Marco Campestrini
email: marco.campestrini@mines-paristech.fr

Francesca Rabino

David Marques

Dyhia Atig

Paolo Stringari

Guido Franzoni

Barbara Picutti

Experimental investigation of the effect of nitrogen on the phase equilibrium behavior of the CH4-CO2 mixture at low temperature for natural and biogas purification

Keywords: Natural gas, biogas, nitrogen, low temperature, phase equilibrium

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Natural gas is expected to play an important two-fold role in the energy transition: supporting coal-switching in fast developing countries and providing a source of low-carbon power when combined with CCUS, either as a direct energy source for power and industrial sectors or to produce blue hydrogen [1]. In addition, biogas and biomethane are going to largely contribute to the low-carbon economy, arriving at covering 20% of current worldwide gas demand at full utilization of the sustainable potential [START_REF]Outlook for biogas and biomethane: Prospects for organic growth[END_REF].

Natural gas is a mixture of hydrocarbons and non-hydrocarbon species, including acid gases (such as CO2 and H2S), water and inert gases (e.g., nitrogen), whose contents vary depending on the characteristics of gas fields [START_REF] De Guido | Cryogenic nitrogen rejection schemes: analysis of their tolerance to CO2[END_REF]. CO2, N2 and water are also the main contaminants of raw biogas [START_REF] Rafiee | Biogas as an energy vector[END_REF], which is basically composed of methane (55-70%), CO2 (30-45%), and nitrogen (up to 15%), with small amounts of oxygen, ammonia, and hydrogen sulfide [START_REF] Awe | A Review of biogas utilisation, purification and upgrading technologies[END_REF].

In both natural gas and biogas, contaminants must be removed before any eventual use. Among the existing purification technologies, cryogenic processes have drawn attention for gas treatment and biogas upgrading applications, especially if coupled with the production of LNG or LBM (liquid biomethane) and with other cryogenic processes (such as Nitrogen Rejection), thanks to the possibility of minimizing energy consumption through the integration of cryogenic purification and gas liquefaction [START_REF] Berstad | Low-temperature CO2 removal from natural gas[END_REF][START_REF] Maqsood | Cryogenic carbon dioxide separation from natural gas: a review based on conventional and novel emerging technologies[END_REF].

At the low operative temperatures of cryogenic purification processes, CO2 solidification can occur, leading to plugging, equipment blockage, ruptures and failures, turning into higher safety risks and plant reduced availability. Accurate and reliable predictions of the thermodynamic behavior of gas mixtures are essential to allow effective and safe design of cryogenic systems in which CO2 freezing conditions are to be avoided. Despite this strong constraint, only few commercial process simulators, widely used for cryogenic units' process design, are able to make solid-fluid equilibria calculations. To overcome this limitation, some simulators include built-in packages which, however, are reported to inaccurately predict solid CO2 formation [START_REF] Eggeman | Beware the pitfalls of CO2 freezing prediction[END_REF][START_REF] Eggeman | Pitfalls of CO2 freezing prediction[END_REF]. Moreover, given binary interaction parameters (kij) may not be adequate to describe the phase equilibria existing in the pressure-temperature range to be simulated [START_REF] Zarenezhad | Prediction of CO2 freezing points for the mixtures of CO2-CH4 at cryogenic conditions of NGL extraction plants[END_REF].

Consequently, the accuracy of thermodynamic models commonly embedded in process simulators in properly representing the thermodynamic behavior of gaseous mixtures containing N2, CH4 and CO2 at the desired conditions, needs to be improved.

In this work, original Vapor-Liquid Equilibrium (VLE), Solid-Liquid Equilibrium (SLE), Solid-Vapor Equilibrium (SVE), and Solid-Liquid-Vapor Equilibrium (SLVE) conditions have been measured for the CH4 + CO2 mixture and two N2 + CH4 + CO2 mixtures. Measurements focused on the description of the phase equilibrium behavior of these mixtures in the low temperature (between 170 K and 210 K) -high pressure (close to the critical-point pressure of methane) region of interest for cryogenic processes. Original measurements have then been compared with modeling results obtained by means of a thermodynamic package given by the Peng-Robinson Equation of State [START_REF] Peng | A new two constant equation of state[END_REF] (PR EoS), as implemented in Aspen HYSYS ® [START_REF]Aspen Plus V10[END_REF], coupled with the classical approach for calculating the fugacity of pure carbon dioxide in the solid state [START_REF] Prausnitz | Molecular thermodynamics of fluidphase equilibria[END_REF].

Modeling results show that, for the natural gas mixtures object of the present investigation, the selected thermodynamic package is inaccurate to predict SLE conditions. The regression of the binary interaction parameters (in particular for the CH4 + CO2 system), could lead to the improvement of the representation of SLE data, keeping at the same time an appropriate representation of the other phase equilibria.

Available equilibrium data for the N2 + CH4 + CO2 system

The first experimental works appeared in the open literature dealing with the phase equilibrium behavior of the N2 + CH4 + CO2 system date back to the beginning of the '70s. VLE data were published by Sarashina et al. [START_REF] Sarashina | Vapor-liquid equilibria for nitrogen-methanecarbon dioxide system[END_REF], Somait and Kidnay [START_REF] Somait | Liquid-vapor equilibriums at 270.00 K for systems containing nitrogen, methane, and carbon dioxide[END_REF], Al-Sahhaf et al. [START_REF] Al-Sahhaf | Liquid + vapor equilibriums in the nitrogen + carbon dioxide + methane system[END_REF],

Trappehl and Knapp [START_REF] Trappehl | Vapour-liquid equilibria in the ternary mixture N2-CH4-CO2 and the quaternary mixture N2-CH4-C2H6-C3H8[END_REF], Al-Sahhaf [START_REF] Al-Sahhaf | Vapor-liquid equilibria for the ternary system N2 + CO2 + CH4 at 230 and 250 K[END_REF], Xu et al. [START_REF] Xu | High pressure vapor liquid equilibria at 293 K for systems containing nitrogen, methane and carbon dioxide[END_REF], and Xu et al. [START_REF] Xu | Vapor Liquid Equilibria for N2-CH4-CO2 system near critical region[END_REF] between 1971 and 1992, and more recently by Ottovy et al. for CO2 capture and storage applications (VLE of carbon dioxide-rich mixtures) [START_REF] Ottøy | Thermodynamics of the carbon dioxide plus nitrogen plus methane (CO2 + N2 +CH4) system: measurements of vapor-liquid equilibrium data at temperatures from 223 to 298 K and verification of EOS-CG-2019 equation of state[END_REF]. SVE data were published in 1972 by Haufe et al. (for nitrogen-rich mixtures) [START_REF] Haufe | Solubility of solid carbon dioxide in a methane-nitrogen mixture[END_REF], in 1974 by Agrawal and Laverman [START_REF] Agrawal | Phase behavior of the methane-carbon dioxide system in the solid-vapor region[END_REF], and more recently by Le and Trebble [START_REF] Le | Measurement of carbon dioxide freezing in mixtures of methane, ethane, and nitrogen in the solid-vapor equilibrium region[END_REF], and Xiong et al. [START_REF] Xiong | Measurement and calculation of CO2 frost points in CH4 + CO2/CH4 + CO2 + N2/CH4 + CO2 + C2H6 mixtures at low temperatures[END_REF].

Shen et al. [START_REF] Shen | Determination of CO2 solubility in saturated liquid CH4 + N2 and CH4 + C2H6 mixtures above atmospheric pressure[END_REF] and Gao et al. [START_REF] Gao | Experimental determination of CO2 solubility in liquid CH4/N2 mixtures at cryogenic temperatures[END_REF] reported the composition of the liquid phase of the mixture at SLVE conditions in 2012, whereas no information about the vapor composition at the SLVE of the mixture was available before Riva and Stringari published some points between 124 K and 146 K in 2018 [START_REF] Riva | Experimental study of the influence of nitrogen and oxygen on the solubility of solid carbon dioxide in liquid and vapor methane at low temperature[END_REF].

All the experimental pressures and temperatures of the aforementioned VLE, SVE, and SLVE data of the ternary system are illustrated in the pressure-temperature diagram shown in Fig. 1; some additional details (like form of the dataset, pressure and temperature ranges, and number of experimental points for each literature work) are gathered in Table 1.

In Fig. 1, symbols represent the literature values: filled circles are related to all the available VLE data [START_REF] Sarashina | Vapor-liquid equilibria for nitrogen-methanecarbon dioxide system[END_REF][START_REF] Somait | Liquid-vapor equilibriums at 270.00 K for systems containing nitrogen, methane, and carbon dioxide[END_REF][START_REF] Al-Sahhaf | Liquid + vapor equilibriums in the nitrogen + carbon dioxide + methane system[END_REF][START_REF] Trappehl | Vapour-liquid equilibria in the ternary mixture N2-CH4-CO2 and the quaternary mixture N2-CH4-C2H6-C3H8[END_REF][START_REF] Al-Sahhaf | Vapor-liquid equilibria for the ternary system N2 + CO2 + CH4 at 230 and 250 K[END_REF][START_REF] Xu | High pressure vapor liquid equilibria at 293 K for systems containing nitrogen, methane and carbon dioxide[END_REF][START_REF] Xu | Vapor Liquid Equilibria for N2-CH4-CO2 system near critical region[END_REF][START_REF] Ottøy | Thermodynamics of the carbon dioxide plus nitrogen plus methane (CO2 + N2 +CH4) system: measurements of vapor-liquid equilibrium data at temperatures from 223 to 298 K and verification of EOS-CG-2019 equation of state[END_REF], empty squares are related to all the available SVE data [START_REF] Haufe | Solubility of solid carbon dioxide in a methane-nitrogen mixture[END_REF][START_REF] Agrawal | Phase behavior of the methane-carbon dioxide system in the solid-vapor region[END_REF][START_REF] Le | Measurement of carbon dioxide freezing in mixtures of methane, ethane, and nitrogen in the solid-vapor equilibrium region[END_REF][START_REF] Xiong | Measurement and calculation of CO2 frost points in CH4 + CO2/CH4 + CO2 + N2/CH4 + CO2 + C2H6 mixtures at low temperatures[END_REF], whereas filled triangles are related to all the available SLVE data [START_REF] Shen | Determination of CO2 solubility in saturated liquid CH4 + N2 and CH4 + C2H6 mixtures above atmospheric pressure[END_REF][START_REF] Gao | Experimental determination of CO2 solubility in liquid CH4/N2 mixtures at cryogenic temperatures[END_REF][START_REF] Riva | Experimental study of the influence of nitrogen and oxygen on the solubility of solid carbon dioxide in liquid and vapor methane at low temperature[END_REF].

Furthermore, the saturation line of nitrogen (dotted line), the saturation and melting lines of methane (bold lines), and the sublimation, melting, and saturation lines of carbon dioxide (dashed lines) calculated from [START_REF] Lemmon | NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties-REFPROP[END_REF] have been drawn in Fig. 1 for the reader convenience.

Fig. 1.

Pressure-temperature conditions of the phase equilibrium values available in the literature for the N2 + CH4 + CO2 system. • : VLE data; □ : SVE data; ▲ : SLVE data. ••• : VLE line of N2 [START_REF] Lemmon | NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties-REFPROP[END_REF]; -: SLE and VLE lines of CH4 [START_REF] Lemmon | NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties-REFPROP[END_REF]; --: SVE, SLE, and VLE lines of CO2 [START_REF] Lemmon | NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties-REFPROP[END_REF].

