In this work, new thermodynamic models for refrigerant mixtures are provided for the binary pairs R-1234yf/134a, R-1234yf/1234ze(E), and R-134a/1234ze(E) based on new reference measurements of speed of sound, density, and bubble-point pressures. Fitting the very accurate liquid-phase speed of sound and density data reproduces the bubble-point pressures to within close to their uncertainty, yielding deviations in density less than 0.1% and speed of sound deviations less than 1% (and less than 0.1% for R-1234yf/134a). Models are also presented for the binary pairs R-125/1234yf, R-1234ze(E)/227ea, and R-1234yf/152a based solely on bubble-point measurements.

The fourth generation of refrigerants largely comprises mixtures containing halogenated olefins,1 for instance, to find non-flammable blends to replace the workhorse refrigerant R-134a.2 In order to reliably assess novel mixtures, new mixture models in the form of equations of state (EOS) are needed for each of the binary pairs included in the candidate mixtures. Of the six binary mixtures we selected for study, current models are all based upon estimation schemes or unpublished models in REFPROP 10.0.3,4 New reference experimental measurements are needed to improve the mixture models. As outlined below, these measurements were obtained as part of the larger project, forming the foundation for the mixture models. This work begins the process of filling in the holes where important mixture models need to be updated or developed.3 

The components considered in this work are all relatively similar and some information about them is presented in Table 1. Two [R-1234yf and R-1234ze(E)] are stereoisomers, and the three main components of the comprehensive measurements have a hydrocarbon backbone with the replacement of four hydrogens by fluorines (as indicated by 4 as the right-most numerical digit). Therefore, we might expect (as we see here) that mixtures of these compounds would behave in a relatively simple way, with nearly ideal mixing behaviors. This makes refrigerant mixtures such as these ideal candidates for the development of highly accurate mixture models.

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. M: molar mass

NameHashaTNBP (K)Tcrit (K)pcrit (MPa)M (kg mol−1)
R-1255  25c5a3a0 225.060 339.173 3.617 70 0.120 021 
R-1234yf6  40377b40 243.692 367.850 3.384 40 0.114 042 
R-134a7  ff1c0560 247.076 374.210 4.059 28 0.102 032 
R-152a8  63f364b0 249.127 386.411 4.516 75 0.066 051 
R-1234ze(E)9  9905ef70 254.177 382.513 3.634 90 0.114 042 
R-227ea10  40091ee0 256.810 374.900 2.925 00 0.170 029 
NameHashaTNBP (K)Tcrit (K)pcrit (MPa)M (kg mol−1)
R-1255  25c5a3a0 225.060 339.173 3.617 70 0.120 021 
R-1234yf6  40377b40 243.692 367.850 3.384 40 0.114 042 
R-134a7  ff1c0560 247.076 374.210 4.059 28 0.102 032 
R-152a8  63f364b0 249.127 386.411 4.516 75 0.066 051 
R-1234ze(E)9  9905ef70 254.177 382.513 3.634 90 0.114 042 
R-227ea10  40091ee0 256.810 374.900 2.925 00 0.170 029 
a

The hash used in REFPROP to define 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.

As part of the larger scope of work within this project, measurements of speed of sound (SOS) (liquid phase), density (gas and liquid phase), and vapor–liquid equilibria (VLE) were carried out for the binary mixtures R-1234yf/134a, R-1234yf/1234ze(E), and R-134a/1234ze(E). These SOS and bubble-point results are presented in a set of papers in the literature.11,12 The datasets are presented in Table 2 and shown graphically in Fig. 1. Most of our measurements extend up to 10 or 20 MPa, with the exception of the SOS data for R-134a/1234ze(E), which extend up to 50 MPa. The interim report from the project includes the full set of measured data,13 though additional analysis and screening of the data was carried out following report preparation. For three additional mixtures [R-125/1234yf, R-1234ze(E)/227ea, and R-1234yf/152a], measurements of bubble points were carried out.11 These datasets form the core of data used in this study because their experimental uncertainties are small and carefully assessed. Other data from the literature are compared with the models developed in this work.

TABLE 2.

Set of comprehensive measurements carried out in the greater NIST study. The mole fractions z1 are the discrete compositions prepared [SOS: speed of sound, PVT: p-v-T, VLE: bubble-point measurement, N: number of data points, and U(χ)̄: mean value of combined expanded (k = 2) relative uncertainty in measured quantity (density in the case of PVT, speed of sound in the case of SOS, and bubble-point pressure for VLE)]

Pair (1/2)KindT (K)z1 (mole frac.)NU(χ)̄ (%)
R-1234yf/1234ze(E) VLE 270–360 0.324, 0.638 48 0.219 
R-1234yf/134a VLE 270–360 0.319 9, 0.646 7 39 0.169 
R-134a/1234ze(E) VLE 270–360 0.334 1, 0.663 1 24 0.17 
R-1234yf/1234ze(E) PVT 230–400 0.335 84, 0.666 6 225 0.0357 
R-1234yf/134a PVT 230–400 0.336 34, 0.667 59 226 0.0375 
R-134a/1234ze(E) PVT 230–400 0.332 5, 0.663 56 175 0.042 
R-1234yf/1234ze(E) SOS 230–345 0.335 84, 0.666 6 131 0.0801 
R-1234yf/134a SOS 230–345 0.336 34, 0.667 59 118 0.0793 
R-134a/1234ze(E) SOS 230–345 0.329 16, 0.671 02 304 0.0624 
Pair (1/2)KindT (K)z1 (mole frac.)NU(χ)̄ (%)
R-1234yf/1234ze(E) VLE 270–360 0.324, 0.638 48 0.219 
R-1234yf/134a VLE 270–360 0.319 9, 0.646 7 39 0.169 
R-134a/1234ze(E) VLE 270–360 0.334 1, 0.663 1 24 0.17 
R-1234yf/1234ze(E) PVT 230–400 0.335 84, 0.666 6 225 0.0357 
R-1234yf/134a PVT 230–400 0.336 34, 0.667 59 226 0.0375 
R-134a/1234ze(E) PVT 230–400 0.332 5, 0.663 56 175 0.042 
R-1234yf/1234ze(E) SOS 230–345 0.335 84, 0.666 6 131 0.0801 
R-1234yf/134a SOS 230–345 0.336 34, 0.667 59 118 0.0793 
R-134a/1234ze(E) SOS 230–345 0.329 16, 0.671 02 304 0.0624 
FIG. 1.

