Colin Leverger
email: colin.leverger@orange.com

Vincent Lemaire

Simon Malinowski

Thomas Guyet

Laurence Rozé

Day-ahead time series forecasting: application to capacity planning

Keywords: Time series, Capacity Planning, Clustering, Markov Models

published or not. The documents may come   L'archive ouverte pluridisciplinaire

Introduction

Capacity planning (CP) is one major preoccupation for todays companies. This ensemble of practices characterises the way an enterprise manages, updates, adds or removes physical/virtualised servers from its infrastructure. A well-made CP helps to reduce operational costs, and improves the quality of the provided services: indeed, one of the major goals is to maintain the best quality of services for the end users. Most of the time, project owners, managers and experts apply CP guidelines to manage their infrastructure manually. This approach is cumbersome and overestimate needs to prevent from any business interruptions. As the manager will have to daily take decisions about its infrastructure, we are interested in forecasting the full time series of the day ahead. Contrary to a lot of time series forecasting techniques, the forecasting horizon is of several steps in this study. We could then rely on the circadian rhythm of the data to improve forecasts.

Bodik et al. [START_REF] Bodik | Automatic exploration of datacenter performance regimes[END_REF] show that data analytics on datacenters key performance indicators (KPI) may be used to do CP (e.g., CPU, number of users, RAM). Indeed, the data generated by servers or user activity could be valuable sources of information. One particular data analytic task that may help resource managers in the daily activity is to forecast the evolution of the KPI in time. That is basically a time series forecasting task. The more accurate are forecasted the KPI, the more informed will be the management decisions regarding CP. In particular, we make the assumption, denoted (A), that KPI time series are driven by a two-order temporal process. The circadian scale drives the daily evolution of the KPI but, at the second order, this daily behaviour is itself driven by some hidden rules. The evolution within a week could be considered. During the weekdays (from Monday to Friday) KPIs have a daily behaviour which is different from the weekend daily behaviour.

The method we propose is to capture the daily behaviour of the KPI by defining some "types of days" and thus to detect second-order behaviours by analysing sequences of typical days. For now and as a primary study, we did not take into account the position of a day in a week (e.g., Mondays, Tuesdays, ...) and only consider types of days regarding shapes of the time series data only. The experiments compare the forecasting performance of our method with different baseline in order to validate our hypothesis A on a large collection of real data coming from a very large infrastructure (large time depth and great variability of KPI).

Day-ahead time series forecasting

In this section, we present our approach to produce one-day-ahead time series forecasting. This approach is composed of three learning steps (i) data normalisation and split, (ii) clustering, (iii) next-day cluster estimation, and a forecasting step (iv) next day forecasting. The learning steps take as input a multivariate time series and build a forecasting model. In the forecasting step, the model is applied to the time series current day in order to forecast the next day. Fig. 1 depicts the learning steps of our approach, i.e. from (i) to (iii), together with the forecasting step (iv). The four steps are presented below.

Let X = X 1 , . . . , X n be a multivariate time series of length n. For all i ∈ [1, n], X i ∈ R p , where p denotes the dimensionality of X (the number of monitored KPI in our application case). Let h be the forecasting horizon that corresponds to one day. (i) Data normalisation and split: First, we normalise each dimension of X using a N (0, 1) normalization. Then, we construct the set D = {D 1 , . . . , D d }, where D i represents the subseries of X corresponding to the i th day of measurements, and d = n h . For sake of simplicity, we assume that n is a multiple of h.

(ii) Clustering: The elements of D are then given to a clustering algorithm. In our case, we use the multidimensional k-means based on the Euclidean distance. The centroids of the created clusters are denoted C 1 , . . . , C k . They correspond to typical day evolution. The choice of the number k of clusters will be discussed in Section 3. (iii) Next-day cluster estimation: The set D is encoded into a sequence c 1 , . . . , c d , where c i ∈ [1, k] corresponds to the index of the cluster that contains the day D i . We model this sequence by a first-order Markov Model of transition matrix Π = (π i,j ), where π i,j is the probability that the next day belongs to cluster j given that the current day belongs to cluster i. These probabilities are estimated using the sequence c 1 , . . . , c d . In other words, this Markov Model enables us to estimate the most probable cluster to which will belong the next day of measurements. (iv) Next-day forecasting: This forecasting step makes use of the model learned above to predict the next day measurements (D T +1 ) given the current day measurements (D T ). First, the index of the closest cluster to D T is computed, and denoted c T . The most probable cluster for the next day is estimated using the transition matrix Π : c T +1 = arg max j∈1,...,k π c T ,j . Finally, the forecasting of the next day is given by the centroid C c T +1 .

Experiments and Results

In this section, we briefly introduce our real capacity planning dataset. Before the results, we present the evaluation protocol of our approach and its underlying assumption stating time series with two-orders temporal scales.

Data

This work has been endorsed by a growing project at Orange: Orange Money (OM) 5 . This latest was established in late 2008. It aims in providing an easy access to bank transfers for African customers. The bank exchange system is hosts by a large Orange infrastructure (162 servers) and is used by more than 31 million customers across all of Africa and Europe.

The datasets are collected from this infrastructure using Nagios supervisor.

It collects technical and operational metrics every five minutes:

-Technical data: data concerning the servers performance such as percentages of CPU/memory use across all servers, -Operational data: data concerning users browsing the service; number of financial transactions every minute, number of people on a website, etc.

Protocol

This study has been conducted in two major steps: (i) We have first experimented using univariate time series, where the day T +1 is forecasted using only past data from current time series. Our multivariate time series have therefore first being exploited as univariate ones (we took each features independently).

(ii) We have also experimented using multivariate time series, where the past of several series is used to forecast future of one KPI. We were curious about the possible improvement of the accuracy using multivariate time series. Indeed, we can assume that having more information describing the datacenter behaviour may help improve forecasts.

