In this work, thermodynamic models based on the corresponding states framework with departure terms are developed for the refrigerant pairs R-32/1234yf, R-32/1234ze(E), R-1234ze(E)/227ea, R-1234yf/152a, and R-125/1234yf. These models are based on new measurements of density, speed of sound, and phase equilibria, combined with the data available in the literature. The model for R-32/1234yf is most comprehensive in its data coverage, with speed of sound deviations within 1%, density deviations within 0.1%, and bubble- and dew-point pressure deviations within 1%. The other mixtures have generally more limited data availability but a similar goodness of fit.

The increasing use of hydrofluoroolefins (HFOs) to replace hydrofluorocarbons (HFCs) in refrigeration applications1 motivates the development of updated thermodynamic models for HFO-containing blends. These models are needed to carry out thermodynamic cycle analysis, design and optimize components, and so on. The previous paper in this series2 developed a new generation of refrigerant mixture models, following in the lineage of similar efforts around the turn of the twenty-first century3,4 and adding modern computational tools (evolutionary global optimization, teqp5 for EOS evaluation), based on new measurements from the National Institute of Standards and Technology (NIST).

The five binary mixtures included in this work [R-32/1234yf, R-32/1234ze(E), R-1234ze(E)/227ea, R-1234yf/152a, and R-125/1234yf] were specified in the project scope based on industrial need.

The mixture of R-32/1234yf is perhaps the most important HFO-containing binary pair for which updated parameters are needed. This binary pair yields the R-454X family of binary mixtures specified in ASHRAE standard 346 (the X indicating a placeholder). This binary pair also appears in other multicomponent mixtures recently added to the standard (e.g., R-448X, R-449X, R-452X, R-457X, and R-468X). The other binary mixtures described in this work also appear in blends recently added to the standard. R-32/1234ze(E) appears in the blend R-444X (and many others, see Ref. 7). R-1234ze(E)/227ea are the components of R-515B and appears in R-471X. R-1234yf/152a appears in the blend R-457B. R-125/1234yf appears in the blend R-448B.

Information on the pure components considered in this study is included in Table 1. A key point to highlight is that R-32 has a markedly higher critical pressure than the other fluids. The critical points of the pure components are shown in Fig. 1, as well as the critical curves for each of the binary mixtures from the models of REFPROP 10.0 and the models developed in this work. With the exception of R-32/1234yf, these models had no critical-point data and are, thus, a test of the reasonableness of the high-pressure phase equilibria behavior.

TABLE 1.

Metadata on the pure components, sorted by normal boiling point temperature. Numerical values are taken from the respective EOS and are rounded for presentation; the unrounded value from the EOS should be used

ASHRAE nameHashaLong namebTNBP (K)Tcrit (K)pcrit (MPa)M (kg mol−1)
R-328  7b05bb60 difluoromethane 221.499 351.255 5.782 00 0.052 024 
R-1259  25c5a3a0 pentafluoroethane 225.060 339.173 3.617 70 0.120 021 
R-1234yf10  40377b40 2,3,3,3-tetrafluoroprop-1-ene 243.692 367.850 3.384 40 0.114 042 
R-152a11  63f364b0 1,1-difluoroethane 249.127 386.411 4.516 75 0.066 051 
R-1234ze(E)12  9905ef70 (E)-1,3,3,3-tetrafluoropropene 254.177 382.513 3.634 90 0.114 042 
R-227ea13  40091ee0 1,1,1,2,3,3,3-heptafluoropropane 256.810 374.900 2.925 00 0.170 029 
ASHRAE nameHashaLong namebTNBP (K)Tcrit (K)pcrit (MPa)M (kg mol−1)
R-328  7b05bb60 difluoromethane 221.499 351.255 5.782 00 0.052 024 
R-1259  25c5a3a0 pentafluoroethane 225.060 339.173 3.617 70 0.120 021 
R-1234yf10  40377b40 2,3,3,3-tetrafluoroprop-1-ene 243.692 367.850 3.384 40 0.114 042 
R-152a11  63f364b0 1,1-difluoroethane 249.127 386.411 4.516 75 0.066 051 
R-1234ze(E)12  9905ef70 (E)-1,3,3,3-tetrafluoropropene 254.177 382.513 3.634 90 0.114 042 
R-227ea13  40091ee0 1,1,1,2,3,3,3-heptafluoropropane 256.810 374.900 2.925 00 0.170 029 
a

The hash used in REFPROP to define the unique identifier used in the mixture models is generated from the SHA256 hash of the standard InChI key from hashlib.sha256(StdInChIkey).hexdigest()[2:9] + “0” with the hashlib module from the Python standard library.

b

The long name used in REFPROP 10.0.

FIG. 1.

Critical curves for the binary pairs in this study. Pure fluid and mixture models of REFPROP were converted to teqp form and integrated. The dashed curves are the REFPROP 10.0 models, and the solid curves are the models developed in this work.

FIG. 1.

Critical curves for the binary pairs in this study. Pure fluid and mixture models of REFPROP were converted to teqp form and integrated. The dashed curves are the REFPROP 10.0 models, and the solid curves are the models developed in this work.

Close modal

The models developed in this work are part of a larger project on the development of new and updated thermodynamic models for HFO-containing refrigerant mixtures. The NIST measurements included the following:

  • Density measurements with a two-sinker densimeter (all five binary pairs except for R-125/1234yf). These measurements are in the gas phase and in the liquid phase.

  • Pulse-echo speed of sound measurements in the liquid phase (all five binary pairs).

  • Bubble-point pressure measurements (all five binary pairs).

For each type of property, measurements were carried out at nominal molar compositions of 2:1 and 1:2 for each binary pair.

All other existing experimental data for these binary pairs we are aware of have been captured in NIST ThermoData Engine. NIST ThermoData Engine has a nearly comprehensive coverage of experimental data captured in the last few decades in the most important journals containing mixture experimental data; data provided via the NIST ThermoData Engine SRD#103a version 10.4.3 were used. Lists of references are provided below for each binary pair. The data used for model optimization are provided in the supplementary material.

The model optimization approach used in this work is the same as in the previous paper in this series2 and so will be covered only very briefly. The model formulation is based on the residual Helmholtz energy,

αr(τ,δ,x)=αCSr+αdepr,
(1)

where αCSr is the corresponding-states contribution given by

αCSr=i=1Nxiαr0,i(τ,δ),
(2)

with τ and δ being the reduced variables defined by τ = Tred(x)/T and δ = ρ/ρred(x). The pure fluid equations of state (EOS) used in this work to obtain the pure fluid contribution αr0,i(τ,δ) are the same as those used in REFPROP 10.0 with the exception of R-1234yf; in REFPROP 10.0, the EOS of Richter et al.14 was used, while this work uses the new EOS from Lemmon and Akasaka.10 

The reducing function is defined by

Yred(x)=i=1Nxi2Ycrit,i+i=1N1j=i+1N2xixjxi+xjβY,ij2xi+xjYij,
(3)

where Y is the parameter of interest, either molar volume v = 1/ρ or temperature T. The necessary parameters are given by

Tij=βT,ijγT,ij(Tcrit,iTcrit,j)0.5,
(4)
vij=18βv,ijγv,ijvcrit,i1/3+vcrit,j1/33,
(5)

where βT,ij, γT,ij, βv,ij, and γv,ij are fitted parameters. The departure contribution in Eq. (1) is given by

αdepr=i=1Nj=i+1NxixjFijαrij(τ,δ),
(6)

where Fij is the scaling factor, normally equal to 1.0 if a departure term has been fitted and zero if not.

