1. Introduction

Using a suitable equation is an essential solution for predicting the thermodynamic properties of fluids in the design and simulation of thermodynamic processes. Although there are many thermodynamic tables and charts that provide detailed information on the thermodynamic properties of fluids, an accurate equation is necessary to correctly predict the thermodynamic properties of fluids in a wide range of temperature and pressure using to design and simulate the thermodynamic cycles.

So far, many of equations of state were suggested such as primary simple equations with a few of parameters and modern equations that are complex but accurate. In the primary equation of state, the pressure is a function of other properties such as temperature and volume so these equations are named the pressure-explicit equation of state while new equations are explicated on Helmholtz energy. These equations that are known as the fundamental equations of state are more complex than pressure-explicit equations but have higher accuracy in the calculation of the thermodynamic properties in a wide range of temperature and pressure. The main advantage of these equations is that there is no need for integration for driving other thermodynamic properties. All required thermodynamic properties can be obtained with differentiating of Helmholtz energy with high accuracy. Due to the applicability of these equations in accurately predicting thermal-physical properties, extensive research is done on the development of these equations. Most of the works were done on this subject are about presenting a model based on Helmholtz energy or using previous models and calculating the thermodynamic properties of fluids and then comparing them with experimental data. Jacobsen et al. presented a thermodynamic property formulation for nitrogen from the freezing line to 2000 K at pressures to 1000 MPa. The given relation can be used for thermodynamic property calculation with uncertainty 0.1% for the pressure and density, 0.2% for thermal capacities and 2% for the values of speed of sound. The formulation is not recommended near the critical point (Jacobsen, Stewart, & Jahangiri, 1986). Dillon and Penoncello presented a formulation for the thermodynamic properties of ethanol (C₂H₅OH) in the liquid, vapor, and saturation. The formulation presented may be used to compute densities with uncertainty 0.2%, heat capacities with uncertainty 3%, and speed of sound with uncertainty 1%. Saturation values of the vapor pressure and saturation densities are represented with uncertainty ±0.5%, except near the critical point (Dillon & Penoncello, 2004). Leachman et al. developed a form of the fundamental equation of state for para-hydrogen, normal hydrogen, and ortho-hydrogen and then calculated the thermodynamic properties of them. The uncertainties of vapor pressures and saturated liquid densities vary from 0.1% to 0.2%. Heat capacities are generally estimated to be accurate to within 1%, while speed-of-sound values are accurate to within 0.5% below 100MPa (Leachman, Jacobsen, Penoncello, & Lemmon, 2009). Bücker and Wagner calculated the thermodynamic properties of ethane in the temperature range of the melting line up to 675 K and the pressure up to 900 MPa. The uncertainty in the calculation of density was less than 0.02%–0.03% from the melting line up to temperatures of 520 K and pressures of 30 MPa. In the gaseous and supercritical region, the high precision speed of sound data is represented generally within less than 0.015%. The primary data, to which the equation was fitted, cover the fluid region from the melting line to temperatures of 675 K and pressures of 900 MPa. Beyond this range, the equation shows reasonable extrapolation behavior up to very high temperatures and pressures (Bücker & Wagner, 2006). Estela-Uribe developed a model based on Helmholtz energy for calculating the thermodynamic properties of a group of refrigerants. He predicted thermodynamic properties of 19 refrigerant and reported following percentage overall absolute average deviations: 0.187 for p-T data, 0.229 for saturation pressures, 0.206 for saturated-liquid densities, 1.053 for saturated-vapor densities, 1.734 for isochoric heat capacities, 1.238 for isobaric heat capacities and 0.662 for speeds of sound (Estela-Uribe, 2014). Estela-Uribe also predicted the thermodynamic properties of non-polar fluids and their mixtures such as natural gases using an improved Helmholtz energy model (Estela-Uribe, 2013a, 2013b, 2013c). Thol et al. predicted the thermodynamic properties of Hexa-methyldisiloxane (Thol, Dubberke, Rutkai, Windmann, Köster, Span, & Vrabec, 2015) and ethylene oxide ( Thol, Rutkai, Köster, Kortmann, Span, & Vrabec, 2015) using fundamental equation of state correlation based on experimental and molecular simulation data. Gernert and Span presented an equation of state and the EOS–CG mixture model for thermodynamic properties of humid gases, combustion gases and CO2-rich mixtures typical for Carbon Capture and Storage processes (Gernert & Span, 2016). They use the mathematical structure of Kunz and Wagner in their work (Kunz & Wagner, 2012).  Grigoriev et al. proposed a new mixture model, explicit in the reduced Helmholtz energy, for modeling the thermodynamic properties and phase equilibria of technological oil fractions. The model covers a broad temperature and pressure range in gaseous, liquid, and supercritical regions (Grigoriev, Alexandrov & Gerasimov, 2018).   

