Predicting vapor-liquid equilibria of CO2+HFC binary mixtures by the PR EOS combined with a group contribution model

Zirui Wu ,Lingfeng Shi ✉,Rui Sun ,Hua Tian ,Xuan Wang ,and Gequn Shu,

1Department of Thermal Science and Energy Engineering, University of Science and Technology of China, Hefei 230027, China;

2State Key Laboratory of Engines, Tianjin University, Tianjin 300072, China

Abstract:CO2+HFC binary mixtures have good performance and environmental friendliness and are considered good alternative working fluids in cooling and power cycle systems.The vapor-liquid phase equilibrium properties are key to the calculation of the enthalpy and entropy of mixtures,which is critical for the analysis of cooling and power cycle systems.To accurately predict the vapor-liquid equilibrium of CO2 and HFC (R23,R32,R41,R125,R134a,R143a,R152a,R161,and R227ea) binary mixtures,a group contribution model based on the excess free energy (GE) mixing rules(PR+MHV1+UNIFAC and PR+LCVM+UNIFAC) is established in this paper.The interaction parameters between groups such as -CO2,-Alkane,-CHF,and -CHF3 are obtained by the vapor-liquid phase equilibrium experiment of CO2 and HFC refrigerants,and these group parameters are critical for predicting their vapor-liquid phase equilibrium properties (the pressures and vapor phase molar fractions).The AARDp value calculated by the PR+LCVM+UNIFAC model is 5.53%,the value of AADy1 is 0.0132,and the AARDp and AADy1 values of the PR+MHV1+UNIFAC model are 7.40% and 0.0229,respectively.However,for the CO2+R32 system,the PR+MHV+UNIFAC prediction model can reproduce the experimental data with lower deviations,and the values of AARDp and AADy1 are 1.53% and 0.0045,respectively.In summary,for CO2+HFC binary mixtures,the PR+LCVM+UNIFAC group contribution model can reproduce the experimental data with lower deviations,but for individual CO2 binary mixtures (such as CO2+R32),the PR+MHV1+UNIFAC model also has unique advantages.According to the prediction results of the group contribution model,the PR+LCVM+UNIFAC model has significantly improved the calculation progress compared with the PR+MHV1+UNIFAC model used in the previous system.

Keywords: cooling and power cycles;CO2+HFC binary mixtures;vapor-liquid equilibrium;the group contribution model;PR+MHV1+UNIFAC model;PR+LCVM+UNIFAC model

Nomenclature

1 Introduction

Carbon dioxide (CO2) is widely used as a working medium in refrigeration cycles,power cycles,supercritical Brayton cycles,and other applications due to its high thermal stability,natural working fluid,safety,environmental protection,and high energy density to achieve miniaturization of components[1,2].However,CO2is a working medium with the disadvantages of high operating pressure and low system thermal efficiency during system operation.Hydrofluorocarbon (HFC) refrigerants are highly thermally efficient in refrigeration cycles and power cycles but have high ozone depletion potential (ODP) and global warming potential (GWP)[3].Mixing CO2with other HFC refrigerants (with low GWP,zero ODP,and no toxicity) is one of the prospective ways to gain the advantages of CO2and minimize its disadvantages.CO2+HFC binary mixtures can not only reduce operating pressure and improve system thermal efficiency but also reduce the ODP value and GWP value[4].Due to this good performance and environmental friendliness,CO2+HFC refrigerant binary mixtures have received extensive attention and research as alternative working media in refrigeration cycles and power cycles[5].

The vapor-liquid phase equilibrium (VLE) data are key to the calculation of the enthalpy and entropy of CO2+HFC refrigerant binary mixtures,which is critical for the analysis of refrigeration cycles and power cycles.Due to the simplicity of the form and the accuracy of the calculation of the cubic equation of state,it is widely used in the simulation and calculation of the thermophysical properties of various working fluids[6].For CO2+HFC refrigerant binary mixtures,the cubic equation (Peng-Robinson equation[7]) of state must be combined with the mixing rules to predict the vapor-liquid phase equilibrium properties.At present,there are many concerns and studies on the mixing rule.In general,the mixing rule is mainly divided into excess free energy mixing rules[8]and van der Waals[9]mixing rules.

