on the top to make all the bees on it fall, allowing a better and safer work on the frame. This simple, key movement would highly benefit a tangible interactor, to reproduce at best the real feeling of the movement, enhancing the immersion feeling and maybe enhance the learning experience of the user as well. A solution could either be a real tracked Dadant hive with real tracked frames, or a re-built simplified version with only the structural needs, as it does not have to look like a hive. We still have to study the feasibility of both ideas.

The vast majority of the beekeeping interaction with the hive requires the handling of a frame, enhancing the needs for tangible interactors. For example, the queen removal action described in the previous section requires to remove a frame from the hive, manipulate it to find the queen, then lift it with an interactor, insert the new queen and then put the frame back into the hive. In the same way, a rotation, circular permutation of all the frames requires to lift a frame, remove the bees on it by hitting it on the top, store it in a dedicated spot near the user, move all the other frames, and then put the stored frame back on the other side of the hive.

Biologists and beekeepers won't have the same needs regarding the interactions and the visualisation of the system, even if they globally interact in the same way with the frames. That's why we intend to implement a system of layers, allowing each user to effectively chose what they want to see. For example, a biologist studying the impact of pheromones on the colony organisation could decide to hide the bees, the frames and the hive, and only see the pheromones. We could even decide to hide or show bees based on their activities, or age, and then display their perception. It's not clear yet, however, how the organisation will be shown to the user. Bringing that abstract concept on the visible layer will require further work, but a first idea would be to create geometries just above each agent, describing what they perceive (with a bar chart) and what they do (for example a square floating above a bee would mean that it's currently a nurse). We could even select an agent and then see its whole history of perception and action with graphs displayed inside the virtual world.

To allow the user to navigate in time, a tangible time controller is considered, which should allow the user to fast forward and see the state of the colony in the future. We are thinking about possible time travel to the past, but we are still unsure if this would be meaningful for the project. Further discussions with biologists and beekeepers are planned.

Finally, a potential multi-verse metaphor could be useful for the user to compare the consequences of multiple gestures upon the same colony. The user would be able at any time to create a new universe, holding a copy of the current state of the colony. Then, he could perform any interaction with the hives in any verse. The verses would all be independent and thus, multiple hives could be watched at the same time. We could, for example, have a "default colony" in the first verse, and never interact with it. Then effectively compare it to the other verses where we would have performed many changes.

Fig. 1 .

 1 Fig. 1. Decoupled architecture. A java server holding the simulation and the interactive and immersive app in Unity3D will talk to each other and synchronise through Web sockets.

Fig. 2 .

 2 Fig. 2. A Unity3D simulation that visualises the results of a Java simulation.

5 Reference to the official publication Springer Nature Switzerland AG 2019 P. Bourdot et al. (Eds.): EuroVR 2019, LNCS 11883, pp. 1-18, 2019. The final authenticated version is available online at: https://doi.org/10.1007/978-3-030-31908-3 23


Alain Valtz

Jamal El Abbadi

Céline Christophe Coquelet

Jamal El-Abbadi

Céline Houriez

Experimental measurements and modelling of vapour-liquid equilibrium of 2,3,3,3-tetrafluoropropene (R-1234yf) + 1,1,1,2,2pentafluoropropane (R-245cb) system

Keywords: Equations of state, vapour-liquid equilibrium, refrigerant fluids, fluorinated compounds cal Calculated property exp Experimental property i, j Molecular species opt Optimized

published or not. The documents may come

Experimental measurements and modelling of vapour-liquid equilibrium of 2,3,3,3-tetrafluoropropene (R-1234yf)-pentafluoropropane (R-245cb) system

Introduction

There is now a broad scientific consensus that anthropogenic greenhouse gas emissions are responsible for a developing pattern of climate change, and that the scale of the problem calls for a multifaceted response, including major reductions in global CO 2 emissions. In the energy and industrial sectors, major reductions could be achieved through a combination of demand reduction, fuel substitution, and carbon capture and storage (CCS) [START_REF] Trusler | Thermophysical Properties and Phase Behavior of Fluids for Application in Carbon Capture and Storage Processes[END_REF]. This also includes refrigerant substitution, through using fluids less harmful to the environment.