The departure term αrij(τ,δ) for the ij pair is defined by exponential terms,

αrij(τ,δ)=knkτtkδdkexpsgn(lk)δlk,
(7)

where sgn is the sign function, 0 for an argument of zero and 1 for positive arguments.

It was highlighted in Ref. 2 that the liquid-phase speed of sound for R-1234ze(E) is not well-represented by the existing EOS.12 The pure-fluid deviations in speed of sound according to the best EOS are shown in Fig. S1 in the supplementary material. This analysis shows that the most accurate speed of sound measurements for R-227ea and R-1234yf are reproduced to near the experimental uncertainty because they were used in model development and that in all other cases, the speed of sound measurements show significant scatter outside the uncertainty bands of the most accurate measurements (with combined expanded uncertainties generally less than 0.1%). The EOS for R-1234ze(E) and R-32 have enough high-quality modern measurements of liquid-phase speed of sound that their refitting would result in a significantly better reproduction of the existing speed of sound measurements, as well as other caloric properties.

A similar approach to the previous paper2 was taken to optimize the model parameters, with the important change of adding the phase equilibria pressures into the optimization process. This change was particularly necessary for R-32/1234yf to better represent the bubble-point pressures measured in our group. The other types of data used in the fit are speed of sound data and density data. A comprehensive description of the optimization approach is not possible to include here, so the complete set of fitting code is provided in the supplementary material, along with the input data files, weights, error metrics, etc., to ensure that the process is reproducible, although the outputs will not be, because the process is stochastic.

The bubble-point pressure is obtained as an iterative calculation starting from guess values for vapor and liquid densities and vapor-phase composition (the liquid-phase composition is specified). In this work, these guess values were obtained from REFPROP 10.0, and the mix_VLE_Tx function of teqp is used to do the iteration to solve for the bubble-point pressure, given the liquid-phase composition and temperature (and with the necessary guess values). The bubble-point pressure is obtained by equating pressures of the co-existing phases, the chemical potentials of each component in both phases, and imposing the liquid-phase composition. The iteration can fail if the model at the given step is not well behaved or if the iterations converge to the trivial solution (for which the liquid and vapor phases are at the same temperatures, densities, and compositions, which trivially satisfies the phase equilibria conditions). The problems of working with “bad” models are exacerbated by global optimization methods that explore the complete parameter space. In the case of failure, a large value is set for the deviation, penalizing “bad” models. The error term for the ith data point that is added into the cost function summation is

rp,i=100×pip meas,i1.
(8)

Plotting the cost function (the sum of each of the contributions to the cost function) as a function of the number of adjustable parameters in the fit gives a sense of how many terms should be used in the departure function. One would like a model with as few terms as possible to minimize the likelihood of non-physical behaviors and overfitting. Figure 2 shows the optimization runs, normalized by the best cost function for the binary pair. The values for Nfitted = 4 are for no terms in the departure function (just a reducing function), and two parameters (nk and tk) are added per term in the departure function.

FIG. 2.

Cost function history in the optimization as a function of number of fitted parameters. The best values of the cost function for each number of fitted parameters is plotted, and each pair is normalized by its best cost function value.

FIG. 2.

Cost function history in the optimization as a function of number of fitted parameters. The best values of the cost function for each number of fitted parameters is plotted, and each pair is normalized by its best cost function value.

Close modal

Table 2 summarizes the interaction parameters to be used in the reducing function, and Tables 37 summarize the coefficients that go into Eq. (7) for each binary pair. R-32/1234yf has more data available and is one of the more asymmetric mixtures studied here.

TABLE 2.

Interaction parameters obtained in this work. The order matters; each pair has been sorted in the increasing normal boiling point temperature order of the components

Pair (1/2)βT,12γT,12βv,12γv,12Fij
R-32/1234ze(E) 1.005 148 0.946 105 0.999 032 1.029 821 1.0 
R-32/1234yf 1.011 011 0.966 965 1.007 852 1.037 663 1.0 
R-1234yf/152a 0.995 987 0.978 058 0.999 710 1.008 608 1.0 
R-1234ze(E)/227ea 1.001 247 0.989 180 0.999 290 1.001 581 1.0 
R-125/1234yf 0.999 969 0.998 028 0.998 226 1.006 068 1.0 
Pair (1/2)βT,12γT,12βv,12γv,12Fij
R-32/1234ze(E) 1.005 148 0.946 105 0.999 032 1.029 821 1.0 
R-32/1234yf 1.011 011 0.966 965 1.007 852 1.037 663 1.0 
R-1234yf/152a 0.995 987 0.978 058 0.999 710 1.008 608 1.0 
R-1234ze(E)/227ea 1.001 247 0.989 180 0.999 290 1.001 581 1.0 
R-125/1234yf 0.999 969 0.998 028 0.998 226 1.006 068 1.0 
TABLE 3.

Coefficients for the R-32/1234yf departure function

knktkdklk
0.449 487 2.538 296 
−0.099 708 3.208 604 
0.098 897 0.456 509 
knktkdklk
0.449 487 2.538 296 
−0.099 708 3.208 604 
0.098 897 0.456 509 
TABLE 4.

Coefficients for the R-32/1234ze(E) departure function

knktkdklk
0.295 192 0.500 457 
−0.737 904 3.104 735 
0.542 962 3.201 720 
−0.201 951 2.567 118 
knktkdklk
0.295 192 0.500 457 
−0.737 904 3.104 735 
0.542 962 3.201 720 
−0.201 951 2.567 118 
TABLE 5.

Coefficients for the R-1234yf/152a departure function

knktkdklk
0.009 165 3.221 227 
0.014 392 1.007 082 
−0.014 246 3.790 953 
knktkdklk
0.009 165 3.221 227 
0.014 392 1.007 082 
−0.014 246 3.790 953 
TABLE 6.

Coefficients for the R-125/1234yf departure function

knktkdklk
−0.032 013 0.008 03 
knktkdklk
−0.032 013 0.008 03 
TABLE 7.

Coefficients for the R-1234ze(E)/227ea departure function

knktkdklk
−0.057 178 1.290 298 
0.031 318 0.038 796 
−0.027 496 2.640 532 
knktkdklk
−0.057 178 1.290 298 
0.031 318 0.038 796 
−0.027 496 2.640 532 

The error metrics are used to quantify the goodness of fit. The deviation between an array of experimental values and an array of model-calculated values is defined by

DEV=100×YexpYEOSYEOS,
(9)

and the overall error metric (the average absolute relative deviation, AARD) is defined by

AARD=mean(abs(DEV)).
(10)

The results for each of the studied binary pairs are included in this section. The binary pairs are presented roughly in order of confidence in the model. R-32/1234yf has been studied by many research groups, and certain categories of experimental data (particularly PTxy data) are well studied. The remaining four binary pairs have decreasing quality and comprehensiveness of their data coverage. The last binary pair [R-1234ze(E)/227ea] is based exclusively on new measurements carried out as part of this project; corroborative measurements from other research groups would be valuable.

TABLE 8.

