The work presents the first wide-range equation of state (EOS) for 3He–4He mixtures based on the reduced Helmholtz free energy multi-fluid approximation model. It covers the temperature range from 2.17 to 300 K and the pressure from the vapor pressure up to 3 MPa for any given mixture 3He mole fraction. In this model, the 4He and 3He reduced Helmholtz free energy equations and departure functions from the literature are employed and only five unknown mixture parameters are needed for each given departure function. The parameters and the best model for the concerned binary mixture were determined by the Levenberg–Marquardt optimization method. With the best developed model, the liquid, gaseous, and saturated thermophysical properties of the mixture can be mostly described with an accuracy better than 5%. Furthermore, a database for the thermophysical properties of 3He–4He mixtures is generated and provided for interpolation in temperature, pressure, and 3He mole fraction. The current EOS and database can be applied to the design and optimization of ultra-low temperature refrigerators.

Since the first liquefaction of 4He in 19081 and the discovery of the superfluidity of pure 3He in 1971,2 the fascination for understanding the properties of 3He–4He quantum fluids at ultra-low temperatures has not ended. The ideal Fermi gas model of 3He–4He solutions was first proposed by Landau and Pomeranchuk3 and improved by Bardeen et al.,4 Radebaugh,5 and Kuerten et al.6 Those calculations were restricted to zero pressure, 3He mole fractions below 8%, and temperatures below 250 mK. Extending these efforts, Chaudhry et al.7,8 published the thermodynamic properties of liquid 3He–4He mixtures over the entire composition range between 0.15 and 1.5 K and up to 10 bars. At higher temperatures, Karnatsevich et al.9 developed an empirical equation of state (EOS) for equimolar 3He–4He mixtures in the temperature range 1.5–14 K by using an expression close to the EOS for 4He proposed by McCarty.10 Sibileva et al.11 improved the empirical EOS for the liquid phase over the whole fraction range. However, there is still no unified EOS for 3He–4He mixtures covering a wide range of temperature, pressure, and composition.

The efforts made by Chaudhry et al.7,8 were to develop sub-kelvin refrigeration cycles that use 3He–4He mixtures as the working fluid, which could prove to be more efficient than 3He–4He dilution refrigerators by eliminating many of the losses associated with the latter.12 These newer cycles use mixtures of 3He and 4He with much higher fractions of 3He.13 The proper design of these machines requires knowledge of 3He–4He mixture properties over the entire fraction range from pure 3He to pure 4He and up to temperatures in excess of 1.2 K, a natural choice for the high-temperature reservoir for a sub-kelvin refrigerator.

Efficient refrigeration from 300 to 1.2 K is the other part of such efforts and is our motivation. Low frequency (less than 2 Hz) pulse-tube refrigeration offers unprecedented reliability with no moving parts in the cold end, together with cryogen-free operation up to 20 000 h before standard maintenance.14 Using 4He as the working fluid, the lowest temperature of a pulse-tube refrigerator is limited by the alpha line of pure 4He at 2.17 K.15–17 Indeed, the base temperature of a commercial 4He pulse-tube refrigerator is 2.5 K. Lower temperature to about 1.3 K can be achieved by replacing 4He with 3He.18 However, the price and availability of 3He limit its large-scale application as a pure gas.19 An alternative solution is the use of 3He–4He mixtures. Figure 1 shows the phase diagram of 3He–4He mixtures at different pressures and the alpha line [the ideal cooling power for a pulse-tube cryocooler is Q̇=TαvVmṅδp̄, as shown in Eq. (5) in Ref. 19, so it only works in the region of αv>0] (αv = 0, where αv is the volumetric thermal expansion coefficient, whose expression can be found in Sec. 3.3) of the 3He–4He mixture. One may note that with a 60% 3He mole fraction, temperatures below 1.2 K may be achieved. This can be compared with the usual 25% mole fraction used for dilution refrigeration. Among our motivations, we aim to introduce a new efficient and reliable 3He–4He 300–1 K Vuilleumier Pulse-Tube Cooler (VPTC) for the space cooling chains of sub-kelvin refrigerators.21,22 This effort would allow reducing the cold mass of present cooling chains by almost half. This would be a breakthrough since the cold mass of a sub-kelvin cooler including redundancy may be by weight about one third of the satellite mass.

FIG. 1.

The 3He–4He mixture phase diagram at the pressures of 2 and 10 bars and the α=0 line at saturated pressure (the phase-separation curve at high pressure is from Ref. 8, and the α=0 line is from Ref. 20).

FIG. 1.

The 3He–4He mixture phase diagram at the pressures of 2 and 10 bars and the α=0 line at saturated pressure (the phase-separation curve at high pressure is from Ref. 8, and the α=0 line is from Ref. 20).

Close modal

Up until now, we have shown that the thermodynamic properties of 3He–4He mixtures are well documented above the superfluid transition temperature of 3He at 2.44 mK and 34.3 MPa, up to 14 K. However, there is still no EOS for 3He–4He gaseous mixtures for a wide range of 3He mole fractions, allowing the calculation of all thermodynamic properties required for designing 1 K class cryocoolers.

In recent years, Kunz and Wagner23,24 published the GERG-2008 mixture model with an EOS based on the Helmholtz free energy, which may be applied to binary mixtures. In this model, the Helmholtz free energy is expressed as an explicit function of density, temperature, and composition, from where all the other thermophysical properties can be accurately derived. We use this model to build the first EOS for the 3He–4He binary mixture using the Helmholtz free energy EOS for pure 3He25 and 4He26,27 and construct the first wide-range EOS for 3He–4He mixtures from 2.17 K to room temperature and from the vapor pressure to pressures higher than 3 MPa.

The remainder of this paper is structured as follows. In Sec. 2, the database used in the present work is summarized. In Sec. 3, the theoretical model is described in detail, in which the fundamental equation, thermophysical property calculation method, and optimization method are presented. In Sec. 4, the fitting results and the fitting error are presented. Finally, a conclusion in Sec. 5 completes the paper.

The critical parameters for pure 3He and 4He have been well studied by many researchers. In the present work, the critical parameters in Refs. 25 and 27 are used, as shown in Table 1. For the critical temperature of the 3He–4He mixture, its value depends on the 3He mole fraction. One can find experimental data for the critical temperature of 3He–4He mixtures in the work of Wallace and Meyer.30 

TABLE 1.

