Accurate Two-Parameter Equation of State for Hydrogen

Abstract

This study developed and validated a new simple and improved two-parameter high-quality equation of state (EOS) for hydrogen that can be used with less computational burden for the accurate description of various processes regarding underground hydrogen storage. Variation in hydrogen gas density with pressure was described by the modified power-law equation by relating density variation to deviations of density from the low-end and high-end limit values of density. The high-end limit density dependence on temperature was described by an Arrhenius-type asymptotic exponential function. The parameters of this EOS were determined by requiring thermodynamic consistency. This simple two-parameter EOS is thermodynamically consistent because the molar density is equal to zero, and its derivative with respect to pressure is equal to 1/(RT), like an ideal gas EOS when pressure approaches zero (T is absolute temperature, and R is the universal gas constant). This new simple EOS correlates hydrogen density efficiently using experimental data over 0–100 MPa and 220–473 K and molecular simulation data over the 14.03–116.064 MPa and 310.9–470 K ranges of pressures and temperatures. The new EOS is very accurate, as indicated by the coefficients of correlation being almost equal to unity (R² = 0.9999), the relative difference between the correlation and measured density values being very close to zero (RMSE = 0.0032), and the percentage average absolute value of the relative deviation of the EOS correlation density values ρ_EOS from the experimental density values ρ_Data being (ρ_Data/ρ_EOS − 1)100 = 0.15% average uncertainty.

1. Introduction

Hydrogen is a clean energy source, and large quantities of hydrogen are stored mostly in underground tanks, salt caverns, and drained hydrocarbon reservoirs (Civan [1], Zhao et al. [2], Bachand et al. [3], Li et al. [4], Luo et al. [5], Malki et al. [6], Zeng et al. [7], Civan [8]). Although hydrogen gas is a beneficial fuel as a clean energy source, the aboveground storage of low-molecular-weight hydrogen gas involves some safety concerns. Therefore, various safe storage materials like atomic and molecular clusters are being explored for storage by adsorbing hydrogen. Investigations of the optimization of hydrogen adsorption conditions can be achieved by phenomenological modeling involving hydrogen gas adsorption kinetics. This requires the application of an accurate hydrogen EOS like the one developed in this paper.

Various virial EOSs have been developed for hydrogen by Sakoda et al. [9], Peng and Long [10], Weiss et al. [11], and Zhang et al. [12], which are computationally complicated high-order polynomial-type virial equations. However, new improved, simple, and highly accurate EOSs are needed, which are applicable over the temperature and pressure conditions required for the underground storage of hydrogen. Well-known EOSs like the Redlich–Kwong (RK) (Redlich and Kwong [13]), Soave–Redlich–Kwong (SRK) (Soave [14]), and Peng–Robinson (PR) (Peng and Robinson [15]) cubic equations of state are mathematically complicated. A concise review of previous EOSs is provided in the next section in terms of their thermodynamical consistency, mathematical complexity, and number of parameters.

A practical and high-quality equation of state (EOS) is required for the accurate description of the various processes concerning underground hydrogen storage (Civan [1], Liu and Civan [16], Barajas-Cortes and Civan [17]). Finding a valid EOS of hydrogen and developing accurate correlations of EOSs are challenging problems. Using molecular simulation data is a convenient option for developing correlations of EOSs rapidly instead of generating experimental data, requiring much more effort and involving some indeterminate measurement errors. However, the question is whether molecular simulation data can replace experimental data reliably. In this paper, this issue was investigated by considering the correlation of the new EOS and comparing the results obtained by using experimental and molecular simulation data. Obtaining high-quality experimental data and developing meaningful correlations are important, but the reduction in the uncertainty errors of experimental data by the averaging effect of the correlation process is generally unexplored (Civan [18]).

In this paper, the variation in hydrogen gas density ρ = ρ (P, T) with pressure P was described by the modified power-law equation (Civan [8,19]), and the high-end limit density dependence on temperature T was described accurately by an Arrhenius [20]-type asymptotic exponential function, which is fundamental and new. This two-parameter EOS is thermodynamically consistent because density and its derivative with respect to pressure satisfy the conditions necessary for the EOS to become an ideal gas EOS when pressure approaches zero. Hydrogen density was correlated with the new simple EOS using the full set of data covering all pressures and temperatures together instead of using the data for various pressures separately at different temperatures, as previously done by Civan [19]. Very accurate correlations of the experimental data of Weber [21] and Sakoda et al. [9] combined and the molecular simulation data of Zhang et al. [12] were accomplished using this new EOS.

In summary, the primary objectives of this paper are the following:

• Review of previous outstanding hydrogen EOSs for thermodynamical consistency and the number of parameters involved.
• Development of a new simple two-parameter improved hydrogen EOS that can be used conveniently for subsurface hydrogen storage problems.
• Comparison of the correlations developed using experimental and molecular simulation data.
• Investigating whether molecular simulation data can be used instead of experimental data for the correlation of the hydrogen EOS.
• Development of a hydrogen compressibility factor chart.

2. Review of Previous EOSs for Thermodynamical Consistency and Number of Parameters Involved

In this section, previous outstanding EOSs are examined for thermodynamical consistency, mathematical complexity, and the number of parameters involved in these EOSs for the justification of developing a new simple and accurate EOS for hydrogen in this paper.