Location of the new experimental data points in the pT plane for each of the three primary blends and for each property. The solid and dashed curves are the vapor pressure curves for the first and second fluids (in order) forming the binary pair.

FIG. 1.

Location of the new experimental data points in the pT plane for each of the three primary blends and for each property. The solid and dashed curves are the vapor pressure curves for the first and second fluids (in order) forming the binary pair.

Close modal

The collection of experimental data was initially based upon the survey of Bell et al.,3 followed by the addition of one dataset from the work of Tomassetti.14 All the datasets are included in the SOURCE database, accessible through NIST TDE,15 and are listed in Tables 7 and 8. Aside from the sources listed, there is additionally one SOS dataset,16 two references reporting critical loci,17,18 and one reporting specific heats.19 

In this work, the relative deviation in an arbitrary quantity χ is defined by

Δ%(χ)=100×χTWχexp1,
(1)

where the subscript TW indicates the value obtained from the model developed in this work and the subscript exp indicates the experimental value. The average absolute relative deviation (AAD) in a quantity χ is defined by

AADχ=mean(abs(Δ%(χ))),
(2)

where Δ%(χ) is the vector of deviations and AAD is therefore on a percentage basis.

The most accurate thermodynamic mixture models available today in the reference software libraries (NIST REFPROP,4 CoolProp,20 and TREND21) are based upon the Helmholtz energy. Derivatives of the Helmholtz energy can be used to obtain all other thermodynamic properties. The Helmholtz energy [divided by the molar gas constant R and the temperature; α = a/(RT)] is given as the sum of residual and ideal gas contributions,

α=αr+α(ig).
(3)

The ideal gas portion does not enter into the fitting and is therefore not further discussed here. The molar gas constant R, which is now an exactly defined value according to CODATA,22 is instead implemented as the mole-fraction-weighted average of the molar gas constants used in developing the EOS for the pure fluids.

The mixture model for the residual portion is given by

αr(τ,δ,z)=αCSr+αdepr,
(4)

where αCSr is the corresponding-state contribution given by

αCSr=i=1Nzjα0,ir(τ,δ),
(5)

where z is the vector of mole fractions. The pure-fluid contributions α0,ir are given at the mixture reduced states τ and δ. The reduced density δ = ρ/ρred(z) and reciprocal reduced temperature τ = Tred(z)/T are defined based on the reducing functions given in a common form by

Yred(z)=i=1Nzi2Ycrit,i+i=1N1j=i+1N2zizjzi+zjβY,ij2zi+zjYij,
(6)

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,
(7)
vij=18βv,ijγv,ijvcrit,i1/3+vcrit,j1/33.
(8)

The departure contribution in Eq. (4) is given by

αdepr=i=1Nj=i+1NzizjFijαijr(τ,δ),
(9)

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

The departure function αijr(τ,δ) for the binary pair ij is, in principle, an arbitrary mathematical function that takes the value of zero at zero density, yields a good representation of the experimental data for the mixture, and has reasonable extrapolation behavior.

As summarized in Ref. 23, the standard thermodynamic properties may be expressed in terms of derivatives of the Helmholtz energy with a concise derivative representation given by

Λij*=τiδj(α*)i+jτiδj,
(10)

where * is one of ig (ideal gas), r (residual), or tot (total). Thermodynamic properties can be given by combinations of derivatives of the Helmholtz energy. For instance, pressure is given by residual contributions only,

p=avT=ρRT(1+Λ01r),
(11)

and the SOS w is obtained from

Mw2RT=1+2Λ01r+Λ02r(1+Λ01rΛ11r)2Λ20(ig)+Λ20r,
(12)

which contains mostly residual properties, except for Λ20(ig) that comes from the ideal gas. The quantity R is the molar gas constant, M is the molar mass, and all quantities on the right-hand side are non-dimensional.

In the course of this project and in the development of routines for tracing critical curves,24 it became clear that a novel and more flexible approach was needed to obtain the thermodynamic derivatives of a mathematical model. The design constraints are that the obtained values should be fast to evaluate but also allow for rapid prototyping of new modeling approaches. Thus, the teqp library was established;23teqp uses automatic differentiation to obtain the derivatives needed to calculate thermodynamic properties without any hand-written derivatives, dramatically speeding up the development process. Even though automatic differentiation is used, the obtained values are in many cases still faster to evaluate than the hand-written derivatives in REFPROP while suffering only a negligible loss in numerical precision. As of publication, teqp does not include any iterative routines, rather the focus is on forward derivatives of the residual Helmholtz energy with respect to temperature, density, and compositions, which meets the needs of this optimization campaign.

The fitting approach for the binary mixtures with comprehensive measurements considered only SOS and densimetry data (no phase equilibria data). SOS and densimetry data are well-suited to optimization because they do not require a full phase equilibrium calculation, the phase equilibrium calculation being an unreliable (and slow) mixture calculation in general. The overall cost function C$ was defined based upon differences between model predictions and the experimental data and is given by

C$=mean(abs(rρ))+Wmean(abs(rw)),
(13)

where the weighting factor W = 0.25 is used to balance deviations between the two kinds of data and the residua rρ,i and rw,i are relative deviations.