The molar mass and critical parameters for pure 4He and 3He

M (g/mol)Tc (K)pc (MPa)ρc (mol/m3)
3He25  3.016 03 3.315 7 0.114 603 9 41.191 
4He27  4.002 602 5.195 3 0.227 46 69.58 
M (g/mol)Tc (K)pc (MPa)ρc (mol/m3)
3He25  3.016 03 3.315 7 0.114 603 9 41.191 
4He27  4.002 602 5.195 3 0.227 46 69.58 

The available experimental data for vapor–liquid equilibrium (VLE) properties of 3He–4He mixtures are summarized in Table 2, which includes the bubble-point pressure, the dew-point pressure, the saturated vapor density, and the saturated liquid density at temperatures above 2.17 K. Eselson and Berezniak28 measured the bubble-point and dew-point pressures in a wide range of 3He mole fraction from 0.04 to 0.908 in the temperature range from 1.2 to 3.5 K. Wallace et al.30,29 measured the bubble-point and dew-point pressures in a range of 3He mole fraction from 0.2 to 0.886 at temperatures up to the critical temperature. Sydoriak and Roberts31 and Sreedhar and Daunt32 measured the bubble-point pressure in the range of 3He mole fraction from 0.1 to 0.9 at temperatures below 2.4 K and from 0.01 to 0.12 at temperatures below 2.6 K, respectively. Sibilyova et al.33 presented the bubble-point pressure with a 3He mole fraction of 0.4 in the temperature range from 2.25 to 3.75 K.

TABLE 2.

The database of the VLE properties for 3He–4He mixtures

SourcesPointsT (K)p (MPa)xUncertainty
Saturated pressure 
Eselson and Berezniak28  140 2.17–3.2 0.007–0.05 0.04–0.908 0.5% in bubble pressure 
2%–3% in dew-point pressure 
Wallace and Meyer30  94a 2.2–4.8 0.006–0.17 0.2–0.866 0.5% 
Wallace et al.29  20 2.5–3.0 0.004–0.07 0.2–0.866 0.5% 
Sydoriak and Roberts31  18 <2.4 0.008–0.02 0.1–0.9 1% in x 
Sreedhar and Daunt32  20 <2.6 0.006–0.018 0.02–0.12 1.2% in vapor pressure 
Sibilyova et al.33  2.25–3.75 0.016–0.11 0.4 ⋯ 
Saturated density 
Ptukha34  49 2.18–3.9 Saturated 0.1–0.85 1%b 
Eselson et al.35  62 2.2–4.2 Saturated 0–1 ⋯ 
Wallace et al.29  20 2.5–3.0 Saturated 0.2–0.866 0.5% 
Sibilyova et al.36  46 2.25–4.2 Saturated 0.2–0.8 T < 3 K, 3% 
T > 3 K, 6% 
Wang et al.37  84 2.2–4.4 Saturated 0–0.494 0.2% in saturated density 
SourcesPointsT (K)p (MPa)xUncertainty
Saturated pressure 
Eselson and Berezniak28  140 2.17–3.2 0.007–0.05 0.04–0.908 0.5% in bubble pressure 
2%–3% in dew-point pressure 
Wallace and Meyer30  94a 2.2–4.8 0.006–0.17 0.2–0.866 0.5% 
Wallace et al.29  20 2.5–3.0 0.004–0.07 0.2–0.866 0.5% 
Sydoriak and Roberts31  18 <2.4 0.008–0.02 0.1–0.9 1% in x 
Sreedhar and Daunt32  20 <2.6 0.006–0.018 0.02–0.12 1.2% in vapor pressure 
Sibilyova et al.33  2.25–3.75 0.016–0.11 0.4 ⋯ 
Saturated density 
Ptukha34  49 2.18–3.9 Saturated 0.1–0.85 1%b 
Eselson et al.35  62 2.2–4.2 Saturated 0–1 ⋯ 
Wallace et al.29  20 2.5–3.0 Saturated 0.2–0.866 0.5% 
Sibilyova et al.36  46 2.25–4.2 Saturated 0.2–0.8 T < 3 K, 3% 
T > 3 K, 6% 
Wang et al.37  84 2.2–4.4 Saturated 0–0.494 0.2% in saturated density 
a

There are no tabular data in the reference, so we selected some data points along the smooth curve in the figure.

b

Systematic error.

For the saturated density, Ptukha34 measured saturated liquid density in a wide range of 3He mole fractions from 0.1 to 0.85 in the temperature range from 1.3 to 3.9 K. Eselson et al.35 measured saturated liquid density with 3He mole fraction from 0 to 1 at temperatures from 1.4 to 4.2 K. Sibilyova et al.36 presented saturated vapor and liquid density with 3He mole fractions of 0.2, 0.6, and 0.8 in the temperature range from 2.25 to 4.2 K. Wang et al.37 measured saturated liquid density with 3He mole fraction from 0 to 0.5 at temperatures from 1.3 to 4.4 K. However, those data were only found after the fitting, so they are only used for checking the performance of the developed EOS in this work.

The only two ρpT experimental data sources for 3He–4He mixtures are the works of Bogoyavlensky et al.,38,39 as shown in Table 3. Those data cover temperatures up to 4.2 K and pressures up to 2.4 MPa for the liquid-phase region and temperatures up to 20 K and pressures up to 3.6 MPa for the gaseous region in a wide range of 3He mole fraction. Besides ρpT data, the virial coefficient is also important for an EOS. Keller40 and Karnatsevic et al.42 reported experimental data for the second virial coefficients for 3He–4He mixtures. Barrufet and Eubank41 reported the second virial coefficients for 3He–4He mixtures at temperatures below 5 K derived from the ρpT data of Wallace and Meyer.

TABLE 3.

The database of ρpT data and virial coefficients for 3He–4He mixtures

SourcesPointsT (K)p (MPa)xUncertainty
ρpT 
Bogoyavlensky and Yuechenko38  432 2.25–4.2 0.1–2.4 0.352–0.651 0.1% in x 
0.2% in p 
0.2% in ρ 
Bogoyavlensky et al.39  1499 4.5–20.2 0.03–3.6 0–1 0.1% in x 
0.2% in p 
10 mK in T 
0.2%–3% in ρ 
Second virial coefficient 
Keller40  2.2–4  0.5475 ⋯ 
Wallace and Meyera 45 3.55–5  0.2–0.8 ⋯ 
Karnatsevic et al.42  35 4.48–8.9  0–1 ⋯ 
Hurly and Moldover43  75 1–10 000  0.5 ⋯ 
Cencek et al.44  79 1–10 000  0.5 0.001%–2.36% 
SourcesPointsT (K)p (MPa)xUncertainty
ρpT 
Bogoyavlensky and Yuechenko38  432 2.25–4.2 0.1–2.4 0.352–0.651 0.1% in x 
0.2% in p 
0.2% in ρ 
Bogoyavlensky et al.39  1499 4.5–20.2 0.03–3.6 0–1 0.1% in x 
0.2% in p 
10 mK in T 
0.2%–3% in ρ 
Second virial coefficient 
Keller40  2.2–4  0.5475 ⋯ 
Wallace and Meyera 45 3.55–5  0.2–0.8 ⋯ 
Karnatsevic et al.42  35 4.48–8.9  0–1 ⋯ 
Hurly and Moldover43  75 1–10 000  0.5 ⋯ 
Cencek et al.44  79 1–10 000  0.5 0.001%–2.36% 
a

The original reference is Wallace and Meyer, “Tabulation of the original pressure-volume-temperature data for 3He-4He mixtures and for 3He,” A technical report from the Department of Physics, Duke University, 1984, but this technical report cannot be found. Here, the data are taken from Ref. 41 that refers to the above report.