The molar density ρ (kmol/m³) of real gases is expressed by the following EOS:

$$\rho ={\displaystyle \frac{1}{V}}={\displaystyle \frac{P}{ZRT}}$$

(1)

where V is volume, P is pressure, T is temperature, R is the universal gas constant, and Z is the compressibility factor. The compressibility factor approaches the unity limit as pressure is lowered, $\underset{P\to P_o=0}{\mathrm{limit}}Z=1.0$, representing the ideal gas condition. Therefore, for thermodynamical consistency, the EOS is required to satisfy the low-end limit boundary conditions for density and the derivative of density with respect to pressure as the pressure approaches zero, expressed, respectively, by the following:

$$\underset{P\to P_o=0}{\mathrm{limit}}{\rho }_T={\rho }_o=0 \mathrm{and} \underset{P\to P_o=0}{\mathrm{limit}}{\left({\displaystyle \frac{\partial \rho }{\partial P}}\right)}_T={\displaystyle \frac{1}{RT}}$$

(2)

Unfortunately, some EOSs presented in the literature do not fulfill these conditions, and therefore they are not thermodynamically consistent, as explained in the following. Many different EOSs have been proposed for describing the PVT properties of real gases. Only outstanding equations are examined in this section. Some symbols were repeated to indicate the specific units used in these equations because some equations reviewed from the literature used units different than the units considered in this paper.

The Redlich–Kwong (RK) (Redlich and Kwong [13]), Soave–Redlich–Kwong (SRK) (Soave [14]), and Peng–Robinson (PR) (Peng and Robinson [15]) cubic EOSs are commonly used for describing the PVT properties of gases. Sakoda et al. [9], Wei et al. [22], Zhang et al. [12], and Civan [19] have developed new EOSs for hydrogen. These and some other EOSs are reviewed here to determine whether they also satisfy the required thermodynamic conditions described by Equation (2) and the number of parameters contained in EOSs as an indication of the complexity of EOSs.

The Redlich–Kwong (RK) EOS (Redlich and Kwong [13]) is given as follows by substituting V = 1/ρ:

$$P={\displaystyle \frac{RT}{V-b}}-{\displaystyle \frac{a}{V\left(V+b\right)\sqrt{T}}}={\displaystyle \frac{\rho RT}{1-b\rho }}-{\displaystyle \frac{a{\rho }^2}{\left(1+b\rho \right)\sqrt{T}}}$$

(3)

where a and b are parameters determined by the following:

$$a=0.42747\left({\displaystyle \frac{R^2{T}_c^{2.5}}{P_c}}\right)$$

(4)

$$b=0.08664\left({\displaystyle \frac{R{T}_c}{P_c}}\right)$$

(5)

where the variables are pressure P in Pa, temperature T in K, density ρ in mol/m³, and molar volume V in m³/mol, and the universal gas constant is R = 8.314 J/(mol K). T_c and P_c are the critical temperature and pressure values for hydrogen.

Equation (3) yields the following:

$$\underset{P\to P_o=0}{\mathrm{limit}}{\rho }_T={\rho }_o=0$$

(6)

The derivative of Equation (3) with respect to pressure at a constant temperature gives the following:

$${\displaystyle \frac{\partial P}{\partial P}}=1={\left[RT{\displaystyle \frac{1-b\rho -\rho \left(-b\right)}{{\left(1-b\rho \right)}^2}}-{\displaystyle \frac{a}{\sqrt{T}}}{\displaystyle \frac{2\rho \left(1+b\rho \right)-b{\rho }^2}{{\left(1+b\rho \right)}^2}}\right]}_T{\left({\displaystyle \frac{\partial \rho }{\partial P}}\right)}_T$$

(7)

Therefore,

$$\underset{P\to P_o=0}{\mathrm{limit}}{\left({\displaystyle \frac{\partial \rho }{\partial P}}\right)}_T={\displaystyle \frac{1}{RT}}$$

(8)

Thus, the Redlich–Kwong (RK) EOS [13] satisfies the required condition expressed by Equation (2) when P = 0.

Similarly, the Soave–Redlich–Kwong (SRK) EOS (Soave [14]) given as the following satisfies the required condition expressed by Equation (2) when P = 0:

$$P={\displaystyle \frac{RT}{V-b}}-{\displaystyle \frac{a}{V\left(V+b\right)}}$$

(9)

where a, k, and b are parameters given by the following:

$$a=0.42747\left({\displaystyle \frac{R^2{T_c}^2}{P_c}}\right){\left[1+k\left(1-\sqrt{{\displaystyle \frac{T}{T_c}}}\right)\right]}^2$$

(10)

$$k=0.48+1.574\omega -0.176{\omega }^2$$

(11)

$$b=0.08664\left({\displaystyle \frac{R{T}_c}{P_c}}\right)$$

(12)

where the variables are pressure P in Pa, temperature T in K, density ρ in mol/m³, and molar volume V in m³/mol, and the universal gas constant is R = 8.314 J/(mol K). T_c and P_c are the critical temperature and pressure values for hydrogen. The acentric factor is ω.