This study aims  to predict the thermodynamic properties of cryogenic refrigerants using fundamental equations of state and calculate the uncertainty in the wide range of temperature and pressure to evaluate the applicability of these equations in the prediction of thermodynamic properties of these refrigerants. Using a developed computer code based on fundamental equations of state, thermodynamic properties of Helium and Neon including density, internal energy, enthalpy, entropy, isobaric specific heat capacity, and isochoric heat capacity are calculated. By comparing the results of predictions with the valid reference data is (Jacobsen, Penoncello & Lemmon, 1997), the scope of this equation to predict the thermodynamic properties of Helium and Neon are offered. In earlier work, not only the prediction of the thermodynamic properties of cryogenic refrigerants such as Helium and Neon not considered, but also providing a diagram that easily presents the applicability of fundamental equation of state in different ranges of temperature and pressure has been neglected. In the following section, a brief description is provided of the types of equations of state. Then the fundamental equations and how to obtain thermodynamic properties have been obtained. Finally, the results of comparing the reference data and predicted thermodynamic properties using fundamental equations of state are presented.

2. Theory of fundamental equation of state

The most common thermodynamic properties that are measured and used in the equations of state are pressure, temperature and specific volume (or its reverse, the density). Primary types of the equation of state make a correlation between the thermodynamic properties in which the pressure is expressed as a function of temperature and density (or specific volume). The general form of the pressure-explicit equation of state is (Jacobsen, Penoncello & Lemmon, 1997):

Virial equation of state (equation 2), cubic equations of state (equation 3) such as Van der Waals and Peng-Robinson and Lee-Kesler equations of state are the most common pressure explicit equations of state. The typical pressure explicit equation of state must be integrated for the calculation of enthalpy and entropy (Jacobsen, Penoncello & Lemmon, 1997).

In the secondary form of the equation of state, Helmholtz energy is used as a dependent variable (Jacobsen, Penoncello & Lemmon, 1997):

This form of the equation of state is named the fundamental equation of state. Therefore, on this basis, equations of state are divided into two general categories:

1. Pressure-explicit equations of state
2. Fundamental equations of state

As mentioned, the fundamental form of the equation of state is implicit in the Helmholtz energy. This form of the equation has some advantages over other forms of the equation of state. An inherent advantage in fundamental equations of state is that all thermodynamic properties can be derived with relatively simple differentiation.

Schmidt and Wagner developed a 32-term fundamental equation in 1985 (Jacobsen, Penoncello, & Lemmon, 1997). Although this form was developed for oxygen, it has been used by other investigators for correlating properties of other fluids. The equation is explicit in dimensionless Helmholtz energy. Jacobsen et al. developed a general form of the fundamental equation similar to that of Schmidt and Wagner which has been used for correlating thermodynamic properties of several cryogens. The functional form of this equation is (Jacobsen, Penoncello & Lemmon, 1997):

where , , and  and  are critical temperature and critical density respectively. The Helmholtz energy consists of two parts: the ideal gas contribution part ( ), and the residual part of the Helmholtz energy .

The ideal gas portion is given by (Jacobsen, Penoncello & Lemmon, 1997):

where  and  are as defined below. Also,  and  are arbitrary values for enthalpy and entropy at standard temperature (  ) and pressure (

The residual part of the Helmholtz energy is given by (Jacobsen, Penoncello & Lemmon, 1997)

where for terms with  and  for terms where . It is generally expected that the  and  are positive integers and the  are real numbers (positive or negative).

3. Prediction of thermodynamic properties using fundamental equation of state

As mentioned, the fundamental equation of states allows computing other thermodynamic properties with differentiation of Helmholtz energy. Table 1 shows examples of functions for obtaining the thermodynamic properties from Helmholtz energy. These functions can be used easily in computer programs.

4. Prediction of thermodynamic properties of Helium and Neon

To calculate the thermodynamic properties of Helium and Neon, coefficients and parameters reported by Jacobsen et al. were used (Jacobsen, Penoncello, & Lemmon, 1997). Thermodynamic properties were calculated at pressures 0.1, 0.2, 0.5, 1, 2, 5, 10, 20, 50, and 100 MPa with temperature step of 10 °C at low temperatures and 100 °C at high temperatures from saturated point up to 1500 °C for Helium and 650 °C for Neon, using a developed computer code in MATLAB software. Since in the fundamental equations of state, Helmholtz energy has been explicit in terms of temperature and density, the equation should be solved numerically to obtain thermodynamic properties at a given temperature and pressure. Therefore, at a given temperature, the Newton-Raphson method was used to achieve the corresponding density to the desired pressure with an approximation of 0.000001 MPa. 