In the past few years, hydrofluoroolefines (HFOs) have been widely introduced as replacement fluids in many industrial applications due to their low global warming potential (GWP), their zero ozone depletion potential (ODP), and their thermophysical properties. In particular, the 1,1,1,2-tetrafluoroethane (HFC-134a or R-134a), which is widely used in automotive air conditioning, is gradually replaced by the 2,3,3,3-tetrafluoropropene (HFO-1234yf or R-1234yf) due to its low GWP equal to 4 [START_REF] Minor | International Refrigeration and Air Conditioning Conference at Purdue[END_REF] and its thermophysical properties, which are similar to those of R134a. In addition, R-1234yf has low toxicity, low flammability and is chemically stable [START_REF] Calm | The next generation of refrigerants -Historical review, considerations, and outlook[END_REF], [START_REF] Yang | Vapor-liquid equilibria of 2,3,3,3-tetrafluoropropene (HFO-1234yf) + 1,1,1,2,2pentafluoropropane (HFC-245cb) system[END_REF]. The 1,1,1,2,2-pentafluropropane (HFC-245cb or R-245cb) is a by-product resulting in the different synthetic routes of producing R-1234yf [START_REF] Yang | Vapor-liquid equilibria of 2,3,3,3-tetrafluoropropene (HFO-1234yf) + 1,1,1,2,2pentafluoropropane (HFC-245cb) system[END_REF]- [START_REF] Merkel | Process for producing 2[END_REF]. In addition, normal boiling points of these two refrigerants are quite close [START_REF] Yang | Vapor-liquid equilibria of 2,3,3,3-tetrafluoropropene (HFO-1234yf) + 1,1,1,2,2pentafluoropropane (HFC-245cb) system[END_REF], [START_REF] Di Nicola | Saturated Pressure Measurements of 2,3,3,3-Tetrafluoroprop-1-ene (HFO-1234yf)[END_REF]- [START_REF] Tanaka | Measurements of the Isobaric Specific Heat Capacity and Density for HFO-1234yf in the Liquid State[END_REF], and we need to separate them during the purification process, that implies to know accurately their vapour-liquid equilibrium (VLE) diagram. Although the vapour pressure and PVT data for single compounds R-1234yf and R-245cb have been measured before [START_REF] Yang | Vapor-liquid equilibria of 2,3,3,3-tetrafluoropropene (HFO-1234yf) + 1,1,1,2,2pentafluoropropane (HFC-245cb) system[END_REF], [START_REF] Di Nicola | Saturated Pressure Measurements of 2,3,3,3-Tetrafluoroprop-1-ene (HFO-1234yf)[END_REF]- [START_REF] Shank | Thermodynamic properties of 1,1,1,2,2-pentafluoropropane (refrigerant 245)[END_REF], VLE data of the mixture R-1234yf + R-245cb are very scarce. And to the best of our knowledge, only one research study [START_REF] Yang | Vapor-liquid equilibria of 2,3,3,3-tetrafluoropropene (HFO-1234yf) + 1,1,1,2,2pentafluoropropane (HFC-245cb) system[END_REF] has reported the VLE data of this binary system.

In this work, the VLE of the binary system R-1234yf + R-245cb were measured using a "static-analytic" apparatus at temperatures between 283.39 and 343.27 K, in order to fill the gap between existing data, and also to compare with these data.

A 3-parameter cubic equation of state (NEoS) was used to model the VLE results. The NEoS was associated with the Mathias-Copeman alpha function, and van der Waals (vdW) mixing rules, and represents very well the experimental data.

2.

Experimental section

Materials

In Table 1 are listed the refrigerants used for the VLE measurements, with the details about their ASHRAE 1 number, chemical formula, CAS 2 number, along with the name of the supplier and the product purity provided by the supplier. No further purification of the chemical products was needed, only degassing was realised when loading the chemicals into the equilibrium cell.

The critical temperature and pressure, and the acentric factor ( ) 3 of these refrigerants are reported in Table 2.

Experimental apparatus

Static-analytic apparatus

The equipment used for the VLE measurements is based on a static-analytic method with liquid and vapour phase sampling using capillary samplers ROLSI™. The equipment can be categorized as "Analytical technique with sampling, isothermal AnT" according to Dohrn et al. classification [START_REF] Dohrn | High-pressure fluid-phase equilibria: Experimental methods and systems investigated (2000-2004)[END_REF]- [START_REF] Christov | High-pressure fluid phase equilibria: Experimental methods and systems investigated (1994-1999)[END_REF].

The main part of the apparatus is the equilibrium cell, where the two-phase equilibrium takes place. The flow diagram of the apparatus is displayed in Figure 1.

The apparatus is equipped with a thermo-regulated liquid bath in which the equilibrium cell is immersed. The bath ensures the control of the temperature within 0.01 K.

The temperature measurement inside the equilibrium cell is performed using two platinum resistance thermometer probes (Pt100) [START_REF] Juntarachat | Experimental measurements and correlation of vapor-liquid equilibrium and critical data for the CO 2 + R1234yf and CO 2 + R1234ze(E) binary mixtures[END_REF]- [START_REF] Valtz | Vapor-liquid equilibrium data for the CO 2 + 1,1,1,2,3,3,3,-heptafluoropropane (R227ea) system at temperatures[END_REF]: one to measure the temperature at the top of the cell, and the other for the temperature at the bottom of the cell. Two other temperature probes are used to control the temperature inside the thermal presses used to load the chemical products into the equilibrium cell.

The Pt100 probes are connected to a data acquisition unit (HP34970A). The Pt100 probes are calibrated against a 25 Ω reference platinum resistance thermometer (Pt25 -Hart Scientific).

The Pt25 reference probe was calibrated by the "Laboratoire National d'Essais de Paris" based on the 1990 International Temperature Scale (ITS 90). The temperature accuracy is estimated to be within ± 0.03 K.

The pressure is measured using one of two pressure transducers (DRUCK, 0-3 MPa, 0-30 MPa) installed on the apparatus. The choice of pressure transducer is dependent upon the maximum pressures generated by the system being characterised. The two pressure transducers are also connected to the data acquisition unit (HP34970A). The pressure transducers were calibrated against a pressure automated calibrator (GE Sensing, model PACE 5000). The pressure accuracy of the transducers is estimated to be within ±0.0004

MPa.