Hurly and Moldover43 presented the second virial coefficients for equimolar 3He–4He mixtures using ab initio calculation. However, the uncertainty of those calculations is not clear. Cencek et al.44 presented the interaction virial B34 for 3He–4He mixtures with very small uncertainty, which can be used to calculate the second virial coefficients at any given 3He mole fraction.

The heat capacity and sound speed can be calculated from the second derivative of the Helmholtz free energy EOS, so they are also important for the fitting of EOS. Table 4 summarizes those data. Dokoupil et al.45 measured saturated heat capacity with 3He mole fraction from 0 to 0.417 at temperatures up to 4 K. Pandorf et al.46 measured heat capacity at constant volume near the freezing pressure with 3He mole fraction from 0.17 to 0.95 at temperatures up to 4.5 K. For sound speed, the main data are from the work of Vignos and Fairbank,47 who measured the sound speed of 3He–4He mixtures with 3He mole fractions of 0.25, 0.75, and 0.98 at pressures up to 7 MPa and temperatures up to 4.5 K. The data from the work of Roberts and Sydoriak48 and Eselson et al.49 were only found after our fitting was complete, so they are only used for checking the performance of the EOS developed in this work.

TABLE 4.

The database of heat capacity and speed of sound for 3He–4He mixtures

SourcesPointsT (K)p (MPa)xUncertainty
Heat capacity 
Dokoupil et al.45  40a 2.2–4 Saturated 0.01–0.417 4% 
Pandorf et al.46  49a 2.5–4.5 5–15 0.17–0.95 ⋯ 
Sound speed 
Vignos and Fairbank47  122 2.5–4 0.1–7 0.25–0.98 0.1%–0.3% in w 
Roberts and Sydoriak48  2.2–2.3 Saturated 0.301 0.3% in w 
Eselson et al.49  43 2.5–4.2 Saturated 0–0.2 0.05% in x 
0.15% in w 
SourcesPointsT (K)p (MPa)xUncertainty
Heat capacity 
Dokoupil et al.45  40a 2.2–4 Saturated 0.01–0.417 4% 
Pandorf et al.46  49a 2.5–4.5 5–15 0.17–0.95 ⋯ 
Sound speed 
Vignos and Fairbank47  122 2.5–4 0.1–7 0.25–0.98 0.1%–0.3% in w 
Roberts and Sydoriak48  2.2–2.3 Saturated 0.301 0.3% in w 
Eselson et al.49  43 2.5–4.2 Saturated 0–0.2 0.05% in x 
0.15% in w 
a

Data points selected from the smooth curve in the figure.

The EOS for 3He–4He mixtures used in the present work is explicitly expressed as the reduced Helmholtz free energy α, which includes an ideal part α0 and a residual part αr,23,24

αδ,τ,x=aρ,T,xRT=α0ρ,T,x+αrδ,τ,x,
(1)

where a is the Helmholtz free energy and ρ, T, and x are the density, temperature, and mole fraction vector of mixture components. R is the universal gas constant. Since 20 May 2019, all SI units are defined in terms of constants that describe the natural world. The Boltzmann constant is fixed as k = 1.380 649 × 10−23 J K−1, and the Avogadro constant is fixed as NA = 6.022 140 76 × 1023 mol−1.50 Then, R is fixed as R=kNA = 8.314 462 618 J mol−1 K−1.

δ and τ are the reduced mixture density and inverse reduced mixture temperature, which are defined as

δ=ρρrandτ=TrT,
(2)

where ρr and Tr are the composition-dependent reducing functions for the mixture density and temperature, respectively. The GERG-2008 reducing functions were used in the present work, which have been widely used as the mixing rule for various refrigerant mixtures.24 Their formulations are

Trx=i=1Nxi2Tc,i+i=1Nj=2N2xixjβT,ijγT,ijxi+xjβT,ij2xi+xj×Tc,iTc,j0.5,
(3)
1ρrx=i=1Nxi21ρc,i+i=1Nj=2N2xixjβν,ijγν,ijxi+xjβν,ij2xi+xj×1ρc,i1/3+1ρc,j1/33,
(4)

where the parameter γ is symmetric and parameter β is asymmetric; then, there are four parameters (βT,12, γT,12, βν,12, γν,12) that need to be fitted.

The ideal gas part and residual part of the reduced Helmholtz free energy for the mixtures are functions of the pure-fluid Helmholtz free energies, which can be expressed as

α0ρ,T,x=i=1Nxiαoi0ρ,T+lnxi,
(5)
αrδ,τ,x=i=1Nxiαoirδ,τ+Δαrδ,τ,x,
(6)

where αoi0 is the ideal part of the reduced Helmholtz free energy of pure 3He and 4He and αoir is the residual part of the reduced Helmholtz free energy of pure 3He and 4He. The pure helium Helmholtz free energies will be introduced in detail in Sec. 3.2.

Δαr is the departure function for the multicomponent mixtures. It is used to describe the non-ideal behavior of mixtures,

Δαrδ,τ,x=i=1N1j=i+1NxixjFijαijrδ,τ,
(7)

where Fij is the interaction parameter for the binary mixture, which is the fifth parameter to be fitted in the present work. αijr is the departure function for the binary pair. In the present work, the purpose is to build a reliable and practical Helmholtz free energy EOS for 3He–4He mixtures covering the limited available data. The equation is based on the thermodynamic relation between the concerned properties and the reduced Helmholtz free energy (for details, see Sec. 3.3). Due to the limited available literature data for this binary mixture, we built the Helmholtz free energy EOS in the common way,51,52 i.e., only optimizing the five parameters by using published and accepted departure functions (using four departure functions can give a check for the fitting quality and sensitivity of the departure function) [hydrocarbon mixtures in GERG 2008 (called KW0 later)24 and departure functions for 4He–Ne, 4He–Ar, and Ne–Ar53] and then comparing the four different combinations to better describe the thermophysical properties of the 3He–4He binary mixture. The formulation of the departure function for KW0 is in Eq. (8), and its coefficients are given in Table 5. The formulation of the departure function for 4He–Ne, 4He–Ar, and Ne–Ar is the same as in Eq. (9), and the coefficients of those departure functions are given in Table 6,

αijrδ,τ=Nkδdkτtkexpsgnlkδlkηkδεk2βkδγk2,
(8)
αijrδ,τ=Nkδdkτtkexpsgnlkδlkηkδεk2βkτγk2.
(9)
TABLE 5.