The performance of our method is compared to four alternative approaches. Two baseline methods are used to evaluate our assumption A on the time series structure in the dataset:

-Mean day is a simple mean day calculation: all days present in learning ensemble D are used to compute the average day. This latest is then given as a forecast result for day T +1. Intuitively, it should give the worst forecasts if there are actually very different types of days. -Omniscient algorithm is an adaptation of the day-ahead forecasting method (with same clusters) with an omniscient prediction of c T +1 instead of the Markov model. This baseline is used to evaluate the Markov Model independently of the clustering step. It assumes that we know to which cluster belong day T + 1. The forecasts given by this method cannot be worse (in terms of prediction error) than the ones given by our approach.

The relative performance of our approach with regard to these two baselines gives indication about the compliance of data with our assumption A. Moreover, two classical forecasting techniques are used to compare the quality of the forecasts given by our approach: 1. Auto-Regressive (AR) model [START_REF] Akaike | Fitting autoregressive models for prediction[END_REF], which is a representation of a random process that can be used to describe some time series, 2. Holt Winters (HW) Triple Exponential Smoothing [START_REF] Winters | Forecasting sales by exponentially weighted moving averages[END_REF], which is a rule of thumb technique for smoothing time series data. It extends the Holts method to capture seasonality.

These two methods have been preferred to more complex models such as ARIMA models [START_REF] Hillmer | An ARIMA-model-based approach to seasonal adjustment[END_REF] or LSTM [START_REF] Hochreiter | Long short-term memory[END_REF] whose parameters are uneasy to configure.

We quantify the prediction error by computing the Mean Square Error (MSE) between the forecasted values and the real values. The dataset is split chronologically into three parts: 70% of the data as training data, 15% as validation data and 15% as test data. The validation dataset is used to select the best number of clusters k (ranging from 2 to 200) based on the MSE.

Results

Table 1 presents the results obtained for 458 univariate time series (both operational and technical coming from 162 servers) of the OM projects. To obtain more significant results, we use all day present on the test ensemble to produce forecasts (i.e. if the test ensemble is composed of 6 days, we produce 5 forecasts (excluding the first day). The mean errors are computed as the average MSE per univariate time series. For each forecast, the methods are ranked by decreasing error, and we compute the mean rank.

Let us first analyse the results to assess our assumption A about the underlying data structure. We observe that forecasting the centroid of the known cluster for day T + 1 (omniscient method) outperforms significantly the daily mean of the whole time series. This means that our assumption about the two temporal scales is satisfied by our dataset. The average number of clusters selected on the validation sets is 16 for 365 days of data. This number is sufficiently low to conclude that there are actually clusters of typical days.

Unsurprisingly, the omniscient method outperforms our method as the former always make the best choice. It also shows that the crucial step of forecasting the type of the day ahead could be improved. In fact, Markov models are very simple models and more advanced approaches could probably help improve our overall approach. But we mainly notice that our algorithm has better performance than AR and HW that is often chosen for seasonal time series forecasting. It can be explained by a weak adequacy of these methods to our task. AR and HW are more used to forecast next few points of the time series while our task requires to forecast the time series for the entire day (96 points). The mean error is higher for AR than the mean-day forecast, but its ranking is better, meaning that AR is better than a mean day on most of the time series, but when it fails, it fails with higher errors. One noticeable thing is the pretty bad performances of the HW model. This could be explained because HW is less efficient in forecasting several points in the future, and often base its forecasts following the last tendency observed.

The previous experiments evaluate our algorithm on univariate time series. Table 2 presents results on multivariate time series with two CPUs from two different servers.

The multivariate mean algorithm is slightly better than our algorithm in terms of rank but not in terms of MSE. It indicates that having more features involved in the process increases the chance that the mean day is more representative for the forecast, and thus better.

One possible explanation is that the number of clusters is too small to extract meaningful groups of days. With multivariate time series, the number of types of day increases but we did not increase the maximum number of cluster (tested during on the validation set). Finally, one noticeable performance of our multivariate algorithms is its low mean error. Nonetheless, experiments on more time series are required to conclude, and but it shows that the method is suitable regarding technical CPU time series.

Discussion

Experimental results show that our approach performs well on our real dataset, but we also know that it is an early proposal that has some limitations and weaknesses.

In the first stage of our model, daily time series are clustered. At the time, we only experimented the k-means algorithm with a Euclidean distance. In practice, if detecting data peaks between 2 PM and 4 PM is something of interest, this distance is to be privileged. But if the goal is to detect peaks in a day without precise information about their timings, DTW [START_REF] Niels | Dynamic time warping[END_REF] is a better candidate. In our mind, there is not a unique good choice but better some choices that fit the data characteristics like Douzal-Chouakria et Amblard [START_REF] Chouakria | Classification trees for time series[END_REF] suggest for a classification task. In addition, the clustering strategy could also be evaluated. One of the weaknesses of the current approach using a k-means algorithm is the critical choice of k. In this study, we find the optimal number of clusters using the validation set. We test various k-means sizes on the training dataset, and select the one which helps our algorithm in having a lower MSE. Less empirical techniques such as David et Bouldin criteria [START_REF] Davies | A cluster separation measure[END_REF] or even Silhouette [START_REF] Rousseeuw | Silhouettes: a graphical aid to the interpretation and validation of cluster analysis[END_REF] could help to select a priori the best number of clusters with a lower computational cost.

An interesting thing would be to know which of the steps described on the Figure 1 is lowering forecast performances. This could help us in improving the chain by tuning very particular and identified parameters. The omniscient algorithm that knows for sure the day T + 1 is useful as it reveals that having an almost perfect clustering algorithm could greatly enhance precision of forecasts (see Tables 1 and2 which show that omniscient technique often outputs forecasts with lower mean error). Enhancing the quality of clustering could then be a key to better results.