According to the reciprocal position of the available literature data and the boundary equilibrium conditions (saturation, melting, and sublimation) of the three pure components, it is possible to state that (i) VLE data have been measured only at temperatures higher than the triple-point temperature of carbon dioxide, (ii) almost all the SVE data are delimited by the saturation line of methane (VLE CH4) and the melting (SLE CO2) and sublimation (SVE CO2) lines of carbon In addition to that, the majority of the data involving solid carbon dioxide (SVE and SLVE data)

have been measured at pressure lower than 2 MPa. As a consequence, in this work authors have focused on the phase equilibrium behavior of the ternary system at a nominal pressure of 4 MPa between 170 K and 210 K. Contrarily to the ternary system, the abundance of literature data allows the evaluation of the phase equilibrium behaviors of the corresponding binary mixtures in the same temperature range (170 K < T < 210 K): the N2 + CH4 system is characterized only by VLE and only the SVE exists for the N2 + CO2 system, whereas the CH4 + CO2 system behaves at VLE, SVE, SLE, or SLVE according to system pressure, temperature, and global composition [START_REF] Riva | Biomethane upgrading process: thermodynamic study of solid-liquid-vapor equilibrium for methane rich mixtures[END_REF].

Materials, apparatus, and protocol

This work dealt with the experimental investigation of the fluid-fluid and solid-fluid equilibrium behavior of the nitrogen + methane + carbon dioxide system between 170 K and 210 K. The solubility of carbon dioxide in the fluid phases at VLE, SVE, and SLE has been measured in three different nitrogen + methane mixtures (hereafter called solvents). The nominal molar compositions of nitrogen and methane in these 3 solvents are gathered in Table 2. The last two columns of Table 2 give the compositions of nitrogen and methane in the gas cylinders (supplied by MESSER) that have been used in the experimental campaigns. for the CH4 + CO2 system with literature data in the same temperature and pressure ranges.

Solvents B and C have been then chosen for evaluating the effect of nitrogen on the phase equilibrium behavior with particular attention to the solubility limits of carbon dioxide in the liquid and vapor phases.

The details concerning the pure chemicals (suppliers, CAS numbers and stated purities) are gathered in Table 3. The equipment used for the evaluation of the phase equilibrium behavior is based on a ″static-analytic″ method with liquid and vapor phase samplings. The equipment is similar to that presented by Campestrini et al. [START_REF] Valtz | Experimental measurements and modelling of vapour-liquid equilibrium of 2,3,3,3-tetrafluoropropene (R-1234yf) + 1,1,1,2,2pentafluoropropane ([END_REF] and its flow diagram is presented in Fig. 2.

The sapphire equilibrium cell is totally immersed into an ethanol bath whose temperature is The equilibrium cell and its loading lines are first evacuated at ambient temperature. The cell is then immersed into the thermo-regulated bath. The temperature of the cell is set to a target value (typically 18°C) by means of the temperature regulator, and adequate stirring is maintained throughout the cell.

A certain pressure of carbon dioxide is then loaded in the cell by opening all the valves between the CO2 cylinder (CO2 in Fig. 2) and the equilibrium cell. Some solvent is then introduced stepby-step into the cell from the corresponding cylinder, S in Fig. 2, during cooling of the cell until reaching the desired pressure (about 4 MPa) at the first target temperature.

At each target temperature, phase equilibrium is assumed to be achieved when the readings of pressure transducers and temperature probes have stabilized to within their instrument uncertainty for at least 10 minutes.

Once equilibrium is reached, sampling of the equilibrium phases starts while temperature and pressure readings are recorded. The liquid and vapor ROLSI ® samplers are used for sampling (i) the liquid phase and vapor phase, respectively, at VLE or SLVE, (ii) the vapor phase at SVE, (iii)

the liquid phase at SLE. The samples are then transferred to the GC and analyzed.

The phase equilibrium is considered properly defined once at least six repeatable molar compositions are obtained for both the liquid and the vapor phases.

The temperature of the cell is then successively decreased and further solvent is loaded in the equilibrium cell for increasing the pressure up to the target value. The procedure is then repeated until the target phase equilibrium behavior has been satisfactorily determined or until the lowest experimental temperature has been reached (about 170 K).

Results and discussion

In this section, tables gather the experimental values measured in this work according to the system and phase equilibrium. In the tables, temperature is given in K, pressure is given in MPa, compositions of the equilibrium phases are expressed in molar fractions (x is used for the liquid phase, y is used for the vapor phase), expanded and combined standard uncertainties (U) are given for molar fractions, and the experimental points are numbered consecutively. Expanded uncertainty on temperature and pressure is estimated to be 0.004 K and 0.002 MPa, respectively.

The approach adopted for estimating the experimental uncertainties is based on the NIST documentation by Taylor and Kuyatt [START_REF] Taylor | Guidelines for evaluating and expressing the uncertainty of NIST measurement results[END_REF]. Readers can refer to the Supporting Material for details related to the calculation of the uncertainties affecting the experimental results presented in this work.

In this section, figures illustrate the evolution of the carbon dioxide composition in the vapor and liquid phases with temperature at the measured VLE, SVE, and SLE conditions. For graphical purposes, qualitative trend lines have been added in all the figures.

For the solvent A (methane) + carbon dioxide system:

• the measured SVE are gathered in Table 4 and compared to some of the literature SVE values measured by Pikaar [START_REF] Pikaar | A study of phase equilibria in hydrocarbon-CO2 system[END_REF] in Fig. 3;

• the measured VLE are gathered in Table 5 and compared to some of the literature VLE values measured by Neumann and Walch [START_REF] Neumann | Vapor-liquid equilibrium for CO2/CH4 in the low temperature region and low CO2-content[END_REF] in Figs. 4 and5;

• the measured SLE are gathered in Table 6 and compared to the literature SLE and SLVE data [START_REF] Donnelly | Phase equilibria in the carbon dioxide -methane system[END_REF][START_REF] Brewer | Freezing points of binary mixtures of methane[END_REF][START_REF] Cheung | Solubility of carbon dioxide and hydrogen sulfide in liquid hydrocarbons at cryogenic temperatures[END_REF][38][39][40][START_REF] Streich | N2 Removal from natural gas[END_REF][START_REF] Davis | Solid-liquid-vapor phase behavior of the methane-carbon dioxide system[END_REF][START_REF] Im | Phase equilibrium of carbon dioxide and light paraffins in presence of solid carbon dioxide[END_REF][START_REF] Sterner | Phase equilibria in the CO2-methane systems[END_REF][START_REF] Langé | Purification of natural gas by means of a new low temperature distillation process[END_REF][START_REF] Sparks | Gas Processors Association Research Report[END_REF] in Fig. 6.

For the methane + carbon dioxide system, a good qualitative agreement is encountered between all the measured VLE, SVE, and SLE data and the selected literature data.

In the low temperature region (188 K < T < 193 K), the measured solubility of solid CO2 in vapor methane is slightly lower than the selected values of Pikaar [START_REF] Pikaar | A study of phase equilibria in hydrocarbon-CO2 system[END_REF] (see Fig. 3). This deviation could be explained by the slight difference between the experimental pressures (the average SVE pressure of the selected values measured by Pikaar [START_REF] Pikaar | A study of phase equilibria in hydrocarbon-CO2 system[END_REF] is 3.970 MPa, whereas the average of the pressure given in Table 4 is 4.019 MPa), but this is in contrast with the fact that the solubility is usually expected to increase with increasing pressures.

Fig. 3.

Experimental solubilities of solid CO2 in the vapor phase at SVE at 4.019 MPa (average value of points n°1-12) and selected literature data at 3.970 MPa (average value) for the solvent A + CO2 system.

• : this work; □ : Pikaar [START_REF] Pikaar | A study of phase equilibria in hydrocarbon-CO2 system[END_REF]; -: trend line.

In Fig. 4, measured VLE compositions of points n°15, n°22, n°23, and n°25 (4.000 MPa average pressure) are compared to a literature value at 4.001 MPa [START_REF] Neumann | Vapor-liquid equilibrium for CO2/CH4 in the low temperature region and low CO2-content[END_REF]. It should be noticed that the filled triangle in Fig. 4 is at the saturation temperature of methane at (4.008 MPa ± 0.004 MPa) measured in this work (186.121 K ± 0.009 K). Similarly in Fig. 5, measured VLE compositions of points n°16, n°21, and n°24 (3.978 MPa average pressure) are compared to a literature value at 3.982

MPa [START_REF] Neumann | Vapor-liquid equilibrium for CO2/CH4 in the low temperature region and low CO2-content[END_REF]. According to these comparisons, it is possible to state that part of the measured VLE data given in Table 5 are in a rather good agreement with the data of Neumann and Walch [START_REF] Neumann | Vapor-liquid equilibrium for CO2/CH4 in the low temperature region and low CO2-content[END_REF]. Measured solubility of carbon dioxide in liquid methane is compared to the solubility of carbon dioxide at SLE and SLVE conditions from the literature [START_REF] Donnelly | Phase equilibria in the carbon dioxide -methane system[END_REF][START_REF] Brewer | Freezing points of binary mixtures of methane[END_REF][START_REF] Cheung | Solubility of carbon dioxide and hydrogen sulfide in liquid hydrocarbons at cryogenic temperatures[END_REF][38][39][40][START_REF] Streich | N2 Removal from natural gas[END_REF][START_REF] Davis | Solid-liquid-vapor phase behavior of the methane-carbon dioxide system[END_REF][START_REF] Im | Phase equilibrium of carbon dioxide and light paraffins in presence of solid carbon dioxide[END_REF][START_REF] Sterner | Phase equilibria in the CO2-methane systems[END_REF][START_REF] Langé | Purification of natural gas by means of a new low temperature distillation process[END_REF][START_REF] Sparks | Gas Processors Association Research Report[END_REF] in the range 170 K < T < 215 K in Fig. 6. For the solvents B and C involving nitrogen, all the VLE and SLVE have been grouped together in Table 11 and 12, respectively. In Tables 11 and12, the molar fraction of nitrogen in the liquid and vapor phase can be calculated as x1 = 1 -x2 -x3 and y1 =1 -y2 -y3, respectively.

As shown in Figs. 7 and8, a first effect of adding nitrogen in the solvent is decreasing the solubility of carbon dioxide in the fluid phase at SVE and SLE. In particular, this effect on the solubility of CO2 in the vapor phase at SVE appears lower than the one in the liquid at SLE. In the temperature range considered in this study, the second effect related to the addition of nitrogen to the solvent is the modification of the temperature ranges interested by the SVE, SLVE, and SLE, as shown also in the modeling section.

Indeed, increasing N2-content in the solvent (nitrogen + methane mixture) makes the SVE exist at lower temperatures with respect to the methane + carbon dioxide system. To illustrate, Fig. 7 shows that the SVE has been measured down to 190.406 K for the solvent A (point n°12 in Table 4), down to 188.509 K for the solvent B (point n°39 in Table 7), and down to 184.521 K for the solvent C (point n°57 in Table 9).

The measurement of the SVE at lower temperatures passing from solvent A to solvent C has been possible seeing that the temperature range interested by the SLVE occurs at lower T / K ρ SOLV / mol×dm -3 L V temperatures for increasing N2-content in the solvent. As a result, the addition of nitrogen makes the SLE appear at lower temperatures for the solvents B and C with respect to the methane + carbon dioxide system. To illustrate, Fig. 8 shows that the SLE has been measured starting at 189.643 K for solvent A (point n°26 in Table 6), 182.565 K for solvent B (point n°40 in Table 8), and 178.765 K for solvent C (point n°58 in Table 10).

Comparison with modeling results

The thermodynamic package used in this work has been obtained coupling a common fluid phase model and the well-known classical approach for solid carbon dioxide; the isofugacity condition at SVE, SLE, or SLVE has been solved under the assumption that the solid phase is made by the pure solid former, as deeply illustrated by Campestrini et al [START_REF] Campestrini | Solubilities of solid n-alkanes in methane: data analysis and models assessment[END_REF].