Figure 1. Flow diagram of the "static-analytic" apparatus [START_REF] Juntarachat | Experimental measurements and correlation of vapor-liquid equilibrium and critical data for the CO 2 + R1234yf and CO 2 + R1234ze(E) binary mixtures[END_REF][START_REF] Williams-Wynn | Experimental determination of the critical loci for R-23+(n-propane or n-hexane) and R-116+n-propane binary mixtures[END_REF][START_REF] Valtz | Vapor-liquid equilibrium data for the CO 2 + 1,1,1,2,3,3,3,-heptafluoropropane (R227ea) system at temperatures[END_REF][START_REF] Madani | Isothermal vapor-liquid equilibrium data for the trifluoromethane (R23) + 2,3,3,3-tetrafluoroprop-1-ene (R1234yf) system at temperatures from 254 to 348[END_REF].

EC: equilibrium cell; LV: loading valve; MS: magnetic stirrer; PP: platinum resistance thermometer probe; PT: pressure transducer; RT: temperature regulator; LB: liquid bath; TP: thermal press; C1: more volatile compound; C2: less volatile compound; V: valve; GC: gas chromatograph; LS: liquid sampler; VS: vapor sampler; SC: sample controlling; PC: personal computer; VP: vacuum pump. The compositions of the phases present within the cell are analysed with a gas chromatograph (PERICHROM, model PR2100) equipped with a thermal conductivity detector (TCD). The analytical column within the gas chromatograph is a RESTEK, 1% RT-1000 on Carboblack B, 60/80 mesh (Silcosteel, from Restek; length: 2.4 m; diameter: 2 mm).

The TCD is calibrated by introducing manually known amounts of each pure compound (for the system studied) into the injector of the gas chromatograph, using an automatic syringe (eVOL XR, from SGE). The calibration equation was fitted to relate the response of the TCD to the number of moles of the compound introduced. The mole number accuracy is estimated to be less than 2% for each of the two pure compounds.

Experimental procedure

At ambient temperature, the equilibrium cell and its loading lines are placed under vacuum. A first thermal press is loaded with one of the compounds, and the second thermal press with the other compound. The liquid bath is set to the temperature desired. When the equilibrium temperature is reached (the equilibrium temperature is reached when the Pt100 probes give the same temperature value within their temperature uncertainty for at least 10 minutes), an amount of about 5 cm 3 of the heaviest component (the component with the lower vapour pressure) is introduced into the equilibrium cell. Its vapour pressure is then measured at this temperature [START_REF] Juntarachat | Experimental measurements and correlation of vapor-liquid equilibrium and critical data for the CO 2 + R1234yf and CO 2 + R1234ze(E) binary mixtures[END_REF]- [START_REF] El Abbadi | Thermodynamic properties of new refrigerants[END_REF].

Then, a given amount of the lightest component (the component with the higher vapour pressure) is introduced step by step in order to increase the pressure inside the cell, leading to successive equilibrium mixtures, in order to have enough points to cover the two-phase envelope.

The equilibrium inside the cell is assumed to be reached when the pressure does not change during a period of 10 minutes within ±0.001 MPa under continuous stirring [START_REF] Juntarachat | Experimental measurements and correlation of vapor-liquid equilibrium and critical data for the CO 2 + R1234yf and CO 2 + R1234ze(E) binary mixtures[END_REF]- [START_REF] El Abbadi | Thermodynamic properties of new refrigerants[END_REF].

For each equilibrium condition, six or more samples of both vapour and liquid phases are taken using the capillary sampler ROLSI ® and analysed in order to verify the repeatability of the measurement. For the complete details about the calibration procedure, the reader can consult the reference [START_REF] El Abbadi | Thermodynamic properties of new refrigerants[END_REF].

Experimental results

Vapour pressure measurements

The vapour pressures of the pure compounds R-245cb and R-1234yf were measured at temperatures from 278.42 to 356.50 K for R-245cb, and 275.81 to 336.67 K for R-1234yf.

The data obtained along with the uncertainties on measurements are listed in Table 3. These latter were calculated following the approach presented in the NIST guidelines [START_REF] Taylor | Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results[END_REF][START_REF] Zhang | Développement d'un dispositif expérimental original et d'un modèle prédictif pour l'étude thermodynamique des composés soufrés[END_REF][START_REF] Soo | Experimental thermodynamic measurements of biofuel-related associating compounds and modeling using the PC-SAFT equation of state[END_REF]. U(T) 4 = 0.06 K; U(P) = 0.0008 MPa.

U(X) = u(X, k=2).

In general, in the DIPPR [START_REF] Daubert | Physical and thermodynamic properties of pure chemicals: data compilation[END_REF], Equation ( 1) is used to correlate the pure compound vapour pressure.

(

At the critical point, T = T C and P = P C , which gives Equation ( 2):

(

Consequently, by considering D = 2, we obtained Equation ( 3):

(3)

Here, Pc is the critical pressure of the considered compound, and T R is the reduced temperature. A, B, and C are three adjustable parameters, whose values are given in Table 4, together with their corresponding deviations. The experimental data were correlated using Equation (3).