In the second stage of our model, the experiment shows that there is possible improvements of the prediction of the type of the next day. A more accurate solution than Markov models can bridge the gap with the omniscient approach. At the time, a Markov model takes only into account the day before the one to predict. This simple model has been preferred to higher order Markov models because of the required quantity of training data. With 7 different days, transition matrix of size 49 are to learn in our case, but of size 343 for a 2-order Markov model. This requires long sequences of days to accurately estimate them. We currently collected 365 days of data history and thus prefer to focus on a simple but sound approach. A study of state-of-the-art approaches of sequence prediction with sparse data will help us to identify good candidates to replace the Markov model.

Finally, as mentioned in the introduction, our model does not take into account the position of the weekday to make forecasts. However, some observations let us believe that it could improve the prediction accuracy. In fact, we noticed that the clustering of daily time series set up with 2 clusters will extract two types of behaviours: A high activity profile (HA) from Mondays to Fridays and a low activity profile (LA) on Saturdays and Sundays. As a consequence, the trained Markov Model will more likely predict a HA day after another HA day (with probability 0.8). This means that every Friday, it will wrongly predict the day with a HA profile. A model with the weekday information will split the next day prediction rule in two different rules: High activity from Monday to Thursday leads to a high activity the next day while high activity on Friday leads to a low activity. It remains interesting to have information about the types of the days as, in real data, there are several different of profiles of days that are not necessarily correlated to the weekday information.

Conclusion

In this work, we presented a general method to address a specific problem of capacity planning, i.e. the forecast of the evolution of KPI indicators a day ahead. The method we propose is a time series forecasting method that is founded on the assumption that the time series are implicitly structured as a sequence of typical days. Our experiments comparing baseline approaches and classical time series forecasting methods to our method show that this assumption is fitted by most of the time series of our dataset. We now have to explore more deeply the characteristics of the data that does not fit it. to improve the overall approach. We observe an interestingly low mean square error for our algorithm on multivariate CPU and find these preliminary results promising. But the proposed method may also benefit from improvements in the two main stages: the clustering of time series and the sequential prediction. We strongly believe that Markov model may be improved by some more recent works on sequence prediction [START_REF] Budhathoki | Causal inference on event sequences[END_REF].

Fig. 1 .

 1 Fig. 1. Illustration of the main steps of the day-ahead time series forecasting: (i) multivariate time-series splitting in daily time series, (ii) time series clustering, (iii) day sequentiality modelling, (iv) forecast on a new daily time series (see details in the text).

Table 1 .

 1 Errors and ranks for univariate forecasting results.

	Algorithm	Mean error ± std Mean rank ± std
	Omniscient algorithm	0.45 ± 0.83	1.51 ± 0.79
	Day-ahead forecasting	0.52 ± 0.87	2.62 ± 0.91
	AR model	0.73 ± 0.92	3.30 ± 1.38
	HW model	314 ± 1775	3 .91 ± 1 .37
	Mean day	0.71 ± 1.03	3.66 ± 0.95

Table 2 .

 2 Errors and ranks for multivariate forecasting results.never had the lower MSE not presented.

	Algorithm	Mean error ± std Mean rank ± std
	Multivariate mean	0.1 ± 0.14	2.07 ± 1.48
	Multivariate day-ahead forecasting	0.044 ± 0.1	2.24 ± 0.86
	Omniscient multivariate algorithm	0.041 ± 0.09	2.44 ± 0.70
	Omniscient univariate algorithm	0.5 ± 0.6	4.03 ± 0.85
	Univariate day-ahead forecasting	0.57 ± 0.63	4.84 ± 0.87
	AR model	0.76 ± 0.7	5.38 ± 1.38

https://orangemoney.orange.fr/


Karima Djebaili

Elise El Ahmar
email: elise.el_ahmar@mines-paristech.fr

Alain Valtz

Hassen Abdeslam

Christophe Meniai

Abdeslam Hassen Meniai

Christophe Coquelet

Vapor-Liquid Equilibrium Data for the Carbon Dioxide (CO 2 ) + 1,1,1,3,3-Pentafluorobutane (R365mfc

Keywords: Experimental VLE data, R365mfc, Peng-Robinson EoS, Binary interaction parameters 1

published or not. The documents may come   L'archive ouverte pluridisciplinaire

Introduction

In the middle of the 20 th century, a collective interest in the human effect on the environment had begun to emerge. In the fields of air conditioning and refrigeration, environmental issues of stratospheric ozone depletion and global warming have been the primary motivations for recent researches. The traditional refrigerants, i.e., chlorofluorocarbon (CFCs) and hydrochlorofluorocarbon (HCFCs) fluids, were banned by the Montreal Protocol (1987) [START_REF]Montreal Protocol on substances that deplete the ozone layer[END_REF] because of their contribution to the depletion of the stratospheric ozone layer. As a consequence, considerable research effort was devoted to the development of environmentally safe refrigeration systems using working fluids the selection of which must be based on three factors: safety (toxicity and flammability), environmental impact (stratospheric ozone and climate change) and performance (cooling and heating with required capacity, energy efficiency, critical and boiling temperatures).