The Peng-Robinson (PR) EoS [START_REF] Peng | A new two constant equation of state[END_REF] as implemented in Aspen HYSYS ® V10 [START_REF]Aspen Plus V10[END_REF] has been selected for calculating the partial molar fugacity of N2, CH4, and CO2 in the liquid and vapor phases. This fluid phase model involves the acentric factor-dependent alpha function [START_REF] Peng | A new two constant equation of state[END_REF] and the Van der Waals one-fluid mixing rules for the calculation of the well-known parameters a and b of cubic EoSs. As a consequence, the only parameters needed for the application of the PR EoS were the critical pressure (PC), the critical temperature (TC), and acentric factor (ω) of the pure fluids of interest in this work (nitrogen, methane, and carbon dioxide), and the binary interaction parameters (kij) of the corresponding binary mixtures (nitrogen + methane, nitrogen + carbon dioxide, and methane + carbon dioxide). For all these parameters, values have been taken from Aspen HYSYS ® V10 database and are given in Table 13.

When applied at temperatures lower than the triple-point temperature of a pure fluid i, the classical approach relates its fugacity in the solid phase ( , ) to its fugacity in the subcooled liquid state ( , ) and to its latent heat of melting (ΔHm = HL -HS), specific heat change upon melting (ΔCpm = CpL -CpS), and its molar volume change upon melting (Δvm = vL -vS) [START_REF] Prausnitz | Molecular thermodynamics of fluidphase equilibria[END_REF].

ln , , , , = ∆ , 1 - , + ∆ , -1 -ln , - ∆ -, 1 29
Except the system temperature T and pressure P, all the properties within Eq. ( 1) are triple point properties (TT,i and PT,i are the triple-point temperature and pressure of component i, respectively).

As a consequence, it should be remembered that constant values of ΔHm, ΔCpm, and Δvm have been adopted in the modeling work, and these values are equal to the values at the triple-point of the fluid. Table 14 summarizes the triple-point properties used in this work for carbon dioxide, corresponding to experimental values taken from the DIPPR database [START_REF]The DIPPR Information and Data Evaluation Manager for the Design Institute for Physical Properties[END_REF]. The solubilities of solid carbon dioxide in vapor and liquid solvent A tabulated in Tables 4 and6 are qualitatively compared to modeling results in Fig. 12, where the calculated VLE has been omitted. The calculated SLVE temperature is about 190.3 K.

3.9 The solubilities of solid carbon dioxide in vapor and liquid solvent B tabulated in Tables 7 and8 are qualitatively compared to modeling results in Fig. 13; calculated VLE and SLVE have been omitted, thus the SVE runs down to the upper SLVE temperature (about 187.3 K), whereas the SLE starts at the lower SLVE temperature (about 184.4 K). The solubilities of solid carbon dioxide in vapor and liquid solvent C tabulated in Tables 9 and10 are qualitatively compared to modeling results in Fig. 14; calculated VLE and SLVE have been omitted, thus the SVE runs down to the upper SLVE temperature (about 184.5 K), whereas the SLE starts at the lower SLVE temperature (about 178.9 K). In Figs. 12 -14, it is possible to observe that the agreement between experimental and calculated solubilities is better in the vapor phase than in the liquid phase. It follows that for a given composition of CO2, the SVE temperature calculated by the model is close to the experimental value, whereas the calculated SLE temperature is lower than the experimental value. As a consequence, the solid appearance temperature of a liquid mixture of known global composition is underestimated by the model. At the same time, at a given temperature, the model overestimates the solubility of CO2 with respect to the value that can be inferred from the experimental values more at SLE than at SVE. This deviation between modeling and experimental results can be related, at least for the Solvent A + carbon dioxide system to an improper value used for the binary interaction parameter of the CH4 + CO2 system, seeing that the experimental solubilities of solid carbon dioxide in liquid methane presented in this work are in a rather good quantitative agreement with the collected literature values, as pointed out in Fig. 6.

Some of the experimental SVE, SLE, VLE, and SLVE gathered in Tables 4 At temperatures higher than the saturation temperature of methane at 4.00 MPa (TSAT = 185.841 K [START_REF] Lemmon | NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties-REFPROP[END_REF]), the nitrogen + methane system is in the vapor/supercritical state, as in Fig. 15. Solid carbon dioxide can be at equilibrium only with this vapor/supercritical phase.

VLE

The only liquid phase that can appear in the ternary diagram at temperatures higher than TSAT develops from the right edge of the diagram, the one representing the phase equilibrium behavior of the methane + carbon dioxide system, as in Fig. 16. In Fig. 16, a SLVE and a VLE then appear due to the presence of this liquid phase, and solid carbon dioxide can be at equilibrium with a liquid phase, or a vapor phase richer in nitrogen than the liquid phase, or both (at SLVE), according to the global composition of the mixture.

As the temperature drops below TSAT, pure methane is stable as a liquid and the VLE and SLVE move from the right to the left of the ternary diagram towards compositions richer in nitrogen. A VLE exists then for the nitrogen + methane mixture, whereas the methane + carbon dioxide mixture is at SLE, as in Fig. 17. This repositioning of the SLVE triangle in the direction of nitrogen related to the change of position of the VLE of the nitrogen + methane mixture proceeds for decreasing temperatures as illustrated in Figs. 18 and19.

It is interesting to notice that cooling the system temperature from about 187 K (Fig. 16) down to about 177 K (Fig. 19) through TSAT entails not only the repositioning of the SLVE triangle from the right to the left of the ternary diagram, but also the thickening or expansion of the surface of the SLVE triangle. Remembering that, at a certain temperature and pressure, all the global compositions of the mixture falling within the SLVE triangle lead to the SLVE, the expansion of the surface of the SLVE triangle with decreasing temperatures means that a larger number of global compositions of the ternary mixture leads to SLVE at about 177 K (Fig. 19) than at 187 K (Fig. 16).

The experimental values tabulated in Tables 4 -12 have been quantitatively compared with modeling results. For all the 89 experimental points (28 VLE, 32 SVE, 17 SLE, and 12 SLVE), the thermodynamic package has been applied at constant temperature and pressure (equal to the experimental values) to calculate the equilibrium compositions successively compared with experimental values. Common statistical indexes (AAD, Bias, and MAD) have been used for the comparison; please refer to the Supporting Material for details related to the calculation of the statistical indexes.

A common FLASH calculation at given global composition of the mixture (CH4 + CO2 or N2 + CH4 + CO2) has then been carried out for all the VLE data.

A "FLASH" calculation has been carried out for the equilibria involving solid carbon dioxide.

For SVE (SLE) data, the CO2 composition of the vapor (liquid) phase at equilibrium with solid carbon dioxide rather than the global composition of the mixture has been given as input value and iterated until satisfying the isofugacity condition between the fugacity of pure carbon dioxide in the solid phase from Eq. (1) and its partial molar fugacity in the vapor (liquid) phase from the PR EoS. During the iteration, the compositions of nitrogen and methane in the fluid phase at equilibrium with solid carbon dioxide have been calculated assuming that the ratio between their molar fractions is equal to the concentrations of the three solvents A, B, and C.

For SLVE data, a common FLASH calculation has been performed at given global composition and the partial molar fugacity of carbon dioxide in the vapor phase (equal to that of the liquid) has been compared to the fugacity of solid carbon dioxide. The global composition has then been iterated until satisfying the SLVE condition.

The quantitative comparison between experimental and calculated VLE and SLVE data is gathered in Table 15. With respect to VLE data of the CH4 + CO2 system, deviations up to about 30% and 20% are encountered for the composition of carbon dioxide in the liquid and vapor phase, respectively, whereas the deviations related to methane are lower than 1% (see 2 nd column of Table 15). The high deviations for carbon dioxide at VLE are due to the low concentration of this component in the mixture when approaching the saturation temperature of methane; the lowest experimental CO2

The higher deviations between modeling and experimental results obtained for SLE data with respect to SVE data confirm the qualitative comparisons illustrated in Fig. 12 (solvent A + CO2 system), Fig. 13 (solvent B + CO2 system), and Fig. 14 (solvent C + CO2 system). As previously stated, this aspect can be related to the use of an unsuitable binary interaction parameter for the CH4 + CO2 system.

This conclusion can be supported by considering also the deviation between calculated and experimental compositions of the liquid and vapor phases at SLVE given in Table 15: deviations for the liquid phase are higher than those for the vapor phase. As illustrated by the reciprocal position of the experimental and calculated compositions of the liquid phase at SLE/SLVE shown in Figs. 17 -19, it clearly appears that the position of the vertex of the SLVE triangle corresponding to the liquid phase is highly dependent on the position of the liquidus line originating at the solubility limit of carbon dioxide in liquid methane in the binary CH4 + CO2 system.

To the contrary, the binary interaction parameter tabulated in Aspen HYSYS ® V10 for the N2 +

CH4 system appears to be suitable in the temperature range of interest seeing that, as illustrated in Figs. 17 -19, the VLE region develops starting from the horizontal edge of the ternary diagram that represents the equilibrium behavior of the N2 + CH4 mixture.

Finally, it seems that a suitable binary interaction parameter has been tabulated in Aspen HYSYS ® V10 for the N2 + CO2 system, even if its effect cannot be sufficiently evaluated owing to the low content of nitrogen in the mixture (about 10% for all the points within Tables 789101112).

Conclusions and perspectives

With respect to the N2 + CH4 + CO2 system, this work presents for the first time (i) VLE data at temperatures lower than the triple-point temperature of carbon dioxide, (ii) SVE data at temperatures lower than the saturation line of methane, and (iii) SLE data.

The experimental analysis of the thermodynamic behavior of three N2 + CH4 + CO2 mixtures reveals that two are the main effects related to the presence of nitrogen in the mixture:

1) the solubility of solid carbon dioxide in both the vapor and liquid phases is reduced. First of all, this can be related to the fact that solid CO2 is generally less soluble in nitrogen (approximately 1.4% at 180 K and 3.1% at 190 K at 4 MPa [START_REF] Sonntag | The solid-vapor equilibrium of carbon dioxide-nitrogen[END_REF]) than in methane.

Secondly, this decrease in the solubility of CO2 in methane due to the addition of nitrogen is more evident in the liquid phase than in the vapor phase, and this can be related to the largest reduction of the density for the liquid phase with respect to the vapor phase;

2) the region of the temperature.vs.CO2-composition diagram affected by the liquid phase is moved towards lower temperatures for increasing contents of nitrogen in the mixture. This is of course due to the fact that nitrogen is lighter than methane, which certainly implies higher costs for the liquefaction of natural or biogas rich in nitrogen.

Furthermore, increasing amounts of nitrogen widen the temperature region of existence of the solid-vapor equilibrium and spread the temperature range of existence of the solidliquid-vapor equilibrium.

As a result and considering values calculated with the model, for the solvent B studied in this work (5% N2 + 95% CH4), the temperature has to be decreased below about 187.3 K before the vapor-liquid equilibrium takes place (about 190.3 K in case of pure methane). At this temperature, carbon dioxide can solidify if its global composition in the mixture is higher than 3.3% (4.1% in case of pure methane).

Similarly, for the solvent C studied in this work (10% N2 + 90% CH4), the temperature has to be decreased below about 184.5 K (6 degrees below the case of pure methane) before the vapor-liquid equilibrium takes place. At this temperature, carbon dioxide can solidify if its global composition in the mixture is higher than 2.6% (1.5% less than in the case of pure methane).

The phase equilibria measured in this work for the methane + carbon dioxide system are in qualitative and quantitative agreement with some of the literature data. In particular, the measured solubility limits of CO2 at SLE agree with the majority of the literature SLE data in the whole investigated temperature range (175 K < T < 190 K).

At the same time and despite the quite good agreement between modeling results and VLE, SVE, and SLVE measurements, higher deviations have been encountered in the representation of measured SLE for the three systems by means of the thermodynamic package. Therefore, the regression of the binary interaction parameters (in particular for the CH4 + CO2 system), could lead to the improvement of the representation of SLE data (both from the literature and from this work) by keeping an appropriate representation of the other kinds of phase equilibrium.