The Peng–Robinson (PR) EOS (Peng and Robinson [15]) is given as follows by substituting V = 1/ρ:

$$P={\displaystyle \frac{RT}{V-b}}-{\displaystyle \frac{a}{V\left(V+b\right)+b\left(V-b\right)}}={\displaystyle \frac{\rho RT}{1-b\rho }}-{\displaystyle \frac{a{\rho }^2}{\left(1+b\rho \right)+b\rho \left(1-b\rho \right)}}=RT{\displaystyle \frac{\rho }{1-b\rho }}-a{\displaystyle \frac{{\rho }^2}{1+2b\rho -b^2{\rho }^2}}$$

(13)

where a, k, and b are parameters given by the following:

$$a=0.45724\left({\displaystyle \frac{R^2{T_c}^2}{P_c}}\right){\left[1+k\left(1-\sqrt{{\displaystyle \frac{T}{T_c}}}\right)\right]}^2$$

(14)

$$k=0.37464+1.5422\omega -0.26992{\omega }^2$$

(15)

$$b=0.7780\left({\displaystyle \frac{R{T}_c}{P_c}}\right)$$

(16)

where the variables are pressure P in Pa, temperature T in K, density ρ in mol/m³, and molar volume V in m³/mol, and the universal gas constant is R = 8.314 J/(mol K). T_c and P_c are the critical temperature and pressure values for hydrogen. The acentric factor is ω.

Equation (13) yields the following:

$$\underset{P\to P_o=0}{\mathrm{limit}}{\rho }_T={\rho }_o=0$$

(17)

The derivative of Equation (13) with respect to pressure P at constant temperature gives the following:

$${\displaystyle \frac{\partial P}{\partial P}}=1={\left[RT{\displaystyle \frac{1-b\rho -\rho \left(-b\right)}{{\left(1-b\rho \right)}^2}}-a{\displaystyle \frac{2\rho \left(1+2b\rho -b^2{\rho }^2\right)-\left(2b-2b^2\rho \right){\rho }^2}{{\left(1+2b\rho -b^2{\rho }^2\right)}^2}}\right]}_T{\left({\displaystyle \frac{\partial \rho }{\partial P}}\right)}_T$$

(18)

Therefore,

$$\underset{P\to P_o=0}{\mathrm{limit}}{\left({\displaystyle \frac{\partial \rho }{\partial P}}\right)}_T={\displaystyle \frac{1}{RT}}$$

(19)

Thus, the Peng–Robinson (PR) EOS [15] satisfies the required conditions expressed by Equation (2) when P = 0.

The Jones–Wilkins–Lee (JWL) EOS (Lee et al. [23], Baudin and Serradeill [24]) is given in the following simplified form by substituting V = 1/ρ:

$$P={\displaystyle \frac{RT}{V}}+A{e}^{-BV}=\rho RT+A{e}^{-B/\rho }$$

(20)

where A and B are empirical constants.

Equation (20) yields the following:

$$\underset{P\to P_o=0}{\mathrm{limit}}{\rho }_T={\rho }_o=0$$

(21)

The derivative of Equation (20) with respect to pressure is given by the following:

$${\displaystyle \frac{\partial P}{\partial P}}=1={\left(RT+A{\displaystyle \frac{-B}{{\rho }^2}}{e}^{-B/\rho }\right)}_T{\left({\displaystyle \frac{\partial \rho }{\partial P}}\right)}_T$$

(22)

Therefore,

$$\underset{P\to P_o=0}{\mathrm{limit}}{\left({\displaystyle \frac{\partial \rho }{\partial P}}\right)}_T={\displaystyle \frac{1}{RT}}$$

(23)

Thus, the Jones–Wilkins–Lee (JWL) EOS [23,24] satisfies both required conditions expressed by Equation (2).

Sakoda et al. [9] correlated the hydrogen gas experimental data of Weber [21] and Sakoda et al. [9] combined in the ranges of 0–100 MPa pressures and 220–473 K temperatures using the following third-order polynomial virial equation:

$$Z = 1 + B(T)\rho + C(T){\rho }^2+ D(T){\rho }^3$$

(24)

Molar density is ρ (mole/dm³), pressure P (MPa), and temperature T (K). The coefficients B, C, and D of this equation (Equation (24)) were correlated with temperature using the following three empirical highly nonlinear equations:

$$B(T)=b_1+ b_2\exp \left(T_r^{-1}\right)+ b_3{T}_r^{-k_1}+ b_4{T}_r^{-k_2}$$

(25)

$$C(T)=c_1+c_2{T}_r^{-k_3}+c_3{T}_r^{-k_4}$$

(26)

$$D(T)=d_1{T}_r^{-k_5}$$

(27)

In Equations (24)–(27), B, C, and D are referred to as the second, third, and fourth virial coefficients, respectively; T_r = T/T_c is the reduced temperature; and T_c = 33.145 K is the critical temperature (Leachman et al. [25]). k_i: i = 1, 2, 3, 4, 5; b_i: i = 1, 2, 3, 4; c_i: i = 1, 2, 3; and d₁ represent thirteen empirical constants.