Natural fluids such as carbon dioxide (CO 2 ) which is a well-known refrigerant is considered as an appropriate alternative to chlorinated substances [START_REF] Bhatkar | Alternative refrigerants in vapour compression refrigeration cycle for sustainable environment: A review of recent research[END_REF] . Since CO 2 is a component of air, obtained from atmosphere itself by fractionation, it would have no impact on global warming (apart from for the energy consumption associated with the fractionation process) and no effect on ozone depletion (Ozone Depletion Potential (ODP) = 0) [START_REF] Aprea | An experimental study on charge optimisation of transcritical CO 2 cycle[END_REF] . CO 2 which is an inert gas, has many excellent advantages in engineering applications, such as no toxicity, no inflammability, higher volumetric capacity, lower pressure ratio, better heat transfer properties, easy availability, lower price, no recycling problem and complete compatibility with normal lubricants [START_REF] Aprea | An experimental study on charge optimisation of transcritical CO 2 cycle[END_REF] . However, and despite its interesting thermodynamic characteristics, the use of CO 2 limits come from its relatively low critical coordinates (T C = 304.21K, P C =73.81 bar) 4 requiring the design of systems operating in a trans-critical cycle reaching very high pressures or a cascading cycle. Furthermore, hydrofluorocarbons (HFCs) which are derived from the partial substitution of hydrogen by fluor in the molecules of hydrocarbons have been mainly chosen to replace CFCs and HCFCs in refrigeration and air conditioning due to high merits in safety, performance and no effect on ozone depletion [START_REF] Bivens | Refrigeration and Air Conditioning with Reduced Environmental Impact[END_REF] . Indeed, according to the Kyoto Protocol (1997), most hydrofluorocarbons have large Global Warming Potentials (GWPs) and therefore provide a non-negligible, direct contribution to global warming when leaked to the atmosphere. HFCs have been also identified for use as separating agents in solvent extraction processes in the petrochemical industry. They are generally low in toxicity, are easily recycled, have high stability and are seen as a safer alternative to challenge organic solvents such as hydrocarbons or alcohols [START_REF] Grenfell | New fluorinated solvent alternatives[END_REF][START_REF] Agopovich | Fluorocarbons and supercritical carbon dioxide serve niche needs[END_REF] .

Mixtures of CO 2 and HFCs may be promising refrigerants since they can contemporarily reduce the problems connected to high operating pressure of CO 2 and large Global Warming Potentials of HFCs. Supercritical CO 2 and HFCs are being considered as ozone-depleting chemicals replacements due to their zero ozone-depletion potential, nonflammability, low toxicity, and high compatibility [START_REF] Agopovich | Fluorocarbons and supercritical carbon dioxide serve niche needs[END_REF] .

Many measurements of binary Vapor-Liquid Equilibrium (VLE) for systems containing CO 2 with HFCs have been previously undertaken by A.Valtz et al. [START_REF] Valtz | Vapour-liquid equilibrium data for the CO 2 + 1, 1, 1, 2, 3, 3, 3-heptafluoropropane (R227ea) mixture at temperatures from[END_REF][START_REF] Valtz | Vapour-liquid equilibrium data for the hexafluoroethane + carbon dioxide mixture at temperatures from 253 to 297 K and pressures up to 6.5 MPa[END_REF][START_REF] Valtz | Isothermal vapour-liquid equilibrium data for the carbon dioxide (R744) + decafluorobutane (R610) mixture at temperatures from 263 to 353 K[END_REF] , F. Rivollet et al. [START_REF] Rivollet | Vapour-liquid equilibrium data for the carbon dioxide (CO 2 ) + difluoromethane (R32) mixture at temperatures from 283[END_REF] , C.

Duran-Valencia et al. [START_REF] Duran-Valencia | Vapour-liquid equilibrium data for the (carbon dioxide + 1, 1,1,2-Tetrafluoroethane) system at temperatures from 252[END_REF] and K. Jeong et al. [START_REF] Jeong | Vapour-liquid equilibria of the carbon dioxide (CO 2 ) + 2, 2-dichloro-1, 1, 1-trifluoroethane (R123) system and carbon dioxide (CO 2 ) + 1-chloro-1, 2, 2, 2-tetrafluoroethane (R124) system[END_REF] . To the best of our knowledge, no vapor-liquid equilibrium data for the binary mixture CO 2 -R365mfc are available in the open literature.

In the present work, this binary mixture was studied. The R365mfc, which is a flammable fluid, was chosen for its non-toxicity, its good environmental compatibility; zero ODP and moderate GWP (GWP = 790 over a time horizon of 10 years) and especially for its relatively high critical conditions (Tc = 460K, Pc = 3.266 MPa) 4 enabling it to adapt, easily, in applications with condensation temperatures above 90°C. This volatile liquid (low boiling temperature T B = 40°C [START_REF] Jeong | Vapour-liquid equilibria of the carbon dioxide (CO 2 ) + 2, 2-dichloro-1, 1, 1-trifluoroethane (R123) system and carbon dioxide (CO 2 ) + 1-chloro-1, 2, 2, 2-tetrafluoroethane (R124) system[END_REF] ) is an uncommon compound, EL Ahmar et al. [START_REF] El Ahmar | Isothermal Vapour-liquid equilibrium data for binary azeotropic systems containing pentafluoro-chemicals at Temperatures from 363 to 413 K[END_REF] and H. Madani et al. [START_REF] Madani | Vapour-liquid equilibrium data concerning refrigerant systems[END_REF] have presented data for the vapor-liquid equilibrium of two binary mixtures involving the R365mfc.

This paper is to provide new sets of data to characterize phase equilibrium of the binary mixture CO 2 + 1, 1, 1, 3, 3-pentafluorobutane (R365mfc) at six isotherms from 283.16 to 337.16 K. The new data have been described with the Peng-Robinson Equation of State (PR EoS) [START_REF] Peng | A new two constant equation of state[END_REF] with the Mathias-Copeman alpha function [START_REF] Mathias | Extension of the Peng-Robinson equation of state to complex mixtures: Evaluation of various forms of the local composition concept[END_REF] and the Wong-Sandler (WS) mixing rules [START_REF] Wong | A theoretically correct mixing rules for cubic equations of state[END_REF] incorporating the Non Random Two Liquids (NRTL) activity coefficient model [START_REF] Renon | Local composition in thermodynamic excess function for liquid mixtures[END_REF] .

This article is organized as follows. A presentation of the experimental setup used to investigate vapor-liquid equilibrium (VLE) of binary mixtures followed by the data treatment, results and discussion.

2.

Experimental Section

Apparatus description

The apparatus used in the measurements is based on the "static-analytic" method. The apparatus is similar to that described by Valtz et al. [START_REF] Valtz | Vapour-liquid equilibrium data for the CO 2 + 1, 1, 1, 2, 3, 3, 3-heptafluoropropane (R227ea) mixture at temperatures from[END_REF][START_REF] Valtz | Vapour-liquid equilibrium data for the propane + 1, 1, 1, 2, 3, 3, 3-heptafluoropropane (R227ea) system at temperatures from[END_REF] .