In addition to the regression of new binary interaction parameters, in the next future authors will focus more deeply on the representation of solid-fluid equilibria by means of common process simulators. In particular, a further study will investigate Aspen HYSYS ® and other commercial process simulators deviations against literature values and experimental values measured in the present work. In perspective, the final aim of the investigation will be the enhancement of the

  (iii) the SLVE data have been measured only between the saturation line of nitrogen (VLE N2) and that of methane (VLE CH4).

  regulated at a certain value by means of a temperature regulator. Two platinum resistance thermometer Pt-100 probes and two pressure transducers (DRUCK type PTX611, range: 0-5 MPa, type PTX611, range: 0-20 MPa) allow temperature and pressure readings via a data acquisition unit (HP34970A). Temperature probes have been calibrated against a 25Ω reference platinum thermometer (Pt25, Hart Scientific), previously calibrated by the Trescal Laboratory following the ITS90 (1990 International Temperature Scale) protocol. Pressure transducers are maintained at constant temperature by means of a heating cartridge and have been calibrated against a Dead Weight Balance (Desgranges & Huot).Samples of the equilibrium phases within the cell are withdrawn thanks to two electromagnetic ROLSI ® samplers, vaporized in the heated chamber of the ROLSI ® samplers where they are mixed with the carrier gas (helium) into the transfer line. The samples are thus directly injected into the analytical circuit toward the head of the column (Porapak Q, 80/100 mesh; 1/8" Silcosteel tube, 2m length, 2mm internal diameter, Restek) located in the oven of the gas chromatograph (Perichrom Model PR2100), and finally analyzed by a Thermal Conductivity Detector (TCD) connected to an acquisition system (WINILAB III).

Fig. 2 .

 2 Fig. 2. Flow diagram of the static-analytic apparatus used in this work. CO2 : carbon dioxide cylinder; DAU : data acquisition unit; EC : equilibrium cell; GC : gas chromatograph; LB : liquid bath (ethanol); LS : liquid ROLSI ® sampler; LV: loading valve; MS : magnetic stirrer; P : Purge; PC : personal computer; PP : platinum resistance thermometer probe; PT : pressure transducer; S : gaseous solvent cylinder; SC : sample controller; SD : stirring device; ST : sapphire tube; TR : temperature regulator; Th : thermocouple; V : valve; VP : vacuum pump; VS : vapor ROLSI ® sampler.

Fig. 7 .Fig. 8 .Fig. 9 .

 789 Fig. 7. Effect of nitrogen on the solubility of solid CO2 in the vapor phase at SVE in the three solvent + CO2 systems. • : solvent A; □ : solvent B; ○ : solvent C. Continuous lines are trend lines.

Fig. 12 .

 12 Fig. 12. Calculated and experimental CO2 compositions at SVE and SLE for the solvent A + carbon dioxide system. Experimental values: vapor phase (□); liquid phase (○). Calculated values: SVE and SLE (-), SLVE (--).

Fig. 13 .

 13 Fig. 13. Calculated and experimental CO2 compositions at SVE and SLE for the solvent B + carbon dioxide system. Experimental values: vapor phase (□); liquid phase (○). Calculated values: SVE and SLE (-), SLVE (--).

Fig. 14 .

 14 Fig. 14. Calculated and experimental CO2 compositions at SVE and SLE for the solvent C + carbon dioxide system. Experimental values: vapor phase (□); liquid phase (○). Calculated values: SVE and SLE (-), SLVE (--).

36 Fig. 15

 3615 Fig. 15 presents the comparison between calculated and experimental values (points n°11 in

Fig. 19 .

 19 Fig. 19. Zoom in the CH4-rich region of the ternary diagram of the N2 + CH4 + CO2 system at average temperature (176.607 K) and pressure (3.976 MPa) of points n°31, n°44, and n°61. Experimental values: point n°31, solvent A (○); point n°44, solvent B (□); point n°61, solvent C (Δ). Solid lines are modeling results, dotted lines refer to N2/CH4 ratios of solvents B and C.

Table 1 .

 1 Literature phase equilibrium values for the (1)N2 + (2)CH4 +(3)CO2.

	Reference	Year	Kind of data	Form	N° data T range / K P range / MPa
	14	1971	VLE	PTxy	48	233 -	6.1 -10.1
	15	1978	VLE	PTxy	56	270	4.6 -11.2
	16	1983	VLE	PTxy	37	220 -	6.0 -12.2
	17	1989	VLE	PTxy	56	220	2.0 -12.0
	18	1990	VLE	PTxy	26	230 -	6.2 -10.3
	19	1992	VLE	PTxy	55	293	6.0 -8.3
	20	1992	VLE	PTxy	30	298	7.2 -8.0
	21	2020	VLE	PTx, PTy	62	223 -	0.8 -9.3
	22	1972	SVE	PTy	5	152 -	1.0 -4.0
	23	1974	SVE	PTy	19	154 -	0.2 -2.4
	24	2007	SVE	PTy	24	174 -	1.2 -2.3
	25	2015	SVE	PTy	77	153 -	0.2 -2.0
	26	2012	SLVE	PTx	27	112 -	0.2 -3.1
	27	2012	SLVE	PTx	31	83 -123	0.1 -2.0
	28	2018	SLVE	PTxy	3	124 -	0.5 -2.0

Table 2 .

 2 Molar fraction of nitrogen and methane in the solvents used in this work.

		Nominal composition	Cylinders' composition
	Solvents	xN2	xCH4	xN2	xCH4
		[%]	[%]	[%]	[%]
	Solvent A	0	100	0	99.995
	Solvent B	5	95	5.009	94.991
	Solvent C	10	90	10.2	89.8
	Solvent A (pure methane) has been chosen for the validation of the experimental protocol and
	the calibration of the experimental devices (temperature probes, pressure transducers, and detector
	of the gas-chromatograph), by comparing the VLE, SLE, and SVE values obtained in this work

Table 3 .

 3 Purities and suppliers of the pure chemicals used in this work.

	Chemical Name	CAS number	Purity [mol/mol %]	Source
	Nitrogen	7727-37-9	99.999	MESSER
	Methane	74-82-8	99.995	MESSER
	Carbon dioxide	124-38-9	99.998	MESSER

Table 4 .

 4 SVE data measured for the solvent A [(1) methane] + (2) carbon dioxide system.

	Point	T [K]	P [MPa]	y1	U(y1)	y2	U(y2)
	1	205.423	4.010	0.8944	0.0039	0.1056	0.0039
	2	202.433	4.010	0.9128	0.0033	0.0872	0.0033
	3	199.445	4.152	0.9273	0.0028	0.0727	0.0028
	4	196.461	4.009	0.9403	0.0023	0.0597	0.0023
	5	194.458	4.008	0.9474	0.0022	0.0526	0.0022
	6	193.497	4.004	0.9504	0.0021	0.0496	0.0021
	7	193.268	4.019	0.9514	0.0020	0.0486	0.0020
	8	191.532	4.005	0.9559	0.0018	0.0441	0.0018
	9	190.823	3.982	0.9581	0.0019	0.0419	0.0019
	10	190.555	4.011	0.9583	0.0015	0.0417	0.0015
	11	190.543	4.015	0.9581	0.0018	0.0419	0.0018
	12	190.406	4.007	0.9586	0.0015	0.0414	0.0015

U(T) = 0.004 K; U(P) = 0.002 MPa

Table 5 .

 5 VLE data measured for the solvent A [(1) methane] + (2) carbon dioxide system.

	Point	T [K]	P [MPa]	x1	U(x1)	x2	U(x2)	y1	U(y1)	y2	U(y2)
	13	190.293	3.994	0.9045	0.0033	0.0955	0.0033	0.9592	0.0014	0.0408	0.0014
	14	190.248	4.015	0.9092	0.0039	0.0908	0.0039	0.9605	0.0016	0.0395	0.0016
	15	189.758	4.004	0.9179	0.0026	0.0821	0.0026	0.9638	0.0012	0.0362	0.0012
	16	189.742	3.969	0.9106	0.0028	0.0894	0.0028	0.9618	0.0013	0.0382	0.0013
	17	189.742	4.031	0.9242	0.0024	0.0758	0.0024	0.9655	0.0012	0.0345	0.0012
	18	189.254	3.991	0.9274	0.0023	0.0726	0.0023	0.9673	0.0011	0.0327	0.0011
	19	188.857	3.992	0.9368	0.0020	0.0632	0.0020	0.9707	0.0010	0.0293	0.0010
	20	188.768	3.991	0.9418	0.0021	0.0582	0.0021	0.9725	0.0011	0.0275	0.0011
	21	188.355	3.977	0.9458	0.0018	0.0542	0.0018	0.9744	0.0009	0.0256	0.0009
	22	188.335	3.996	0.9502	0.0019	0.0498	0.0019	0.9759	0.0015	0.0241	0.0015
	23	187.553	4.000	0.9688	0.0012	0.0312	0.0012	0.9841	0.0006	0.0159	0.0006
	24	187.428	3.988	0.9691	0.0012	0.0309	0.0012	0.9842	0.0007	0.0158	0.0007
	25	186.936	3.999	0.9822	0.0007	0.0178	0.0007	0.9905	0.0004	0.0095	0.0004

U(T) = 0.004 K; U(P) = 0.002 MPa Fig.

4

. Experimental compositions of CO2 in the liquid and vapor phases at VLE at 4.000 MPa (average value of points n°15, n°22, n°23, and n°25) and a selected literature data at 4.001 MPa for the solvent A + CO2 system.

•,▲ : this work; □ : Neumann and Walch

[START_REF] Neumann | Vapor-liquid equilibrium for CO2/CH4 in the low temperature region and low CO2-content[END_REF]

; -: trend lines.

V L VLE Fig.

5

. Experimental compositions of CO2 in the liquid and vapor phases at VLE at 3.978 MPa (average value of points n°16, n°21, and n°24) and a selected literature data at 3.982 MPa for the solvent A + CO2 system. • : this work; □ : Neumann and Walch

[START_REF] Neumann | Vapor-liquid equilibrium for CO2/CH4 in the low temperature region and low CO2-content[END_REF]

; -: trend lines.

Table 6 .

 6 SLE data measured for the solvent A [(1) methane] + (2) carbon dioxide system.

	Point	T [K]	P [MPa]	x1	U(x1)	x2	U(x2)
	26	189.643	3.969	0.9059	0.0040	0.0941	0.0040
	27	188.756	3.996	0.9103	0.0035	0.0897	0.0035
	28	185.513	3.984	0.9244	0.0036	0.0756	0.0036
	29	182.553	3.993	0.9360	0.0024	0.0640	0.0024
	30	179.590	4.006	0.9467	0.0020	0.0533	0.0020
	31	176.612	3.931	0.9551	0.0017	0.0449	0.0017
	U(T) = 0.004 K; U(P) = 0.002 MPa

V L VLE Fig.

6

. Experimental solubilities of solid CO2 in the liquid phase at SLE at 3.980 MPa (average value of points n°26-31) and selected literature SLE and SLVE data in the range 170 K < T < 215 K for the solvent A + CO2 system. • : this work; □ : SLE and SLVE data

[START_REF] Donnelly | Phase equilibria in the carbon dioxide -methane system[END_REF][START_REF] Brewer | Freezing points of binary mixtures of methane[END_REF][START_REF] Cheung | Solubility of carbon dioxide and hydrogen sulfide in liquid hydrocarbons at cryogenic temperatures[END_REF][38][39][40][START_REF] Streich | N2 Removal from natural gas[END_REF][START_REF] Davis | Solid-liquid-vapor phase behavior of the methane-carbon dioxide system[END_REF][START_REF] Im | Phase equilibrium of carbon dioxide and light paraffins in presence of solid carbon dioxide[END_REF][START_REF] Sterner | Phase equilibria in the CO2-methane systems[END_REF][START_REF] Langé | Purification of natural gas by means of a new low temperature distillation process[END_REF][START_REF] Sparks | Gas Processors Association Research Report[END_REF]

; -: trend line.