Concerning R-245cb, we have included the pure compound vapour pressure data from Shank [START_REF] Shank | Thermodynamic properties of 1,1,1,2,2-pentafluoropropane (refrigerant 245)[END_REF] in the data treatment, in order to cover the maximum range of temperature. According to

VLE measurements

The binary system R-1234yf + R-245cb was studied at four isotherms ranging between T = 283.39 and 343.27 K. The experimental results for this system, along with the number of samples (n), the standard deviations ( x 1 and y 1 ) and the uncertainties on measurements are reported in Table 5. Expanded uncertainties are: U(T) = 0.06 K; U(P) = 0.0008 MPa; U(x 1 ) = 0.008; U(y 1 ) = 0.008.

Modelling & discussions

The modelling of vapour pressures and VLE was performed using a newly introduced equation of state (NEoS) [START_REF] Coquelet | Prediction of thermodynamic properties of refrigerant fluids with a new three-parameter cubic equation of state[END_REF] associated with the well-known Mathias-Copeman alpha function [START_REF] Mathias | Extension of the Peng-Robinson equation of state to complex mixtures: Evaluation of the various forms of the local composition concept[END_REF]. The mixing rules used are the classical vdW mixing and combining rules. More details about this model can be found in Appendix or in references [START_REF] El Abbadi | Thermodynamic properties of new refrigerants[END_REF], [START_REF] Coquelet | Prediction of thermodynamic properties of refrigerant fluids with a new three-parameter cubic equation of state[END_REF]. For comparison purpose, the Peng-Robinson (PR) EoS [START_REF] Peng | A new two-constant equation of state[END_REF] with the same alpha function and the same mixing rules is also used to model the obtained data.

Modelling of pure compound properties

For the NEoS, the alpha function parameters m 1 , m 2 and m 3 , along with the critical compressibility factor Z c , were fitted on the pure compound data to accurately represent the vapour pressures of these single compounds. The parameters of the NEoS for the R-1234yf come from reference [START_REF] El Abbadi | Thermodynamic properties of new refrigerants[END_REF]. The average relative deviation (ARD) and the bias are calculated by using Equation ( 4), relative to data predicted using REFPROP software (labelled asX exp in the equation):

For the R-1234yf, the ARD is less than 1.4% (bias equal to -1.4%) for vapour pressure and less than 2.2% (bias equal to -0.6%) for liquid densities at saturation. Concerning R-245cb, the parameters were fitted using saturated density data from Shank [START_REF] Shank | Thermodynamic properties of 1,1,1,2,2-pentafluoropropane (refrigerant 245)[END_REF], and their corresponding vapour pressures calculated through the correlation given in Equation (3). The parameters of the NEoS are presented in Table 6. The Figure 2 shows the result of the adjustment for the R-245cb. The ARD is less than 0.3% (bias equal to 0.1%) for vapour pressure, and less than 3.8% (bias equal to 0.6%) for the liquid densities at saturation.

For the PR EoS, the parameters m 1 , m 2 and m 3 of the alpha function were determined by using the same protocol and are also presented in Table 6. These results are supported by the evaluation of the ARD and bias for the vapour pressures of these two refrigerants (Table 7), for the two models considered in the present paper (NEoS and PR EoS), relative to experimental data. The NEoS reproduces quite accurately both sets of data, as can be seen from Figure 3.

VLE modelling

The NEoS and PR EoS were used to correlate the experimental data. The binary interaction parameter (BIP) was fitted on experimental VLE data to take into account the effect of intermolecular interactions (Table 8). This system shows a small deviation from Raoult's law and exhibits no azeotropic behaviour in the investigated temperature range, as can be seen on Figure 4.

From comparing the modelling and experiment results (Figure 4 for the VLE and Figure 5 for the relative volatility), it appears that the NEoS reproduces quite accurately the experimental data of both liquid and gas phases for each of the four isotherms studied, with a good representation of the single compound vapour pressures. From the values of ARD and bias presented in Table 8, we can note that the modelling using the NEoS doesn't improve significantly the results, relative to the PR EoS. However, with the PR EoS, the BIP exhibits a stronger temperature dependency, as compared to NEoS (Figure 6), although the BIP values are small for the two models regardless of the temperature considered (maximum of 0.0081 for the BIP). We have also repeated the data treatment using the NEoS with no temperature dependency of the BIP. Indeed, we previously show in previous paper that an interesting advantage of the NEoS is a non-dependency of the BIP on the temperature, besides a better prediction of densities (Coquelet et al. [START_REF] Coquelet | Prediction of thermodynamic properties of refrigerant fluids with a new three-parameter cubic equation of state[END_REF]), see in

Appendix. The NeoS has three parameters adjustable on vapour pressure and density data. So, the intermolecular interactions, i.e. repulsive and attractive interactions, are better taken into account. In consequence, for the mixture, it is not necessary to correct strongly with high value of BIP. The value we used for the BIP is k 12 = -0.00095. The values obtained for ARD on bubble pressure and vapour composition are less than 0.38% (bias 0.3%) and less than 1.15% (bias -0.78%), respectively.

