Crossover volume translation cubic equation of state and its high-order crossover functions

The critical point of a pure fluid is the end point of the vapor-liquid coexistence curve, at which both the temperature and the pressure are highest. At the critical point, the vapor and liquid phases become indistinguishable; thermodynamic properties present asymptotic singularity and scaling laws, that is, they strictly diverge according to the asymptotic exponential behavior. The equation of state (EoS) is a powerful tool for modelling fluid thermodynamic properties in various aspects of engineering. Classical EoSs, such as the cubic EoSs, can accurately represent thermodynamic properties in wide ranges of temperature and density, but fail to reproduce the critical behavior. The renormalization group theory provides an exact description of the critical behavior but is not applicable except for in the vicinity of the critical point. The crossover method connects the renormalization group theory at the critical point with the classical thermodynamic property models that are applicable far away from the critical point in a semi-theoretical manner. This work presents a crossover volume translation (VT) Soave-Redlich-Kwong (SRK) EoS based on the Kiselev crossover method and features a systematic investigation of the effect of the order of the crossover function on the performance of thermodynamic property modelling. The procedure is demonstrated on 7 typical nonpolar fluids (methane, ethane, propane, n-butane, n-pentane, n-hexane, and carbon dioxide). The crossover function dictates the convergence behavior of the crossover EoS when the system is restored from critical singularity to classical rules. The vapor pressure, saturated liquid density, and saturated vapor density are calculated with the crossover VTSRK EoSs using 2nd to 8th order crossover functions, and the results are compared with the reference data. The effect of the order of the crossover functions on the accuracy of thermodynamic property calculations and on the convergence behavior of the crossover EoSs from critical singularity to classical behavior as the system departs from the critical point are investigated. This work finds that in general, the crossover VTSRK EoS using the 3rd order crossover function presents the best accuracy among the crossover VTSRK EoSs. Regardless of the order of the crossover function, the crossover VTSRK EoS is more accurate than the SRK and VTSRK EoSs. Because of the characteristics of the fluids and the classical EoSs, the crossover EoSs that use overly high-order crossover functions present lower accuracy than those that use crossover functions of appropriate orders, especially for the saturated liquid density. This work proposes that, to achieve high accuracy both within and beyond the near-critical region with the smallest number of adjustable parameters, future work on the crossover EoSs should investigate and determine the order of the crossover function according to the characteristics of both the fluids and the classical EoSs. With only 1 adjustable parameter, the present model accurately represents thermodynamic properties both within and beyond the near-critical region. Average absolute relative deviation between the results calculated by the crossover VTSRK EoS using the 3rd order crossover function and the reference data is 0.72% for the vapor pressure, 1.5% for the saturated liquid density, and 0.98% for the saturated vapor density. Furthermore, the crossover VTSRK EoS correctly reproduces the theoretical critical exponents that are determined by the renormalization group theory. In contrast, the classical SRK EoS presents values determined by the mean-field theory, which deviate largely from the correct critical exponent values.