According to Fig.

6

, it is possible to state that measured compositions qualitatively agree with the majority of the literature values as evidenced by the trend line. Unlike the high-temperature region (190 K < T < triple-point temperature of CO2), regarding all the points shown in Fig.

6

it can be stated that the solubility of solid carbon dioxide in liquid methane almost linearly changes with temperature, from about 4% at 175 K up to about 10% at 190 K.

Table 7 .

 7 SVE

		data measured for the solvent B [(1) 5%nitrogen + (2) 95%methane] + (3) carbon
	dioxide system.
	Point	T [K]	P [MPa]	y1	U(y1)	y2	U(y2)	y3	U(y3)
	32	205.480 3.999	0.0437	0.0020	0.8522	0.0050	0.1041	0.0035
	33	202.415 3.995	0.0446	0.0021	0.8698	0.0045	0.0856	0.0029
	34	199.406 4.000	0.0452	0.0021	0.8839	0.0041	0.0709	0.0025
	35	196.423 3.996	0.0459	0.0021	0.8959	0.0037	0.0582	0.0020
	36	193.458 4.006	0.0466	0.0022	0.9052	0.0035	0.0482	0.0017
	37	190.462 4.002	0.0469	0.0022	0.9131	0.0033	0.0400	0.0014
	38	189.514 3.997	0.0469	0.0022	0.9150	0.0032	0.0381	0.0013
	39	188.509 3.999	0.0473	0.0022	0.9165	0.0032	0.0362	0.0013

U(T) = 0.004 K; U(P) = 0.002 MPa

Table 8 .

 8 SLE

		data measured for the solvent B [(1) 5%nitrogen + (2) 95%methane] + (3) carbon
	dioxide system.
	Point	T [K]	P [MPa]	x1	U(x1)	x2	U(x2)	x3	U(x3)
	40	182.565 4.002	0.0458	0.0022	0.8939	0.0038	0.0603	0.0021
	41	181.576 4.002	0.0458	0.0022	0.8969	0.0037	0.0573	0.0020
	42	179.551 4.001	0.0462	0.0022	0.9028	0.0035	0.0510	0.0018
	43	177.548 3.998	0.0464	0.0022	0.9077	0.0034	0.0459	0.0016
	44	176.558 4.001	0.0459	0.0021	0.9109	0.0033	0.0432	0.0015
	45	175.767 4.000	0.0460	0.0021	0.9123	0.0033	0.0417	0.0015
	U(T) = 0.004 K; U(P) = 0.002 MPa
	For the solvent C (10%nitrogen + 90%methane) + carbon dioxide system the measured SVE and

SLE are gathered in

Table 9 and 10, respectively.

Table 9 .

 9 SVE data measured for the solvent C [(1) 10%nitrogen + (2) 90%methane] + (3) carbon dioxide system.

	Point	T [K]	P [MPa]	y1	U(y1)	y2	U(y2)	y3	U(y3)
	46	205.408 3.995	0.0882	0.0037	0.8073	0.0059	0.1045	0.0031
	47	202.418 3.993	0.0896	0.0038	0.8248	0.0056	0.0855	0.0026
	48	199.413 4.000	0.0911	0.0038	0.8385	0.0052	0.0704	0.0021
	49	196.845 3.993	0.0926	0.0039	0.8483	0.0051	0.0591	0.0018
	50	196.438 3.997	0.0924	0.0039	0.8499	0.0050	0.0577	0.0018
	51	193.467 4.010	0.0935	0.0040	0.8591	0.0049	0.0474	0.0015
	52	190.579 3.994	0.0952	0.0045	0.8658	0.0054	0.0390	0.0014
	53	190.479 4.004	0.0944	0.0040	0.8670	0.0048	0.0387	0.0012
	54	187.496 3.999	0.0956	0.0041	0.8726	0.0047	0.0318	0.0010
	55	186.501 4.001	0.0957	0.0041	0.8745	0.0047	0.0298	0.0010
	56	185.506 4.001	0.0960	0.0044	0.8758	0.0050	0.0282	0.0010
	57	184.521 3.999	0.0968	0.0043	0.8766	0.0048	0.0266	0.0009

U(T) = 0.004 K; U(P) = 0.002 MPa

Table 10 .

 10 SLE

			data measured for the solvent C [(1) 10%nitrogen + (2) 90%methane] + (3) carbon
	dioxide system.
	Point	T [K]	P [MPa]	x1	U(x1)	x2	U(x2)	x3	U(x3)
	58	178.765 4.003	0.0903	0.0045	0.8625	0.0051	0.0462	0.0016
	59	178.496 3.992	0.0906	0.0041	0.8638	0.0051	0.0456	0.0015
	60	177.573 4.007	0.0909	0.0039	0.8656	0.0048	0.0435	0.0014
	61	176.652 3.995	0.0911	0.0039	0.8673	0.0047	0.0416	0.0013
	62	175.721 3.996	0.0920	0.0039	0.8685	0.0047	0.0395	0.0013
	U(T) = 0.004 K; U(P) = 0.002 MPa

Table 11 .

 11 VLE data measured for the (1) nitrogen + (2) methane + (3) carbon dioxide system using the solvents B and C.

	Point	T [K]	P [MPa]	U(x1)	x2	U(x2)	x3	U(x3)	U(y1)	y2	U(y2)	y3	U(y3)
	63	183.532 4.003	0.0026	0.8956	0.0044	0.0552	0.0023	0.0049	0.8713	0.0054	0.0232	0.0008
	64	182.545 4.001	0.0041	0.8921	0.0068	0.0504	0.0030	0.0055	0.8599	0.0059	0.0202	0.0007
	65	182.542 3.998	0.0025	0.9110	0.0035	0.0369	0.0014	0.0050	0.8761	0.0053	0.0158	0.0006
	66	181.561 4.000	0.0039	0.8917	0.0056	0.0448	0.0023	0.0062	0.8467	0.0065	0.0179	0.0006
	67	181.557 3.996	0.0035	0.9129	0.0047	0.0283	0.0014	0.0058	0.8676	0.0060	0.0123	0.0005
	68	181.554 4.002	0.0027	0.9277	0.0031	0.0173	0.0007	0.0058	0.8807	0.0060	0.0079	0.0003
	69	180.623 4.007	0.0046	0.8854	0.0063	0.0414	0.0023	0.0069	0.8310	0.0072	0.0163	0.0005
	70	180.571 4.003	0.0031	0.9201	0.0035	0.0156	0.0006	0.0067	0.8631	0.0068	0.0070	0.0003
	71	180.275 4.001	0.0042	0.9111	0.0049	0.0198	0.0010	0.0069	0.8519	0.0071	0.0086	0.0003
	72	179.634 3.998	0.0044	0.8818	0.0058	0.0367	0.0023	0.0076	0.8170	0.0078	0.0148	0.0005
	73	179.589 3.998	0.0035	0.9091	0.0040	0.0167	0.0006	0.0073	0.8431	0.0075	0.0072	0.0003
	74	179.589 4.010	0.0036	0.9117	0.0040	0.0137	0.0005	0.0077	0.8447	0.0078	0.0061	0.0002
	75	179.145 4.003	0.0044	0.8776	0.0054	0.0364	0.0014	0.0077	0.8092	0.0079	0.0142	0.0005
	76	179.092 4.000	0.0040	0.9081	0.0043	0.0133	0.0005	0.0081	0.8367	0.0082	0.0058	0.0002
	77	178.602 4.006	0.0041	0.9035	0.0045	0.0127	0.0005	0.0086	0.8272	0.0087	0.0055	0.0002
	U(T) = 0.004 K; U(P) = 0.002 MPa

Table 12 .

 12 SLVE data measured for the (1) nitrogen + (2) methane + (3) carbon dioxide system using the solvents B and C.

	Point	T [K]	P [MPa]	U(x1)]	x2	U(x2)	x3	U(x3)	U(y1)	y2	U(y2)	y3	U(y3)
	78 1 183.560 4.002						0.0049	0.8646	0.0053	0.0247	0.0008
	79 1 183.537 4.040						0.0050	0.8620	0.0055	0.0246	0.0008
	80 1 183.521 4.014						0.0049	0.8622	0.0053	0.0246	0.0008
	81 1 182.571 3.998						0.0055	0.8504	0.0059	0.0229	0.0007
	82	182.533 4.025	0.0035	0.8795	0.0059	0.0585	0.0030	0.0056	0.8471	0.0060	0.0227	0.0007
	83	181.540 4.000	0.0031	0.8760	0.0044	0.0553	0.0019	0.0061	0.8344	0.0065	0.0212	0.0007
	84	181.521 4.006	0.0038	0.8757	0.0054	0.0554	0.0044	0.0063	0.8330	0.0067	0.0210	0.0007
	85 1 181.055 4.003						0.0068	0.8264	0.0071	0.0203	0.0007
	86	181.039 3.997	0.0033	0.8736	0.0045	0.0535	0.0018	0.0065	0.8266	0.0069	0.0203	0.0006
	87	180.541 3.998	0.0034	0.8710	0.0046	0.0518	0.0017	0.0069	0.8187	0.0072	0.0195	0.0006
	88 1 180.058 3.998						0.0074	0.8109	0.0077	0.0188	0.0006
	89	180.043 3.998	0.0036	0.8682	0.0048	0.0502	0.0017	0.0075	0.8106	0.0077	0.0188	0.0006

Table 13 :

 13 Critical-point properties and acentric factors of N2, CH4, and CO2, and binary interaction parameters of the corresponding binary mixtures[START_REF]Aspen Plus V10[END_REF].

		Pure Fluids
	Fluid	TC [K]	PC [MPa]	Ω
	N2	126.194	3.39437	0.04
	CH4	190.699	4.64068	0.0115
	CO2	304.100	7.37	0.2389
		Binary Mixtures
	Mixture		kij
	N2 + CH4		0.036
	N2 + CO2		-0.02
	CH4 + CO2		0.1

Table 14 :

 14 Triple-point properties of CO2 used in this work[START_REF]The DIPPR Information and Data Evaluation Manager for the Design Institute for Physical Properties[END_REF].All the VLE measurements gathered in Table5are compared to the calculated phase equilibrium behavior at 4 MPa in Fig.10. In Fig.10, the x-axis representing the molar fraction of carbon dioxide has been limited to a CO2-content of 11%. The horizontal line represents the calculated SLVE condition of the mixture at 4 MPa, whereas symbols are the values measured between 3.969 MPa and 4.031 MPa. vapor phase owing to the fact that, in the pressure-composition diagram, the slope of the liquid curve is lower than that of the vapor when the global composition of the mixture approaches pure methane, as pointed out by the modeling (and experimental) results illustrated in Fig.11.

	Fluid	TT [K]	PT [MPa]	ΔHm [kJ/mol]	ΔCpm [kJ/mol/K]	Δvm [dm 3 /mol]
	CO2	216.592	0.49952	8.65	0.0138	0.00858
	In the following, experimental measurements are qualitatively compared with the calculated
	phase behaviors.

the CO2-content in the equilibrium phases at VLE. This decrease is more pronounced in the liquid Fig. 11. Experimental and calculated phase equilibrium behavior of the CH4+CO2 system at 188.345 K and 189.747 K from 3.9 MPa up to 4.6 MPa for CO2-contents lower than 11%. Experimental values at 189.747 K (average value): • : point n°17 at 4.031 MPa; □ : point n°15 at 4.004 MPa; ■ : point n°16 at 3.969 MPa. Experimental values at 188.345 K (average value): ◊ : point n°22 at 3.996 MPa; ♦ : point n°21 at 3.977 MPa. Calculated values: -.