Due to the simplicity of the form and the accuracy of the calculation,the cubic equation of state with mixing rules is widely studied in VLE properties.In the mixing rules,the cubic equations of state have a continuous and consistent description of supercritical and subcritical regional components,but the prediction accuracy of complex systems (such as highly nonideal systems and strong polarity systems) is not advanced,while the activity coefficient model has very good descriptive ability for highly nonideal systems and strong polarity systems but only for low-pressure conditions.Therefore,combining these two approaches to form an EOS/GEmodel can play to their respective advantages and establish a broader range of mixing rules[6].

The mixing rules of the excess gibbs free energy mainly include the HV[10]mixing rules,WS[11]mixing rules,MHV1[12]mixing rules,MHV2[13]mixing rules,and LCVM[14]mixing rules.The activity coefficient is the key to calculating the mixing rules.Many activity coefficient models have been proposed,such as Wilson[15],NRTL[16],and the group contribution model (such as UNIQUAC[15]and UNIFAC[17]). The group contribution model mainly includes original UNIFAC and modified UNIFAC (Dortmund)[18–21],PSRK[22–25],and VTPR[26–28],which have been widely used to predict the VLE properties of binary mixtures[6].

At present,there are some studies on the group contribution theory based on CO2-based binary mixtures.Hou et al.[29]used the MHV1+UNIFAC group contribution model to predict the vapor-liquid phase equilibrium properties of CO2and five HFC refrigerants,with high prediction accuracy.Gao et al.[30]used the MHV1+UNIFAC and MHV2+UNIFAC group contribution models to predict the VLE properties of CO2+hydrocarbons,but the prediction accuracy was poor.Horstman et al.[31]used the PRSK group contribution model to predict the VLE properties of CO2+n-alkane.Schmid et al.[28]used the VTPR group contribution model to predict the vaporliquid phase equilibrium properties of CO2+CH4.The prediction accuracy was high,but only the CO2+CH4working medium was not representative.In summary,the LCVM+UNIFAC group contribution model has not been used to predict the vapor-liquid phase equilibrium properties of CO2+HFC binary mixtures.Since vapor-liquid phase equilibrium experiments require considerable time and effort,it is also of great significance to propose a prediction model suitable for CO2binary mixtures.

In this work,based on the background of the power cycle of CO2binary mixtures,the experimental vapor-liquid phase equilibrium data of 9 CO2+HFC binary mixtures were collected,including CO2+HFC (R23,R32,R41,R125,R134a,R143a,R152a,R161,R227ea).Based on the group contribution model (UNIFAC),the VLE data of these 9 binary mixtures were predicted by using the PR equation combined with two different forms of mixing rules (MHV1 and LCVM[14]).By comparing the predicted values with the experimental values published in the literature,the relative deviation between the prediction results and the experimental results is obtained to analyze and discuss the applicability of different mixing rules to various types of CO2+HFC binary mixtures.

2 Thermodynamic equations

2.1 PR equation of state

Peng and Robinson[7]proposed the PR equation of state in 1976,which can be used to predict the VLE properties of CO2+HFC refrigerant binary mixtures.

whereRrepresents the universal gas constant (8.314472 J·mol-1·K-1);Tis the temperature (K);vis the molar volume(cm3·mol-1);andpis the pressure (MPa).

The two parametersaandbrepresent the energy parameter (J·cm3·mol-2) and covolume (cm3·mol-1),and their expressions are shown:

whereTcandpcdenote the critical temperature (K),and critical pressure (MPa),respectively.α (T) represents the temperaturedependent alpha function,ωis the acentric factor,andTr=T/Tcis the reduced temperature.

2.2 EOS/GE mixing rules

2.2.1 HV mixing rule

The first EOS/GEmixing rule was proposed by Huron and Vidal[10],

whereCEOSis a constant related to the state equation;for the PR equation,CEOS=-0.62323.GErepresents the excess Gibbs free energy under infinite pressure,which can be calculated by the activity coefficient model.

2.2.2 MHV1 mixing rule

The MHV1 mixing rule was proposed by Michelse[12],which is an improved mixing rule based on the HV mixing rule.

whereq1is a constant related to the state equation;for the PR equation,q1=-0.528.

2.2.3 LCVM mixing rule

Boukouvalas[14]found that for some binary systems,the HV mixing rule tends to give a negative bias,while the MHV1 mixing rule gives a positive deviation.Boukouvalas linearly combines two mixing rules to form a new LCVM mixing rule,combined with the UNIFAC activity coefficient model,which gives highly accurate predictions for many systems.

where the value ofλis generally 0.36.CHVandCMHV1are constants related to the state equation,CHV=-0.62323 andCMHV1=-0.528.