The equilibrium cell was immersed inside a temperature regulated liquid bath. The temperature of the equilibrium cell was measured with two platinum resistance thermometer probes (Pt-100) inserted into the walls of the equilibrium cell. These Pt100 probes were calibrated against a 25 Ω reference probe (TINSLEY Precision Instrument) already certified by the Laboratoire National d'Essai (Paris). Pressures were measured using a Druck pressure transducer (Type PTX611, range: 0-16 MPa). This sensor was calibrated against a Dead Weight Pressure Balance (Model 5202S supplied by Desgranges and Huot). Pressure and temperature data acquisition was performed with a computer linked to an HP unit (HP34970A). The analysis of the phases (liquid and vapor) composition was carried out using a gas chromatograph (Perichom, Model: PR-2100) equipped with a Thermal Conductivity Detector (TCD) connected to a data acquisition system. The analytical column used for the separation was a 5% Krytox on CarboBlack B model with 60/80 mesh supplied by Restek,

France. The TCD was calibrated by repeated injections of known amounts of each pure component into the injector of the gas chromatograph using a gas tight syringe. To estimate the uncertainties in temperature, pressure and composition, the guidelines of NIST [START_REF] Taylor | Guidelines for evaluating and expressing the uncertainty of NIST measurement results[END_REF] were followed. The uncertainty estimation procedure is fully described by Zhang et al. [START_REF] Zhang | An improved static-analytic apparatus for vapour-liquid equilibrium (PTxy) measurement using modified in-situ samplers[END_REF] . Both measurements repeatability and calibration uncertainties were taken into account. All the expanded uncertainties (U(x 1 ), U(y 1 ), coverage factor, k=2) for experimental data are reported later on. The resulting accuracies of pressure and temperature in this work were ±0.004 MPa and ±0.04 K, respectively.

Experimental procedure

At room temperature, the equilibrium cell and its associated loading lines were evacuated with a vacuum pump.. The cell was then loaded with given quantity (approximately 5 cm 3 ) of the compound having the lowest vapor pressure (R365mfc). The liquid was degassed by periodically removing vapor phase through an overhead valve, while heating to the desired temperature. Then CO 2 was introduced stepwise, leading to successive equilibrium mixtures of increasing overall CO 2 content. After each new CO 2 loading, equilibrium was assumed when the total pressure remains unchanged to within ±1.0 kPa during a period of 30 min under efficient stirring.

For each equilibrium condition, at least five samples of each phase (vapor and liquid)

were withdrawn using the cappillary samplers ROLSI ®, [START_REF] Guilbot | Rapid on-line Sampler-injector: a reliable tool for HT-HP sampling and on-line GC analysis[END_REF] and analyzed in order to check the repeatability of that measurement.

Modeling

For binary mixture modeling, the critical parameters (temperature (T C ) and pressure (P C )), and acentric factor () for each pure component are provided in Table 1 4 . With a φ -φ symmetric approach, experimental data were correlated using in-house software developed at CTP. The thermodynamic model which is presented in appendix A is based on the Peng

Robinson cubic equation of state (PR EoS) [START_REF] Peng | A new two constant equation of state[END_REF] with the Mathias-Copeman alpha function [START_REF] Mathias | Extension of the Peng-Robinson equation of state to complex mixtures: Evaluation of various forms of the local composition concept[END_REF] , the Wong-Sandler mixing rules [START_REF] Wong | A theoretically correct mixing rules for cubic equations of state[END_REF] incorporating the NRTL activity coefficient model [START_REF] Renon | Local composition in thermodynamic excess function for liquid mixtures[END_REF] . The Mathias-Copeman coefficients for the PR EoS are presented in Table 2 4,14 . The interaction parameters ( ij  and ji  ) are adjusted directly utilizing the measured VLE data through a modified Simplex algorithm [START_REF] Aberg | Design and evaluation of modified simplex methods[END_REF] using the following objective function: Where N is the number of experimental data points, x, y are liquid and vapor phase mole fractions, respectively, exp and cal are subscripts denoting experimental and calculated, respectively.

                             

4.

Results and discussion

Vapor-Liquid Equilibrium data

Vapor-liquid equilibrum data for the binary mixture, CO 3 and plotted in Figure 1, the mixture shows a strong negative deviation from the ideality (the vapor pressure of the mixture is lower than expected from Raoult's law). This can be attributed to the large dipole moment of the R365mfc (μ 365mfc = 3.807 Debyes 4 ) and quadrupole moment of CO 2 which cause the increase of intermolecular attractive forces between the two components of the mixture. The phase diagram of the CO 2 (1) -R365mfc (2) mixture is a type I in the Scott and van Konynenburg six types classification with one critical line joining the CO 2 and R365mfc critical points.

To quantify the fit of the model to the experimental data, the deviations, AADU (average absolute deviation), RDU (relative deviation) and ARDU (average relative deviation) were determined for both the liquid and vapor phase mole fractions. The deviations are defined by:

          exp 1 exp , , exp , exp / 100 N i i cal i i U U U N BIASU (eq.02)         exp 1 exp , , exp / 1 N i i cal i U U N AADU (eq.03)   exp , exp , , / . 100 i i cal i U U U RDU   (eq.04)         exp 1 exp , exp , , exp / / 100 N i i i cal i U U U N ARDU (eq.05)
Where N, exp, cal as defined previously and U= x 1 or y 1 .

These indicators which give information about the agreement between model and experimental results are presented in Table 4. Generally, the model gives a good fit of the experimental data for both liquid and vapor compositions as shown in Table 4.

The relative volatilities were computed by means of the model and compared to the experimental values. Figure 2 shows the composition dependency of the relative volatility for the six measured isotherms. It can be seen that the separation factor α 12 for the mixture CO 2

(1) -R365mfc (2) decreases with increasing the temperature and the CO 2 composition.