Table 4 ,

 4 n°37 in Table7, and n°53 in Table9) at average temperature and pressure of 190.495 K and 4.007 MPa, respectively.

	1.18
		20%			CO 2
	1.13
		SVE
	1.08		10%
	1.03
		V
	N 2						CH 4
	0.98
	1.7	1.75	1.8	1.85	1.9	1.95	2	2.05
	Fig. 15. Zoom in the CH4-rich region of the ternary diagram of the N2 + CH4 + CO2 system at
	average temperature (190.495 K) and pressure (4.007 MPa) of points n°11, n°37, and n°53.
	Experimental values: point n°11, solvent A (○); point n°37, solvent B (□); point n°53, solvent C
	(Δ). Solid lines are modeling results, dotted lines refer to N2/CH4 ratios of solvents B and C.

Table 5 ,

 5 and n°54 in Table9) at average temperature and pressure of 187.525 K and 4.000 MPa, respectively.

	1.18
		20%			CO 2
	1.13
		SVE
	1.08		10%
						L
						VLE
	1.03
		V
	N 2						CH 4
	0.98
	1.7	1.75	1.8	1.85	1.9	1.95	2	2.05
	Fig. 16. Zoom in the CH4-rich region of the ternary diagram of the N2 + CH4 + CO2 system at
	average temperature (187.525 K) and pressure (4.000 MPa) of points n°23 and n°54.
	Experimental values: point n°23, solvent A (○); point n°54, solvent C (Δ). Solid lines are modeling
	results, dotted line refers to N2/CH4 ratio of solvent C.

Table 6 ,

 6 n°40 in Table 8, n°64 and n°65 in Table 11, and n°82 in Table 12) at average temperature and pressure of 182.548 K and 4.004 MPa, respectively.

	1.18
		20%			CO 2
	1.13			SLVE
		SVE			SLE
	1.08		10%
	1.03				VLE	L
		V
	N 2						CH 4
	0.98
	1.7	1.75	1.8	1.85	1.9	1.95	2	2.05
	Fig. 17. Zoom in the CH4-rich region of the ternary diagram of the N2 + CH4 + CO2 system at
	average temperature (182.548 K) and pressure (4.004 MPa) of points n°29, n°40, n°64, n°65, and
	n°82.
	Experimental values: point n°29, solvent A (○); point n°40, solvent B (□); points n°64 and n°65
	(•); point n° 82 (■). Solid lines are modeling results, dotted lines refer to N2/CH4 ratio of solvent
	B.

Table 8 ,

 8 n°66-n°68 inTable 11, and n°84 in Table12) at average temperature and pressure of 181.554 K and 4.001 MPa, respectively.

	1.18
		20%			CO 2
	1.13
		SVE			SLE
			SLVE
	1.08		10%
	1.03					L
					VLE
		V
	N 2						CH 4
	0.98
	1.7	1.75	1.8	1.85	1.9	1.95	2	2.05
	Fig. 18. Zoom in the CH4-rich region of the ternary diagram of the N2 + CH4 + CO2 system at
	average temperature (181.554 K) and pressure (4.001 MPa) of points n°41, n°66, n°67, n°68, and
	n°84.
	Experimental values: point n°41, solvent B (□); points n°66-n°68 (•); point n° 84 (■). Solid lines
	are modeling results, dotted line refers to N2/CH4 ratio of solvent B.

Table 6 ,

 6 n°44 in Table8, and n°61 in Table10) at average temperature and pressure of at 176.607 K and 3.976 MPa.

Table 15 :

 15 Quantitative comparison between VLE and SLVE data and modeling results for

	the (1)N2 + (2)CH4 + (3)CO2 mixtures.
	Equilibrium	VLE		SLVE
	Mixture	CH4 + CO2	N2 + CH4 + CO2	N2 + CH4 + CO2
	AAD(x1)%		1.87	2.72
	Bias(x1)%		1.87	2.72
	MAD(x1)%		3.61	3.40
	AAD(x2)%	0.39	0.22	0.62
	Bias(x2)%	-0.39	-0.22	-0.62
	MAD(x2)%	0.61	0.31	0.65
	AAD(x3)%	7.95	2.90	6.45
	Bias(x3)%	7.95	2.90	6.45
	MAD(x3)%	27.23	4.77	7.01
	AAD(y1)%		1.83	1.84
	Bias(y1)%		-1.83	1.84
	MAD(y1)%		3.81	4.26
	AAD(y2)%	0.06	0.35	0.28
	Bias(y2)%	-0.05	0.35	-0.28
	MAD(y2)%	0.20	0.49	0.56
	AAD(y3)%	3.59	4.07	0.82
	Bias(y3)%	3.51	-4.07	-0.80
	MAD(y3)%	20.94	6.70	1.54

Because of the global composition of the mixture, the amount of the liquid phase within the equilibrium cell was not enough developed to be sampled by the liquid ROLSI ® sampler. U(T) = 0.004 K; U(P) = 0.002 MPa

Acknowledgement

The Centre of Thermodynamics of Processes of MINES ParisTech, PSL University, is thankful to Maire Tecnimont for the financial contribution.

Nomenclature

VLE, Vapor-Liquid Equilibrium; SVE, Solid-Vapor Equilibrium; SLE, Solid-Liquid Equilibrium; SLVE, Solid-Liquid-Vapor Equilibrium; AAD, Absolute Average Deviation; MAD, Maximum Absolute Deviation

molar fraction in the liquid and vapor phases are 1.78% and 0.95%, respectively (point n°25 in Table 5).

For VLE data of the N2 + CH4 + CO2 system, deviations are lower than 4%, 0.5%, and 7% for methane, nitrogen, and carbon dioxide, respectively (see 3 rd column of Table 15). For SLVE data, deviations are lower than 5%, 1%, and 7% for methane, nitrogen, and carbon dioxide, respectively (see 4 th column of Table 15). All these deviations are within the experimental uncertainties.

The quantitative comparison between experimental and calculated SVE and SLE data is gathered in Table 16.

For the solvent A, the deviations given in Table 16 are of about 0.2% and 2% for the CH4composition and CO2-composition, respectively, in the vapor phase at SVE (2 nd column). These values are within the experimental uncertainties (a maximum value of 0.4% and 6.2% for CH4 and CO2, respectively). Deviations are of about 0.8% and 15% for the CH4-composition and CO2composition in the liquid phase at SLE (5 th column), respectively.

For the solvent B, the deviations given in Table 16 are of about 3%, 0.5%, and 3% for the N2composition, CH4-composition, and CO2-composition, respectively, in the vapor phase at SVE (3 rd column). These values are within the experimental uncertainties (a maximum value of 4.8%, 0.6% and 3.5% for N2, CH4, and CO2, respectively). Deviations are of about 4%, 1%, and 14% for the N2-composition, CH4-composition, and CO2-composition, respectively, in the liquid phase at SLE (6 th column). These values are similar to the experimental uncertainties only for nitrogen (maximum of 4.8%) and methane (maximum of 0.4%), but much higher for carbon dioxide (maximum of 4%).

For the solvent C, the deviations given in Table 16 are of about 2%, 0.5%, and 3% for the N2composition, CH4-composition, and CO2-composition, respectively, in the vapor phase at SVE (4 th column). These values are similar to the experimental uncertainties (a maximum value of 4.7%, 0.7% and 3.6% for N2, CH4, and CO2, respectively). Deviations are of about 5%, 2%, and 34% for the N2-composition, CH4-composition, and CO2-composition, respectively, in the liquid phase at SLE (7 th column). These values are similar to the experimental uncertainties only for nitrogen (maximum of 5.0%) and methane (maximum of 0.6%), but much higher for carbon dioxide (maximum of 3.4%).

Declaration of competing interests

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

CRediT authorship contribution statement


Derek Noël

Philippe Rivière

Odile Cauret

Dominique Marchio

Cédric Teuillières

Vapour quality determination for heat pumps using two-phase suction

Keywords:

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Title

Vapour quality determination for heat pumps using two-phase suction

Author names and affiliations

• Derek Noël, affiliation: 1 and 2, corresponding author, email: derek.noel@edf.fr, phone number: +33 6 26 54 84 81, fax number: none • Philippe Rivière, affiliation: 1, email: philippe.riviere@mines-paristech.fr • Odile Cauret, affiliation: 2, email: odile.cauret@edf.fr • Dominique Marchio, affiliation: 1, email: dominique.marchio@mines-paristech.fr • Cédric Teuillières, affiliation: 2, email: cedric.teuillieres@edf.fr

Introduction

Thanks to their high efficiency, heat pumps play a major role in greenhouse gas emission and energy consumption reduction in buildings in Europe. In order to reduce the impact of the refrigerant on climate change, the fluids with the highest global warming potentials (GWP) will progressively be phased out, as a consequence of the F-Gas regulation (EU) No 2014/517. Therefore, in particular for residential heat pumps, R410A is being replaced with R32, which has a 3-time lower GWP. R32 can operate under the same temperature conditions than R410A without any major change in the thermodynamic cycle structure and can also improve the heat pump performances. Xu et al. (2013) experimentally demonstrated that the COP improvement using R32 instead of R410A on the same single-stage heat pump was between 2 and 9 %. However, controlling the expansion valve on suction superheat in the same way would lead to much higher discharge temperatures that could damage the compressor. Xu et al. (2013) showed that the discharge temperature using R32 was significantly higher than when using R410A in every operating condition, up to + 34 °C for an outdoor temperature of -18 °C.

Thus, several patents were published with new control strategies. Using liquid injection cycles would allow to reduce the discharge temperature, but it would increase the manufacturing cost of the heat pump. Taira et al. (2003) recommended a method using the electronic expansion valve (EEV) to control directly the discharge temperature, and [START_REF] Yamada | Mitsubishi Heavy Industries Thermal Systems[END_REF] suggested to use the EEV to control the discharge superheat value, depending on the compressor frequency and the compression ratio. For both solutions, it leads to two-phase fluid at the compressor suction port in most operating conditions, which reduces significantly the discharge temperature. Taira et al. (2003) states that this wet compression cannot harm rotary-type compressors (rolling piston) as long as the suction vapour quality is higher than 0.75. Nowadays, two-phase suction is widely used in order to lower the discharge temperature in R32 air conditioners and heat pumps using a rotary compressor.

The effects of wet compression on the heat pump performances have been studied, but the effect of the discharge temperature control of heat pumps using two-phase suction on vapour quality is unknown. For example, [START_REF] Seong | The Effects of Wet Compression by the Electronic Expansion Valve Opening on the Performance of a Heat Pump System[END_REF] showed that this process can improve the heat pump performances by lowering the compression ratio and increasing the isentropic efficiency. However, for that purpose, the suction vapour quality should be controlled to an optimal value depending on the operating conditions. In order to optimise the performances of heat pumps using two-phase suction, the real-time estimation of the suction vapour quality on an operating heat pump is necessary. Huang et al. (2018) presented a method based on Quasi Saturation Isentropic Compression in order to control the discharge temperature. With this method, the refrigerant can also be diphasic at the compressor suction, so the suction vapour quality was estimated using a fixed compressor global efficiency value. This method allows to compare the suction quality values on the same compressor under different operating conditions, but as the method precision is not established by the authors, it is not known if it is accurate enough for wet compression optimisation. Endoh (2014) presented a method to determine the suction vapour quality from the mass flow rate measurement and the compressor energy balance. It gives very good results, with a 0.5 % accuracy. However, using a mass flow meter on an operating heat pump is not always possible and can be expensive, so this method is not adapted for a real-time in-situ optimisation purpose.