2.3 Activity coefficient model

In 1968,Renon and Prausnitz[16]modified the local composition expression of the solution and proposed the NRTL activity coefficient model on the basis of two-fluid theory,which can well describe the phase equilibrium properties of binary systems,and the expressions for common binary systems are

The UNIFAC[32]model is proposed on the basis of the UNIQUAC[33]model,which is used to calculate the activity coefficients for CO2+HFC binary mixtures.The UNIFAC model obtains the interaction parameters between groups by fitting the experimental data and then calculates the activity coefficient of the binary mixture through the interaction parameters.

where γiis the activity coefficient of componenti;is the combinatorial part;andis the residual part.

The combinatorial part of the activity coefficient is the same as the UNIQUAC model,which depends on the molecule’s size and shape.

where ϕiis a section fraction similar to the volume fraction;xidenotes the mole fraction;θirepresents the area fraction;Zis the number of coordinates;is the number of groupk;riandqiare structural parameters of the pure component;andRkandQkare volume parameters and area parameters of the group,respectively.

The residual part of the activity coefficient is the basis for the group contribution model.

whereΓkrepresents the group activity coefficient;amnandanmare the group partner energy parameters;Qmis the area parameter;Θmis the area fraction;andXmdenotes the molar fraction of groupm.

2.4 Thermodynamic conditions

The thermodynamic condition for the VLE is the equality of the fugacities of each component in each phase,which is expressed as

Fig.1 shows the group contribution prediction model that predicts the VLE properties of the binary mixture.Based on the known temperature and liquid phase components and the assumed pressure and vapor phase components,the fugacity,specific pressure and vapor phase components are calculated by the cubic equation of state (PR EOS) combined with the mixing rule (MHV1 and LCVM) and the group contribution activity coefficient model (UNIFAC).

Fig.1. Flow diagram of the VLE calculation.

2.5 Objective function

The objective function (OF) is expressed as follows:

where the subscript “cal” indicates the calculated value;the subscript “exp” indicates the experimental value;the subscript “i” indicates componenti;pis the pressure;andyis the vapor phase molar fraction.The bubble-point pressure and the vapor phase molar fraction are obtained according to the temperature and liquid phase molar fraction.

where AARDpis the average absolute relative deviation of pressure;AADy1is the average absolute deviation of vapor phase molar fractions;andNis the number of experimental data.

3 Results and discussion

3.1 Prediction model comparison

Table 1 lists the thermodynamic parameters of the working fluid,mainly including critical pressure,temperature,and acentric factors.By referring to the group assignments law of the UNIFAC model,the group assignments of all working fluids are illustrated in Table 2[29].

Table 1. Thermodynamic parameters of the working fluid[6].

Table 2. Group assignment for refrigerants[29].

The group parameters of CO2and the group interaction parameters among -CO2,-Alkane,-CHF,and -CHF3are determined by regressing the VLE data,as shown in Table 3.

Table 3. The group interaction parameters[6].

The VLE properties of 9 CO2+HFCs (R23,R32,R41,R125,R134a,R143a,R152a,R161,and R227ea) were predicted by the PR+MHV1+UNIFAC model and the PR+LCVM+UNIFAC model.Table 4 lists the predicted values and deviations for the group contribution model.Using the PR+MHV1+UNIFAC group contribution model,the value of AARDpis 7.40%,and the value of AADy1 is0.0229;using the PR+LCVM+UNIFAC group contribution model,the value of AARDpis 5.53%,and the value of AADy1is 0.0132.According to the prediction results of the group contribution model,the PR+LCVM+UNIFAC group contribution model has significantly improved the calculation progress compared with the PR+MHV1+UNIFAC group contribution model used in the previous system.However,for CO2+R32 binary mixtures,the PR+MHV+UNIFAC prediction model has higher calculation accuracy,and the values of AARDpand AADy1are 1.53% and 0.0045,respectively.In summary,for CO2+HFC binary mixtures,the PR LCVM+UNIFAC group contribution model has higher prediction accuracy overall,but for individual CO2binary mixtures (such as CO2+R32),the MHV1+UNIFAC model also has unique advantages.