For density data, it would be ideal to use the residuum

rρ,i=ρexpρmodel(T,p,z)ρexp,
(14)

but a downside of this approach is that the calculation ρmodel(T, p, z) requires computationally costly iteration (which also can fail if the initial guess is not accurate enough). Instead, we would like a non-iterative proxy that is a good stand-in for the iterative calculation. Starting from an isothermal series expansion of molar density around the experimental density yields

ρ=ρexp+Δpρp(Texp,ρexp)T,fit+.
(15)

Truncation of higher-order terms yields the difference in density of

Δρ=ρρexp=Δpρp(Texp,ρexp)T,fit,
(16)

and the proxy residue for p-ρ-T data is therefore

rρ,i=p(T,ρ)pexpρexpρp(Texp,ρexp)T,fitΔρρexp
(17)

in which p(T, ρ) and the isothermal derivative of p with respect to ρ are evaluated simultaneously (and directly) from the equation of state. This cost function contribution has a similar behavior to Eq. (14), though it is non-iterative and, thus, very fast to evaluate (on the order of a few μs/evaluation in C++).

Data points along the curve where p/ρT approaches zero (it is zero at the critical point for a pure species) result in large contributions to the cost function, but most of the data points are relatively far from this locus.

The multi-fluid Helmholtz-explicit model has as independent variables temperature T, density ρ, and the vector of mole fractions z. Unfortunately, SOS is usually measured quasi-isochorically with temperature, pressure, and composition as independent variables, so internal iteration is required to solve for the mixture density, given the temperature, pressure, and composition (what was attempted to be avoided in the density deviation term). In the case of SOS data, the initial guess density was that obtained from the model in REFPROP 10.0.4 The deviations between the guess density and the final density were very small, small enough to serve as a reliable-enough starting value in any case.

Once the density has been obtained from one step of iteration, the SOS w is obtained from Eq. (12). In teqp , Λ01r and Λ02r are obtained from a single call and Λ11r and Λ20r are obtained from two further calls. The ideal gas contribution Λ20(ig) is not implemented in teqp as of publication, and so this quantity was taken from the implementation in the HEOS backend of CoolProp.20 The quantity Λ20(ig) does not depend on the departure function or the interaction parameters.

The SOS residue is therefore defined by

rw,i=100×wmodelwexp1.
(18)

As has been successfully applied by the author previously,25 stochastic (random) global optimization provides a reliable approach that is robust to failures and computationally efficient enough for practical application.

While experiments with new model formulations were carried out (for instance, the invariant reducing functions from the GERG-2004 model26 were added to teqp), the end goal was always to develop thermodynamic models compatible with REFPROP 10.0.4 The full set of departure functions available in REFPROP 10 is relatively narrow, with terms of the following kinds available:

  • nkτtkδdk,

  • nkτtkδdkexp(δlk) (see Ref. 27),

  • nkτtkδdkexp(ηk(δεk)2βk(δγk)) (see Eq. 7.8 from Ref. 26), and

  • nkτtkδdkexp(ηk(δεk)2βk(τγk)2) (Gaussian terms like those used for pure fluids).

The parameters lk and dk must be integers in order to yield finite contributions to the second (and higher) virial coefficients. Thus, optimization of the departure function is a mixed-integer optimization problem in which the values of some parameters are integers and others are floating point values. The conventional approach for this problem is to “fit” the dk values by manual optimization while allowing the optimizer to fit the floating point values. A similar approach was taken here. The values of dk were set to dk = k + 1 for k starting at 0 and the same for lk: lk = k + 1.

The cost function defined above is minimized with global optimization. The variables βT,ij, γT,ij, βv,ij, and γv,ij were all bounded to be in [0.75, 1.25], nk was bounded to be in [−3, 3], and tk was bounded to be in [0, 4]. The CEGO library28 was specifically designed for this particular problem, but the widely available differential evolution algorithm implemented in scipy29,30 proved to be adequate for the optimization task and was used for the optimization.

The departure term used was very simple,

αijr=knkτtkδdkexp(sgn(lk)δlk),
(19)

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

In a single evaluation of the cost function, first a Python data structure is constructed that converts an array of double-precision numbers (the variables in the optimization) to the necessary values of βT,ij, γT,ij, n, d, and so on in Eqs. (19) and (6). This Python data structure is dumped to a string in the JSON format, which is then unpacked in the mutant construction routines of teqp. One important implementation note is that the mutant models only hold references to the pure fluid EOS because construction of the pure fluid EOS is the most costly part of the mixture model construction. Thus, users should be careful to ensure that the pure fluid models do not fall out of scope and get prematurely de-allocated as this will result in a dangling reference to the pure fluid EOS. Construction of the mutant usually takes on the order of tens of microseconds.

The first set of model results are those for the binary pairs R-1234yf/134a, R-1234yf/1234ze(E), and R-134a/1234ze(E). For each of these binary pairs, sufficient data were available to fit a reducing function in the form of Eq. (6) and a small departure function in the form of Eq. (19). In order to avoid model over-fitting, a graduated optimization approach was taken for each mixture model. The departure function defined above was given an increasing number of terms, and the statistics of fit were monitored in order to determine how many terms would be required. In all cases, the variables βT,ij, γT,ij, βv,ij, and γv,ij were fitted. This process was done in a deterministic fashion. For each of the three binary pairs with the full complement of data, 0 to Ndep terms were added to the departure function for the pair. For each term count in the departure term, the optimization was repeated five times to hopefully find the global minimum of the cost function (although that cannot be guaranteed in general).

The optimized cost function as a function of the number of terms in the departure function is shown in Fig. 2 for each of the binary pairs. The cost function values for Ndep = 0 are obtained by optimizing only βT,ij, γT,ij, βv,ij, and γv,ij. It is difficult to make out at the scale of the figure without zooming in, but all five replicates of the optimization result are shown, highlighting that the method quite reliably finds close to the same minimum of the cost function. There is a relatively large step decrease when adding a single term to the departure function. This result demonstrates that adjusting the four interaction parameters alone is not sufficient to obtain a good representation of the data; the departure function is needed. The reduction in cost function slows down after more than two terms are included, so the (somewhat arbitrary) decision was made to use two terms in the departure function.

FIG. 2.

Cost function C$ versus the number of terms in the departure function Ndep.

FIG. 2.

Cost function C$ versus the number of terms in the departure function Ndep.

Close modal

The obtained values of the interaction parameters are shown in Table 3, and the departure functions are given in Tables 46. The optimization result with two terms in the departure function appears to be a good compromise of flexibility and model fidelity.

TABLE 3.