Departure function coefficients for the KW0 model24 

kNkdktklkηkεkβkγk
2.557 477 684 411 8 
−7.984 635 713 635 3 1.55 
4.785 913 146 580 6 1.7 
−0.732 653 924 000 0 0.25 
1.380 547 134 531 2 1.35 
0.283 496 035 000 0 
−0.490 873 859 000 0 1.25 
−0.102 918 889 000 0 
0.118 363 147 000 0 0.7 
10 0.000 055 527 385 7 5.4 
kNkdktklkηkεkβkγk
2.557 477 684 411 8 
−7.984 635 713 635 3 1.55 
4.785 913 146 580 6 1.7 
−0.732 653 924 000 0 0.25 
1.380 547 134 531 2 1.35 
0.283 496 035 000 0 
−0.490 873 859 000 0 1.25 
−0.102 918 889 000 0 
0.118 363 147 000 0 0.7 
10 0.000 055 527 385 7 5.4 
TABLE 6.

Departure function coefficients for 4He mixtures53 

kNktkdkηkβkγkεk
4He–Ne 
−4.346 85 1.195 
−0.884 38 1.587 
0.258 416 1.434 
3.502 188 1.341 −0.157 −0.173 1.31 1.032 
0.831 33 1.189 −0.931 −1.07 1.356 1.978 
2.740 495 1.169 −0.882 −0.695 1.596 1.966 
−1.582 23 0.944 −0.868 −0.862 1.632 1.709 
−0.304 9 1.874 −0.543 −0.971 0.766 0.583 
4He–Ar 
−2.643 65 1.03 
−0.347 5 0.288 
0.201 207 0.572 
1.171 326 1.425 −0.371 −0.32 1.409 0.378 
0.216 379 1.987 −0.081 −1.247 1.709 0.741 
0.561 37 0.024 −0.375 −1.152 0.705 0.322 
0.182 57 1.434 −0.978 −0.245 1.162 1.427 
0.017 879 0.27 −0.971 −1.03 0.869 2.088 
Ne–Ar 
−1.039 69 0.723 
0.593 776 1.689 
−0.186 53 1.365 
−0.223 32 0.201 −1.018 −0.36 1.119 2.49 
0.160 847 0.164 −0.556 −0.373 1.395 1.202 
0.405 228 0.939 −0.221 −0.582 1.01 2.468 
−0.264 56 1.69 −0.862 −0.319 1.227 0.837 
−0.033 57 1.545 −0.809 −0.56 1.321 2.144 
kNktkdkηkβkγkεk
4He–Ne 
−4.346 85 1.195 
−0.884 38 1.587 
0.258 416 1.434 
3.502 188 1.341 −0.157 −0.173 1.31 1.032 
0.831 33 1.189 −0.931 −1.07 1.356 1.978 
2.740 495 1.169 −0.882 −0.695 1.596 1.966 
−1.582 23 0.944 −0.868 −0.862 1.632 1.709 
−0.304 9 1.874 −0.543 −0.971 0.766 0.583 
4He–Ar 
−2.643 65 1.03 
−0.347 5 0.288 
0.201 207 0.572 
1.171 326 1.425 −0.371 −0.32 1.409 0.378 
0.216 379 1.987 −0.081 −1.247 1.709 0.741 
0.561 37 0.024 −0.375 −1.152 0.705 0.322 
0.182 57 1.434 −0.978 −0.245 1.162 1.427 
0.017 879 0.27 −0.971 −1.03 0.869 2.088 
Ne–Ar 
−1.039 69 0.723 
0.593 776 1.689 
−0.186 53 1.365 
−0.223 32 0.201 −1.018 −0.36 1.119 2.49 
0.160 847 0.164 −0.556 −0.373 1.395 1.202 
0.405 228 0.939 −0.221 −0.582 1.01 2.468 
−0.264 56 1.69 −0.862 −0.319 1.227 0.837 
−0.033 57 1.545 −0.809 −0.56 1.321 2.144 

To build an accurate EOS for 3He–4He mixtures, a high-accuracy Helmholtz free energy EOS for pure helium is required. For 4He, its Helmholtz free energy EOS has been well developed26,27 and widely used in software such as REFPROP.54 It can be expressed as27 

αo0δ,τ=lnρrρcδ+a01lnTcTrτ+a1+a2TcTrτ,
(10)
αo,He4rδ,τ=1kNkδdkτtk×expsgnlkδlkηkδεk2βkτγk2,
(11)

where a0 = 2.5, a1 = 0.173 348 642 2, and a2 = 0.467 452 363 8 for 4He. Tc and ρc are the critical temperature and density, as shown in Table 1. Nk, dk, tk, lk, ηk, εk, βk, and γk are the coefficients and exponents for 4He. The values of these parameters are shown in Table 7. The model has a very good performance on the density and speed of sound with errors less than 0.5%, and the error of heat capacity is about 2%.

TABLE 7.

The Helmholtz free energy EOS coefficients and exponents for 4He (Ref. 27)

kNktkdklkηkβkγkεk
0.015 559 
3.063 893 0.425 
−4.242 08 0.63 
0.054 418 0.69 
−0.189 72 1.83 
0.087 856 0.575 
2.283 357 0.925 
−0.533 32 1.585 
−0.532 97 1.69 
10 0.994 449 1.51 
11 −0.300 79 2.9 
12 −1.643 26 0.8 
13 0.802 91 1.26 1.5497 0.2471 3.15 0.596 
14 0.026 839 3.51 9.245 0.0983 2.54505 0.3423 
15 0.046 877 2.785 4.76323 0.1556 1.2513 0.761 
16 −0.148 33 6.3826 2.6782 1.9416 0.9747 
17 0.030 162 4.22 8.7023 2.7077 0.5984 0.5868 
18 −0.019 99 0.83 0.255 0.6621 2.2282 0.5627 
19 0.142 835 1.575 0.3523 0.1775 1.606 2.5346 
20 0.007 418 3.447 0.1492 0.4821 3.815 3.6763 
21 −0.229 9 0.73 0.05 0.3069 1.61958 4.5245 
22 0.792 248 1.634 0.1668 0.1758 0.6407 5.039 
23 −0.049 39 6.13 42.2358 1357.658 1.076 0.959 
kNktkdklkηkβkγkεk
0.015 559 
3.063 893 0.425 
−4.242 08 0.63 
0.054 418 0.69 
−0.189 72 1.83 
0.087 856 0.575 
2.283 357 0.925 
−0.533 32 1.585 
−0.532 97 1.69 
10 0.994 449 1.51 
11 −0.300 79 2.9 
12 −1.643 26 0.8 
13 0.802 91 1.26 1.5497 0.2471 3.15 0.596 
14 0.026 839 3.51 9.245 0.0983 2.54505 0.3423 
15 0.046 877 2.785 4.76323 0.1556 1.2513 0.761 
16 −0.148 33 6.3826 2.6782 1.9416 0.9747 
17 0.030 162 4.22 8.7023 2.7077 0.5984 0.5868 
18 −0.019 99 0.83 0.255 0.6621 2.2282 0.5627 
19 0.142 835 1.575 0.3523 0.1775 1.606 2.5346 
20 0.007 418 3.447 0.1492 0.4821 3.815 3.6763 
21 −0.229 9 0.73 0.05 0.3069 1.61958 4.5245 
22 0.792 248 1.634 0.1668 0.1758 0.6407 5.039 
23 −0.049 39 6.13 42.2358 1357.658 1.076 0.959 