When the fluid is in two-phase state at the compressor suction port, making the hypothesis that it is saturated leads the compressor energy balance to highly overestimate the mass flow rate. On the contrary, making this assumption to calculate the mass flow rate using the volumetric efficiency or the global efficiency gives underestimated results. The real mass flow rate value can thus be bounded using the compressor energy balance and the volumetric or the global efficiency with a vapour quality of 1. This paper presents two similar methods using iterations on suction vapour quality to make these calculations converge, allowing to obtain the suction vapour quality and the refrigerant mass flow rate without using a flow meter. An experimental study of an air-to-air heat pump using R32 in climatic chambers is then presented. The first objective is to measure the suction vapour quality from the mass flow rate measurement to obtain a correlation as a function of the operating conditions. The second one is to experimentally validate the iterative method, using the compressor energy balance and the volumetric or global efficiency of the compressor, and to compare the results with the reference measurement. The results and the main uncertainty factors of the method are discussed.

Compressor energy balance

For heat pumps with a controlled positive suction superheat value, a method has been previously developed to measure the mass flow rate and heating capacity on field using a light instrumentation. To avoid using a flowmeter, the refrigerant mass flow rate can be indirectly measured using only temperature and electrical power measurements. This method, developed by [START_REF] Tran | Méthodes de mesure in situ des performances annuelles des pompes à chaleur air/air résidentielles[END_REF], relies on the compressor energy balance (equation 1).

= ℎ

, -ℎ ,"# $ + & ''(' (Eq. 1)

The suction and discharge temperatures are measured, as well as the evaporating and condensing temperatures. The suction and discharge enthalpy, respectively ℎ ,"# and ℎ , , can then be computed from these measurements, as the fluid is superheated vapour. The compressor power input is also measured. & ''(' represents the compressor heat losses for a hermetic compressor. In order to improve the compressor heat losses estimation, Niznik (2017) presented a measurement method using two temperature measurements for rotary-type compressors. One is placed on the compressor shell surface ' * , and the other measures the ambient temperature + , . [START_REF] Niznik | Improvment and integration of the in-situ heat pump performance assessment method[END_REF] showed that the temperature of the air entering the outdoor unit was the best representation of the compressor ambient temperature, and that the temperature of the compressor shell is uniform enough to use only one sensor for rotary-type compressors. & ''(' is thus calculated using equation 2, considering the convective and radiative heat losses. The emissivity of the compressor shell surface is supposed to be equal to 1.

& ''(' = - . . + /,0 / + /,1 / 2 ' * -+ , $ + ' * 3 -+ , 3 $ (Eq. 2)
Where is the Nusselt number for the different sides of the compressor, k is the thermal conductivity of the air, L, D and A stand respectively for the length (height), diameter (1 and 2 for the top and the bottom surface respectively) and area for the lateral side, the top and the bottom of the compressor, is the Stefan-Boltzmann constant for radiative heat transfer. The air properties are calculated at film temperature. 2018) experimentally validated the compressor energy balance method on an air-to-water heat pump with a controlled positive suction superheat in stationary state, and concluded that the refrigerant mass flow rate and the heating capacity are obtained with an average 4.4 % accuracy compared to the water-side measurement.

Noel et al. (

However, for heat pumps allowing two-phase fluid at the compressor suction port to reduce the discharge temperature, the suction vapour quality and enthalpy are unknown. By comparing the calculation with an hypothesis of saturated fluid and using various theoretical vapour quality values, an estimation has been made on a few working conditions that the hypothesis of saturated fluid leads to a very high overestimation of the mass flow rate that can reach 20 % for a 0.97 vapour quality and about 40 % for a 0.94 vapour quality. The mass flow rate is needed to know the suction quality and enthalpy, or the vapour quality is necessary to calculate the suction enthalpy and the mass flow rate with the compressor energy balance. Therefore, another equation is needed to be able to make iterations on these variables.

In-situ vapour quality calculation method

When the fluid is in two-phase state at the compressor suction port, its vapour quality cannot be directly measured. Moreover, its evaluation through the void fraction measurement would not be accurate enough [START_REF] Tran | Méthodes de mesure in situ des performances annuelles des pompes à chaleur air/air résidentielles[END_REF]) and would need additional instrumentation. The compressor energy balance needs to be completed with another equation to be able to calculate the refrigerant mass flow rate and the suction vapour quality. The following paragraphs show how it can be done using the mass flow rate calculation from the volumetric or global compressor efficiency.

Mass flow rate calculation using volumetric efficiency

From the volumetric efficiency

& , the refrigerant mass flow rate can be calculated using equation 3.

' is the refrigerant suction density, 9 the displacement volume of the compressor, and : the compressor rotating frequency. In order to calculate ' , the suction vapour quality is needed. Thus, its value can be initialized to ,; = 1, considering that the fluid is saturated vapour. Making this hypothesis when the fluid is actually under two-phase state underestimates the suction density, thus equation 3 underestimates the refrigerant mass flow rate. The subscript "0" is used for the chosen or calculated initial values of vapour quality or mass flow rate. 4 567 ,; $ < (Eq. 4)

Mass flow rate calculation using global efficiency

The global efficiency is the ratio between the actual compression work required for the compression and the isentropic work. It considers both the isentropic losses and the heat losses. It is more adapted for studying a real compression than the isentropic efficiency, which only includes isentropic losses under adiabatic process. The refrigerant mass flow rate can be calculated from the compressor global efficiency > using equation 5.

4 ? = > . @A B ℎ , ,"' -ℎ ,"# (Eq. 5)
is the measured compressor power input. The suction vapour quality is needed to calculate the suction enthalpy, ℎ ,"# , and the corresponding discharge isentropic enthalpy, ℎ , ,"' . As for the volumetric efficiency equation, the calculation can be initialised considering the refrigerant as saturated, with a vapour quality of ,; = 1. With this hypothesis, the mass flow rate is underestimated because the slope of the isentropic lines increases with a decreasing vapour quality, so the isentropic enthalpy difference decreases.

?

,; $ < (Eq. 6)

Iterations on vapour quality

Using the volumetric or global efficiency with the hypothesis that the fluid is saturated underestimates the refrigerant mass flow rate. As seen in paragraph 2, the mass flow rate can also be calculated using the compressor energy balance using the same vapour quality initial value (equation 7).

CD ,; = @A B -EAFFGF ℎ , -ℎ ,"# ( ,; ) (Eq. 7)

However, it overestimates the suction enthalpy, so equation 7 overestimates the mass flow rate.

Since the calculation method is the same whether the volumetric efficiency or the global efficiency is used, the mass flow rate calculated with one or the other is called 4 . Its value calculated with the hypothesis that the fluid is saturated ( ,; = 1) is called 4,; .

Thus, the real mass flow rate value is necessarily between the efficiency and the compressor energy balance results (equation 8).

4,; < < CD ,;

(Eq. 8)

As the suction vapour quality is needed to calculate the mass flow rate and vice-versa, it is necessary to make iterations using two different equations. Equation 8shows how the mass flow rate value is bounded, so iterations can then be made by progressively decreasing the vapour quality value to make the efficiency and the compressor energy balance calculation converge. The iterations are stopped when the difference between the compressor energy balance and the volumetric (or global efficiency) is smaller than the measurement uncertainty.

Figure 1 illustrates the principle of these iterations on vapour quality, where 4 is the mass flow rate calculated with one of those efficiencies and CD is the mass flow rate calculated with the compressor energy balance. Their initialised values calculated for ,; = 1 are respectively 4,; and CD ,; .

Experimental set-up

A commercial 3.5 kW air-to-air heat pump has been instrumented and tested in order to study its twophase suction control and the iterative methods presented before. The heat pump is composed of an inverter-driven rotary compressor, an electronic expansion valve, an evaporator and a condenser. The working fluid is R32. In order to reduce the compressor discharge temperature, two-phase fluid is allowed at the compressor suction port.

Metrology

The heat pump was instrumented in order to have a reference measurement of the mass flow rate which can be used to obtain reference values of suction vapour quality and heating capacity.

The refrigerant temperature is measured with surface Pt100 sensors. Their measurement uncertainty is considered to be 0.8 K, taking into account the uncertainty caused by the surface measurement [START_REF] Lne | Guide d'utilisation des capteurs de température de surface[END_REF]. The air temperature is measured with Pt 100 sensors with a 0.1 K measurement uncertainty.

The compressor input power and the total heat pump power are measured with power meters with a 0.5 % relative measurement uncertainty. Suction and discharge pressures are measured with a relative measurement uncertainty of 0.25 %. The refrigerant mass flow rate is measured with a Coriolis-effect flow meter placed at the condenser outlet, where the fluid is liquid. Its relative measurement uncertainty is 0.5 % of the mass flow rate value. The thermodynamic properties are calculated with CoolProp Library [START_REF] Bell | Pure and Pseudo-pure Fluid Thermophysical Property Evaluation and the Open-Source Thermophysical Property Library CoolProp[END_REF]. According to Casari et al. ( 2019), the relative uncertainty on enthalpy values calculated using CoolProp is below 0.8 % for R410A. It is supposed to be equivalent for R32 considering the similarities between those fluids. A fraction of the compressor lubricating oil migrates into the refrigerant cycle. [START_REF] Niznik | Improvment and integration of the in-situ heat pump performance assessment method[END_REF], assumed that this oil circulation ratio represents 0.5 % of the refrigerant mass under any working condition, and experimentally verified that it leads to accurate mass flow rate results. The effects of the miscibility of the lubricant in the refrigerant are neglected. This value is used to calculate the refrigerant enthalpy values, according to equation 9.

ℎ = (1 -9 "& ). ℎ (* + 9 "& . ℎ "& (Eq. 9)

Figure 2: Instrumentation installed on the air-to-air heat pump

Operating conditions

Different outdoor and heat pump set point temperatures were tested in order to make the air-to-air heat pump operate under various functioning temperatures, compressor frequencies and thermal loads. 60 combinations of operating conditions have been studied. The outdoor temperature ranged between -10 °C and + 10 °C, and the compressor rotating frequency ranged between 30 and 100 Hz.

For each operating condition, some dynamic phases could occur. In particular, for outdoor temperatures below 5 °C, the heat pump needed to engage defrosting sequences, with an inverted cycle operation during a few minutes. These phases, as well as the compressor starting phases, were excluded from the results. This study focuses only on steady-state operating phases, and the values of the variables of interest represented after are the mean values calculated during each of the 60 steadystate phases. Each operating condition was set during at least 2 hours in order to let the heat pump parameters stabilise.

Reference Values

Heating capacity

Thanks to the mass flow rate, temperature and pressure measurements, the reference heating capacity of the heat pump can be calculated with equation 10.

#J, (* = . (ℎ #J,"# -ℎ #J, ) (Eq. 10)

is the measured mass flow rate, ℎ #J,"# the refrigerant enthalpy at the condenser inlet, and ℎ #J, the enthalpy at the condenser outlet.

Considering the different measurement and calculation uncertainties, the uncertainty of the reference heating power can be calculated with the uncertainty propagation equation (equation 11).

K #J = L M - N @AOP, Q N " 2 1 R S 1 " (Eq. 11)
For each operating condition, the relative measurement uncertainty of the reference heating capacity is below 1 % of its value.

Suction vapour quality

In order to obtain a reference value of the suction vapour quality, the compressor energy balance is used as well as the measured mass flow rate (equation 1, recalled below).

= (ℎ , -ℎ ,"# ) + & ''(' (Eq. 1)

The compressor power input and its heat losses & ''(' are estimated [START_REF] Niznik | Improvment and integration of the in-situ heat pump performance assessment method[END_REF]. The discharge temperature and pressure are measured, allowing to calculate the discharge enthalpy value ℎ , . The suction enthalpy is then deduced from the compressor energy balance, according to equation 12. (Eq. 12)

The suction vapour quality can then be determined from the measured suction pressure ' and enthalpy values (Eq 13), using CoolProp library.