Interaction parameters obtained from fitting SOS and density data. Departure functions were fit for each binary pair. Components in each binary pair are sorted by normal boiling point temperatures, and the order matters

Pair (1/2)βT,ijγT,ijβv,ijγv,ijFij
R-1234yf/1234ze(E) 0.998 886 0.993 309 0.999 302 0.998 590 1.0 
R-1234yf/134a 1.000 026 0.987 057 1.000 272 1.003 747 1.0 
R-134a/1234ze(E) 0.998 593 0.992 009 0.998 995 0.998 621 1.0 
Pair (1/2)βT,ijγT,ijβv,ijγv,ijFij
R-1234yf/1234ze(E) 0.998 886 0.993 309 0.999 302 0.998 590 1.0 
R-1234yf/134a 1.000 026 0.987 057 1.000 272 1.003 747 1.0 
R-134a/1234ze(E) 0.998 593 0.992 009 0.998 995 0.998 621 1.0 
TABLE 4.

Departure function for R-1234yf/134a

kntdl
0.051 900 2.477 314 
−0.011 472 0.070 541 
kntdl
0.051 900 2.477 314 
−0.011 472 0.070 541 
TABLE 5.

Departure function for R-1234yf/1234ze(E)

kntdl
0.072 640 0.012 643 
−0.024 746 3.992 829 
kntdl
0.072 640 0.012 643 
−0.024 746 3.992 829 
TABLE 6.

Departure function for R-134a/1234ze(E)

kntdl
0.068 889 3.184 446 
−0.004 831 2.034 344 
kntdl
0.068 889 3.184 446 
−0.004 831 2.034 344 

First, we consider deviations in the SOS data in Fig. 3. For the mixture R-1234yf/134a, the SOS deviations are mostly significantly below 0.1%. To be sure, some of the quality of fit can be attributed to the symmetry of the mixture interactions, as partially evidenced by the near perfect overlaying of their vapor pressure curves. For the two binary pairs containing R-1234ze(E), the deviations are significantly larger and appear to, in general, increase as the amount of R-1234ze(E) increases. Model deviations are only meaningful when compared with the measurement uncertainty, and in this case we have experimental uncertainties that have been carefully assessed. Figure 4 shows the model deviations compared with the combined relative uncertainties of the experimental measurements at each state point. The model deviations for R-1234yf/134a are mostly within two times the combined expanded relative uncertainties, even in the critical region, which is excellent agreement, especially considering that there is still uncertainty in the pure fluid equation of state’s representation of the SOS. As before, the SOS deviations for the binary mixtures containing R-1234ze(E) are much larger than the experimental uncertainty in the binary mixture measurements.

FIG. 3.

Relative deviations in SOS from the comprehensive measurements taken from Ref. 12. The average combined expanded uncertainty is shown by dashed lines and given in the figure.

FIG. 3.

Relative deviations in SOS from the comprehensive measurements taken from Ref. 12. The average combined expanded uncertainty is shown by dashed lines and given in the figure.

Close modal
FIG. 4.

Relative deviations in SOS divided by combined expanded relative uncertainty at the respective state point from the comprehensive measurements taken from Ref. 12. The ±2 band is shown by dashed lines.

FIG. 4.

Relative deviations in SOS divided by combined expanded relative uncertainty at the respective state point from the comprehensive measurements taken from Ref. 12. The ±2 band is shown by dashed lines.

Close modal

In order to understand the deviations for R-134a/1234ze(E) and R-1234yf/1234ze(E), the SOS data were collected for pure R-1234ze(E). In total, three datasets were used to fit the pure fluid EOS of Thol and Lemmon:9 Lago et al.31 (liquid phase), Perkins and McLinden32 (vapor phase), and Kano et al.33 (vapor phase). A recent liquid-phase dataset is available from our group.34Figure 5 shows the deviations for the liquid-phase data. For the data obtained in McLinden and Perkins,34 the path length obtained from direct measurement of the spacer length at atmospheric pressure and 293 K agreed with that obtained from SOS measurements with propane to within 0.03% at the same conditions, giving confidence in the obtained values. These two datasets show that while the EoS fits the single liquid-phase dataset available at the time (except for the highest temperature of 360 K), the new data from our group contradict the data from Lago et al.31 Indeed, the same problem is seen with the measurements taken from Ref. 31 for R-1234yf; their R-1234yf measurements deviate by more than 2.5% from the new EoS for R-1234yf,6 calling into question their liquid-phase SOS data for pure R-1234ze(E) that appear to deviate systematically.

FIG. 5.

Relative deviations between experimental liquid-phase SOS data for pure R-1234ze(E) (unpublished data from McLinden and Perkins34 and data from Lago et al.31) and the EOS.9 

FIG. 5.

Relative deviations between experimental liquid-phase SOS data for pure R-1234ze(E) (unpublished data from McLinden and Perkins34 and data from Lago et al.31) and the EOS.9 

Close modal

Figure 6 shows the deviations in density for the three mixtures. Away from the critical region (see Sec. 5.2.3), the deviations are nearly all less than 0.1%, with AAD all significantly less than 0.1%. The pure-fluid densities are in general not better reproduced than 0.1% by the pure-fluid EOS, so this is about the best that can be achieved without overfitting. Just like for SOS, deviations relative to the uncertainty are considered, and shown in Fig. 7. Most of the density data are represented within five times the experimental uncertainty, except for in the critical region. The density deviations appear to have less sensitivity to defects in the pure-fluid EoS for R-1234ze(E) than for SOS.

FIG. 6.

Relative deviations in density from the comprehensive measurements.13 The average combined expanded uncertainty band is shown by dashed lines and given in the figure.

FIG. 6.

Relative deviations in density from the comprehensive measurements.13 The average combined expanded uncertainty band is shown by dashed lines and given in the figure.

Close modal
FIG. 7.

Relative deviations in density divided by combined expanded relative uncertainty at the respective state point from the comprehensive measurements.13 The ±2 band is shown by dashed lines.

FIG. 7.

Relative deviations in density divided by combined expanded relative uncertainty at the respective state point from the comprehensive measurements.13 The ±2 band is shown by dashed lines.