The values of these thirteen coefficients were determined by Sakoda et al. [9] by correlating the hydrogen gas experimental data of Weber [21] and Sakoda et al. [9] combined. Thus, substituting Equation (24) into Equation (1) gives the following:

$$\rho ={\displaystyle \frac{P}{\left[1 + B(T)\rho + C(T){\rho }^2+ D(T){\rho }^3\right]RT}}$$

(28)

Equation (28) yields the following:

$$\underset{P\to P_o=0}{\mathrm{limit}}{\rho }_T={\rho }_o=0$$

(29)

The derivative of Equation (28) with respect to pressure is given by the following:

$${\left\{\left[1 + B(T)\rho + C(T){\rho }^2+ D(T){\rho }^3\right]+\rho \left[ B(T)+ C(T)2\rho + D(T)3{\rho }^2\right]\right\}}_T{\left({\displaystyle \frac{\partial \rho }{\partial P}}\right)}_T={\displaystyle \frac{1}{RT}}$$

(30)

Therefore,

$$\underset{P\to P_o=0}{\mathrm{limit}}{\left({\displaystyle \frac{\partial \rho }{\partial P}}\right)}_T={\displaystyle \frac{1}{RT}}$$

(31)

Thus, both density and its derivative satisfy the required conditions expressed by Equation (2) when P = 0.

Wei et al.’s [22] EOS contains seventeen empirical constants (c_i: i = 1, 2, …, 17):

$$Z={\displaystyle \frac{MP}{\rho RT}}=A+{\displaystyle \frac{1-A}{e^B}}+C{P_{pr}}^D$$

(32)

where Z is the compressibility factor (dimensionless), M is molecular weight (kg/mol), P is pressure (Pa), ρ is hydrogen density (kg/m³), R is the molar gas constant (8.314472 J/(mol.K)), and T is the absolute temperature (K). The parameters A, B, C, and D are expressed as follows:

$$A=c_1{\left(T_{pr}-c_2\right)}^{0.5}-c_3{T}_{pr}-c_4$$

(33)

$$B=\left(c_5-c_6{T}_{pr}\right)P_{pr}+\left[{\displaystyle \frac{c_7}{\left(T_{pr}-c_8\right)}}-c_9+{\displaystyle \frac{c_{10}}{{10}^{c_{11}\left(T_{pr}-c_{12}\right)}}}\right]{P_{pr}}^2$$

(34)

$$C=c_{13}-c_{14}\log \left(T_{pr}\right)$$

(35)

$$D={10}^{c_{15}-c_{16}{T}_{pr}+c_{17}{T_{pr}}^2}$$

(36)

where the pseudo-reduced temperature T_pr and pressure P_pr are defined by the following:

$$T_{pr}={\displaystyle \frac{T}{T_c}},P_r={\displaystyle \frac{P}{P_c}}$$

(37)

where T_c and P_c are the critical temperature and pressure values for hydrogen.

Equation (32) can be rearranged as follows:

$${\displaystyle \frac{MP}{RT}}=\left[A+\left(1-A\right)e^{-B}+C{P_{pr}}^D\right]\rho$$

(38)

For P = 0 and constant temperature T, Equation (33) gives A as a constant, Equation (34) gives B = 0, Equation (35) gives C as a constant, and Equation (36) gives D as a constant. Thus, Equation (38) yields the following:

$$\underset{P\to P_o=0}{\mathrm{limit}}{\rho }_T={\rho }_o=0$$

(39)

The derivative of Equation (34) with respect to pressure is given by the following:

$${\displaystyle \frac{\partial B}{\partial P_{pr}}}=\left(c_5-c_6{T}_{pr}\right)+\left[{\displaystyle \frac{c_7}{\left(T_{pr}-c_8\right)}}-c_9+{\displaystyle \frac{c_{10}}{{10}^{c_{11}\left(T_{pr}-c_{12}\right)}}}\right]2P_{pr}$$

(40)

Thus, the derivative of Equation (38) with respect to pressure P at constant temperature T gives the following:

$$\begin{array}{l}{\displaystyle \frac{M}{RT}}={\left\{{\displaystyle \frac{\partial }{\partial P}}\left[A+\left(1-A\right)e^{-B}+C{P_{pr}}^D\right]\rho \right\}}_T+{\left[A+\left(1-A\right)e^{-B}+C{P_{pr}}^D\right]}_T{\left({\displaystyle \frac{\partial \rho }{\partial P}}\right)}_T\\ ={\left\{\left[\left(1-A\right)\left(-B{e}^{-B-1}{\displaystyle \frac{\partial B}{\partial P_{pr}}}{\displaystyle \frac{\partial P_{pr}}{\partial P}}\right)+CD{P_{pr}}^{D-1}{\displaystyle \frac{\partial P_{pr}}{\partial P}}\right]\rho \right\}}_T+{\left[A+\left(1-A\right)e^{-B}+C{P_{pr}}^D\right]}_T{\left({\displaystyle \frac{\partial \rho }{\partial P}}\right)}_T\end{array}$$

(41)

Equation (40) simplifies as follows for P = 0 and constant temperature T:

$$\underset{P\to P_o=0}{\mathrm{limit}}{\left({\displaystyle \frac{\partial B}{\partial P_{pr}}}\right)}_T=c_5-c_6{T}_{pr}$$

(42)

Substituting Equations (39) and (42) into Equation (41) yields the following for derivative of molar density with respect to pressure:

$$\underset{P\to P_o=0}{\mathrm{limit}}{\left({\displaystyle \frac{\partial \rho }{\partial P}}\right)}_T={\displaystyle \frac{1}{RT}}$$

(43)

Therefore, Equation (32) satisfies both required conditions expressed by Equation (2), proving that this equation is thermodynamically consistent.