Comparison with literature data

In order to compare our results with literature data, we modelled the data for the system R-1234yf + R-245cb from Yang et al. [START_REF] Yang | Vapor-liquid equilibria of 2,3,3,3-tetrafluoropropene (HFO-1234yf) + 1,1,1,2,2pentafluoropropane (HFC-245cb) system[END_REF] by using the NEoS, as NEoS and PR-EoS results are quite similar. We used the NEoS with a BIP independent of the temperature.

We calculated the ARD and bias for bubble pressures (0.80% and 0.61%) and vapour mole fraction (1.82% and 1.13%).

On Figure 7, we have plotted the literature data from Yang et al. with our data. Albeit both VLE measurement sets are in good agreement (Figure 7), an unexpected behaviour was observed for the relative volatility calculated from Yang et al. data (figure 8) and the prediction using our NEoS model.

In fact, it is well known that the relative volatility is supposed to vary as a decreasing function (exponential or polynomial) of the liquid molar fraction, that can be verified from the results of our data (as can be seen on Figure 5). On the other hand, the relative volatility calculated from Yang et al. data is varying as an increasing function of the liquid molar fraction, which is somehow unexpected. From analysing further, the data reported by Yang et al. [START_REF] Yang | Vapor-liquid equilibria of 2,3,3,3-tetrafluoropropene (HFO-1234yf) + 1,1,1,2,2pentafluoropropane (HFC-245cb) system[END_REF], one can notice that for each isotherm, the molar fractions obtained for both vapour and liquid phases are almost the same. And this is somehow strange, given the fact that the measurements are performed through a static-analytic method, and that the odds of obtaining the same composition every time are pretty rare. That leads to think that maybe the experimental procedure followed is not the one announced by the authors, but rather they proceed by keeping the global composition constant, and they vary the temperature. But, even if this is the case, there should be, in our opinion, a greater difference between the compositions of the points at different temperatures.

Conclusions

In this paper, the isothermal VLE of the binary system R-1234yf + R-245cb were measured using a static-analytic apparatus at temperatures from 278.15 to 333.15 K. This system shows a small deviation from Raoult's law and exhibits no azeotropic behaviour in the investigated temperature range.

A 3-parameter cubic equation of state (NEoS) associated with the Mathias-Copeman alpha function, and with vdW classical mixing rules was used for the modelling calculations, and leads to a good description of the experimental results. This work aimed to complete the VLE data for this binary system provided in the literature [START_REF] Yang | Vapor-liquid equilibria of 2,3,3,3-tetrafluoropropene (HFO-1234yf) + 1,1,1,2,2pentafluoropropane (HFC-245cb) system[END_REF], and in the same time check their consistency. Although these latter two VLE result sets are in good accordance, we found a discrepancy when calculating the relative volatility. Nevertheless, more investigations are needed for VLE data of this system in order to provide data in an extended range of temperatures, and compare the validity of the available data.

The parameters a c , b and c of Equations ( 1) to (3) can conventionally be obtained from the thermodynamic conditions at the critical point, defined as follows (Eq. 4).

(

) 4
Or from the mathematical constraint (Eq. 5).

(

) 5
where v c is the optimized critical volume.

After rewriting Equation ( 5), we obtain Equation ( 6).

The parameters a c , b and c are calculated according to the relations (Eq. 7):

where Ω a , Ω b and Ω c are factors depending on the substance [START_REF] Segura | An accurate direct technique for parameterizing cubic equations of state[END_REF]. T c and P c are respectively the experimental critical temperature and pressure. We set T = T c and P = P c in Equation ( 6), then the comparison with Equation ( 5) results in Equation ( 8).

Z c,opt is an apparent optimized critical compressibility factor. It is different from the experimental critical compressibility factor Z c , and adjusted from the experimental VLE data [START_REF] Ji | A systematic study of cubic three-parameter equations of state for deriving a structurally optimized PVT relation[END_REF], [START_REF] Forero | The Patel-Teja and the Peng-Robinson EoSs performance when Soave alpha function is replaced by an exponential function[END_REF], [START_REF] Ji | Extrapolation of VLE data and simultaneous representation of caloric and volumetric properties by means of a cubic 3-parameter equation of state[END_REF] in order to improve the prediction of liquid densities. Here, as u + w = 0, we can simplify Equation ( 8) to obtain Equation ( 9).

By including the critical compressibility factor in the calculations, better results can be obtained, even though the apparent critical compressibility factor Z c,opt is larger than the experimental one, Z c .

Mathias-Copeman alpha function

The alpha function used in this work is Mathias-Copeman alpha function [START_REF] Mathias | Extension of the Peng-Robinson equation of state to complex mixtures: Evaluation of the various forms of the local composition concept[END_REF], defined as follows:

where T and T c are respectively the temperature and the critical temperature. m 1 , m 2 and m 3 are three adjustable parameters fitted on the experimental data, and depending on the acentric factor ω. The Mathias-Copeman alpha function is defined by two relations depending if we are above or below the critical temperature, as shown in Eq. 10. In our case, the temperatures of the isotherms are lower than the critical temperature of each pure compound. The analysis of alpha functions is very important and many papers have been published on this topic. We can cite the works from Segura et al. [START_REF] Segura | Unnoticed pitfalls of soave-type alpha functions in cubic equations of state[END_REF] and Pina-Martinez et al. [START_REF] Pina-Martinez | Updated versions of the generalized Soave α-function suitable for the Redlich-Kwong and Peng-Robinson equations of state[END_REF]. In these two papers, the authors noticed pitfalls of the Soave-type alpha functions and proposed new mathematical expressions. However, we do not use them in this work and prefer using a common alpha function available in simulation softwares.