Although the shape of the curves appears to be the same, the alpha values calculated by the model are lower than those determined from the experimental data, especially at low CO 2 concentrations because of the uncertainties on vapor composition caused by the large differences in volatilities between the two substances.

Binary interaction parameters τ 12 , τ 21 and k ij for the WS mixing rules are given in Table 5 and plotted in Figures 3,4 and 5. As it was observed in previous studies [START_REF] Valtz | Vapour-liquid equilibrium data for the CO 2 + 1, 1, 1, 2, 3, 3, 3-heptafluoropropane (R227ea) mixture at temperatures from[END_REF][START_REF] Valtz | Vapour-liquid equilibrium data for the hexafluoroethane + carbon dioxide mixture at temperatures from 253 to 297 K and pressures up to 6.5 MPa[END_REF][START_REF] Valtz | Isothermal vapour-liquid equilibrium data for the carbon dioxide (R744) + decafluorobutane (R610) mixture at temperatures from 263 to 353 K[END_REF][START_REF] Rivollet | Vapour-liquid equilibrium data for the carbon dioxide (CO 2 ) + difluoromethane (R32) mixture at temperatures from 283[END_REF][START_REF] Duran-Valencia | Vapour-liquid equilibrium data for the (carbon dioxide + 1, 1,1,2-Tetrafluoroethane) system at temperatures from 252[END_REF][START_REF] Jeong | Vapour-liquid equilibria of the carbon dioxide (CO 2 ) + 2, 2-dichloro-1, 1, 1-trifluoroethane (R123) system and carbon dioxide (CO 2 ) + 1-chloro-1, 2, 2, 2-tetrafluoroethane (R124) system[END_REF] , the parameters τ 12 , τ 21 and k ij behave differently below and above the critical temperature of the volatile component (CO 2 ). So it would be totally wrong to extrapolate temperature dependent binary parameters (adjusted only at temperatures above or below the lighter component's critical temperature) to temperatures at the other side of the critical point. Also, parameters have been adjusted using all the data for T>T C of CO 2 . The results are presented in Table 5. These parameters are used to compute the critical line.

Critical point determination

Methods to calculate critical points were proposed by Heidemann and Khalil 25 in 1980 and Michelsen and Heidemann [START_REF] Michelsen | Calculation of critical points from cubic 2 constant equations of state[END_REF] in 1981. These authors assumed that the stability criterion for an isothermal variation between an initial stable point and a very close new one can be explained with a minimum of the Helmholtz energy (eq.06) and they considered that the volume variation at the critical point was constant.

    0 0 0 0 0 0 0 0             i i i i i i i n A A n n V V P A A   (eq.06)
In this method the Taylor series expansion of the Helmholtz energy A was used around the stable state at constant total volume V (eq.07). with the critical point corresponding to the limit of stability. They developed an algorithm to calculate the critical point with a van der waals type EoS, combined with the classical mixing rules. In 1998, Stockfleth and Dohrn 27 improved this method by generalizing the algorithm of Heidemann and Khalil 25 to a generalized equation of state. The Stockfleth and Dohrn 27 method is chosen in this work to calculate the critical line using the PR EoS with WS mixing rules involving the NRTL model. The binary parameters used in this method were those obtained by correlating the VLE data in the CO 2 supercritical domain. The predicted critical locus line is in good agreement with the experimental isothermal phase envelopes (Figure1). For four temperatures above the critical temperature of pure CO 2 , the critical phase behaviors of mixture were estimated at each temperature by using power laws with asymptotic behavior at critical point [28][START_REF] Rainwater | Vapor-liquid equilibrium of near-critical binary alkane mixtures[END_REF][START_REF] Moldover | Interfacial tension and vapor-liquid equilibria in the critical region of mixtures[END_REF][START_REF] Ungerer | Applications of molecular simulation in the oil and gas industry: Monte Carlo methods[END_REF] with the experimental data highly close to the critical point. The obtained results are presented in Table 6.

k j i V T i j k k j i j i V T i j j i n n n n n n A n n n n A                               

Conclusion

In this paper, Vapor-Liquid Equilibrium (VLE) data for the mixture (CO 2 + C 4 H 5 F 5 ) are presented at six isotherms: three isotherms below and three above the critical temperature of CO 2 . This mixture has not been previously studied and the measurements represent novel data in the open literature. The experimental setup using a "static-analytic" method is completely described and the uncertainties in the experimental data are also given for the temperature, the pressure and the equilibrium phase moles fractions. The experimental data were well correlated using in-house thermodynamic software based on the Peng-Robinson equation of state combined with the Mathias-Copeman alpha function and the Wong-Sandler mixing rule incorporating the NRTL model. Critical conditions were reported for the mixture under study and the binary critical point line was also calculated using this model.

The R365mfc is not principally used as a refrigerant but as blowing agent [START_REF] Marrocho | Vapor-phase thermal conductivity, vapor pressure, and liquid density of R365mfc[END_REF] . Its use as a refrigerant mixture requires that it be used with another refrigerant in the liquid state at ambient temperature and pressure. With the refrigerant mixture (R365mfc-CO 2 ), a glide of pressure / temperature is observed. The application potentially targeted by this mixture would be heat pumps with thermal water as heat source, for example.

Prospective work on similar binary pairs with very different volatilities should be conducted.

Appendix A

The thermodynamic model used in this study is based on the Peng Robinson cubic equation of state (PR EoS) [START_REF] Peng | A new two constant equation of state[END_REF] with the Mathias-Copeman alpha function [START_REF] Mathias | Extension of the Peng-Robinson equation of state to complex mixtures: Evaluation of various forms of the local composition concept[END_REF] , the Wong-Sandler mixing rules [START_REF] Wong | A theoretically correct mixing rules for cubic equations of state[END_REF] incorporating the NRTL activity coefficient model [START_REF] Renon | Local composition in thermodynamic excess function for liquid mixtures[END_REF] . The PR EoS and the Mathias-Copeman alpha function are expressed as follows:

      b V b b V V T a b V T R P      
(eq.01)

C C P T R b 07780 . 0  (eq.02)   2 3 2 1 3 2 2 1 2 2 1 1 1 1 1 1                                R R R T c T c T c T  for r T < 1 (eq.03)   2 2 1 1 1 1                R T c T  for r T > 1 (eq.04)
where T C and P C are the critical temperature and pressure, respectively and T R is the reduced temperature.