Zhang et al.’s [12] EOS expressing pressure as an explicit function of density, which contains six empirical constants (c₁, c₂, c₃ and k₁, k₂, k₃), is given as the following with pressure P (MPa), temperature T (K), ρ (kg/m³), and molar volume V (L/mol):

$$P=K\left(\rho \right)T+C\left(\rho \right)$$

(44)

where the fitting parameters K and C are correlated by the following:

$$K\left(\rho \right)=k_1\rho +k_2{\rho }^2+k_3{\rho }^3$$

(45)

$$C\left(\rho \right)=c_1\rho +c_2{\rho }^2+c_3{\rho }^3$$

(46)

When ρ = 0, Equations (44)–(46) give the following:

$$K\left(\rho \right)=0,C\left(\rho \right)=0,P=0$$

(47)

So, Equation (44) satisfies the first required condition expressed by Equation (2) because

$$\underset{P\to P_o=0}{\mathrm{limit}}{\rho }_T={\rho }_o=0$$

(48)

But the derivative of Equation (44) with respect to pressure P at constant temperature T gives the following:

$$\begin{array}{l}{\displaystyle \frac{\partial P}{\partial P}}=1={\left({\displaystyle \frac{\partial \rho }{\partial P}}\right)}_T{\left[{\displaystyle \frac{\partial K\left(\rho \right)}{\partial \rho }}T+{\displaystyle \frac{\partial C\left(\rho \right)}{\partial \rho }}\right]}_T\\ ={\left({\displaystyle \frac{\partial \rho }{\partial P}}\right)}_T{\left[\left(k_1+k_{2}2\rho +k_{3}3{\rho }^2\right)T+c_1+c_2\rho +c_{3}3{\rho }^2\right]}_T\end{array}$$

(49)

Unfortunately, Equation (44) of Zhang et al.’s [9] EOS does not satisfy the second required condition expressed by Equation (2) when ρ = 0 for P = 0 because k₁ = 0.00417 ≠ R and c₁ = 0.00704 ≠ 0, and therefore Equation (49) gives the following:

$$\underset{P\to P_o=0}{\mathrm{limit}}{\left({\displaystyle \frac{\partial \rho }{\partial P}}\right)}_T={\displaystyle \frac{1}{k_{1}T+c_1}}\ne {\displaystyle \frac{1}{RT}}$$

(50)

The EOS by Xue and Guo [26] contains two dimensionless constants α and β:

$$Z={\displaystyle \frac{P}{\rho RT}}=\alpha \left(1+{\displaystyle \frac{1}{\beta }}\right){\displaystyle \frac{C_V}{R}}-\alpha {\displaystyle \frac{C_{\alpha }}{R}}{\displaystyle \frac{P}{T}}$$

(51)

where C_α is the work capacity, C_V is the specific heat at constant volume, and C_S is the specific work at constant entropy expressed, respectively, as follows:

$$C_{\alpha }=\left(1+{\displaystyle \frac{1}{\alpha }}\right)C_S-{\displaystyle \frac{1}{\alpha }}V$$

(52)

$$C_V={\left({\displaystyle \frac{\partial U}{\partial T}}\right)}_V=T{\left({\displaystyle \frac{\partial S}{\partial T}}\right)}_V$$

(53)

$$C_S={\left({\displaystyle \frac{\partial U}{\partial P}}\right)}_S=-P{\left({\displaystyle \frac{\partial V}{\partial P}}\right)}_S$$

(54)

U is internal energy, S is entropy, α and β are dimensionless coefficients, Z is the compressibility factor (dimensionless), M is molecular weight (kg/mol), P is pressure (Pa), ρ is hydrogen molar density (mol/m³), R is the molar gas constant (8.314472 J/(mol.K)), and T is the absolute temperature (K). Equation (51) satisfies the first required condition expressed by Equation (2) because

$$\underset{P\to P_o=0}{\mathrm{limit}}{\rho }_T={\rho }_o=0$$

(55)

The derivative of Equation (51) with respect to pressure is as follows:

$${\displaystyle \frac{1}{RT}}={\left({\displaystyle \frac{\partial \rho }{\partial P}}\right)}_T{\left[\alpha \left(1+{\displaystyle \frac{1}{\beta }}\right){\displaystyle \frac{C_V}{R}}-\alpha {\displaystyle \frac{C_{\alpha }}{R}}{\displaystyle \frac{P}{T}}\right]}_T+\rho {\left\{{\displaystyle \frac{\partial }{\partial P}}\left[\alpha \left(1+{\displaystyle \frac{1}{\beta }}\right){\displaystyle \frac{C_V}{R}}-\alpha {\displaystyle \frac{C_{\alpha }}{R}}{\displaystyle \frac{P}{T}}\right]\right\}}_T$$

(56)

Unfortunately, Equation (51) of Xue and Guo’s [26] EOS does not satisfy the second required condition expressed by Equation (2) when ρ = 0 for P = 0 because Equation (56) gives the following:

$$\underset{P\to P_o=0}{\mathrm{limit}}{\left({\displaystyle \frac{\partial \rho }{\partial P}}\right)}_T={\displaystyle \frac{\frac{1}{RT}}{\alpha \left(1+\frac{1}{\beta }\right)\frac{C_V}{R}}}\ne {\displaystyle \frac{1}{RT}}$$

(57)

Therefore, Xue and Guo’s [26] EOS does not simplify as the ideal gas EOS at the low-density limit.