Table 4. The predicted values and deviations for the group contribution model.

3.2 Prediction of the vapor-liquid phase equilibrium properties of CO2+HFC binary mixtures

Based on the group contribution prediction model(PR+MHV1+UNIFAC and PR+LCVM+UNIFAC),the vaporliquid phase equilibrium properties of CO2+HFC binary mixtures are predicted.Figs.2–4 show the vapor-liquid phase equilibrium of CO2+R32[35],CO2+R143a[38],and CO2+R152a[39]binary mixtures,respectively.It can be seen from the figure that the predictive model (PR+LCVM+UNIFAC) and the known experimental data have a good match.It can be seen from Fig.2 that the vapor-liquid phase equilibrium properties of the CO2+R32 binary mixture are predicted well by the PR+MHV1+UNIFAC group contribution model,which is consistent with the previous calculation results.From Fig.3 and Fig.4,the vapor-liquid phase equilibrium of the CO2+R143a and CO2+R152a binary mixtures predicted by the PR+LCVM+UNIFAC group contribution model is better than that predicted by PR+MHV1+UNIFAC.

Fig.2. Prediction of the vapor-liquid phase equilibrium properties for CO2+R32 at three temperatures.Experimental data[35]: T=283.12 K (●,○);T=303.13 K (,□).Solid red line represents the PR+MHV1+UNIFAC model;solid blue line represents the PR+LCVM+UNIFAC model.

Fig.3. Prediction of the vapor-liquid phase equilibrium properties for CO2+R143a at three temperatures.Experimental data[38]: T=273.15 K(●,○);T=303.15 K (,□).Solid red line represents the PR+MHV1+UNIFAC model;solid blue line represents the PR+LCVM+UNIFAC model.

Fig.4. Prediction of the vapor-liquid phase equilibrium properties for CO2+R152a at three temperatures.Experimental data[39]: T=278.25 K(▲,△);T=298.84 K (●,○);T=3323.30 K (,□).Solid red line represents the PR+MHV1+UNIFAC model; solid blue line represents the PR+LCVM+UNIFAC model.

4 Conclusions

In this work,to accurately predict the VLE of CO2and HFC(R23,R32,R41,R125,R134a,R143a,R152a,R161,and R227ea) binary mixtures,a group contribution model based on the excess free energy (GE) mixing rules(PR+MHV1+UNIFAC and PR+LCVM+UNIFAC) is established.Based on the interaction parameters between groups such as -CO2,-Alkane,-CHF,and -CHF3,the vaporliquid phase equilibrium properties of CO2and HFC refrigerants were predicted,and the value of AARDpcalculated by the PR+LCVM+UNIFAC model was 5.53%,the value of AADy1was 0.0132,and the AARDpvalue and AADy1values of the PR+MHV1+UNIFAC model were 7.40% and 0.0229,respectively. The prediction accuracy of the PR+LCVM+UNIFAC model for CO2+HFC binary mixtures is more accurate.However,for CO2+R32 binary mixtures,the PR+MHV+UNIFAC prediction model has higher calculation accuracy,and the values of AADRpand AADyare 1.53%and 0.0045,respectively.In summary,for CO2+HFC binary mixtures,the PR+LCVM+UNIFAC group contribution model has higher prediction accuracy overall,but for individual CO2binary mixtures (such as CO2+R32),the PR MHV1+UNIFAC model also has unique advantages.It can be seen from the figure that the predictive model(PR+LCVM+UNIFAC) and the known experimental data have a good match.It is further explained that for CO2+HFC binary systems with very different molecular structures,the LCVM mixing rule obtains better results than the MHV1 mixing rule.

Acknowledgements

This work was supported by the USTC Research Funds of the Double First-Class Initiative (YD2090002008) and the Fundamental Research Funds for the Central Universities(WK2090000032).

Conflict of interest

The authors declare that they have no conflict of interest.

Biographies

Zirui Wuis a doctoral candidate under the supervision of Prof.Gequn Shu at the University of Science and Technology of China.His current research is focused on CO2-based binary mixtures physical properties.

Lingfeng Shiis an Associate Researcher at department of Thermal Science and Energy Engineering,University of Science and Technology of China.He received the B.S.degree in Xi’an Jiaotong University and the Ph.D.degree in Tianjin University.His research interests focus on thermal cooling and power conversion cycle,design of heat exchanger components and mixtures physical properties.