Wong-Sandler mixing rules is represented as:

                                 i E i i i i ij j j i m CRT G T R b a x T R a b x x b  1 (eq.05)            i E i i i m m C G b a x b a  (eq.06)   ij j i ij k T R a b T R a b T R a b                                       1 2 1
(eq.07)

Where

  2 / 1 2 ln   C
for the PR EoS, and ij k is the adjustable binary interaction parameter.

In this study, we have used the NRTL activity coefficient model to calculate the excess Gibbs energy:

  ji j k ki ki k ji ji j i i i E T R x T R x x T R x P T g                                      exp exp , , (eq.08)
ji  is the non-randomness parameter which is fixed to 0.3 for such mixtures. ij  and ji  are the interaction parameters.

List of figures

List of tables

Table 1. Critical properties and acentric factors for each pure component 4 . Table 2. Mathias-Copeman coefficients for the PR EoS. Table 3. Vapor-liquid equilibrium pressure and phase compositions for the CO 2 (1) + C 4 H 5 F 5 (2) binary mixture. composition: y for vapor phase and x for liquid phase, n: samples number). U(T)= 0.04 K, U(P)=0.004 MPa., U(x), U(y) (U: expanded uncertainties for a 0.95 confidence level (k=2)). Temperature /K

1 +

 1 Residual term > 0 (eq.07)

Figure3. 12

 12 Figure3. 12 binary parameter as a function of temperature. Vertical dashed: CO 2 critical temperature. Horizontal dashed line: parameter value obtained after considering all the data for T > T c (CO 2 ). Figure4. 21 binary parameter as a function of temperature. Vertical dashed: CO 2 critical temperature. Horizontal dashed line: parameter value obtained after considering all the data for T > T c (CO 2 ). Figure5. k 12 binary parameter as a function of temperature. Vertical dashed line: CO 2 critical temperature. Horizontal dashed line: parameter value obtained after considering all the data for T > T c (CO 2 ).

Figure 5 .

 5 Figure3. 12 binary parameter as a function of temperature. Vertical dashed: CO 2 critical temperature. Horizontal dashed line: parameter value obtained after considering all the data for T > T c (CO 2 ). Figure4. 21 binary parameter as a function of temperature. Vertical dashed: CO 2 critical temperature. Horizontal dashed line: parameter value obtained after considering all the data for T > T c (CO 2 ). Figure5. k 12 binary parameter as a function of temperature. Vertical dashed line: CO 2 critical temperature. Horizontal dashed line: parameter value obtained after considering all the data for T > T c (CO 2 ).

1 Figure3. 12 Figure4. 21 Figure 5 .

 112215 Figure3. 12 binary parameter as a function of temperature. Vertical dashed: CO 2 critical temperature. Horizontal dashed line: parameter value obtained after considering all the data for T > T c (CO 2 ).

Figure 1 .

 1 Phase diagram (P-x-y) for CO 2 (1) + C 4 H 5 F 5 (2) mixture. Symbols: experimental work (△:283.19 K, ○:293.23 K, □: 303.32 K, ▲:317.73 K, •: 327.90 K, ■:337.81 K); Black lines: PR EoS; dashed line: critical locus. Figure 2. Plot of relative volatility (α 12 ) against mole fraction for for CO 2 (1) + C 4 H 5 F 5 (2) mixture. Symbols: values obtained from experimental work (△:283.19 K, ○:293.23 K, □: 303.32 K, ▲:317.73 K, •: 327.90 K, ■:337.81 K); Black lines: values obtained from PR EoS and are decreasing function of temperature.

Table 4 .

 4 Deviations BIAS, AADU, ARDU and MRDU (maximum RDU) obtained in fitting experimental VLE data with PR EoS, Mathias-Copeman alpha function and WS mixing rules involving NRTL model for the CO 2 (1) + C 4 H 5 F 5 (2) binary mixture.

Table 5 .

 5 Model parameters regressed for the PR EoS with the Mathias-Copeman alpha function and the WS mixing rules incorporating the NRTL activity coefficient model for the CO 2 (1) + C 4 H 5 F 5 (2) binary mixture.

Table 6 .

 6 Determination of critical points using power laws with asymptotic behavior for the CO 2 (1) + C 4 H 5 F 5 (2) binary mixture.

Table 1 .

 1 Critical properties and acentric factors for each pure component4 .

	Compound	T C /K	P C / MPa	ω
	CO 2	304.2	7.377	0.22394
	C 4 H 5 F 5	460	3.266	0.38

Table 2 .

 2 Mathias-Copeman coefficients for the PR EoS.

	Coefficients	CO 2 4	C 4 H 5 F 5	14
	c 1	0.704	1.014
	c 2	-0.314	-1.214
	c 3	1.890	3.749

Table 3 .

 3 Vapor-liquid equilibrium pressure and phase compositions for the CO 2 (1) + C 4 H 5 F 5 (2) binary mixture. composition: y for vapor phase and x for liquid phase, n: samples number). U(T)= 0.04 K, U(P)=0.004 MPa., U(x), U(y) (U: expanded uncertainties for a 0.95 confidence level (k=2)).