3He has not been widely studied and does not have a reference formulation like 4He. The most accurate 3He Helmholtz free energy was developed based on Debye phonon theory by Huang et al.,25 which has been used in the software He3Pak55 and REGEN.56 This Debye model is valid from 0.01 K to room temperature with pressures from the vapor pressure to the melting pressure. The ρpT, heat capacity, and sound speed calculated by this model have a good accuracy with errors within 1%, except the error of heat capacity in the gaseous phase region, which is up to 6.32%. The Helmholtz free energy based on the Debye model is expressed as25 

aHe3δ,1/τ=ΘH0+11+eC11/τ1i=14δiC2i+51/τ2+C2i+61/τ41+eiC2δ1+i=13Ci+14δi+i=12δi1+eC3δ1C2i+16+C2i+1711+eC41/τ1+1eC5/τ21+eC61/δ1i=14C5/τi3C3i+19+C3i+20δ+C3i+21δ2C34/τ+C35,
(12)

where Θ is the Debye theta and ΘH0 is the ideal Helmholtz free energy from the basic Debye phonon theory. In the present work, the values of coefficients Ci take the same value as in the software REGEN 3.3, as shown in Table 8. One also needs to be careful in changing the form of Eq. (12), especially when using its derivative to calculate the other thermophysical properties. To be convenient, the  Appendix shows the changes of its derivative.

TABLE 8.

The Helmholtz free energy EOS coefficients for 3He (Refs. 25 and 56)

C110C1120C2130C3138
2.671 780 514 1.312 844 165 4.881 756 64 0.028 348 498 
0.991 964 789 −2.259 725 062 0.917 917 318 −0.396 573 697 
2.296 872 88 −0.459 896 431 −6.583 662 879 0.061 367 227 
2.584 609 302 0.715 119 974 1.273 447 716 7.815 876 899 
0.110 712 677 −0.873 794 229 2.153 080 193 6.729 311 6 
0.236 350 211 −2.663 910 299 −6.662 204 915 3.272 801 522 
2.859 386 169 1.071 179 613 1.282 675 91 0.304 642 607 
−1.453 876 695 −2.347 570 003 0.833 196 39 0.061 983 903 
−4.341 750 761 −2.574 785 443 −2.290 583 478 
3.168 840 649 −0.029 477 671 0.366537464 
C110C1120C2130C3138
2.671 780 514 1.312 844 165 4.881 756 64 0.028 348 498 
0.991 964 789 −2.259 725 062 0.917 917 318 −0.396 573 697 
2.296 872 88 −0.459 896 431 −6.583 662 879 0.061 367 227 
2.584 609 302 0.715 119 974 1.273 447 716 7.815 876 899 
0.110 712 677 −0.873 794 229 2.153 080 193 6.729 311 6 
0.236 350 211 −2.663 910 299 −6.662 204 915 3.272 801 522 
2.859 386 169 1.071 179 613 1.282 675 91 0.304 642 607 
−1.453 876 695 −2.347 570 003 0.833 196 39 0.061 983 903 
−4.341 750 761 −2.574 785 443 −2.290 583 478 
3.168 840 649 −0.029 477 671 0.366537464 

In order to develop the Helmholtz free energy EOS for mixtures, Eq. (12) is rewritten as the ideal part and reduced part like 4He. To be convenient, the same ideal part of the reduced Helmholtz free energy [same as Eq. (10)] is used for pure 3He, and the residual part of the reduced Helmholtz free energy can be expressed as

αo,He3rδ,τ=τRTcaHe3δ,1/ταo0δ,τ.
(13)

From the above reduced Helmholtz free energy of 3He–4He mixtures, all the other thermodynamic properties can be calculated, as shown in Table 9 (which only presents the main properties for analyzing cryocoolers; one can find all the others in Ref. 24). However, in the practical application, because the pressure is more easily measured than density, pressure is usually used as the input parameter instead of density. To obtain the value of density, the Newton–Raphson method can be used to solve the pressure equation in Table 9. The detailed solution process can be found in Ref. 52.

TABLE 9.

The thermodynamic properties derived from the reduced Helmholtz free energy

PropertyRelation to αPropertyRelation to α
Pressure pρRT=1+δαδr Compression factor Z=1+δαδr 
Internal energy uRT=τατ0+ατr Enthalpy hRT=1+τατ0+ατr+δαδr 
Isochoric heat capacity CvR=τ2αττ0+αττr Speed of sound Mw2RT=1+2δαδr+δ2αδδr+1+δαδrδταδτr2τ2αττ0+αττr 
Isobaric heat capacity CpR=τ2αττ0+αττr+1+δαδrδταδτr21+2δαδr+δ2αδδr Volumetric thermal expansion coefficient αvT=1+δαδrδταδτr1+2δαδr+δ2αδδr 
PropertyRelation to αPropertyRelation to α
Pressure pρRT=1+δαδr Compression factor Z=1+δαδr 
Internal energy uRT=τατ0+ατr Enthalpy hRT=1+τατ0+ατr+δαδr 
Isochoric heat capacity CvR=τ2αττ0+αττr Speed of sound Mw2RT=1+2δαδr+δ2αδδr+1+δαδrδταδτr2τ2αττ0+αττr 
Isobaric heat capacity CpR=τ2αττ0+αττr+1+δαδrδταδτr21+2δαδr+δ2αδδr Volumetric thermal expansion coefficient αvT=1+δαδrδταδτr1+2δαδr+δ2αδδr 

In addition to the above thermodynamic properties, VLE properties can provide more input data, which help obtain the optimal fitting parameters. From the reduced Helmholtz free energy of 3He–4He mixtures, the VLE can be obtained by solving the equilibrium conditions

Tliquid=Tvapor,pliquid=pvapor,fliquid,i=fvapor,i,
(14)

where f is the fugacity of component i, which can be calculated from the fugacity coefficient52 

lnφi=αr+δαδrln1+δαδr+1xiδαδrρrρrxi+τατrTrTrxi+αxir.
(15)

There are several methods that can be used to solve the equilibrium conditions. Here, the common Newton–Raphson method is used.57 One can find other methods in the literature.58,59 By solving the equilibrium conditions using the reduced Helmholtz free energy of 3He–4He mixtures, the bubble-point pressure, dew-point pressure, saturated liquid density, and saturated vapor density of 3He–4He mixtures can be obtained.