The Mathias-Copeman alpha function must satisfy the required conditions of an alpha function, as suggested by several authors [START_REF] Forero | The Patel-Teja and the Peng-Robinson EoSs performance when Soave alpha function is replaced by an exponential function[END_REF], [START_REF] Coquelet | Development of a New Alpha Function for the Peng-Robinson Equation of State: Comparative Study of Alpha Function Models for Pure Gases (Natural Gas Components) and Water-Gas Systems[END_REF]:

-It has to be real and positive at all temperatures;

-It has to be a decreasing function, approaching a positive value when the temperature tends to infinity;

-It has to be equal to one at the critical temperature;

Density prediction comparison

The 6. Dotted line: PR EoS prediction with parameters from Table 6. As expected, the NEOS shows better predictive capabilities for densities than PR EoS.

Mixing rules

For the mixtures, the classical vdW mixing and combining rules [START_REF] Peng | A new two-constant equation of state[END_REF] were used for the calculations. They are defined as follows: [START_REF] Weber | Vapor Pressure of 1,1,1,2,2-Pentafluoropropane[END_REF] where x i is the mole fraction of the component i, a i is the energy parameter, b i and c i are the co-volume parameters of the component i, and k ij is the binary interaction parameter. N is the number of components of the system.

The vdW mixing rules were chosen for their simplicity, and ease of computing and also to be able to use a predictive binary interaction parameter k ij .

The binary interaction parameter k ij is fitted on the VLE data of bubble pressure and vapour molar fraction according to the following objective function: [START_REF] Shank | Thermodynamic properties of 1,1,1,2,2-pentafluoropropane (refrigerant 245)[END_REF] where N is the number of data points, P exp the experimental bubble pressure, P cal the calculated bubble pressure, y exp the experimental vapour molar fraction and y cal the calculated vapour molar fraction.

Figure 2 :

 2 Figure 2: Pressure-density diagram of the R-245cb. Solid line: NEoS prediction with parameters from table 6. (Δ): Data from Shank [13].

3 Figure 3 .

 33 Figure 3. Experimental and calculated vapour pressures of the pure compounds R-1234yf and R-245cb. R-1234yf: (○) This work; (∆) Hu et al. [30]. R-245cb: (•) This work; (▲) Shank [13]; (×) Weber and Defibaugh [12]. (-) NEoS.

Figure 4 :

 4 Figure 4: VLE experimental data and modelling for the binary mixture R-1234yf + R-245cb. (Δ) 283.39 K; () 303.15 K; (○) 323.30 K; (×) 343.27 K; (-) NEoS.

Figure 5 :

 5 Figure 5: Relative volatility for the binary mixture R-1234yf + R-245cb. (Δ) 283.39 K; () 303.15 K; (○) 323.30 K; (×) 343.27 K; (-) NEoS. Error bars: 8%.

Figure 6 :

 6 Figure 6: Variation of the Binary Interaction Parameter (BIP) k 12 as a function of temperature, for the two EoSs. (▲): PR EoS; (Δ): NEoS. Solid line: BIP for NEoS without temperature dependency.

Figure 7 :

 7 Figure 7: VLE experimental data for the binary system R-1234yf + R-245cb: comparison between this work (empty symbols) and literature data of Yang et al. [4] (filled-in symbols), for isotherms at 283.15 K (squares) and 303.15 K (triangles).

Figure 8 :

 8 Figure 8: Relative volatility calculated from literature data of Yang et al. [4] for the binary system R-1234yf + R-245cb. () 283.15 K; (∆) 293.15 K; (○) 303.15 K; (×) 313.15 K. (-) NEoS.

  figure A presents a comparison between density predicted by REFPROP, PR EoS and the NEOS for the R245cb. The same comparison for R-1234yf is shown on figure B.

Figure A :

 A Figure A: Pressure-density diagram of the R-245cb. Solid line: NEoS prediction with parameters from Table6. Dotted line: PR EoS prediction with parameters from Table6.

Figure B :

 B Figure B: Pressure-density diagram of the R-1234yf. Solid line: NEoS prediction with parameters from Table6. Dotted line: PR EoS prediction with parameters from Table6. Dashed line: REFPROP 10.0 prediction.

Table 1 . Refrigerants used for the experiments.

 1

Table 2 . Critical properties and acentric factors of R-1234yf and R-245cb.

 2

	Compounds	ASHRAE Number	Formula	CAS Number	Supplier	Purity (vol%)
	2,3,3,3-tetrafluoropropene	R-1234yf	C 3 H 2 F 4	754-12-1	Honeywell	> 99.5%
	1,1,1,2,2-pentafluropropane	R-245cb	C 3 H 3 F 5	1814-88-6	Synquest	> 99%
	Compounds	T c /K		P c /MPa	ω
	R-1234yf a	367.85 a		3.3822 a	0.276 a
	R-245cb b	380.38 b		3.1483 b	0.297 b

a : Lemmon et al.