Civan [19] previously developed an EOS for hydrogen containing only two constants a and b in the form of a modified power-law equation as follows:

$$\rho \left(P,T\right)={\left({\displaystyle \frac{1}{{\rho }_{\infty }\left(T\right)}}+{\displaystyle \frac{RT}{P}}\right)}^{-1}={\displaystyle \frac{P}{\frac{P}{{\rho }_{\infty }\left(T\right)}+RT}}={\left({\displaystyle \frac{1}{a{T}^b}}+{\displaystyle \frac{RT}{P}}\right)}^{-1}$$

(58)

Note that Civan [19] correlated Equation (58) for density ρ (kg/m³) instead of (kmol/m³) by pressure P (MPa) at various temperatures T (K) and determined the value of the high-end limit density ρ_∞ = ρ_∞(T) at each temperature T. Then, the high-end limit density ρ_∞ vs. temperature T was correlated by the following power-law equation for determining the best estimate values of two constants a and b:

$${\rho }_{\infty }\left(T\right)=a{T}^b$$

(59)

Equation (58) was developed such that both density and its derivative satisfy the conditions required by Equation (2) when P = 0. Thus, Equation (58) gives the following:

$$\underset{P\to P_o=0}{\mathrm{limit}}{\rho }_T={\rho }_o=0$$

(60)

Also, the derivative of Equation (58) is given by the following:

$${\left({\displaystyle \frac{\partial \rho }{\partial P}}\right)}_T={\displaystyle \frac{\frac{P}{{\rho }_{\infty }}+RT-\frac{P}{{\rho }_{\infty }}}{{\left(\frac{P}{{\rho }_{\infty }}+RT\right)}^2}}={\displaystyle \frac{RT}{{\left(\frac{P}{{\rho }_{\infty }}+RT\right)}^2}}$$

(61)

Therefore,

$$\underset{P\to P_o=0}{\mathrm{limit}}{\left({\displaystyle \frac{\partial \rho }{\partial P}}\right)}_T={\displaystyle \frac{1}{RT}}$$

(62)

Thus, both density and its derivative satisfy the conditions required by Equation (2) when P = 0.

Table 1. Comparison of various EOSs for thermodynamical consistency and number of empirical parameters present.

EOS	$\underset{\mathit{P}\to {\mathit{P}}_{\mathit{o}}=0}{\mathbf{limit}}{\mathit{\rho }}_{\mathit{T}}={\mathit{\rho }}_{\mathit{o}}$	$\underset{\mathit{P}\to {\mathit{P}}_{\mathit{o}}=0}{\mathbf{limit}}{\left(\frac{\partial \mathit{\rho }}{\partial \mathit{P}}\right)}_{\mathit{T}}$	Is the EOS Thermodynamically Consistent?	Number of Empirical Parameters
Redlich–Kwong (RK) [13]	0	${\displaystyle \frac{1}{RT}}$	Yes	2
Soave–Redlich–Kwong (SRK) [14]	0	${\displaystyle \frac{1}{RT}}$	Yes	5
Peng–Robinson (PR) [15]	0	${\displaystyle \frac{1}{RT}}$	Yes	5
Jones–Wilkins–Lee (JWL) [23,24]	0	${\displaystyle \frac{1}{RT}}$	Yes	2
Sakoda et al. [9]	0	${\displaystyle \frac{1}{RT}}$	Yes	13
Wei et al. [22]	0	${\displaystyle \frac{1}{RT}}$	Yes	17
Zhang et al. [12]	0	${\displaystyle \frac{1}{k_{1}T+c_1}}$	No	6
Xue and Guo [26]	0	${\displaystyle \frac{\frac{1}{RT}}{\alpha \left(1+\frac{1}{\beta }\right)\frac{C_V}{R}}}$	No	2
Civan [19]	0	${\displaystyle \frac{1}{RT}}$	Yes	2
Present paper	0	${\displaystyle \frac{1}{RT}}$	Yes	2

presents a summary of the various EOSs examined in this section, indicating whether density and its derivative with respect to pressure satisfy the required conditions for thermodynamical consistency expressed by Equation (2) when P = 0 and the number of empirical parameters involved in the EOS. As indicated by the results presented in Table 1, the Redlich–Kwong EOS (RK EOS) [13], Jones–Wilkins–Lee (JWL) EOS [23], and Civan EOS [19] are thermodynamically consistent and contain only two empirical parameters. The Soave–Redlich–Kwong EOS (SRK EOS) [14] and Peng–Robinson EOS (PR EOS) [15] are thermodynamically consistent but contain five empirical parameters. Sakoda et al.’s [9] EOS is thermodynamically consistent but contains thirteen empirical parameters. Wei et al.’s [22] EOS is thermodynamically consistent but contains seventeen empirical parameters. Xue and Guo’s [26] EOS is not thermodynamically consistent and contains two empirical parameters. Zhang et al.’s [12] EOS is not thermodynamically consistent and contains six empirical parameters. The mathematical complexity of EOSs increases as the number of empirical parameters increases.

3. Development of New Simple Improved Two-Parameter EOS for Hydrogen

Civan [19] previously correlated the high-end limit density ρ_∞(T) dependence on temperature T using the power-law equation (Equation (59)) with the correlation coefficient R² being lower than 1.0. In this section, an Arrhenius [20]-type asymptotic exponential function is applied for the accurate correlation of the high-end limit density dependence on temperature T with the correlation coefficient R² being almost equal to 1.0.