The collection of existing data for R-32/1234yf (PVT: pvT data, SOS: speed of sound, VLE: vapor–liquid equilibria, N: number of data points, CRIT: critical point, CP: constant-pressure specific heat, and x1: composition of the first component)

KindFirst author (year)Nx1 (mole frac.)T (K)
PVT Kobayashi (2011)15  295 0.35–0.69 310.0–395.0 
PVT Akasaka (2013)16  40 0.20–0.69 332.8–363.6 
PVT Dang (2015)17  26 0.48–0.84 283.4–318.6 
PVT Cai (2018)18  153 0.39–0.94 279.8–347.9 
PVT Yang (2019)19  492 0.12–0.87 298.8–383.1 
PVT Jia (2020)20  485 0.19–0.90 283.5–363.2 
PVT Tomassetti (2020)21  217 0.12–0.95 263.1–383.1 
PVT Fortin (2023)22  217 0.34–0.67 230.1–400.2 
SOS Shimoura (2011)23  92 0.63–0.63 283.1–313.1 
SOS Rowane (2022)24  94 0.34–0.67 230.0–345.0 
VLE Kamiaka (2010)25  35 0.34–0.84 273.3–333.1 
VLE Kobayashi (2011)15  11 0.35–0.69 347.8–359.7 
VLE Shimoura (2011)23  0.63–0.63 283.1–313.1 
VLE Kamiaka (2013)26  79 0.00–1.00 273.3–333.1 
VLE Hu (2017)27  55 0.00–1.00 283.1–323.1 
VLE Al Ghafri (2019)28  0.79–0.80 273.7–340.9 
VLE Yamada (2020)29  53 0.09–0.76 283.1–329.9 
VLE Li (2021)30  77 0.00–1.00 273.1–358.1 
VLE Outcalt (2022)31  30 0.37–0.69 270.0–350.0 
CRIT Kobayashi (2011)15  0.35–0.69 352.4–359.0 
CRIT Akasaka (2013)16  0.20–0.69 352.4–363.1 
CP Al Ghafri (2019)28  0.50–0.50 283.2–283.2 
KindFirst author (year)Nx1 (mole frac.)T (K)
PVT Kobayashi (2011)15  295 0.35–0.69 310.0–395.0 
PVT Akasaka (2013)16  40 0.20–0.69 332.8–363.6 
PVT Dang (2015)17  26 0.48–0.84 283.4–318.6 
PVT Cai (2018)18  153 0.39–0.94 279.8–347.9 
PVT Yang (2019)19  492 0.12–0.87 298.8–383.1 
PVT Jia (2020)20  485 0.19–0.90 283.5–363.2 
PVT Tomassetti (2020)21  217 0.12–0.95 263.1–383.1 
PVT Fortin (2023)22  217 0.34–0.67 230.1–400.2 
SOS Shimoura (2011)23  92 0.63–0.63 283.1–313.1 
SOS Rowane (2022)24  94 0.34–0.67 230.0–345.0 
VLE Kamiaka (2010)25  35 0.34–0.84 273.3–333.1 
VLE Kobayashi (2011)15  11 0.35–0.69 347.8–359.7 
VLE Shimoura (2011)23  0.63–0.63 283.1–313.1 
VLE Kamiaka (2013)26  79 0.00–1.00 273.3–333.1 
VLE Hu (2017)27  55 0.00–1.00 283.1–323.1 
VLE Al Ghafri (2019)28  0.79–0.80 273.7–340.9 
VLE Yamada (2020)29  53 0.09–0.76 283.1–329.9 
VLE Li (2021)30  77 0.00–1.00 273.1–358.1 
VLE Outcalt (2022)31  30 0.37–0.69 270.0–350.0 
CRIT Kobayashi (2011)15  0.35–0.69 352.4–359.0 
CRIT Akasaka (2013)16  0.20–0.69 352.4–363.1 
CP Al Ghafri (2019)28  0.50–0.50 283.2–283.2 
TABLE 9.

The collection of existing data for R-32/1234ze(E). (PVT: pvT data, SOS: speed of sound, VLE: vapor–liquid equilibria, N: number of data points, CRIT: critical point, CP: constant-pressure specific heat, CV: constant-volume specific heat, and x1: composition of the first component)

KindFirst author (year)Nx1 (mole frac.)T (K)
PVT Kobayashi (2010)40  23 0.42–0.69 341.2–371.8 
PVT Kobayashi (2011)41  330 0.42–0.69 310.0–400.0 
PVT Tanaka (2011)42  81 0.43–0.67 310.0–350.0 
PVT Yamaya (2011)43  47 0.70–0.70 280.1–460.1 
PVT Jia (2016)44  540 0.17–1.00 283.5–362.7 
PVT Tomassetti (2020)45  182 0.17–0.95 263.1–373.1 
PVT Fortin (2023)22  237 0.34–0.67 230.0–400.0 
SOS Rowane (2022)24  341 0.34–0.67 230.0–345.0 
VLE Kobayashi (2011)41  14 0.42–0.69 358.1–371.5 
VLE Tanaka (2011)42  10 0.43–0.67 310.0–350.0 
VLE Akasaka (2013)33  151 0.06–0.94 272.9–314.0 
VLE Hu (2017)46  65 0.00–1.00 283.1–323.1 
VLE Hu (2017)27  15 0.15–0.73 283.1–323.1 
VLE Kou (2019)47  36 0.00–1.00 293.1–323.1 
VLE Yang (2021)48  25 0.00–1.00 263.1–323.1 
VLE Outcalt (2022)31  37 0.34–0.64 270.0–360.0 
CRIT Kobayashi (2010)40  0.42–0.69 362.2–370.8 
CRIT Kobayashi (2011)41  0.42–0.69 362.2–370.8 
CP Tanaka (2011)42  38 0.43–0.67 310.0–350.0 
CV Yamaya (2011)43  47 0.70–0.70 280.1–460.1 
CP Gao (2018)49  25 0.23–0.95 310.1–330.1 
KindFirst author (year)Nx1 (mole frac.)T (K)
PVT Kobayashi (2010)40  23 0.42–0.69 341.2–371.8 
PVT Kobayashi (2011)41  330 0.42–0.69 310.0–400.0 
PVT Tanaka (2011)42  81 0.43–0.67 310.0–350.0 
PVT Yamaya (2011)43  47 0.70–0.70 280.1–460.1 
PVT Jia (2016)44  540 0.17–1.00 283.5–362.7 
PVT Tomassetti (2020)45  182 0.17–0.95 263.1–373.1 
PVT Fortin (2023)22  237 0.34–0.67 230.0–400.0 
SOS Rowane (2022)24  341 0.34–0.67 230.0–345.0 
VLE Kobayashi (2011)41  14 0.42–0.69 358.1–371.5 
VLE Tanaka (2011)42  10 0.43–0.67 310.0–350.0 
VLE Akasaka (2013)33  151 0.06–0.94 272.9–314.0 
VLE Hu (2017)46  65 0.00–1.00 283.1–323.1 
VLE Hu (2017)27  15 0.15–0.73 283.1–323.1 
VLE Kou (2019)47  36 0.00–1.00 293.1–323.1 
VLE Yang (2021)48  25 0.00–1.00 263.1–323.1 
VLE Outcalt (2022)31  37 0.34–0.64 270.0–360.0 
CRIT Kobayashi (2010)40  0.42–0.69 362.2–370.8 
CRIT Kobayashi (2011)41  0.42–0.69 362.2–370.8 
CP Tanaka (2011)42  38 0.43–0.67 310.0–350.0 
CV Yamaya (2011)43  47 0.70–0.70 280.1–460.1 
CP Gao (2018)49  25 0.23–0.95 310.1–330.1 

The mixture of R-32/1234yf is the best experimentally studied mixture of those included in this study. Although the coverage of speed of sound data remains somewhat sparse (all the mixture data are in the liquid phase), there are multiple overlapping datasets for phase equilibria and density, providing a quite good coverage of the phase diagram. The datasets are listed in Table 8. The data in Ref. 16 were previously presented in Ref. 32, and the newer reference was retained as it is more broadly accessible.