[START_REF] Lemmon | REFPROP, Reference Fluid Thermodynamic and Transport Properties[END_REF]

, b : Weber and Defibaugh

[START_REF] Weber | Vapor Pressure of 1,1,1,2,2-Pentafluoropropane[END_REF]

1 ASHRAE: American Society of Heating, Refrigerating and Air-Conditioning Engineers 2 CAS: Chemical Abstracts Service

3

The acentric factor is a conceptual number introduced by Kenneth Pitzer in 1955, and it describes the nonsphericity (centricity) of molecules.

Table 2 . Experimental vapour pressures of the pure compounds R-245cb and R-1234yf.

 2

	R-245cb		R-1234yf
	T/K	P/MPa	T/K	P/MPa
	278.42	0.2446	275.81	0.3445
	283.05	0.2864	275.82	0.3446
	283.25	0.2884	278.21	0.3727
	287.61	0.3321	278.21	0.3727
	287.65	0.3326	283.03	0.4355
	292.85	0.3919	283.04	0.4353
	292.86	0.3921	287.94	0.5065
	297.57	0.4523	287.94	0.5064
	297.60	0.4527	292.76	0.5846
	302.75	0.5264	292.82	0.5857
	307.05	0.5941	297.70	0.6736
	307.11	0.595	297.71	0.6740
	312.49	0.6901	302.59	0.7717
	312.49	0.6902	302.60	0.7719
	317.09	0.7793	307.39	0.8780
	317.13	0.7802	307.43	0.8789
	322.23	0.8894	312.22	0.9951
	322.25	0.8894	312.25	0.9961
	327.07	1.0025	317.25	1.1298
	327.07	1.0026	322.11	1.2720
	331.95	1.1278	322.12	1.2727
	331.95	1.1277	327.00	1.4286
	336.83	1.2641	327.00	1.4288
	336.84	1.2644	331.82	1.5970
	341.72	1.4129	331.83	1.5971
	341.74	1.4135	336.65	1.7793
	346.30	1.5613	336.67	1.7800
	346.49	1.5684
	351.62	1.7527
	351.60	1.7522
	356.36	1.9359
	356.45	1.9409
	356.50	1.9427

Table 4

 4

, the experimental data are very well correlated by Equation (

3

) and the corresponding parameters.

Table 4 : Equation (3) parameters for the R-1234yf and R-245cb.

 4

	Compound	A	u(A)	B	u(B)	C	u(C) Tmin/K Tmax/K ARD% Bias%
	R-1234yf	-19.7088	1.03 -22.3328 1.75 4.9048 0.37	275	337	0.26	0.08
	R-245cb	-15.8426	0.74 -15.9154 1.25 3.7025 0.26	232	380	0.28	0.10

Table 5 . VLE data of the system R-1234yf (1) + R-245cb (2).

 5

	P/MPa	n	x 1	x 1	n	y 1	y 1
				T = 283.39 K
	0.2891	-	0	-	-	0	-
	0.3140		0.1550	0.0010	6	0.2110	0.0010
	0.3391		0.3216	0.0009	7	0.4010	0.0010
	0.3723		0.5490	0.0010	7	0.6320	0.0030
	0.4063		0.7670	0.0010	6	0.8190	0.0010
	0.4203	11	0.8670	0.0020	9	0.8950	0.0020
	0.4402	-	1	-	-	1	-
				T = 303.15 K
	0.5322	-	0	-	-	0	-
	0.5747		0.1652	0.0007	12	0.2080	0.0010
	0.6131		0.3187	0.0009	6	0.3858	0.0009
	0.6561		0.4890	0.0020	6	0.5580	0.0010
	0.6969		0.6530	0.0010	6	0.7100	0.0030
	0.7368		0.8110	0.0030	6	0.8460	0.0020
	0.7839	-	1	-	-	1	-
				T = 323.30 K
	0.9125	-	0	-	-	0	-
	0.9871		0.1860	0.0007	7	0.2229	0.0008
	1.0535		0.3617	0.0006	5	0.4160	0.0030
	1.0978	13	0.4760	0.0010	9	0.5340	0.0030
	1.1593		0.6304	0.0009	5	0.6798	0.0006
	1.2192		0.7920	0.0020	6	0.8256	0.0006
	1.3087	-	1	-	-	1	-
				T = 343.27 K
	1.4615	-	0	-	-	0	-
	1.5658		0.1830	0.0010	6	0.2160	0.0030
	1.6653		0.3570	0.0020	5	0.4090	0.0040
	1.7393		0.4840	0.0020	6	0.5280	0.0040
	1.8083	11	0.6050	0.0030	6	0.6540	0.0020
	1.9079		0.7756	0.0006	7	0.8053	0.0009
	2.0577	-	1	-	-	1	-

Table 6 . Pure compound EoS parameters.

 6

	Thermodynamic model	Compound	m 1	m 2	m 3	Z c
		R-1234yf	0.4636	0.4700	-0.0800	0.28085
	NEoS
		R-245cb	0.5625	-0.4543	3.0585	0.28072
		R-1234yf	0.8049	-0.4242	1.7469	-
	PR EoS
		R-245cb	0.8920	-1.2280	4.7966	-
	Following the modelling of the vapour pressures for the single compounds R-1234yf and R-
	245cb, a comparison with our measured data and literature data was carried out.