Close modal

Finally, Fig. 8 shows the bubble-point deviations. Almost all of the data are represented within 1%, except for two data points for R-134a/1234ze(E) for the composition more rich in R-1234ze(E). The deviations divided by the respective combined expanded uncertainties at each individual state point are shown in Fig. 9. The deviations in pressure are mostly within two times the combined expanded uncertainties, again except for the mixture R-134a/1234ze(E). These deviations are remarkable because the bubble-point pressures were not included in the optimization procedure; only the SOS and density data were included.

FIG. 8.

Deviations in bubble-point pressure from the comprehensive measurements taken from Ref. 11. The average combined expanded uncertainty band is shown by dashed lines and given in the figure.

FIG. 8.

Deviations in bubble-point pressure from the comprehensive measurements taken from Ref. 11. The average combined expanded uncertainty band is shown by dashed lines and given in the figure.

Close modal
FIG. 9.

Relative deviations in bubble-point pressure divided by combined expanded relative uncertainty at the respective state point from the comprehensive measurements taken from Ref. 11. The ±1 band is shown by dashed lines, when visible.

FIG. 9.

Relative deviations in bubble-point pressure divided by combined expanded relative uncertainty at the respective state point from the comprehensive measurements taken from Ref. 11. The ±1 band is shown by dashed lines, when visible.

Close modal
FIG. 10.

Location of data (left panel) and relative deviations in dew- and bubble-point pressure (right panels) for literature data (Kou et al.,38 Al Ghafri et al.,19 Kamiaka et al.,35 Ye et al.,37 and Chen et al.36) for VLE. The subscript TW indicates the model in this work. Open markers are dew points and filled markers are bubble points. Each marker type corresponds to a given dataset.

FIG. 10.

Location of data (left panel) and relative deviations in dew- and bubble-point pressure (right panels) for literature data (Kou et al.,38 Al Ghafri et al.,19 Kamiaka et al.,35 Ye et al.,37 and Chen et al.36) for VLE. The subscript TW indicates the model in this work. Open markers are dew points and filled markers are bubble points. Each marker type corresponds to a given dataset.

Close modal

Overall, the set of new measurements appear to be very consistent, as evidenced by the fact that fitting the SOS and densimetry data allows the vapor–liquid-equilibrium data to be represented to within close to their experimental uncertainties. From a certain standpoint, this is not terribly surprising, as a brief thought experiment will indicate. Let us suppose that we can measure the pressure of a pure fluid along a subcritical isotherm as a function of density, including the unstable portion of the isotherm between the spinodals. This pressure curve is all that is needed to obtain the vapor pressure after applying the Maxwell conditions for phase equilibrium. Therefore, for a pure fluid, perfect knowledge of the density along an isotherm would yield the correct vapor pressure by default. The extension of this argument to mixtures is somewhat more challenging because the energetic portion of the phase equilibrium is no longer defined by the lever rule (alternatively, equality of the Gibbs energy); rather, it is necessary to equate chemical potentials of each component in each phase.

Aside from the data from our group (see above), there are a few other datasets in the literature for the properties we measured. Some significant discrepancies can be identified. In each case, our VLE data (not included in the fitting) agrees with one of the datasets, but not the other one. The datasets are listed in Tables 7 and 8, and the statistics of the fit are also included.

TABLE 7.

Existing literature sources for VLE for the three primary binary pairs included in this work. The AAD includes the saturated vapor and liquid points; the mole fractions z1 combine mole fractions of the first component in the liquid and vapor phases to indicate concisely the composition coverage

Pair (1/2)AuthorNz1 (mole frac.)T (K)AADpσ (%)
R-1234yf/134a Kamiaka et al.35  67 0.00–1.00 273–333 0.17 
R-1234yf/134a Chen et al.36  41 0.48–0.58 268–323 4.10 
R-1234yf/1234ze(E) Al Ghafri et al.19  0.66–0.73 274–342 3.58 
R-1234yf/1234ze(E) Ye et al.37  77 0.00–1.00 284–333 0.34 
R-134a/1234ze(E) Al Ghafri et al.19  0.49–0.57 274–341 1.79 
R-134a/1234ze(E) Kou et al.38  40 0.00–1.00 293–323 0.89 
Pair (1/2)AuthorNz1 (mole frac.)T (K)AADpσ (%)
R-1234yf/134a Kamiaka et al.35  67 0.00–1.00 273–333 0.17 
R-1234yf/134a Chen et al.36  41 0.48–0.58 268–323 4.10 
R-1234yf/1234ze(E) Al Ghafri et al.19  0.66–0.73 274–342 3.58 
R-1234yf/1234ze(E) Ye et al.37  77 0.00–1.00 284–333 0.34 
R-134a/1234ze(E) Al Ghafri et al.19  0.49–0.57 274–341 1.79 
R-134a/1234ze(E) Kou et al.38  40 0.00–1.00 293–323 0.89 
TABLE 8.

Existing literature sources for PVT for the three primary binary pairs included in this work. The AAD includes only points for which the iterative calculations in REFPROP 10.0 were successful

Pair (1/2)AuthorNz1 (mole frac.)T (K)AADρ (%)Nfail
R-1234yf/134a Yotsumoto et al.39  575 0.00–0.82 263–323 0.02 
R-1234yf/134a Akasaka et al.17  22 0.32–0.72 350–371 1.66 
R-1234yf/134a Chen et al.40  94 0.04–0.86 299–403 0.39 
R-1234yf/1234ze(E) Higashi18  14 0.50–0.50 355–374 2.53 
R-1234yf/1234ze(E) Higashi41  52 0.50–0.50 340–430 1.56 
R-1234yf/1234ze(E) Al Ghafri et al.19  37 0.50–0.50 252–404 0.18 
R-134a/1234ze(E) Zhang et al.42  101 0.36–0.57 270–300 0.24 
R-134a/1234ze(E) Al Ghafri et al.19  59 0.50–0.50 252–403 0.11 
Pair (1/2)AuthorNz1 (mole frac.)T (K)AADρ (%)Nfail
R-1234yf/134a Yotsumoto et al.39  575 0.00–0.82 263–323 0.02 
R-1234yf/134a Akasaka et al.17  22 0.32–0.72 350–371 1.66 
R-1234yf/134a Chen et al.40  94 0.04–0.86 299–403 0.39 
R-1234yf/1234ze(E) Higashi18  14 0.50–0.50 355–374 2.53 
R-1234yf/1234ze(E) Higashi41  52 0.50–0.50 340–430 1.56 
R-1234yf/1234ze(E) Al Ghafri et al.19  37 0.50–0.50 252–404 0.18 
R-134a/1234ze(E) Zhang et al.42  101 0.36–0.57 270–300 0.24 
R-134a/1234ze(E) Al Ghafri et al.19  59 0.50–0.50 252–403 0.11 