The baseline mixture model from REFPROP 10.0 is based on the EOS for R-32 from Ref. 8, the EOS for R-1234yf from Ref. 14, and the mixing model from Ref. 33.

4.1.1. VLE

In total, there are nine datasets with VLE data and another with critical loci from Ref. 16. Most of the data cover the temperature range of practical refrigeration, and the only datasets covering the higher-temperature region are from the works of Li et al.,30 Kobayashi et al.15 and Outcalt and Rowane.34 The measurements are largely PTxy where the phase compositions are obtained by sampling from the co-existing phases, an exception being that of Ref. 34, which are “non-invasive” bubble-point measurements for gravimetrically prepared mixtures.

Figure 3 joins all the VLE plots into one figure. The upper part of the figure is a px diagram showing the experimental data as well as traced isotherms for all of the experimentally measured temperatures. The qualitative behavior is excellent; there are no spurious wiggles or obviously physically incorrect behaviors. The isotherms were traced with the teqp library, starting from pure component 2. The critical locus was also traced with teqp with the algorithm from Deiters and Bell,35 showing that it is a type I mixture (has a critical locus smoothly connecting the pure components).36 Although it is not particularly obvious in the figure, the slight dip in the R-32-rich end of the critical curve indicates that the two-phase equilibria portions of the phase diagram are separated for a very narrow range of temperatures around the critical temperature of R-32, analogous to the case of R-1234yf/134a.2 

FIG. 3.

VLE plots for R-32/1234yf. Experimental datasets for R-32/1234yf are listed in Table 8. Top: A px diagram with points and curves colored by temperature. Bottom left: Deviation plots for bubble-point data. Bottom right: Deviation plots for dew-point data. For the deviation plots, the AARD is listed in the legend, and the portions of the plot shaded in gray are on a logarithmic scale. Thin vertical gray lines indicate values of the independent variable at the failure of an iterative calculation.

FIG. 3.

VLE plots for R-32/1234yf. Experimental datasets for R-32/1234yf are listed in Table 8. Top: A px diagram with points and curves colored by temperature. Bottom left: Deviation plots for bubble-point data. Bottom right: Deviation plots for dew-point data. For the deviation plots, the AARD is listed in the legend, and the portions of the plot shaded in gray are on a logarithmic scale. Thin vertical gray lines indicate values of the independent variable at the failure of an iterative calculation.

Close modal

The lower half of the figure shows the bubble- and dew-point deviations as a function of temperature and composition for this model and the model of Akasaka. The model from Akasaka33 implemented in REFPROP 10.0 represents the experimental data available with an overall AARD in bubble-point pressure of 1.05% and an overall AARD in dew-point pressure of 1.49%. The new model improves the deviations, and the AARD for the bubble-point pressures are 0.45% and 0.54% for the dew-point pressures.

The dataset of Shimoura et al.23 is biased systematically by ∼10%. Otherwise, the datasets are largely in good agreement, and the data are represented to within mostly 1%, except for some of the higher temperature data from Li et al.30 and a point from Al Ghafri et al.28 

Overall, the data quality and availability for this binary pair appear to be suitable for the development of an accurate PTxy surface. The datasets appear to be largely consistent and the model reproduces the data within ∼1% in phase equilibrium pressure.

4.1.2. PVT

As has been mentioned before,2,7 assessing the quality of fit for density data from a thermodynamic model is non-trivial. A complicating factor is that iterative solutions for density may disappear as small changes are made to the model; the problem is especially pronounced for saturated state points in the critical region. Statistics of fit quality, such as the AARD, may be deceptive because functions may become quite flat, and a large value of AARD for a density dataset does not necessarily mean that the data and model are in meaningful disagreement. Taking this preamble into account, overall density deviation plots for the model for R-32/1234yf are presented in Fig. 4. The model developed in this work yields deviations that are sometimes tens of percents in the worst case, but this error metric is deceptive, as was discussed previously.7 The portions of the phase diagram will be investigated individually.

FIG. 4.

PVT deviations for the model in this work and that of REFPROP 10.0 for R-32/1234yf. The references are as listed in Table 8, and the AARD for each dataset are listed in the legends. The portions of the plot shaded in gray are on a logarithmic scale. Thin vertical gray lines indicate values of the independent variable at the failure of an iterative calculation.

FIG. 4.

PVT deviations for the model in this work and that of REFPROP 10.0 for R-32/1234yf. The references are as listed in Table 8, and the AARD for each dataset are listed in the legends. The portions of the plot shaded in gray are on a logarithmic scale. Thin vertical gray lines indicate values of the independent variable at the failure of an iterative calculation.

Close modal

The gas phase has been studied by a number of groups. An expanded view of the gas-phase deviations are presented in Fig. S3 in the supplementary material. The quality of the low-density gas data and models can be assessed from the virial coefficients. The virial expansion for low-density gases reads

ZpρRT=1+Bρ+Cρ2+,
(11)

where B and C are the second and third virial coefficients, respectively, and each have composition and temperature dependence. The second virial coefficient for a binary mixture is given by

B(T,x)=x12B1(T)+2x1x2B12(T)+x22B2(T),
(12)

where B1 and B2 are the second virial coefficients of the first and second component, respectively, and x are mole fractions. For each binary mixture, low-density constant-composition quasi-isothermal data allow for the calculation of B(T, x) and, given the composition, B12. According to rigorous theory, the value of B12 should not depend on composition and, thus, serves as a sensitive consistency check for gas-phase isothermal data.

The derivation of B12 from the experimental data is more involved than it might appear at first glance. The process used in this work, including the important correction term described by Moldover and McLinden,37 is described in Sec. 2 in the supplementary material. The calculated values for B12 from the experimental gas-phase measurements of Fortin and McLinden22 are shown in Fig. 5. The experimental data show a very small amount of scatter. The model developed in this work dramatically improves the representation of the cross second virial coefficient as compared with the model of Akasaka.33 Both models yield a composition-dependent B12, which is a consequence of the corresponding-states-based model formulation.38 The improvement in the cross second virial coefficients is related to the improvement in the accuracy of the dew-point curve because the second virial coefficient largely governs the thermodynamics in the low-density gas.

FIG. 5.

Values of B12 for R-32/1234yf gas-phase data. Experimental data are from Fortin and McLinden22 with the data analysis in this work, and the models are the model in this work and that of the model in REFPROP 10.0.

FIG. 5.

Values of B12 for R-32/1234yf gas-phase data. Experimental data are from Fortin and McLinden22 with the data analysis in this work, and the models are the model in this work and that of the model in REFPROP 10.0.

Close modal

The measurements of Cai et al.18 are also quasi-isothermal measurements at constant composition in the low-density gas. Values of B12 were obtained from all nominal isotherms with at least four datapoints for a given mixture composition. The calculated values are in Fig. 6; there is an offset of ∼50 cm3 mol−1 from the values in Fig. 5 at 293.15 K. Given the consistency of the results in Fig. 5, the NIST data seem to be more reliable than those of Cai et al.18 The other low-density gas-phase datasets (Tomassetti39 and Yang et al.19) are not amenable to the same sort of analysis.

FIG. 6.

Other literature values of B12 for low-density gas data for R-32/1234yf. Experimental data from Ref.18.