The previous EOS developed by Civan [19] using Equation (58) and included in Table 1 is improved here by representing the high-end limit density ρ_∞(T) dependence on temperature T more accurately by using an Arrhenius [20]-type asymptotic exponential function and applying a one-step full range temperature and pressure data correlation method. For this purpose, the end-limit density values ρ_o and ρ_∞(T) at the low-end and high-end limit pressure conditions denoted respectively by P_o and P_∞ are described as follows:

$$\rho \left(P,T\right)={\rho }_o=0,P=P_o=0$$

(63)

$$\rho \left(P,T\right)={\rho }_{\infty }\left(T\right),P=P_{\infty }\to \infty$$

(64)

The formulation of the new EOS is based on expressing the variation in hydrogen gas density ρ = ρ (P, T) by pressure P at any temperature T with the power-law functions of the deviations of the density value ρ = ρ (P, T) from the low-end and high-end limit density values ρ_o and ρ_∞(T), respectively, as follows (see Appendix A for details):

$$\begin{array}{l}{\displaystyle \frac{\partial \rho \left(P,T\right)}{\partial P}}=k\left(T\right){\left[\rho \left(P,T\right)-{\rho }_o\right]}^{m_o\left(T\right)}{\left[{\rho }_{\infty }\left(T\right)-\rho \left(P,T\right)\right]}^{m_{\infty }\left(T\right)}\\ ={\displaystyle \frac{{\left[A\left(T\right)\right]}^{1/B\left(T\right)}B\left(T\right)}{{\rho }_{\infty }\left(T\right)-{\rho }_o}}{\left[\rho \left(P,T\right)-{\rho }_o\right]}^{1-1/B\left(T\right)}{\left[{\rho }_{\infty }\left(T\right)-\rho \left(P,T\right)\right]}^{1+1/B\left(T\right)}\end{array}$$

(65)

For a convenient analytical solution of Equation (65), the proportionality coefficient is expressed as k(T) = ${\displaystyle \frac{{\left[A\left(T\right)\right]}^{1/B\left(T\right)}B\left(T\right)}{{\rho }_{\infty }\left(T\right)-{\rho }_o}}$, and the exponents of intensities are expressed as m_o(T) = 1 − 1/B(T) and m_∞(T) = 1 + 1/B(T). A = A (T) and B = B (T) are temperature-dependent parameters. The analytical solution of Equation (65) provides the following modified power-law equation (see Appendix A for details):

$${\displaystyle \frac{\rho \left(P,T\right)-{\rho }_o}{{\rho }_{\infty }\left(T\right)-\rho \left(P,T\right)}}=A\left(T\right){\left(P-P_o\right)}^{B\left(T\right)}$$

(66)

The density ρ given by Equation (66) is expressed explicitly as follows:

$$\rho \left(P,T\right)={\rho }_{\infty }\left(T\right)+{\displaystyle \frac{{\rho }_o-{\rho }_{\infty }\left(T\right)}{1+A\left(T\right){\left(P-P_o\right)}^{B\left(T\right)}}}={\displaystyle \frac{{\rho }_o+{\rho }_{\infty }\left(T\right)A\left(T\right){\left(P-P_o\right)}^{B\left(T\right)}}{1+A\left(T\right){\left(P-P_o\right)}^{B\left(T\right)}}}$$

(67)

The density expression determined by Equation (67) is required to satisfy the following conditions for thermodynamical consistency according to Equation (2):

$$\underset{p\to p_o=0}{\mathrm{limit}}{\rho }_T={\rho }_o=\underset{p\to p_o=0}{\mathrm{limit}}\left[{\rho }_{\infty }\left(T\right)+{\displaystyle \frac{{\rho }_o-{\rho }_{\infty }\left(T\right)}{1+A\left(T\right){\left(P-P_o\right)}^{B\left(T\right)}}}\right]=0$$

(68)

and

$$\underset{P\to P_o=0}{\mathrm{limit}}{\left({\displaystyle \frac{\partial \rho }{\partial P}}\right)}_T=\underset{P\to P_o=0}{\mathrm{limit}}\left\{{\displaystyle \frac{A\left(T\right)B\left(T\right)\left({\rho }_{\infty }\left(T\right)-{\rho }_o\right){\left(P-P_o\right)}^{B\left(T\right)-1}}{{\left[1+A\left(T\right){\left(P-P_o\right)}^{B\left(T\right)}\right]}^2}}\right\}={\displaystyle \frac{1}{RT}}$$

(69)

Therefore, the parameters A and B are determined as follows by solving Equations (68) and (69) simultaneously:

$$A\left(T\right)={\displaystyle \frac{1}{{\rho }_{\infty }\left(T\right)RT}},B\left(T\right)=1$$

(70)

Thus, a new simple EOS can be derived as follows by substituting the expressions of the parameters A and B provided in Equation (70) into Equation (67):

$$\rho \left(P,T\right)={\left({\displaystyle \frac{1}{{\rho }_{\infty }\left(T\right)}}+{\displaystyle \frac{RT}{P}}\right)}^{-1}={\displaystyle \frac{{\rho }_{\infty }\left(T\right)P}{{\rho }_{\infty }\left(T\right)RT+P}}$$

(71)

The high-end limit density ρ_∞(T) decreases when temperature increases. The high-end limit density dependence on temperature is described by an Arrhenius [20]-type asymptotic exponential function:

$${\displaystyle \frac{1}{{\rho }_{\infty }\left(T\right)}}=\exp \left({\displaystyle \frac{{\alpha }_1}{T}}+{\alpha }_2\right)$$

(72)

where α₁ and α₂ are empirical constants.