In the above model, the unknown five parameters are the four binary parameters (βT,12, γT,12, βν,12, γν,12) in Eqs. (3) and (4) and the interaction parameter (F12) in Eq. (7). As mentioned before, four departure functions (KW0, 4He–Ne, 4He–Ar, and Ne–Ar) were tested. For each of them, the departure function coefficients were fixed and the unknown five parameters, βT,12, γT,12, βν,12, γν,12, F12, were optimized with the Levenberg–Marquardt method. The goal is to minimize the residual sum of squares of calculated and experimental data, and the objective function can be written as

χ2=1NcWcCv,calCv,expCv,exp2+1NwWwwcalwexpwexp2+1NpWppcalpexppexp2+1NρWρρcalρexpρexp2,
(16)

where N is the total number of experimental data, W is a weighting coefficient, Cv is the isochoric heat capacity, w is the speed of sound, and p and ρ include the pressure and density at gaseous, liquid, and VLE states. Because the heat capacity from Ref.45 is at the saturated state, it was not used in the fitting process and only used to check the performance of the developed EOS. In the optimization process, to improve the convergence, the VLE data are first fitted and then the gaseous and liquid data are added to improve the fitting parameters.

Finally, the performances of fitting results are evaluated by the average absolute relative deviation, which is defined as

AARD=100Ni=1NXcalXexpXexp,
(17)

where X represents the property from the literature and N is the total number of data points of X. The subscripts cal and exp denote the calculated and experimental values of property X.

Based on the above method, an optimization code was written using the MATLAB software. The optimized parameters and objective function values of different departure functions are shown in Table 10. One can see that the KW0 model shows the minimum calculated objective function values among the four models used. To further evaluate the effectiveness of each departure function, the average absolute relative deviations for different thermophysical properties are shown in Table 11. Bold and italic fonts are used to indicate the minimum and maximum deviation values of different models. One can see that the KW0 model performs better on the heat capacity and all the pressures, but it is a little worse for the saturated density calculation. The Ne–Ar model gives a medium outcome among the different models. The 4He–Ne and 4He–Ar models are not as good as the others, but all the average absolute relative deviations are within 6%.

TABLE 10.

Optimized parameters and objective function values for 3He–4He mixtures

βT,12γT,12βν,12γν,12F12χ2
KW0 1.010 348 12 1.013 693 13 1.012 189 49 0.977 416 78 0.059 846 8 0.061 8 
4He–Ar 1.017 886 12 1.001 223 20 1.029 022 11 0.974 895 06 0.001 569 35 0.068 5 
Ne–Ar 1.026 592 40 1.003 488 58 1.018 375 17 0.975 521 68 0.005 208 43 0.075 8 
4He–Ne 1.030 465 95 1.002 108 03 1.016 524 58 0.985 061 54 0.003 342 57 0.091 7 
βT,12γT,12βν,12γν,12F12χ2
KW0 1.010 348 12 1.013 693 13 1.012 189 49 0.977 416 78 0.059 846 8 0.061 8 
4He–Ar 1.017 886 12 1.001 223 20 1.029 022 11 0.974 895 06 0.001 569 35 0.068 5 
Ne–Ar 1.026 592 40 1.003 488 58 1.018 375 17 0.975 521 68 0.005 208 43 0.075 8 
4He–Ne 1.030 465 95 1.002 108 03 1.016 524 58 0.985 061 54 0.003 342 57 0.091 7 
TABLE 11.

The average absolute relative deviations of different departure functions (the bold font numbers indicate the best; the italic numbers indicate the worst)

KW04He–Ne4He–ArNe–Ar
AARD (%)
Cv 3.245 4.191 3.873 3.941 
w 1.218 1.149 1.193 1.216 
pbubble 3.763 5.214 5.884 5.220 
pdew 2.398 2.962 2.624 2.712 
ρsat,l 0.967 1.103 1.413 0.963 
ρsat,v 5.106 4.601 4.629 4.014 
pl 2.704 3.691 3.820 2.757 
ρl 0.329 0.420 0.357 0.331 
pg 1.469 1.964 1.644 1.630 
ρg 1.212 1.327 1.219 1.213 
KW04He–Ne4He–ArNe–Ar
AARD (%)
Cv 3.245 4.191 3.873 3.941 
w 1.218 1.149 1.193 1.216 
pbubble 3.763 5.214 5.884 5.220 
pdew 2.398 2.962 2.624 2.712 
ρsat,l 0.967 1.103 1.413 0.963 
ρsat,v 5.106 4.601 4.629 4.014 
pl 2.704 3.691 3.820 2.757 
ρl 0.329 0.420 0.357 0.331 
pg 1.469 1.964 1.644 1.630 
ρg 1.212 1.327 1.219 1.213 

Figures 2 and 3 show the calculated isotherms in the gaseous and liquid phase region. The experimental data cover a range of temperatures from 4.52 to 13 K with pressures from 0.03 to 3.6 MPa for gas and from 2.25 to 4.2 K with pressures from 0.1 to 2.4 MPa for liquid. One can see that the calculated results are very consistent with the experimental data. For the gaseous state results at 4.52 K, the calculated isothermal line with 3He mole fractions of 0.1708 and 0.3518 can also predict the behavior of the gas–liquid transition near the critical region.

FIG. 2.

Comparison of the calculated isotherms (KW0 model) with the experimental data39 in the gaseous region. Circles are experimental data; the lines and dashed lines are calculated data. The x-axis is a logarithmic coordinate to clearly show the low-pressure data.

FIG. 2.

Comparison of the calculated isotherms (KW0 model) with the experimental data39 in the gaseous region. Circles are experimental data; the lines and dashed lines are calculated data. The x-axis is a logarithmic coordinate to clearly show the low-pressure data.

Close modal
FIG. 3.

Comparison of the calculated isotherms with the experimental data38 in the liquid region. Circles are experimental data; the lines and dashed lines are calculated data. The x-axis is a logarithmic coordinate to clearly show the low-pressure data.

FIG. 3.

Comparison of the calculated isotherms with the experimental data38 in the liquid region. Circles are experimental data; the lines and dashed lines are calculated data. The x-axis is a logarithmic coordinate to clearly show the low-pressure data.

Close modal

The fitting residuals of all the ρpT isothermal data are shown in Fig. 4. One can see that the residuals are well within 1% for the liquid region. For the gaseous region data, most residuals are within 2%, but for the data at low pressure, especially near the critical region, the residual could be higher than 4%.

FIG. 4.

The fitting residuals of all the ρpT isothermal data.