FIG. 6.

Other literature values of B12 for low-density gas data for R-32/1234yf. Experimental data from Ref.18.

Close modal

At the other end of the phase diagram is the liquid phase, which is also well-represented by the model. In the liquid phase, here defined by ρ > 800 kg m−3, the fit was anchored by the data from NIST, which are represented with an AARD of 0.07%. An expanded view of the liquid-phase data is shown in Fig. S4 in the supplementary material. Systematic deviations relative to the NIST data can be seen for the datasets of Dang et al.17 and Jia et al.20 

In between the gas-like and liquid-like densities is the extended critical region. In this region, the physics become more complicated, and the assessment of model performance for PVT data also becomes more challenging. As a demonstration of one problem, the saturated liquid and vapor density data of Akasaka et al.16 and the model results for the experimentally studied isopleths are shown in Fig. 7. Although the deviations are tens of percents in some cases, the performance of the model is qualitatively correct, the problem being that the isopleths are quite flat near their maximum temperature, which results in large density deviations.

FIG. 7.

Saturated isopleths for R-32/1234yf. The curves are from the new models with color matching the data points and the experimental data points for saturated phases are from Ref. 16.

FIG. 7.

Saturated isopleths for R-32/1234yf. The curves are from the new models with color matching the data points and the experimental data points for saturated phases are from Ref. 16.

Close modal

More generally, for points in the extended critical region (roughly 200 < ρ < 800 kg m−3), the density deviation is no longer meaningful because the derivative (∂ρ/∂p)T becomes very large and it is more reasonable to consider the deviations in the pressure calculated from the given temperature and density. Those results are shown in Fig. S2 in the supplementary material, which show that the deviations in terms of pressure are less than 1%, which is probably as good as can be expected.

Overall, the density data appear to be well captured by the model. Even though the deviation plots in Fig. 4 show some initially worrying deviations, after deeper assessment, the problems appear to be artifacts of the error metric rather than problems in the data or the model. This highlights the challenge in assessing mixture model performance: A deviation of 20% in density does not necessarily indicate fatal flaws of the model and it is only by deeper investigation that PVT deviations can be more completely understood.

4.1.3. Caloric properties

There are two datasets for speed of sound23,24 and a single data point of isobaric specific heat capacity cp of Al Ghafri et al.28Figure 8 presents the deviations for the speed of sound data. Both speed of sound datasets systematically deviate from the model of Akasaka by approximately +2%. The updated model in this work included the data from Ref. 24 in the fit, and it is therefore no surprise that the deviations of these data points are mostly within 1% with a clear temperature dependence in the deviations. The data of Shimoura et al.23 are likewise well-represented by the new model; the two datasets appear to be consistent. This stands in contrast to the VLE data of Shimoura et al.,23 which deviate systematically and significantly from the other VLE data.

FIG. 8.

Speed of sound deviations for model in this work and that of REFPROP 10.0 for R-32/1234yf. The references are as listed in Table 8.

FIG. 8.

Speed of sound deviations for model in this work and that of REFPROP 10.0 for R-32/1234yf. The references are as listed in Table 8.

Close modal

As was mentioned above, the deviations between the pure fluid liquid-phase speed of sound data and the respective EOS are not negligible. In the case of R-32, those deviations have a systematic trend in temperature, while the new EOS for R-1234yf largely corrects the defects in the reproduction of the pure fluid speed of sound. The mixture model probably fits the experimental speed of sound data too well. The deviations for pure R-32 should range from −0.25% to +0.75% in these coordinates, and R-1234yf should be represented to within near the experimental uncertainty. From the new model, the limit of pure R-32 seems to be a deviation with a positive offset of ∼0.5%.

The single datapoint for cp is from Al Ghafri et al.;28 the AARD with the new model developed in this work is 1.7% and that with the REFPROP 10.0 model is 2.1%.

The mixture of R-32/1234ze(E) is almost as well studied experimentally as R-32/1234yf. There are multiple overlapping datasets for phase equilibria and density, providing a quite good coverage of the phase diagram, except perhaps at extremes in temperature. The datasets are listed in Table 9. Unlike R-32/1234yf, there are multiple datasets of specific heat measurements.

The baseline mixture model from REFPROP 10.0 is based on the EOS for R-32 of Tillner-Roth and Yokozeki,8 the EOS for R-1234ze(E) from Thol and Lemmon,12 and the mixing model from Akasaka.33 

4.2.1. VLE

In total, there are eight datasets with VLE data and another with critical loci from Ref. 41, which presents the same data points as Kobayashi et al.40 As for R-32/1234yf, the measurements are largely PTxy, where the phase compositions are obtained by sampling from the co-existing phases, but Outcalt and Rowane34 measured the VLE of gravimetrically prepared mixtures.

Figure 9 joins all the VLE plots into one figure. The discussion largely follows that for R-32/1234yf, although the experimental data end at a lower temperature relative to the critical locus than that of R-32/1234yf. As such, there is more uncertainty in the correct shape of the critical locus. The data of Kobayashi et al.41 suggest that the critical locus is monotonic in pressure-composition coordinates, but neither the REFPROP 10.0 model of Akasaka33 nor the model from this work shows that behavior. The lower half of the figure shows the bubble- and dew-point deviations as a function of temperature and composition for this model and the model of Akasaka. The model of Akasaka implemented in REFPROP 10.0 represents the experimental data available with an overall AARD in bubble-point pressure of 1.88% and an overall AARD in dew-point pressure of 1.80%. The new model improves the deviations, and the AARD for the bubble-point pressures are 0.77% and 1.05% for the dew-point pressures. The work of Akasaka33 also showed that the mixture models fitted to experimental data for R-32/1234ze(E) tend to have more scatter and larger deviations than those for R-32/1234yf. The origin of this scatter is unclear. Overall, the datasets are largely in good agreement, and the data are represented to within mostly 2% in pressure.

FIG. 9.

VLE plots for R-32/1234ze(E). Experimental datasets for R-32/1234ze(E) are listed in Table 9. Top: A px diagram with points and curves colored by temperature. Bottom left: Deviation plots for bubble-point data. Bottom right: Deviation plots for dew-point data. For the deviation plots, the AARD is listed in the legend, and the portions of the plot shaded in gray are on a logarithmic scale.

FIG. 9.

VLE plots for R-32/1234ze(E). Experimental datasets for R-32/1234ze(E) are listed in Table 9. Top: A px diagram with points and curves colored by temperature. Bottom left: Deviation plots for bubble-point data. Bottom right: Deviation plots for dew-point data. For the deviation plots, the AARD is listed in the legend, and the portions of the plot shaded in gray are on a logarithmic scale.

Close modal

Overall, the data quality and availability for this binary pair appear to be suitable for the development of an accurate PTxy surface. The datasets appear to be largely consistent, and the model reproduces the data within ∼1% in phase equilibrium pressure.

4.2.2. PVT

The cross second virial coefficients are in Fig. 10. The deviations are generally within 20 cm3/mol, except for at the lowest temperature of 253.15 K. The deviations in the low-density region are shown in Fig. S7 in the supplementary material and are mostly within 0.05% in density.

FIG. 10.

Values of B12 for R-32/1234ze(E) gas-phase data. Experimental data from Fortin and McLinden22 with the data analysis in this work, and models are the model in this work and that of the model in REFPROP 10.0.

FIG. 10.

Values of B12 for R-32/1234ze(E) gas-phase data. Experimental data from Fortin and McLinden22 with the data analysis in this work, and models are the model in this work and that of the model in REFPROP 10.0.