Table 7 : ARD and bias for the two EoSs, relative to our measured vapour pressures of R-1234yf and R-245cb.

 7

	Compound	Model	ARD (%)	Bias (%)
		NEoS	-0.46	0.46
	R-1234yf
		PR EoS	0.02	0.01
		NEoS	0.26	0.08
	R-245cb	PR EoS	0.12	-0.01

Table 8 . ARD and bias for the two EoSs, relative to the measured data of the binary mixture R- 1234yf (1) + R-245cb (2).

 8

	Model	T/K	k 12	ARD P (%) ARD y1(%)	Bias P (%)	Bias y1(%)
		283.39	0.00003	0.18	0.91	-0.13	-0.72
		303.15	-0.00241	0.65	1.31	0.65	-1.31
	NEoS	323.30	-0.00244	0.72	1.30	0.68	-1.30
		343.27	0.00148	0.23	0.57	-0.07	0.22
		283.39	0.0081	-1.00	-1.60	1.00	1.60
	PR-EoS	303.15 323.30	0.0034 0.0014	-0.19 -0.003	-1.85 -1.57	0.19 0.15	1.85 1.57
		343.27	0.0007	-0.06	0.14	0.14	0.58

Appendix

Description of the NEoS

In order to predict accurately the thermodynamic properties of refrigerants (both pure compounds and mixtures), a new EoS (denoted by NEoS) was developed, based on the modification of the well-known Patel-Teja (PT)EoS [START_REF] Patel | A new cubic equation of state for fluids and fluid mixtures[END_REF].

The NEoS is a 3-parameter cubic EoS and is defined by Equation ( 1).

(

where P is the pressure, T the temperature, v the volume, and R the universal constant for ideal gases. b is the volumetric parameter and a(T) the cohesive energy parameter. u and w are two parameters defined in order to have: u + w = 0, which was shown to be the optimal combination for liquid density calculations by cubic EoSs [START_REF] Ji | A systematic study of cubic three-parameter equations of state for deriving a structurally optimized PVT relation[END_REF]. Note that other authors such as Segura et al. [START_REF] Segura | An accurate direct technique for parameterizing cubic equations of state[END_REF] work on similar approaches, by the parameterization of u and w, without fixing a relation between them.

Here, u and w are defined as follows (Eq. 2).

(2) While the PT EoS [START_REF] Patel | A new cubic equation of state for fluids and fluid mixtures[END_REF], [START_REF] Ji | A systematic study of cubic three-parameter equations of state for deriving a structurally optimized PVT relation[END_REF] [34] and the NEoS have the same definition for u, a different definition for w is chosen in the NEoS in order to fulfil the conditions defined by Ji and Lempe [START_REF] Ji | A systematic study of cubic three-parameter equations of state for deriving a structurally optimized PVT relation[END_REF], i.e. u + w = 0 (note that in the case of the PT EoS, u + w = 1).

The cohesive energy parameter a(T) depends on the temperature and is defined as follows (Eq. 3):

(3) α(T) is the alpha function that will be defined below, and which depends on both the temperature and the substance.

-It has to be continuous, as well as its first and second derivatives.

We can verify the conditions of the continuity of the Mathias-Copeman alpha function and its first derivative at the limit of the critical temperature (where the alpha function is defined from two relations). This latter condition is important in order to ensure the continuity of the thermodynamic properties (Coquelet et al. [START_REF] Pina-Martinez | Updated versions of the generalized Soave α-function suitable for the Redlich-Kwong and Peng-Robinson equations of state[END_REF]), see Equation [START_REF] Fedele | Saturated Pressure Measurements of 2,3,3,3-Tetrafluoroprop-1-ene (R1234yf) for Reduced Temperatures Ranging from 0[END_REF]. [START_REF] Fedele | Saturated Pressure Measurements of 2,3,3,3-Tetrafluoroprop-1-ene (R1234yf) for Reduced Temperatures Ranging from 0[END_REF] We can notice however that the second derivative of the Mathias-Copeman alpha function presents a discontinuity at the critical point. In fact, a recent paper from Le Guennec et al. [START_REF] Guennec | A consistency test for αfunctions of cubic equations of state[END_REF] defines the different conditions that an alpha function should satisfy (referred to as a consistency test): the alpha function must be positive, decreasing, convex with negative third derivative. According to this study no alpha function from the literature satisfies all the conditions of the consistency test. However, we can notice that in their very recent papers (Le Guennec et al. [START_REF] Guennec | On the imperative need to use a consistent α-function for the prediction of pure-compound supercritical properties with a cubic equation of state[END_REF], Pina Martinez et al. [START_REF] Pina-Martinez | Analysis of the Combinations of Property Data That Are Suitable for a Safe Estimation of Consistent Twu α-Function Parameters: Updated Parameter Values for the Translated-Consistent tc-PR and tc-RK Cubic Equations of State[END_REF]), these authors present consistent alpha functions. The Mathias-Copeman alpha function can be associated with any cubic EoS. In this work it was associated with the NEoS.