5.2.1. VLE

For the mixture R-1234yf/134a, the VLE dataset from Shimoura et al.16 has been dropped because the deviations are greater than 30%. These large deviations suggest a systematic error in the measurements and also call into question the SOS data from the same reference, perhaps explaining the significant deviations from the measurements of Rowane and Perkins.12 Otherwise, the VLE dataset from Chen et al.36 shows large deviations, some greater than 10% compared with the model from this work, while that of Kamiaka et al.35 agrees well with the model developed in this work, with deviations mostly significantly less than 1% and an AAD of 0.2%.

For R-1234yf/1234ze(E), the measurements of Ye et al.37 are in good agreement with the model; deviations are all almost less than 1% and the AAD is 0.34%. The data from Al Ghafri et al.19 do not agree with either Ye et al.37 or our new model.

For R-134a/1234ze(E), the measurements of Kou et al.38 are consistent with the new model (AAD of 0.84%), while those of Al Ghafri et al.19 again deviate systematically.

5.2.2. PVT

The comparisons against existing experimental data are more challenging for density because of the near-critical region. These calculations are complicated by the fact that a solution may not exist thermodynamically or that the initial guesses may not be adequate to converge to the correct solution.

As is discussed in the work of Bell et al.,3 mixture models may cause problems for the iterative calculations in the critical region. Figure 11 provides a demonstration. The isopleth of the phase envelope of the new mixture model (the solid curve) falls just below the points in the critical region (worst deviations on the order of 0.3 K in the temperature direction). Saturation temperatures above the maximum of the curve are not accessible by the mixture model. The model from this work but with the departure term disabled (with Fij = 0), the dashed curve, is in better agreement with the data in the critical region but provides a worse representation of the other high-accuracy data.

FIG. 11.

Isopleth for equimolar composition of the binary mixture R-1234yf/1234ze(E). The solid curve is the new model, and the dashed curve is with Fij = 0. Points are the saturated vapor and liquid points from Higashi.41 Isopleth was obtained via CoolProp’s phase envelope low-level method, calling REFPROP. Failures of a saturation calculation are indicated by × markers.

FIG. 11.

Isopleth for equimolar composition of the binary mixture R-1234yf/1234ze(E). The solid curve is the new model, and the dashed curve is with Fij = 0. Points are the saturated vapor and liquid points from Higashi.41 Isopleth was obtained via CoolProp’s phase envelope low-level method, calling REFPROP. Failures of a saturation calculation are indicated by × markers.

Close modal

Taking into consideration the challenges of assessing density data, deviation plots for the existing data are shown in Fig. 12.

FIG. 12.

Relative deviations in density for existing data (Higashi,41 Zhang et al.,42 Yotsumoto et al.,39 Akasaka et al.,17 Higashi,18 Al Ghafri et al.,19 and Chen et al.40) as a function of temperature and composition for the three primary mixtures. The subscript TW indicates the model in this work. Thin vertical lines indicate temperatures or compositions corresponding to a failure of an iterative calculation in REFPROP that is not plotted.

FIG. 12.

Relative deviations in density for existing data (Higashi,41 Zhang et al.,42 Yotsumoto et al.,39 Akasaka et al.,17 Higashi,18 Al Ghafri et al.,19 and Chen et al.40) as a function of temperature and composition for the three primary mixtures. The subscript TW indicates the model in this work. Thin vertical lines indicate temperatures or compositions corresponding to a failure of an iterative calculation in REFPROP that is not plotted.

Close modal

For R-1234yf/134a, the datasets of Yotsumoto et al.39 and Chen et al.40 are in good agreement with the new model, although systematic composition-dependent errors can be seen for Chen et al.40 Although the deviations for Akasaka et al.17 are generally larger, this error metric exaggerates the errors3 as a consequence of the flatness of the saturation curve near the critical point (see Fig. 11).

For R-1234yf/1234ze(E), all the measurements are at the same bulk composition. The deviations of Al Ghafri et al.19 are small, while Higashi41 scatters more, though the same discussion about the exaggeration of the error metric applies.

For R-134a/1234ze(E), the two other datasets19,42 are in good agreement with the data. The datasets cover a relatively limited range of composition.

5.2.3. Critical region

Although mixture critical locus data were not included in the model development, there are a few experimental data points available in the literature.17,41 These critical loci data are shown in Fig. 13, along with curves calculated from the new mixture model in this work. The curves were traced with the teqp23 library. No VLE data were included in the model development for either binary pair, so it is comforting to see that the temperature and pressure of the critical loci are well represented with the new mixture models according to the admittedly very limited data in the literature.

FIG. 13.

Critical loci for R-1234yf/134a (with starred data points from Akasaka et al.17), for R-1234yf/1234ze(E) (with starred data point from Higashi41), and for R-134a/1234ze(E). The filled black circles indicate the pure-fluid values.

FIG. 13.

Critical loci for R-1234yf/134a (with starred data points from Akasaka et al.17), for R-1234yf/1234ze(E) (with starred data point from Higashi41), and for R-134a/1234ze(E). The filled black circles indicate the pure-fluid values.