Close modal

The liquid-phase densities are also well-represented by the model. The fit in this region was based on the NIST data, which are represented with an AARD of 0.02% and are shown in Fig. 11. An expanded view of the liquid-phase data is shown in Fig. S7 in the supplementary material. Random deviations relative to the NIST data can be seen for the dataset from Jia et al.44 The combined expanded uncertainties of Jia et al.44 are ∼0.3% in density, and this dataset shows a systematic shift in density deviations at ∼320 K.

FIG. 11.

PVT deviations for the model in this work and that of REFPROP 10.0 for R-32/1234ze(E). The references are as listed in Table 9, and the AARD for each dataset are listed in the legends. The portions of the plot shaded in gray are on a logarithmic scale. Thin vertical gray lines indicate values of the independent variable at the failure of an iterative calculation.

FIG. 11.

PVT deviations for the model in this work and that of REFPROP 10.0 for R-32/1234ze(E). The references are as listed in Table 9, and the AARD for each dataset are listed in the legends. The portions of the plot shaded in gray are on a logarithmic scale. Thin vertical gray lines indicate values of the independent variable at the failure of an iterative calculation.

Close modal

The results in terms of pressure deviations in the so-called extended critical region are shown in Fig. S2 in the supplementary material, showing that the deviations in pressure are less than 2%, which is probably as good as can be expected, and much smaller than deviations in terms of density.

In summary, as for R-32/1234yf, the presence of experimental measurements in the extended critical region complicates the assessment of the model quality. When considering the density in the gas and liquid phases appropriately, it is shown that the model is a faithful representation of the experimental data.

4.2.3. Caloric properties

There is a single dataset of speed of sound data of Rowane and Perkins24 and three datasets with specific heats of Tanaka et al.,42 Yamaya et al.,43 and Gao et al.49 The coverage of specific heats is by far the most comprehensive of any of the binary pairs studied in this work.

The speed of sound deviations with the new model are shown in Fig. 12. There are no other corroborative data in the literature, so it is not possible to ascertain whether systematic deviations in the data are present. The low-temperature data were weighted more heavily in the fit such that the overall representation of the speed of sound would be improved. As was described above, the pure-fluid EOS for R-32 and R-1234ze(E) need to be refit to better reproduce the liquid-phase speed of sound.

FIG. 12.

Speed of sound deviations for the model in this work and that of REFPROP 10.0 for R-32/1234ze(E). The references are as listed in Table 9.

FIG. 12.

Speed of sound deviations for the model in this work and that of REFPROP 10.0 for R-32/1234ze(E). The references are as listed in Table 9.

Close modal

Deviation plots for the three heat capacity datasets are shown in Fig. 13, and overall, the improvements in the thermodynamic model do not result in significant reductions in the deviations. Tanaka et al.42 claimed an uncertainty of 5%, which is similar to the systematic offset of their data, and the AARD for the data of Gao et al.49 is similar for the two models. The isochoric heat capacity data of Tanaka et al.42 extend into the critical region, causing failures of iterative routines, albeit fewer than with the model of Akasaka.33 

FIG. 13.

Heat capacity deviations for the model in this work and that of REFPROP 10.0 for R-32/1234ze(E). The references are as listed in Table 9, and the AARD of each dataset is listed in the legend. The portions of the plot shaded in gray are on a logarithmic scale. Thin vertical gray lines indicate values of the independent variable at the failure of an iterative calculation.

FIG. 13.

Heat capacity deviations for the model in this work and that of REFPROP 10.0 for R-32/1234ze(E). The references are as listed in Table 9, and the AARD of each dataset is listed in the legend. The portions of the plot shaded in gray are on a logarithmic scale. Thin vertical gray lines indicate values of the independent variable at the failure of an iterative calculation.

Close modal

The datasets from the literature for R-1234yf/152a are shown in Table 10. Three of the datasets were measured in the last two years at NIST (Outcalt and Rowane,34 Rowane and Perkins,50 and Fortin and McLinden22). Additionally, there are phase equilibria datasets measured in 2014 and 2018 (those of Hu et al.51 and Yang et al.52) and a gas-phase dataset from Tomassetti.39 The NIST datasets cover both liquid and vapor densities, and the speed of sound data are liquid-phase.

TABLE 10.

The collection of existing data for R-1234yf/152a. (PVT: pvT data, SOS: speed of sound, VLE: vapor–liquid equilibria, N: number of data points, and x1: composition of the first component)

KindFirst author (year)Nx1 (mole frac.)T (K)
PVT Tomassetti (2021)39  161 0.13–0.90 263.1–373.1 
PVT Fortin (2023)22  165 0.33–0.67 230.0–400.0 
SOS Rowane (2022)50  138 0.33–0.67 230.0–345.0 
VLE Hu (2014)51  60 0.00–1.00 283.1–323.1 
VLE Yang (2018)52  25 0.00–1.00 283.1–323.1 
VLE Outcalt (2021)34  28 0.37–0.69 270.0–360.0 
VLE El Abbadi (2022)53  47 0.00–1.00 278.2–333.3 
KindFirst author (year)Nx1 (mole frac.)T (K)
PVT Tomassetti (2021)39  161 0.13–0.90 263.1–373.1 
PVT Fortin (2023)22  165 0.33–0.67 230.0–400.0 
SOS Rowane (2022)50  138 0.33–0.67 230.0–345.0 
VLE Hu (2014)51  60 0.00–1.00 283.1–323.1 
VLE Yang (2018)52  25 0.00–1.00 283.1–323.1 
VLE Outcalt (2021)34  28 0.37–0.69 270.0–360.0 
VLE El Abbadi (2022)53  47 0.00–1.00 278.2–333.3 

The baseline model from REFPROP 10.0 uses the older pure-fluid EOS for R-1234yf of Richter et al.14 and the same pure-fluid EOS for R-152a.11 The mixture model in REFPROP 10.0 is based on a simple fitting approach,7,54 yielding the interaction parameters βT,12 = 0.9963 and γT,12 = 0.981 12 with γv,12 = βv,12 = 1 and Fij = 0. The model parameters in REFPROP 10.0 are very similar to those in Table 2.

4.3.1. VLE

The VLE assessment is shown in Fig. 14. The datasets of Hu et al.51 and Yang et al.52 are in good agreement. Both the new model from this work and the model from REFPROP 10.0 yield AARD for bubble- and dew-points less than 0.2%, which is on the order of experimental uncertainty for a sampling-based PTxy measurement. The dataset of Outcalt and Rowane34 is a tale of two compositions. The composition richer in R-1234yf agrees with the other two datasets to within their mutual uncertainties. On the other hand, the composition less rich in R-1234yf shows a systematic bias on the order of 1.5%. The origin of this shift is unknown. In any case, the deviations are still less than 2%, and the corroborating data allow for a reliable mixture model. The good accuracies of the pure fluid speeds of sound are particularly helpful in this case to pin down the thermodynamics, as Ref. 2 showed that highly accurate speed of sound and pvT data can be sufficient in some cases to develop thermodynamic models without any phase equilibria data if the mixture thermodynamics are simple enough.

FIG. 14.

VLE plots for R-1234yf/152a. The references are as listed in Table 10. Top: A px diagram with points and curves colored by temperature. Bottom left: Deviation plots for bubble-point data. Bottom right: Deviation plots for dew-point data. For the deviation plots, the AARD is listed in the legend.

FIG. 14.

