Equations of state for the thermodynamic properties of binary mixtures for helium-4, neon, and argon Equations of state for the thermodynamic properties of binary mixtures for helium-4, neon, and argon Jakub Tkaczuk,1, a) Eric W. Lemmon,2, b) Ian H. Bell,2, c) Nicolas Luchier,1 and François Millet1 1)Université Grenoble Alpes, CEA IRIG-dSBT, F-38000 Grenoble, France 2)National Institute of Standards and Technology, Boulder, Colorado 80305, United States (Dated: 12 March 2020) Based upon the conceptual design reports for the Future Circular Collider cryogenic system, the need for more accurate thermodynamic property models of cryogenic mixtures of noble gases was identified. Both academic institutes and industries have identified the lack of a reliable equation of state for mixtures used at very low temperatures. Detailed cryogenic architecture modeling and design cannot be carried out without accurate fluid properties. Therefore, the helium-neon equation was the first goal of this work and it was further extended to other fluids beneficial for scientific and industrial applications beyond the particle physics needs. The properties of the noble gas mixtures of helium-neon, neon-argon, and helium-argon are accurately modeled with the equations of state explicit in the Helmholtz energy. Key words: helium-4; neon; argon; equation of state; fundamental equation; thermodynamic properties. CONTENTS LIST OF FIGURES 1 Introduction . .2 1 The phase envelopes for three binary mixtures 2 Fundamental equation of state . .2 shown in three-dimensional space of p − T − x 2.1 Optimization algorithm . .3 with isobars in red, and isotherms in black (a, c, 2.2 Available data used for the equation of state de- e). Their projection on 2D plane of p − x (b, d, velopment . .3 f) constructed with the algorithm for equilibria of 2.3 Fluid properties scaling . .6 binary mixtures1............................7 3 Results . .6 2 Schematic representation of the orthogonal er- 3.1 Uncertainty discussion for single-phase . .6 ror calculations for the phase envelopes. Two 3.2 Uncertainty discussion at phase boundary . .6 experimental compositions for liquid and vapor 3.3 Ternary mixture . 11 from one measurement are marked with circles 3.4 Validation data . 11 (the same pressure and temperature) and the re- 4 Conclusions . 11 spective isotherm evaluated with the EOS is plot- 5 Supplementary materials . 11 ted in two-dimensional space of reduced pressure Acknowledgments . 11 pˆ = p=pr and mole fraction in liquid x and vapor y.8 6 References . 11 3 Relative deviations of the 4He − Ne equation to data.......................................9 4 Relative deviations of the 4He − Ar equation to data9 LIST OF TABLES 5 Relative deviations of the Ne − Ar equation to data9 6 Three mixture phase envelopes with color scales 1 Molar mass and critical parameters for pure fluids2 representing the absolute orthogonal error values 2 Reducing temperature and density parameters . .2 plotted for 99:99% of data points for 4He − Ne, 3 Available and used data for the development of 91:36% of data points for 4He − Ar, and 99:23% 4 − the equations of state for He Ne . .4 of data points for Ne − Ar. The dotted line in 4 Available and used data for the development of each figure represents the reducing pressure scale 4 − the equations of state for He Ar . .4 from Eq. (11).............................. 10 5 Available and used data for the development of 7 Vapor-liquid and gas-gas equilibria orthogonal the equations of state for Ne − Ar . .5 error values for three binary mixtures. Color 6 Binary specific parameters obtained in this work .6 intensity increases with increasing number of 7 Departure function coefficients . .8 points in the hexagonal unitary area. The 8 Data points at equimolar (0.5/0.5) composition marginal histograms represent the number of data for EOS validation . 11 points in different data bins in whole pressure range (x-axis) and orthogonal error range (y-axis) 10 8 Cumulative percentage of points with error less than the specified value of the VLE orthogonal error for each EOS. 4He − Ar errors are con- a)Electronic mail: [email protected] sidered for two pressure ranges: complete data b)Electronic mail: [email protected] availability pressure range and up to 100 MPa c)Electronic mail: [email protected] with smaller orthogonal error . 