Close modal

An interesting point about the critical locus for R-1234yf/134a is that it has a temperature minimum at 367.69 K, which is ∼0.1 K below the critical point of R-1234yf. This means that for temperatures between the temperature minimum of the critical curve and the critical temperature of R-1234yf, there are two critical points and the px curve is in two unconnected portions originating from the pure fluids. This behavior is not uncommon;43 the mixture carbon dioxide + ethane also shows this behavior.44 

A new means of assessing the extrapolation behavior of EOS is that of the effective hardness of interaction,45 which we denote as neff. In the dilute-gas limit, this quantity is defined by second virial coefficients B2 and their temperature derivatives,

limρ0neff=3TdB2dT+B2T2d2B2dT2+2TdB2dT,
(20)

and is plotted for two models in Fig. 14 in the dilute-gas limit as a function of temperature and composition. While the EOS for R-134a has the wrong infinite temperature limit (which should approach 3/2 for all potentials that are finitely valued at all separations; derivation in appendix of Ref. 46), the mixture models appear to smoothly transition between the two pure fluids. This is as expected, but comforting to see nonetheless. The EOS for R-1234ze(E) and R-1234yf demonstrate a small bump in the vicinity of 200 K, which is suspect because neither the Stockmayer fluid (a common model potential for molecules with dipole–dipole interactions) nor molecular models for water47 (a molecule with strong gas-phase association) show the bumps.45 Otherwise, the shapes of the neff curves are qualitatively similar to those of the small rigid molecules for which ab initio data are available.45 

FIG. 14.

Values of the effective hardness of interaction as a function of temperature and linearly spaced molar compositions in the dilute gas limit.

FIG. 14.

Values of the effective hardness of interaction as a function of temperature and linearly spaced molar compositions in the dilute gas limit.

Close modal

The second set of model results are for the binary pairs for which we included in the optimization only bubble-point pressure measurements from our group. In this case, an alternative model fitting procedure was employed. The method described in detail in Ref. 25 was used, which uses an evolutionary optimization approach in concert with iterative bubble-point pressure calculations from REFPROP 10.0.4 The interaction parameters βT,ij and γT,ij were optimized, the values of βv,ij and γv,ij were set to 1.0, and no departure function was used. The obtained parameters are in Table 9, and the deviations are shown in Fig. 15. The AAD for the bubble-point pressures from our measurements are all less than 0.2%. For the binary pair R-125/R-1234yf, the model is able to fully capture the bubble-point data by fitting βT,ij and γT,ij, showing no systematic errors and an AAD less than 0.1% (which is much smaller than the experimental uncertainty). For the other two binary pairs, fitting only βT,ij and γT,ij does not appear to be sufficient to remove the systematic temperature deviation, although the AAD are all still below 0.2%.

TABLE 9.

Interaction parameters obtained from fitting bubble-point data only. Note that there is no departure function. Components in each binary pair are sorted by normal boiling point temperatures and the order matters

Pair (1/2)βT,ijγT,ijβv,ijγv,ijFij
R-125/1234yf 0.999 637 0.999 356 1.0 1.0 0.0 
R-1234yf/152a 1.002 918 0.983 928 1.0 1.0 0.0 
R-1234ze(E)/227ea 1.000 895 0.993 523 1.0 1.0 0.0 
Pair (1/2)βT,ijγT,ijβv,ijγv,ijFij
R-125/1234yf 0.999 637 0.999 356 1.0 1.0 0.0 
R-1234yf/152a 1.002 918 0.983 928 1.0 1.0 0.0 
R-1234ze(E)/227ea 1.000 895 0.993 523 1.0 1.0 0.0 
FIG. 15.

Deviations for VLE-only models compared with literature data (Tomassetti,14 Hu et al.,51 Yang et al.,52 Al Ghafri et al.,19 Kamiaka et al.,49 Yang et al.,50 Dang et al.,48 and Kamiaka et al.35) as open markers. Solid markers are the fitted VLE data from Ref. 11, with the AAD as indicated in the figure. The two-phase data from Tomassetti14 have been dropped.

FIG. 15.

Deviations for VLE-only models compared with literature data (Tomassetti,14 Hu et al.,51 Yang et al.,52 Al Ghafri et al.,19 Kamiaka et al.,49 Yang et al.,50 Dang et al.,48 and Kamiaka et al.35) as open markers. Solid markers are the fitted VLE data from Ref. 11, with the AAD as indicated in the figure. The two-phase data from Tomassetti14 have been dropped.

Close modal

The other datasets from the literature are listed in Table 10 and their deviations are plotted in Fig. 15. For R-125/1234yf and R-1234yf/152a, the existing data do not completely agree with our measured data, demonstrating scatter in pressure up to 2%.

TABLE 10.

Literature sources for the binary pairs with VLE-only measurements included in this work. For PVT points, the AAD is a deviation in density, and for VLE points, it is the deviation in bubble-point pressure. The two-phase data from Tomassetti14 have been dropped in the calculation of AAD

Pair (1/2)KindAuthorNT (K)AAD (%)
R-125/1234yf VLE Kamiaka et al.49  28 273–333 0.55 
R-125/1234yf VLE Kamiaka et al.35  84 273–333 0.28 
R-125/1234yf VLE Yang et al.50  35 283–323 0.16 
R-125/1234yf PVT Dang et al.48  27 284–318 0.47 
R-125/1234yf PVT Al Ghafri et al.19  40 252–383 0.26 
R-1234yf/152a VLE Hu et al.51  60 283–323 0.70 
R-1234yf/152a VLE Yang et al.52  25 283–323 0.60 
R-1234yf/152a PVT Tomassetti14  136 268–373 0.16 
Pair (1/2)KindAuthorNT (K)AAD (%)
R-125/1234yf VLE Kamiaka et al.49  28 273–333 0.55 
R-125/1234yf VLE Kamiaka et al.35  84 273–333 0.28 
R-125/1234yf VLE Yang et al.50  35 283–323 0.16 
R-125/1234yf PVT Dang et al.48  27 284–318 0.47 
R-125/1234yf PVT Al Ghafri et al.19  40 252–383 0.26 
R-1234yf/152a VLE Hu et al.51  60 283–323 0.70 
R-1234yf/152a VLE Yang et al.52  25 283–323 0.60 
R-1234yf/152a PVT Tomassetti14  136 268–373 0.16 

The existing density datasets for these three binary mixtures are those of Dang et al.,48 Al Ghafri et al.,19 and Tomassetti.14 The data from Dang et al.48 deviate up to 1% in density. Al Ghafri et al.19 claimed a combined expanded uncertainty of 0.2% in the liquid phase and 2% in the gas phase; their density data are reproduced within this band. Tomassetti14 included some two-phase points, resulting in nonsensical density deviation values. The two-phase points were dropped, and the phase was specified to be gas.