VLE plots for R-1234yf/152a. The references are as listed in Table 10. Top: A px diagram with points and curves colored by temperature. Bottom left: Deviation plots for bubble-point data. Bottom right: Deviation plots for dew-point data. For the deviation plots, the AARD is listed in the legend.

Close modal

4.3.2. PVT

The density data cover the gas phase, the liquid phase, and some points at intermediate densities (the extended critical region). The deviations are plotted in Fig. 15. The fitted model yields excellent agreement with the experimental data used to fit it, with an AARD of 0.03% for the data of Fortin and McLinden22 and an AARD of 0.18% for the data of Tomassetti.39 The model from REFPROP 10.0 does not reproduce the liquid-phase densities as well as the new model does; the REFPROP 10.0 model yields essentially unchanged statistics in the gas phase and a systematic bias of −0.25% in the liquid phase. The new model provides a good representation of the experimental density data where they exist.

FIG. 15.

Density deviations for the model in this work and that of REFPROP 10.0 for R-1234yf/152a. The references are as listed in Table 10.

FIG. 15.

Density deviations for the model in this work and that of REFPROP 10.0 for R-1234yf/152a. The references are as listed in Table 10.

Close modal

The model also performs well in the dilute gas phase. The cross second virial coefficients from the data from Fortin and McLinden22 are plotted in Fig. 16. The two models have essentially unchanged cross second virial coefficients.

FIG. 16.

Values of B12 for the model in this work and that of REFPROP 10.0 for R-1234yf/152a. Experimental data of Fortin and McLinden22 with the data analysis in this work; models are the model in this work and that of the model in REFPROP 10.0.

FIG. 16.

Values of B12 for the model in this work and that of REFPROP 10.0 for R-1234yf/152a. Experimental data of Fortin and McLinden22 with the data analysis in this work; models are the model in this work and that of the model in REFPROP 10.0.

Close modal

4.3.3. Caloric properties

R-1234yf/152a is one of the mixtures studied in this work with a good representation of the pure-fluid speed of sound for both pure-fluid EOS. As a result, the highly accurate mixture speed of sound data can be used to full effect to constrain the mixture model because the mixture model does not need to correct for the erroneous pure-fluid EOS. Unsurprisingly, the representation of the mixture speed of sound after fitting is therefore also good. Figure 17 shows the deviations obtained from the model and that in REFPROP 10.0. The mixture speed of sound data are reproduced within 0.4%, with an AARD of 0.06%, which is approximately equal to the combined expanded uncertainty of the measurements. The model from REFPROP 10.0 yields much larger deviations, up to 1.5% in the worst case, and an AARD of 0.35%.

FIG. 17.

Speed of sound deviations for the model in this work and that of REFPROP 10.0 for R-1234yf/152a. The references are as listed in Table 10.

FIG. 17.

Speed of sound deviations for the model in this work and that of REFPROP 10.0 for R-1234yf/152a. The references are as listed in Table 10.

Close modal

The mixture model for R-125/1234yf is based on VLE data of Outcalt and Rowane34 and the speed of sound data from Rowane and Perkins.50 The density data are used for comparison only. The existing data in the literature are summarized in Table 11.

TABLE 11.

The collection of existing data for R-125/1234yf. (PVT: pvT data, SOS: speed of sound, VLE: vapor–liquid equilibria, N: number of data points, CP: constant-pressure specific heat, and x1: composition of the first component)

KindFirst author (year)Nx1 (mole frac.)T (K)
PVT Dang (2015)17  27 0.29–0.69 283.6–318.0 
PVT Al Ghafri (2019)28  40 0.50–0.50 252.1–382.9 
SOS Rowane (2022)50  133 0.33–0.67 230.0–345.0 
VLE Kamiaka (2010)25  28 0.19–0.49 273.3–333.1 
VLE Kamiaka (2013)26  84 0.00–1.00 273.3–333.2 
VLE Yang (2020)55  35 0.00–1.00 283.1–323.1 
VLE Outcalt (2021)34  28 0.35–0.66 270.0–335.0 
VLE Peng (2022)56  66 0.00–1.00 263.1–313.1 
CP Al Ghafri (2019)28  0.50–0.50 293.2–293.2 
KindFirst author (year)Nx1 (mole frac.)T (K)
PVT Dang (2015)17  27 0.29–0.69 283.6–318.0 
PVT Al Ghafri (2019)28  40 0.50–0.50 252.1–382.9 
SOS Rowane (2022)50  133 0.33–0.67 230.0–345.0 
VLE Kamiaka (2010)25  28 0.19–0.49 273.3–333.1 
VLE Kamiaka (2013)26  84 0.00–1.00 273.3–333.2 
VLE Yang (2020)55  35 0.00–1.00 283.1–323.1 
VLE Outcalt (2021)34  28 0.35–0.66 270.0–335.0 
VLE Peng (2022)56  66 0.00–1.00 263.1–313.1 
CP Al Ghafri (2019)28  0.50–0.50 293.2–293.2 

The mixture model from REFPROP 10.0 is based on the EOS for R-125 of Lemmon and Jacobsen,9 the EOS for R-1234yf of Richter et al.,14 and unpublished mixing parameters.7 

The available VLE data are plotted in Fig. 18. Aside from the data from Refs. 25 and 26, the bubble-point data are represented within 0.5%. The representation of dew points is similar; the same datasets Refs. 25 and 26 show larger deviations, and the remaining datasets are mostly represented within 0.5%. Interestingly, the updated mixture model has only a very small impact on the VLE. The critical curve traced with teqp is smooth and shows no anomalous behavior.

FIG. 18.

VLE plots for R-125/1234yf. The references are as listed in Table 11. Top: A px diagram with points and curves colored by temperature. Bottom left: Deviation plots for bubble-point data. Bottom right: Deviation plots for dew-point data. For the deviation plots, the AARD is listed in the legend.

FIG. 18.

VLE plots for R-125/1234yf. The references are as listed in Table 11. Top: A px diagram with points and curves colored by temperature. Bottom left: Deviation plots for bubble-point data. Bottom right: Deviation plots for dew-point data. For the deviation plots, the AARD is listed in the legend.

Close modal

The data availability for PVT is more sparse; additional gas-phase density measurements for this mixture would be beneficial. The data in the liquid phase come from Dang et al.17 and Al Ghafri et al.28 and the gas-phase data come from Al Ghafri et al.28 The deviations are plotted in Fig. S8 in the supplementary material. Like the VLE, the PVT deviations are improved with the new mixture model but only slightly.

The deviations between the experimental speed of sound data and the model are shown in Fig. S9 in the supplementary material. Like for R-1234yf/152a, the pure-fluid EOS provide good values for the speed of sound for both components, and therefore, the mixture model also yields highly accurate values for the mixture speed of sound. The mixture speed of sound values are reproduced within 0.2% with an AARD of 0.1%. The mixture model from REFPROP 10.0 yields errors of up to 1.4%, with an AARD of 0.64%. The single data point for cp is from Al Ghafri et al.;28 the AARD with the new model developed in this work is 1.0% and that with the REFPROP 10.0 model is 0.9%.

The data for R-1234ze(E)/227ea used for model development all come from new measurements conducted at NIST and the datasets are summarized in Table 12. All the data were used in the model fitting, and as a consequence, all the data are well-represented by the model. Because of the limited data coverage, the figures have been placed in the supplementary material, and the data analysis is presented here.

TABLE 12.