FIG. 4.

The fitting residuals of all the ρpT isothermal data.

Close modal

Figure 5 shows the x = 0.507 isobaric line in the gaseous phase region. It covers a range of temperatures from 4.3 to 20.2 K and pressures from 0.1 to 1.61 MPa. One can see that the calculated results from the EOS developed in the present work are consistent with the experimental data. Here, experimental data at temperatures from 4.35 to 4.99 K at 0.195 MPa from Ref. 39 are not used because those points are too close to the two-phase region and the calculation model is unable to predict them.

FIG. 5.

Comparison of the calculated isobaric line in the gaseous region using the KW0 model with the experimental data39 and its residuals. Circles are experimental data; the lines and dashed lines are calculated data.

FIG. 5.

Comparison of the calculated isobaric line in the gaseous region using the KW0 model with the experimental data39 and its residuals. Circles are experimental data; the lines and dashed lines are calculated data.

Close modal

Figures 69 show comparisons of calculated VLE properties with experimental data from the literature.28–36 One can see that the present EOS can also predict well most of the VLE properties. The errors of calculated saturated liquid density and bubble-point pressure are as good as 2%, except for points near the critical region where the errors of calculated bubble-point pressure increase to above 5%. The errors of calculated dew-point pressure show a little higher error of 5%, mainly because the experimental data have a larger uncertainty. However, for the saturated gas density, the present EOS shows relatively poor prediction in which the calculation errors increase up to 8%.

FIG. 6.

Comparison of the calculated vapor pressure using the KW0 model with the experimental data28–33 and its residuals.

FIG. 6.

Comparison of the calculated vapor pressure using the KW0 model with the experimental data28–33 and its residuals.

Close modal
FIG. 7.

Comparison of the calculated dew-point pressure using the KW0 model with the experimental data28–29 and its residuals.

FIG. 7.

Comparison of the calculated dew-point pressure using the KW0 model with the experimental data28–29 and its residuals.

Close modal
FIG. 8.

Comparison of the calculated saturated liquid density using the KW0 model with the experimental data30,34,35 and its residuals.

FIG. 8.

Comparison of the calculated saturated liquid density using the KW0 model with the experimental data30,34,35 and its residuals.

Close modal
FIG. 9.

Comparison of the calculated saturated vapor density using the KW0 model with the experimental data30,33,36 and its residuals.

FIG. 9.

Comparison of the calculated saturated vapor density using the KW0 model with the experimental data30,33,36 and its residuals.

Close modal

As mentioned before, the saturated liquid density data from Ref.37 were only found after the optimization of the five parameters given in Table 10. Figure 10 shows the comparison of the molar volume between the present EOS and the data of Wang et al.37 The present results show excellent agreement with the experimental data, and the deviation is generally less than 1%.

FIG. 10.

Comparison of the calculated molar volume using the KW0 model with the experimental data37 and its residuals. The line and dashed line are the calculated data.

FIG. 10.

Comparison of the calculated molar volume using the KW0 model with the experimental data37 and its residuals. The line and dashed line are the calculated data.

Close modal

Figures 11 and 13 compare the calculated isochoric and saturated heat capacity with the experimental data from the literature.45,46 The available experimental data are limited to isochoric heat capacities at high pressure near the solidification region and saturated heat capacity at the saturated state. Most isochoric heat capacity data can be predicted with an accuracy of 5%, as shown in Fig. 12. However, for the predicted saturated heat capacity [the saturated heat capacity is the heat capacity along the saturation line, which can be calculated from the equation Csat=CpTdVdTdPdTsat], the present EOS has a predictive trend with the maximum error higher than 10%, as shown in Fig. 13. There are two reasons for the weak prediction of saturated heat capacity. One is the Helmholtz free energy model for pure 3He, whose maximum error for heat capacity can be as high as 6.32%. The second reason is that the experimental data of the heat capacity are at the saturated state with a relatively large uncertainty of about 4%. In the calculation, one needs first to obtain the saturated density and then to calculate the heat capacity, which also increases the calculated error. In the future, more accurate experimental data for heat capacity are needed to check and improve the present EOS.

FIG. 11.

Comparison of the calculated isochoric heat capacity using the KW0 model with the experimental data at four molar volumes.46 

FIG. 11.

Comparison of the calculated isochoric heat capacity using the KW0 model with the experimental data at four molar volumes.46 

Close modal
FIG. 12.

The fitting residuals of all the Cv data.

FIG. 12.

The fitting residuals of all the Cv data.

Close modal
FIG. 13.

Comparison of the calculated isochoric heat capacity using the KW0 model with the experimental data.45 The solid lines are the calculated data.

FIG. 13.

Comparison of the calculated isochoric heat capacity using the KW0 model with the experimental data.45 The solid lines are the calculated data.

Close modal

Figures 14 and 15 compare the calculated sound speed in the liquid mixture and the experimental data from Ref. 47. In the wide 3He mole fraction range of 0.25–0.98 and pressures from 0.1 to 7 MPa, the present EOS has a good prediction with errors within 2%. Table 12 presents the comparison results of the two calculated speeds of sound at vapor pressure with experimental data from Ref. 48. It is evident that the present model can also predict the speed of sound well at the vapor pressure with an accuracy around 1%. In addition, the present calculated sound speeds at the saturated liquid state are compared with the experimental results from Ref.49, as shown in Fig. 16. The residual errors are better than 3%. The data from REFPROP software for pure 4He are also compared in Fig. 16; one can see that the data from Ref.49 also deviate from the REFPROP values.

FIG. 14.

Comparison of the calculated speed of sound using the KW0 model with the experimental data.47 

FIG. 14.

Comparison of the calculated speed of sound using the KW0 model with the experimental data.47 

Close modal
FIG. 15.

The fitting residuals of all the sound-speed data.

FIG. 15.

The fitting residuals of all the sound-speed data.

Close modal
TABLE 12.

Comparison of the calculated isochoric heat capacity using the KW0 model with the experimental data48 

T (K)xwexp (m/s)wcal (m/s)Error (%)
2.305 0.301 203.33 205.40 1.02 
2.206 0.301 204.34 205.24 0.44 
T (K)xwexp (m/s)wcal (m/s)Error (%)
2.305 0.301 203.33 205.40 1.02 
2.206 0.301 204.34 205.24 0.44 
FIG. 16.

Comparison of the calculated speed of sound using the KW0 model with the experimental data49 and its residuals. The solid lines in the left figure are the fitting results.

FIG. 16.

Comparison of the calculated speed of sound using the KW0 model with the experimental data49 and its residuals. The solid lines in the left figure are the fitting results.