, (* = T( F @ , ℎ ,"# )

(Eq. 13)

The vapour quality relative measurement uncertainty, calculated from table 1 and equation 10, is below 1.2 % of its value for every steady-state operating condition considered. The uncertainties caused by the assumptions needed to calculate the compressor heat losses are difficult to estimate. However, from 6 operating conditions, [START_REF] Niznik | Improvment and integration of the in-situ heat pump performance assessment method[END_REF], experimentally estimated the compressor heat losses uncertainty to be 10.4 %, making the hypothesis that it is equal to the experimental root mean square error. A 10.4 % uncertainty on the compressor heat losses leads to a 0.3 % uncertainty on the suction vapour quality calculated with equations 12 and 13.

At first sight, the main parameters leading to a high discharge temperature should be a high compression ratio, a high compressor frequency and a high heating capacity. Therefore, these conditions should lead to the lowest vapour quality values. Figure 3 shows the suction vapour quality as a function of the compression ratio.

Figure 3: Reference suction vapour quality as a function of the compression ratio

Figure 3 shows that the suction vapour quality tends to decrease with the compression ratio, but it also shows that it cannot be resumed to it. To have a better modelling of the suction vapour quality, the evaporating temperature, the condensing temperature and the compressor frequency are taken into account to determine a correlation (equation 14).

= 1,008 -1,491. 10 YZ ( + + 1,852. 10 YZ #J -1,787. (Eq. 14)

Figure 4 represents the vapour quality calculated with equation 14 compared to the vapour quality measurement.

This relationship between the suction vapour quality and the operating conditions could be useful to have a simple estimation of the suction vapour quality in steady-state operation of a heat pump using two-phase suction. However, its accuracy needs to be evaluated when used on another heat pump that might have a different control logic of the discharge temperature.

Volumetric and global compressor efficiencies

The compressor volumetric efficiency can be calculated from equation 15.

& = ' . 9. :

(Eq. 15)

With the measured mass flow rate, ' the suction density, 9 the compressor displacement, and : the compressor frequency. ' is calculated from the measured suction pressure and vapour quality, calculated from equation 13.

The compressor global efficiency considers the irreversibility and the heat losses of the compression. It can be calculated using equation 16. With the measured mass flow rate and the measured compressor power input. The suction enthalpy ℎ ,"# is calculated from the measured suction pressure and vapour quality, calculated from equation 13. The corresponding isentropic discharge enthalpy ℎ , ,"' is then deduced from ℎ ,"# and the measured discharge pressure.

Figure 5 represents the mean volumetric and global compressor efficiencies as a function of the mean compression ratio for every steady-state operating condition studied.

Considering every other parameter fixed, the volumetric efficiency decreases for high compression ratios. However, in this case, the compressor rotating frequency and the suction vapour quality depend on the heat pump control and are influenced by the compression ratio. A high compressor rotating frequency has a positive effect on volumetric efficiency, and [START_REF] Seong | The Effects of Wet Compression by the Electronic Expansion Valve Opening on the Performance of a Heat Pump System[END_REF] showed that a decreasing vapour quality can increase volumetric efficiency to a certain extent. Therefore, figure 5 is a consequence of the combined effects of these parameters.

As for the compressor volumetric efficiency, figure 5 does not represent a usual compressor performance map for global efficiency, which would be established with a fixed frequency in specified operating conditions. It is a consequence of the combined effects of the evolution of the compression ratio, the compressor frequency and the suction vapour quality.

Results and discussion

Experimental results

The iterative method presented in paragraph 3 is applied to the 60 steady-state phases set on the airto-air heat pump, using the volumetric efficiency (equation 3) or the global efficiency (equation 5), and the compressor energy balance (equation 1). In every operating condition, it converges to a mass flow , (* (Eq. 17)

Figure 6 shows the estimated suction vapour quality using the volumetric efficiency (blue) and the global efficiency (orange), as a function of the reference vapour quality calculated with equation 13.

In the 60 operating conditions tested, the estimated vapour quality from the compressor energy balance and the volumetric or global efficiency is always within 0.4 % of the suction vapour quality obtained from the mass flow rate measurement.

These methods also provide mass flow rate values 4 , so the heating capacity can be calculated according to equation 18.

#J,4 = 4 . (ℎ #J,"# -ℎ #J, ) (Eq. 18)

Where #J,4 is the heating capacity calculated with the iterative methods using either the volumetric or the global efficiency. Figure 7 shows #J,4 compared to the reference heating capacity Both iterative methods give results within 1 % of the reference heating capacity.

Therefore, the presented iterative method gives very good results of vapour quality and mass flow rate compared to the reference measurements when the compressor volumetric or global efficiencies are known.

Main uncertainty factors

The volumetric efficiency equation (Eq. 3) requires also the knowledge of the compressor displacement and the rotating frequency of the compressor. The latter can be measured on field by measuring the electrical frequency of the compressor power input. If the method is implemented by the manufacturer, it is also possible to use the frequency value set by the compressor control. The manufacturer also knows the compressor displacement. However, for an independent implementation of the method, the displacement is not always easily available. In that case, it would be very complicated to make an estimation of the compressor displacement, and it would be preferable to use the global efficiency equation.

Concerning the compressor global efficiency equation (Eq. 5), it requires the compressor power input measurement, which is also necessary for the compressor energy balance.

In the results presented on figures 6 and 7, the compressor efficiencies were known from the mass flow rate measurement. Compressor manufacturers generally provide performance maps, giving the compressor efficiency as a function of the functioning temperatures, for given rotating frequencies and fixed suction superheat conditions. It would be possible to use these correlations to estimate the volumetric or global efficiency on field, and use the methods presented in this paper. However, the difference in efficiencies with real operation under wet compression will thus not be considered properly, leading to additional bias as compared to the results presented in Section 5.1. Generic correlations or methods could not be found to consider the effect of wet compression.

Figure 7: Heating capacity estimated using the iterative method compared to the reference value

In the case of an independent on-field measurement, where compressor maps are not always available, it could be possible to use a published correlation made for a similar compressor. The uncertainty on the obtained vapour quality value would probably be larger than using the specific compressor data, but it could at least give a good representation of the effect of the control logic on suction vapour quality.

In order to have an estimation of the sensitivity of the iterative method, the calculation have been made introducing an under and overestimation of the efficiency values of 10 %. Figure 8 shows the results of the method on suction vapour quality, using volumetric or global efficiency with a 10 % relative error, compared to the reference measurement. Figure 8 shows that even if a 10 % error is made on the estimation of the compressor efficiencies, the results on suction vapour quality are kept within 2.5 % of the reference values. It can be noted that an underestimation is slightly preferable than an overestimation. Also, the error is reduced when the vapour quality gets close to 1, as the errors made on suction enthalpy and density at the first iteration are smaller.

The effects of a 10 % efficiency error on the heating capacity estimation are represented on figure 9.

The relationship between an error on the efficiency values and the error on heating capacity is more direct than for suction vapour quality (i.e. 10 % error on the efficiency leads to about 10 % maximum error on the capacity). Therefore, in a performance assessment purpose, the compressor efficiencies estimation needs to be accurate enough to have a correct heating capacity evaluation. In any case, even with a 10 % error on the efficiencies, this iterative method gives far better results that those of the compressor energy balance used with the saturated vapour hypothesis when the fluid is in twophase state at the compressor suction port.

Conclusion

A method has been presented in order to calculate the suction vapour quality and the refrigerant mass flow rate using the compressor volumetric or global efficiency and the compressor energy balance, for a heat pump using two-phase suction to reduce discharge temperature. An experimental study has been carried out to measure the suction vapour quality and to test the method on a recent air-to-air heat pump using R32.

First, a correlation giving the suction vapour quality in steady-state operating conditions was presented. It can be used to have a simple model of the two-phase suction control from the evaporating and condensing temperatures and the compressor frequency. The possibility to generalise it to other similar heat pumps needs to be evaluated.

Then, the experimental study showed that it is possible to obtain the suction vapour quality and the refrigerant mass flow rate making iterations on suction vapour quality and using the compressor energy balance. This method could allow compressor manufacturers to control precisely the twophase suction process, in order to optimise the heat pump performances.

The method requires the compressor efficiencies knowledge, which can be obtained using the compressor performance maps provided by the manufacturer. Ideally, these maps should also be corrected to account for the effect of wet suction, but presently this requires refrigerant mass flow measurement as generic correlations or methods to account for the impact of wet compression on global and volumetric efficiencies could not be found. These maps might however not always be available for an independent implementation, so the sensitivity of the method to the efficiency accuracy was estimated using 10 % under and overestimated volumetric and global efficiency values. The estimated suction vapour quality is kept within 2.5 % of the reference value, while the heating capacity estimation is close to 10 %. In any case, this method allows to make a much better estimation of the heat pump heating capacity compared to the compressor energy balance when the fluid is in two-phase state at the compressor suction port.

Therefore, in addition of two-phase suction optimisation, the proposed method to evaluate the mass flow rate could make in-situ performance monitoring possible for these heat pumps without needing a mass flow meter.

The experimental testing of the method was realized in steady-state operating conditions of an air-toair heat pump in heating mode. It would be interesting to extend this study to the cooling mode. Moreover, its extension to transitory operating phases could allow to better understand the suction vapour quality variation after a compressor start-up, and to have a more accurate performance assessment during these phases.

Figure 1 :

 1 Figure 1: Principle of the Iterative method on suction vapour quality

Figure 4 :

 4 Figure 4: Comparison of vapour quality from the model (Eq. 14) and from the measurement

Figure 5 :

 5 Figure 5: Volumetric and global efficiency as a function of the compression ratio

Figure 6 :

 6 Figure 6: Suction vapour quality estimated with the iterative method compared to the reference measurement

Figure 8 :

 8 Figure 8: Vapour quality results with a 10 % error on volumetric and global efficiency

Figure 9 :

 9 Figure 9: Heating capacity results with a 10 % error on volumetric and global efficiency

  Affiliation 1: Center for Energy efficient Systems (CES), Mines ParisTech, PSL Research University, 60 boulevard Saint Michel, 75272 Paris Cedex 6, Paris, France.

	cond	Condenser
	comp	Compressor
	evap	Evaporator
	D,1	Superior compressor shell surface
	D,2	Inferior compressor shell surface
	g	Global
	in	Inlet
	is	Isentropic
	L	Lateral side of the compressor
	losses	Compressor thermal losses
	out	Outlet
	ref	Reference
	suc	Suction
	surf	Compressor shell surface
	tot	Total
	vol	Volumetric
	Affiliation 2: Department Technologies & Research for Energy Efficiency (TREE), EDF R&D, 77818,
	Moret-sur-Loing, France
	Nomenclature
		Compressor shell area	(m²)
	C	Compressor displacement	(m 3 )
		Diameter	(m)
		Relative error
	F	Compressor rotating frequency	(Hz)
	ℎ	Enthalpy Thermal conductivity	(J/kg) (W/(m.K))
		Compressor height	(m)
		Mass flow rate	(kg/s)
		Nusselt number
		Pressure	(Pa)
		Thermal power	(W)
		Temperature	(K ou °C)
		Electric power	(W)
		Vapour quality
		Efficiency
		Mass density	(kg/m 3 )
		Uncertainty
		Stefan-Boltzmann coefficient	(W/(m².K 4 ))
	Subscripts
	0	Initialized value
	amb	Ambient air

  Table1sums up the tools used for measurement and calculation and their associated uncertainties. The acquisition time step is 10 s.

	Measurement	Product	Measurement uncertainty
	Mass flow rate	Coriolis 83F15-AD2S	0.5 %
	Fluid temperature	Surface Pt100	0.8 K
	Air temperature	Pt100	0.1 K
	Pressure	Pressure transmitter	0.25 %
	Power	Power meter	0.5 %
	Thermodynamic properties	CoolProp	0.8 %
	Table 1: Measurement and calculation uncertainties

& . ' . 9. : (Eq. 3)

Funding

This work was supported by EdF R&D, TREE department -Technologies & Research for Energy Efficiency.