The collection of existing data for R-1234ze(E)/227ea. (PVT: pvT data, SOS: speed of sound, VLE: vapor–liquid equilibria, N: number of data points, and x1: composition of the first component)

KindFirst author (year)Nx1 (mole frac.)T (K)
PVT Fortin (2023)22  164 0.33–0.67 230.0–400.0 
SOS Rowane (2022)50  313 0.33–0.67 230.0–345.0 
VLE Outcalt (2021)34  29 0.33–0.68 270.0–360.0 
KindFirst author (year)Nx1 (mole frac.)T (K)
PVT Fortin (2023)22  164 0.33–0.67 230.0–400.0 
SOS Rowane (2022)50  313 0.33–0.67 230.0–345.0 
VLE Outcalt (2021)34  29 0.33–0.68 270.0–360.0 

The model in REFPROP 10.0 for the binary pair is the pure-fluid EOS for R-1234ze(E) of Thol and Lemmon12 and the pure-fluid EOS for R-227ea of Lemmon and Span,13 and the mixture model uses the parameters from R-1234yf/227ea7,54 because no experimental data were previously available.

The bubble-point measurements show systematic deviations on the order of +2% from the model in REFPROP 10.0, whereas the new model fits the bubble-point measurements with an AARD of 0.05%. The bubble-point data were included in the fit.

The density measurements cover the full density range and are reproduced with an AARD of 0.03% and a maximum deviation of 0.5%. The density deviations from the model in REFPROP 10.0 were represented with an AARD of 0.47%, so the new model represents a significant improvement in accuracy.

The speed of sound measurements show the characteristic temperature-dependent deviations shown for all studied mixtures containing R-1234ze(E) (see Ref. 2). The deviations increase in magnitude as the amount of R-1234ze(E) increases but are still within 1.5% for all the measurements.

The models presented in this work were implemented in REFPROP and teqp. The necessary files are provided in the supplementary material. Calculated values are provided in Table 13, and it was confirmed that the upcoming version of REFPROP and teqp yielded the same results to 13 digits. The new equation of state for R-1234yf of Lemmon and Akasaka10 is also provided in REFPROP and teqp formats. An updated fluid file for REFPROP for R-152a is also provided, with the coefficients rounded to agree with the original publication11 in order to yield the precise values in the table.

TABLE 13.

Check values for the fitted model. Values obtained with teqpa from nominal values of ρ/ρred = 0.8 and Tred/T = 0.8. The first set of lines are check values for the pure fluids followed by one check value per binary pair. Names match the fluid file in REFPROP, are case-sensitive, and are compatible with CoolProp and teqp

Names (1/2)z1 (mole frac.)T (K)ρ (mol/m3)Tred (K)ρred (mol/m3)αr
R32 1.0 439.0 6520.0 351.255 000 000 000 0 8150.084 599 999 999 2 −0.540 274 653 742 97 
NEWR1234YF 1.0 460.0 3344.0 367.850 000 000 000 0 4180.000 000 000 000 0 −0.468 353 705 968 76 
R125 1.0 424.0 3823.0 339.173 000 000 000 0 4779.000 000 000 000 0 −0.455 060 052 344 49 
R152A 1.0 483.0 4457.0 386.411 000 000 000 0 5571.449 999 999 999 8 −0.507 421 495 701 51 
R1234ZEE 1.0 478.0 3432.0 382.513 000 000 000 0 4290.000 000 000 000 0 −0.463 409 784 472 30 
R227EA 1.0 469.0 2796.0 374.900 000 000 000 0 3495.000 000 000 000 0 −0.442 385 761 979 82 
R32/NEWR1234YF 0.4 445.0 4149.0 355.822 371 896 262 3 5186.140 360 148 214 3 −0.473 110 647 439 11 
R32/R1234ZEE 0.4 451.0 4242.0 360.536 339 642 561 0 5302.989 447 905 131 8 −0.485 761 867 602 31 
R125/NEWR1234YF 0.4 445.0 3513.0 355.904 171 448 396 3 4391.874 275 580 102 2 −0.465 763 074 794 47 
NEWR1234YF/R152A 0.4 469.0 3930.0 374.817 327 199 672 7 4912.718 419 059 895 5 −0.489 675 489 166 38 
R1234ZEE/R227EA 0.4 470.0 3023.0 376.013 953 032 751 7 3778.458 494 637 484 8 −0.453 788 347 707 36 
Names (1/2)z1 (mole frac.)T (K)ρ (mol/m3)Tred (K)ρred (mol/m3)αr
R32 1.0 439.0 6520.0 351.255 000 000 000 0 8150.084 599 999 999 2 −0.540 274 653 742 97 
NEWR1234YF 1.0 460.0 3344.0 367.850 000 000 000 0 4180.000 000 000 000 0 −0.468 353 705 968 76 
R125 1.0 424.0 3823.0 339.173 000 000 000 0 4779.000 000 000 000 0 −0.455 060 052 344 49 
R152A 1.0 483.0 4457.0 386.411 000 000 000 0 5571.449 999 999 999 8 −0.507 421 495 701 51 
R1234ZEE 1.0 478.0 3432.0 382.513 000 000 000 0 4290.000 000 000 000 0 −0.463 409 784 472 30 
R227EA 1.0 469.0 2796.0 374.900 000 000 000 0 3495.000 000 000 000 0 −0.442 385 761 979 82 
R32/NEWR1234YF 0.4 445.0 4149.0 355.822 371 896 262 3 5186.140 360 148 214 3 −0.473 110 647 439 11 
R32/R1234ZEE 0.4 451.0 4242.0 360.536 339 642 561 0 5302.989 447 905 131 8 −0.485 761 867 602 31 
R125/NEWR1234YF 0.4 445.0 3513.0 355.904 171 448 396 3 4391.874 275 580 102 2 −0.465 763 074 794 47 
NEWR1234YF/R152A 0.4 469.0 3930.0 374.817 327 199 672 7 4912.718 419 059 895 5 −0.489 675 489 166 38 
R1234ZEE/R227EA 0.4 470.0 3023.0 376.013 953 032 751 7 3778.458 494 637 484 8 −0.453 788 347 707 36 
a

teqp version 0.14.1.

In this study, mixture models were developed for five binary pairs of varying experimental databases. The model for R-32/1234yf is based on data from multiple overlapping datasets that are well-represented. On the other end of the spectrum is the model for R-1234ze(E)/227ea based exclusively on new data. In all cases, the models provide as good a model as can be justified without over-fitting the existing data.

As in the previous study,2 it is again highlighted that deficiencies in the pure-fluid EOS, particularly in their representation of liquid-phase speed of sound, cause difficulty in the development of the highest-accuracy mixture models. If the mixture models were not overfit, the mixture models will hopefully not need to be updated when the EOS for R-32 and R-1234ze(E) are updated (but success cannot be assessed until the EOS are updated at some point in the future).

In order to ensure reproducibility of the results, the supplementary material includes (1) the fitting code and the experimental data used in the fitting process, (2) the Python code used to generate the check values as well as the fluid files needed for the new EOS for R-1234yf, and (3) model implementations in the formats needed for REFPROP/TREND and CoolProp/teqp. For REFPROP, this includes the HMX.BNC file.

The author would like to thank Demian Riccardi (of NIST) for help in accessing data of TDE. This work was partially supported by the U.S. Department of Energy, Building Technologies Office, under Agreement No. 892434-19-S-EE000031.

The author has no conflicts to disclose.

The data that support the findings of this study are available within the article and its supplementary material and from the corresponding author upon reasonable request.

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Supplementary Material