Close modal

From the above comparisons, one can see that the present EOS shows good prediction of the ρpT relation at temperatures below 20 K. However, for practical application, the 3He–4He mixture normally works from room temperature to the low temperature. We found no experimental data for 3He–4He mixtures at temperatures above 20 K. In order to check the present EOS at a higher temperature, a feasible way is to compare it with the virial EOS (VEOS). VEOS is a well-known EOS that is a polynomial series in the density, is explicit in pressure, and can be derived from statistical mechanics,60 

pρRT=1+BTρ+CTρ2+,

where B(T), C(T), …, are the second, third, …, virial coefficients. For 3He–4He mixtures, the second virial coefficient has been accurately computed by Hurly and Moldover43 and Cencek et al.44 using ab initio calculation, and there are no available data for the third or higher order mixture virial coefficients. Because the calculated interaction virial B34 for 3He–4He mixtures from Ref.44 has a very small uncertainty, it is used to calculate the density and compared with the experimental data from Ref.39. As shown in Fig. 17, the second-order VEOS does not predict the gas density well at temperatures below 15 K. That is because the third and higher virial coefficients become significant at a low temperature and high pressure. At temperatures above 15 K, the VEOS with a second virial coefficient shows good agreement with the experimental data, even at pressures up to 1 MPa. Therefore, it is reasonable to use the above VEOS to check the present EOS at temperatures above 20 K.

FIG. 17.

Comparison of the calculated density using the virial EOS (only using second virial coefficient) with the experimental data from Ref.39.

FIG. 17.

Comparison of the calculated density using the virial EOS (only using second virial coefficient) with the experimental data from Ref.39.

Close modal

Figure 18 shows the calculated density by using the ideal gas equation, the VEOS with the second virial coefficient, and the present EOS. One can also see that the present model agrees with the second-order virial EOS very well, with the maximum deviation of about 0.6%. It indicates that the present model has a good extrapolation performance at temperatures from about 20 K to room temperature, which once again reflects that the EOS developed in the present work is reliable.

FIG. 18.

Comparisons of the present model with the virial EOS and the ideal gas EOS (x = 0.5, p = 1 MPa).

FIG. 18.

Comparisons of the present model with the virial EOS and the ideal gas EOS (x = 0.5, p = 1 MPa).

Close modal

The present work developed the first wide-range EOS for 3He–4He mixtures based on the Helmholtz free energy, which is reliable for temperatures from 2.17 K to room temperature and pressures from the vapor pressure to higher than 3 MPa. To obtain an accurate calculation model, four different departure functions from KW0, 4He–Ne, 4He–Ar, and Ne–Ar multi-fluid models were tested. For each of them, five parameters were optimized to find the minimum deviation from the available experimental data by using the Levenberg–Marquardt method. The results showed that the KW0 model was the best one to predict the thermodynamic properties for 3He–4He mixtures.

Comparisons between the present model and the available experimental data show that the present model has a good predictive performance not only for the liquid and gas ρpT relation but also for the VLE properties of 3He–4He mixtures. For most ρpT data, saturated liquid density, and speed of sound, the error of the present EOS is less than 2%. For most data of bubble-point pressure, dew-point pressure, saturated vapor density, and isochoric heat capacity, the error of the present EOS is less than 5%, except for some points near the critical region or the λ-point, the error of which can be higher than 8%. Although the present model gives a relatively poor prediction of the saturated heat capacity, it can be improved in the future if more accurate experimental data are available. Furthermore, by comparing with the virial EOS, it also shows that the current model has good extrapolation performance at temperatures from above 20 K to room temperature.

The supplementary material contains files with the original data used in the fitting and the database calculated by the present EOS. The calculated tabulated database covers the thermophysical properties of ρpT relation, entropy, enthalpy, heat capacity, volumetric thermal expansion, and compression factor at pressures from saturation up to 3 MPa, temperatures from 2.2 to 350 K, and 3He mole fractions from 0 to 1. In addition, it also includes a calculator developed by using MATLAB graphical user interfaces (GUIs). One can use it to calculate the thermophysical properties at a given point.

This work was supported by funding from the European Union’s Horizon 2020 research and innovation programme under Marie Skłodowska-Curie Grant Agreement No. 834024 and supported by the National Natural Science Foundation of China (Grant No. 52006231) and the Scientific Instrument Developing Project of the Chinese Academy of Sciences (Grant No. ZDKYYQ20210001).

The authors are grateful to Professor Yonghua Huang from Shanghai Jiao Tong University for providing the software He3Pak for trial. They are also grateful to Professor V. K. Chagovets from the Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine for providing the text of Ref. 39. They are also grateful to Dr. Allan Harvey for help and supplying more data for 3He–4He mixtures.

The data that support the findings of this study are available within the article and its supplementary material.

Abbreviations
3He

helium-3

4He

helium-4

AARD

average absolute relative deviation

EOS

equation of state

KW0

departure function coefficients for hydrocarbon mixtures

VEOS

virial equation of state

VLE

vapor–liquid equilibrium

VPTC

Vuilleumier pulse-tube cryocooler

Symbols
a or A

Helmholtz free energy

Cp

isobaric heat capacity

Cv

isochoric heat capacity

f

fugacity

F12

interaction parameter in departure function

h

enthalpy

k

Boltzmann constant/index of the fitted coefficients

m

mass

NA

Avogadro constant

N

number of experimental data/number of components inthe mixture

p

pressure

R

universal gas constant

T

temperature

u

internal energy/velocity

w

speed of sound

W

weighting coefficients

x

mole fraction vector of mixture constituents

Z

compression factor

Greek letters
α

reduced Helmholtz free energy

αv

volumetric thermal expansion coefficient

βT,12, γT,12

binary parameters in reducing mixture temperaturefunction

βν,12, γν,12

binary parameters in reducing mixture densityfunction

δ

reduced density

Δαr

departure function

Θ

Debye theta

ρ

density

τ

inverse reduced temperature

φ

fugacity coefficient

χ2

objective function

Superscript / Subscripts
c

critical value

cal

calculated value

exp

experimental value

i

component index/experimental data index

j

component index

o

ideal part

r

residual part

In the work of Huang et al.,25 the reduced temperature is defined as the inverse of the one used in the common Helmholtz free energy EOS. To use it as the input of our model, the form and its derivative need to be changed. The following equations show those changes for the equations of Huang et al.

For reduced Helmholtz free energy,

ατ,δ=A1/τ,δRT=τA1/τ,δRTc.

The derivative of reduced Helmholtz free energy is given as follows:

ατ,δδ=1RTA1/τ,δδ=τRTcA1/τ,δδ,
2ατ,δδ2=τRTc2A1/τ,δδ2,
ατ,δτ=A1/τ,δRTc1RTcτA1/τ,δτ,
2ατ,δτ2=1RTcτ32A1/τ,δτ2,
2ατ,δδτ=1RTcA1/τ,δδ1RTcτ2A1/τ,δδτ,

where A is the Helmholtz free energy in Ref. 25.

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Supplementary Material