Table 1. functions for calculation of thermodynamic properties from Helmholtz (Jacobsen, Penoncello, & Lemmon, 1997)

5. Result and discussion

Using a developed computational code based on fundamental equation of state, thermodynamic properties of gases Helium and Neon were calculated as two cryogenic refrigerants. By comparison, these calculated properties with reference (Jacobsen, Penoncello & Lemmon, 1997), the applicability diagrams of the fundamental equation of state were provided for these two refrigerants. Also, forecast errors of thermodynamic properties inside the applicability region are presented. The thermodynamic properties were calculated for pressures up to 100 MPa and in the temperature range 10 to 1500 K and 30 to 650 K, respectively for Helium and Neon. A standard reverse-Brayton cryocooler cycle for long-length High-Temperature Superconducting cable systems is one of the most application of the range studied in this investigation (Hou, Zhao, Chen, & Xiong, 2006) and (Maboudi & Zia Bashrhagh, 2015).

5.1 Thermodynamic properties of Helium

As an example, Table 2 shows a part of the thermodynamic property table obtained for Helium using the fundamental equation of state. Figure 1 shows a comparison between the entropy of Helium gas calculated using the fundamental equations of state and reference data (Jacobsen, Penoncello & Lemmon, 1997) in various temperatures and pressures. As can be seen in this figure, the difference between the results obtained from the fundamental equations of state and the reference data is negligible, which indicates the capability of the fundamental equations of states in the calculation of the thermodynamic properties of gases. These comparisons were done about other thermodynamic properties of Helium and adaptation of results with reference data was investigated. In Figure 2, the applicability region of the fundamental equation of state for predicting the thermodynamic properties of Helium can be seen on the temperature-entropy graph for different pressures. Forecast errors of thermodynamic properties of Helium in high accuracy region are presented in Table 3.

Table 2. Thermodynamic properties of Helium

Figure 1. Entropy of Helium vs temperature in different pressures

Figure 2. Applicability area of fundamental equation of state in predicting the thermodynamic properties of Helium.

Table 3. Relative error of predicting thermodynamic properties of Helium in high accuracy area shown in figure 2

Figure 3 shows the relative error percentage of the results on the boundary of the applicability area of the fundamental equation of states (the line between the green and brown areas in Figure 2) in predicting the thermodynamic properties of Helium. According to the graph, it is shown that relative error of density, unlike other thermodynamic properties, decreases with increasing pressure. The maximum error is related to the isobaric heat capacity and isochoric heat capacity which is 3.37% and 0.94% respectively. The relative error of other thermodynamic properties on the boundary between high and low accuracy areas does not exceed 0.6%. The relative error in the low accuracy area shown in Figure 2, is above 5%.

Figure 3. Relative error of thermodynamic properties of Helium using fundamental equation of states on the boundary between high and low accuracy areas shown in figure 2

5.2 Thermodynamic properties of Neon

Table 4 shows a selection of thermodynamic properties of Neon that were calculated using the fundamental equation of state. Figure 4 compares the entropy of Neon gas calculated using the fundamental equations of state with entropy obtained from reference data(Jacobsen, Penoncello & Lemmon, 1997) in various temperatures and pressures. As can be seen in this figure, the difference between the results calculated using the fundamental equations of state and the reference data is negligible. These comparisons were done for other thermodynamic properties of Neon and adaptation of results with reference data was investigated.Figure 5 shows the areas with acceptable and high accuracy of the fundamental equations of state to predict the thermodynamic properties of Neon gas. The relative error of calculating the thermodynamic properties in the area with high accuracy can be seen in Table 5. For Neon in the acceptable accuracy area of the fundamental equations of state, the maximum error in the calculation of the thermodynamic properties was 5.42 %, which is related to the isobaric heat capacity.

Figure 6 shows the relative error percentage of the results on the boundary of the applicability area of the fundamental equation of states (the line between the green and yellow areas shown in Figure 5) in predicting the thermodynamic properties of Neon. According to the graph, the relative error percentage of thermodynamic properties on the boundary between high and acceptable accuracy areas of the fundamental equation of states does not exceed 1% except density that its relative error percentage is little more than 1 %. The relative error of all thermodynamic properties in acceptable accuracy area in figure 5 is in the range of 1% to 5%, except isobaric heat capacity that its relative error percentage is little more than 5 %. The relative error in the low accuracy area, shown in Figure 5, is above 5%.

Table 4. Thermodynamic properties of Helium

Figure 4. Entropy of Neon vs temperature in different pressures

Figure 5. Applicability area of fundamental equation of state in predicting the thermodynamic properties of Neon

Table 5. Relative error of predicting thermodynamic properties of Neonin high accuracy area shown in figure 5

Figure 6. Relative error of thermodynamic properties of Neon on the boundary between high and acceptable accuracy areas of fundamental equation of states