			T/K = 283.19
	P/MPa	x 1	y 1	U (x 1 )	U(y 1 )	nx 1 ; ny 1
	0.7067	0.22	0.966	1E-02	2E-03	4;5
	1.5355	0.439	0.985	2E-02	1E-03	5;5
	2.2290	0.61	0.9915	2E-02	6E-04	4;4
	2.8605	0.75	0.9934	2E-02	5E-04	4;4
	3.2524	0.82	0.9953	1E-02	4E-04	4;4
	3.4038	0.853	0.9958	9E-03	4E-04	5;4
	3.6726	0.898	0.9967	7E-03	3E-04	5;4
	3.9840	0.946	0.9972	4E-03	5E-04	5;4
	4.2939	0.984	0.9991	1E-03	1E-04	4;5
			T/K = 293.23
	P/MPa	x 1	y 1	U (x 1 )	U(y 1 )	nx 1 ; ny 1
	0.7442	0.19	0.949	1E-02	3E-03	4;4
	1.5745	0.38	0.976	2E-02	2E-03	5;4
	2.1774	0.50	0.983	2E-02	1E-03	6;4
	2.8301	0.62	0.9868	2E-02	9E-04	5;5
	3.5419	0.75	0.9912	1E-02	7E-04	5;4
	4.2497	0.853	0.9922	1E-02	6E-04	5;5
	4.9454	0.944	0.9948	4E-03	4E-04	6;5
	5.1753	0.962	0.9967	3E-03	4E-04	5;4
			T/K = 303.32
	P/MPa	x 1	y 1	U (x 1 )	U(y 1 )	nx 1 ; ny 1
	0.7263	0.146	0.916	9E-03	5E-03	5;4
	1.1478	0.23	0.947	1E-02	4E-03	6;6
	1.8411	0.37	0.968	2E-02	2E-03	6;6
	2.4958	0.48	0.975	2E-02	2E-03	6;5
	3.3739	0.61	0.979	2E-02	2E-03	4;5
	3.9206	0.69	0.984	2E-02	1E-03	4;4
	4.8458	0.812	0.987	1E-02	1E-03	4;5
	5.4825	0.883	0.99	7E-03	7E-04	5;6
	5.6522	0.901	0.989	6E-03	1E-03	4;4
	6.3139	0.957	0.992	3E-03	6E-04	5;4

Table 4 .

 4 Deviations BIAS, AADU, ARDU and MRDU (maximum RDU) obtained in fitting experimental VLE data with PR EoS, Mathias-Copeman alpha function and WS mixing rules involving NRTL model for the CO 2 (1) + C 4 H 5 F 5 (2) binary mixture.

			T/K = 317.70
	P/MPa	x 1	y 1	U (x 1 )	U(y 1 )	nx 1 ; ny 1
	0.7021	0.112	0.846	7E-03	9E-03	5;4
	1.5377	0.25	0.929	1E-02	5E-03	4;4
	2.3358	0.37	0.953	2E-02	3E-03	4;6
	3.4718	0.51	0.966	2E-02	2E-03	6;6
	4.6955	0.66	0.971	2E-02	3E-03	5;5
	5.7744	0.77	0.976	1E-02	2E-03	5;5
	6.5165	0.83	0.975	1E-02	4E-03	4;6
	7.0504	0.885	0.979	7E-03	7E-03	5;7
	7.4811	0.911	0.979	6E-03	1E-03	4;4
	7.8999	0.939	0.978	4E-03	2E-03	5;5
			T/K = 337.81
	P/MPa	x 1	y 1	U (x 1 )	U(y 1 )	nx 1 ; ny 1
	0.6602	0.062	0.65	4E-03	2E-02	4;5
	1.1754	0.136	0.81	8E-03	1E-02	6;8
	2.1828	0.26	0.895	1E-02	7E-03	7;5
	2.8915	0.34	0.916	2E-02	5E-03	4;5
	3.4900	0.41	0.928	2E-02	6E-03	5;6
	3.9530	0.46	0.935	2E-02	4E-03	5;5
	4.8282	0.54	0.943	2E-02	4E-03	6;6
	5.6248	0.61	0.944	2E-02	5E-03	6;6
	6.1989	0.66	0.946	2E-02	6E-03	6;6
	7.002	0.71	0.948	1E-02	4E-03	6;7
	7.8516	0.78	0.945	1E-02	8E-03	5;8
	8.5800	0.84	0.945	1E-02	4E-03	7;5
	9.0945	0.878	0.935	7E-03	4E-03	5;6

Table 5 .

 5 Model parameters regressed for the PR EoS with the Mathias-Copeman alpha function and the WS mixing rules incorporating the NRTL activity coefficient model for the CO 2 (1) + C 4 H 5 F 5 (2) binary mixture.

	T/K	τ 12 /J.mol -1 τ 21 /J.mol -1	k 12
	283.19	10278	-3666.4	0.2263
	293.23	-1426.4	1310.4	0.3627
	303.32	-1051.8	994.64	0.3628
	317.73	-1498.4	1379.8	0.3591
	327.90	2106.9	-2355.1	0.3919
	337.81	3358.9	-2822.8	0.4027
	T>T c CO 2 * with no T dependency	-1630.8	1383.9	0.3863
	*T c CO 2 : critical temperature of CO 2 .

Table 6 .

 6 Determination of critical points using power laws with asymptotic behavior for the CO 2 (1) + C 4 H 5 F 5(2) binary mixture. Phase diagram (P-x-y) for CO 2 (1) + C 4 H 5 F 5 (2) mixture. Symbols: experimental work (△:283.19 K, ○:293.23 K, □: 303.32 K, ▲:317.73 K, •: 327.90 K, ■:337.81 K); Black lines: PR EoS; dashed line: critical locus. Figure 2. Plot of relative volatility (α 12 ) against mole fraction for for CO 2 (1) + C 4 H 5 F 5 (2) mixture. Symbols: values obtained from experimental work (△:283.19 K, ○:293.23 K, □: 303.32 K, ▲:317.73 K, •: 327.90 K, ■:337.81 K); Black lines: values obtained from PR EoS and are decreasing function of temperature.

	T/K	P C /MPa	x 1C
	317.70	8.3647	0.9795
	327.90	9.0977	0.9629
	337.81	9.3703	0.9169