11 Equations of state for the thermodynamic properties of binary mixtures for helium-4, neon, and argon 2 1. INTRODUCTION TABLE 1: Molar mass and critical parameters for pure fluids −1 −3 The need for equations of state (EOS) capable of accu- M = kg mol Tc = K pc = MPa rc = mol dm 4 12 −3 rate representation of thermodynamic properties for cryogenic He 4:0026 × 10 5:195 0:22832 17:384 Ne13 20:179 × 10−3 44:400 2:66163 24:100 mixtures increases as more complex and more efficient cryo- 14 −3 genic systems are being investigated and developed. Mixtures Ar 39:948 × 10 150:687 4:86300 13:407 are presently considered as working fluids not only for large cryogenic systems but also for smaller, space-ready refrigera- TABLE 2: Reducing temperature and density parameters tors. The potential applications of equations of state for noble gases include cooling cycles for fusion reactors or particle ac- 2 Yr Yc;i bY;i j Yi j celerators such as the Future Circular Collider, (FCC) space 0:5 refrigerators, and modeling of welding processes with inert Tr Tc;i bT;i j bT;i jgT;i j Tc;iTc; j 3 gases. 1 1 1 1 vr bv;i j bv;i jgv;i j + rc;i 8 r1=3 r1=3 No accurate model has been developed so far for binary c;i c; j mixtures of helium-4, neon, and argon.3 Until now, either lin- ear or Lorentz-Berthelot4,5 classical rules for combining fluids have been used.6,7 Neither of them, however, represents quan- 2. FUNDAMENTAL EQUATION OF STATE tum gas behavior correctly.8 Following the best practices from the domain of the equations of state, the quantum effects were Among the available formulations of equations of state, the included in the empirical model development with no special Helmholtz energy a(T;r) has two important advantages. Un- treatment. like internal energy u(v;s) and enthalpy h(p;s), it is a func- The mixture models presented in this paper are explicit in tion of measurable properties - temperature T and density r. the Helmholtz energy and use reducing parameters dependent Moreover, unlike the Gibbs energy g(T; p), it is continuous on the mole fractions of mixture constituents and the critical within the phase boundary, where v is specific volume, s is parameters of the pure fluids. The Helmholtz energy a is de- specific entropy, and p is pressure. fined as follows The equation of state is used in a reduced form and is a function of non-dimensional quantities - reduced density and reciprocal reduced temperature15,16 a = u − Ts; (1) T r t = r and d = ; (2) T rr where u is internal energy, T is temperature and s is en- where Tr and rr are the reducing temperature and density. tropy. Since it is a fundamental value, all other thermody- The most recent formulation16,17 allows for the use of a com- namic properties can be calculated from analytical deriva- mon form for reducing parameters for the mixture (Tr and tives. The Helmholtz energy formulation takes advantage of vr = 1=rr) the consistent description of all fluid states because tempera- ture and density are the independent variables. The pure fluid N N N contributions, and the contribution from mixing for each bi- 2 xi + x j Yr (x¯) = ∑ xi Yc;i + ∑ ∑ 2xix j 2 Yi j; (3) nary mixture use an empirical formulation, which is temper- i=1 i=1 j=i+1 bY;i jxi + x j ature, density, and molar composition-dependent. The pure fluid contributions take advantage of the highly accurate EOS where Y is the parameter of interest, temperature T or mo- of pure fluids. In this work, each mixture-specific departure lar specific volume v given in Table2. Tc;i, rc;i are critical function consists of three exponential terms and four to five temperature and density of the pure constituents. bT;i j, gT;i j, Gaussian terms. Comparisons to available experimental data bv;i j, and gv;i j are the fitted parameters with both b parameters are given to validate the accuracy of thermodynamic proper- −1 −1 being asymmetric, that is bT;i j = bT; ji, and bv;i j = bv; ji . ties calculated with each equation of state. The total reduced Helmholtz energy a can be divided into o r The accuracy of mixture models is dependent on the pure an ideal gas part a and a residual part a fluid models. The equation of state for the considered mix- ture constituents, helium-4, neon, and argon have state-of-the- a(r;T;x¯) art accuracy. This important starting point allows for the de- a (d;t;x¯) = = ao (r;T;x¯) + ar (d;t;x¯); (4) RT velopment of mixture models that will become the industrial standard. The critical parameters of the pure fluids, used in wherex ¯ is the mole fraction vector, and R = Helmholtz energy equations of state (HEOS) are presented in 8:314 462 618 J mol−1 K−1 is the gas constant.18 The Table1. Additionally, the equations, in the form presented ideal and residual contributions represent the Helmholtz in this paper, can be implemented in fluid property modeling energy for the mixture. Both are functions of the pure fluid software, such as REFPROP,9 CoolProp,10 or TREND.11 Helmholtz energies Equations of state for the thermodynamic properties of binary mixtures for helium-4, neon, and argon 3 where Wx is the weight assigned to each data point of the ther- modynamic property x, and Fx is the function used to min- N o o imize the relative deviation between the data points and the a (r;T;x¯) = ∑ xi [aoi (r;T) + lnxi]; (5) 21 i=1 equation of state.