The models presented in this work were implemented in REFPROP and CoolProp. The necessary files are provided in the supplementary material. Calculated values are provided in Table 11, and it was confirmed that the upcoming version of REFPROP and CoolProp version 6.4.2 yielded the same results to 13 digits. The new equation of state for R-1234yf of Lemmon and Akasaka6 is also provided in REFPROP and CoolProp formats. An updated fluid file for REFPROP for R-152a is also provided, with the coefficients rounded to agree with the original publication8 in order to yield the precise values in the table.

TABLE 11.

Check values obtained with CoolPropa for the model obtained 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
R125 1.0 424 3823 339.173 000 000 000 0 4779.000 000 000 000 0 −0.455 060 052 344 49 
NEWR1234YF 1.0 460 3344 367.850 000 000 000 0 4180.000 000 000 000 0 −0.468 353 699 040 81 
R134A 1.0 468 3983 374.180 000 000 000 0 4978.830 171 000 001 4 −0.466 824 484 145 93 
R152A 1.0 483 4457 386.411 000 000 000 0 5571.449 999 999 999 8 −0.507 421 495 701 51 
R1234ZEE 1.0 478 3432 382.513 000 000 000 0 4290.000 000 000 000 0 −0.463 409 784 472 30 
R227EA 1.0 469 2796 374.900 000 000 000 0 3495.000 000 000 000 0 −0.442 385 761 979 82 
NEWR1234YF+R1234ZEE 0.4 469 3399 375.368 708 235 417 6 4248.595 802 001 349 5 −0.460 594 641 762 52 
NEWR1234YF+R134A 0.4 462 3698 369.337 535 207 333 2 4622.435 740 472 200 5 −0.465 508 591 288 31 
R134A+R1234ZEE 0.4 472 3639 377.666 767 145 286 7 4548.679 872 122 646 9 −0.462 451 303 341 93 
R125+NEWR1234YF 0.4 445 3523 356.118 063 976 404 5 4403.741 839 301 608 7 −0.463 908 051 082 07 
NEWR1234YF+R152A 0.4 470 3947 376.126 334 562 794 2 4933.399 297 446 838 3 −0.492 524 178 122 31 
R1234ZEE+R227 EA 0.4 471 3025 376.790 938 517 181 3 3781.010 485 718 356 4 −0.451 298 816 594 40 
Names (1/2)z1 (mole frac.)T (K)ρ (mol/m3)Tred (K)ρred (mol/m3)αr
R125 1.0 424 3823 339.173 000 000 000 0 4779.000 000 000 000 0 −0.455 060 052 344 49 
NEWR1234YF 1.0 460 3344 367.850 000 000 000 0 4180.000 000 000 000 0 −0.468 353 699 040 81 
R134A 1.0 468 3983 374.180 000 000 000 0 4978.830 171 000 001 4 −0.466 824 484 145 93 
R152A 1.0 483 4457 386.411 000 000 000 0 5571.449 999 999 999 8 −0.507 421 495 701 51 
R1234ZEE 1.0 478 3432 382.513 000 000 000 0 4290.000 000 000 000 0 −0.463 409 784 472 30 
R227EA 1.0 469 2796 374.900 000 000 000 0 3495.000 000 000 000 0 −0.442 385 761 979 82 
NEWR1234YF+R1234ZEE 0.4 469 3399 375.368 708 235 417 6 4248.595 802 001 349 5 −0.460 594 641 762 52 
NEWR1234YF+R134A 0.4 462 3698 369.337 535 207 333 2 4622.435 740 472 200 5 −0.465 508 591 288 31 
R134A+R1234ZEE 0.4 472 3639 377.666 767 145 286 7 4548.679 872 122 646 9 −0.462 451 303 341 93 
R125+NEWR1234YF 0.4 445 3523 356.118 063 976 404 5 4403.741 839 301 608 7 −0.463 908 051 082 07 
NEWR1234YF+R152A 0.4 470 3947 376.126 334 562 794 2 4933.399 297 446 838 3 −0.492 524 178 122 31 
R1234ZEE+R227 EA 0.4 471 3025 376.790 938 517 181 3 3781.010 485 718 356 4 −0.451 298 816 594 40 
a

CoolProp version 6.4.2dev, revision be204ce8d5b877246c2954a9948c3d72302ff587 (SHA1 hash).

Most importantly, this study demonstrates that high-accuracy liquid-phase SOS and density data are largely sufficient to fit highly accurate thermodynamic models for mixtures of chemically similar HFO- and HFC-containing binary mixtures. With the fitting of four interaction parameters and small departure functions, the experimental data can be reproduced to nearly their experimental uncertainty. In addition, the availability of highly accurate reference mixture measurements allows the quality of the datasets in the literature to be assessed, identifying a number of discrepancies in the mixture data and recommending which datasets should be considered. The highly accurate mixture data show that the equation of state for R-1234ze(E) needs to be refit in order to better reproduce the new liquid-phase SOS data.

In a parallel project, further measurements are under way on some of the binary mixtures for which only VLE measurements were available in this work. These measurements will allow refinement of the interim models up to the level of the other mixture models obtained in this work.

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

This work was partially supported by the Strategic Environmental Research and Development Program (Project WP19-1385: WP-2740 Follow-On: Low-GWP Alternative Refrigerant Blends for HFC-134a). The author would like to thank Demian Riccardi (of NIST) for help accessing data of TDE.

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. Additional information is available from the corresponding author upon reasonable request.

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Supplementary Material