Substituting Equation (72) into Equation (71) yields an EOS as follows:

$${\displaystyle \frac{1}{\rho \left(P,T\right)}}={\displaystyle \frac{1}{{\rho }_{\infty }\left(T\right)}}+{\displaystyle \frac{RT}{P}}={\displaystyle \frac{RT}{P}}+\exp \left({\displaystyle \frac{{\alpha }_1}{T}}+{\alpha }_2\right)$$

(73)

The new variables x and y are defined as follows so that Equation (73) can be transformed into a simple algebraic equation:

$$x=T,y=\ln \left({\displaystyle \frac{1}{\rho \left(P,T\right)}}-{\displaystyle \frac{RT}{P}}\right)$$

(74)

Thus, Equation (73) can be expressed in a simple form as follows:

$$xy={\alpha }_1+{\alpha }_{2}x$$

(75)

The values of the α₁ and α₂ constants are determined for the best correlation of all experimental or simulation data by applying the method of least squares. If the various experimental or simulation data of hydrogen density acquired under different pressure and temperature conditions are sequenced in any order by the index i = 1, 2, …, n, then the objective function is expressed by the following:

$$J\left({\alpha }_1,{\alpha }_2\right)={\displaystyle \sum _{i=1}^n{\left({\alpha }_1+{\alpha }_2{x}_i-x_i{y}_i\right)}^2}$$

(76)

The best estimate values of the α₁ and α₂ constants are determined by applying the following conditions to minimize the objective function:

$${\displaystyle \frac{\partial J\left({\alpha }_1,{\alpha }_2\right)}{\partial {\alpha }_1^{\prime }}}=2{\displaystyle \sum _{i=1}^n\left({\alpha }_1+{\alpha }_2{x}_i-x_i{y}_i\right)}=0$$

(77)

$${\displaystyle \frac{\partial J\left({\alpha }_1,{\alpha }_2\right)}{\partial {\alpha }_2}}=2{\displaystyle \sum _{i=1}^n{x}_i\left({\alpha }_1+{\alpha }_2{x}_i-x_i{y}_i\right)}=0$$

(78)

Equations (77) and (78) are combined as the following matrix equation:

$$\left[\begin{array}{cc}n& {\displaystyle \sum _{i=1}^n{x}_i}\\ {\displaystyle \sum _{i=1}^n{x}_i}& {\displaystyle \sum _{i=1}^n\left({x_i}^2\right)}\end{array}\right]\left[\begin{array}{c}{\alpha }_1\\ {\alpha }_2\end{array}\right]=\left[\begin{array}{c}{\displaystyle \sum _{i=1}^n{x}_i{y}_i}\\ {\displaystyle \sum _{i=1}^n\left({x_i}^2{y}_i\right)}\end{array}\right]$$

(79)

The solution of Equation (79) provides the best estimate values of the α₁ and α₂ constants. The quality of the correlation of the hydrogen density data is determined by means of the coefficient of correlation (R²), the root mean square of the relative difference (RMSE) between the correlation and measured or simulation density values, and the percentage average absolute value of the relative deviation (ρ_Data/ρ_EOS − 1)100 of the EOS correlation density values ρ_EOS from the measured or simulation density values ρ_Data.

4. Correlations of New EOS for Hydrogen with Experimental and Molecular Simulation Data

The units are considered for mass density ρ as kg/m³, pressure P as MPa, and temperature T as K. The molecular weight of hydrogen M is 2.01568 g/mole. The universal gas constant is R = 8.314462 × 10⁻³ MPa.dm³/mol/K.

First, the experimental data of Weber [21] for parahydrogen and Sakoda et al. [9] for normal hydrogen are considered. The differences in density between the normal hydrogen and parahydrogen are negligible under the temperature and pressure conditions of these data, as indicated by Sakoda et al. [9]. Therefore, these data can be combined and used together as done by Sakoda et al. [9].

The uncertainties involved in the experimental measurements of the pressure, temperature, and density were 28 kPa, 20 mK, and 0.07 to 0.24%, respectively, as reported by Sakoda et al. [9]. The uncertainties involved in the experimental measurements of the pressure, temperature, and density were 0.01% pressure gage accuracy, 0. 002 to 0. 028 K, and about 0.02%, respectively, as reported by Weber [21].

The hydrogen gas density experimental data of Weber [21] and Sakoda et al. [9] over 0–100 MPa pressures and 220–473 K temperatures were combined for the solution of Equation (79) to obtain the optimum values of the α₁ and α₂ constants, as shown in

Table 2. Best estimate parameter values determined using experimental data of Weber [21] and Sakoda et al. [9] and molecular simulation data of Zhang et al. [12].

Data Used for Correlation	Experimental Data of Weber [21] and Sakoda et al. [9] Combined Over 0–100 MPa and 220–473 K	Experimental Data of Sakoda et al. [9] Over 0–100 MPa and 353–473 K	Experimental Data of Weber [21] Over 0–80 MPa and 220–300 K	Molecular Simulation Data of Zhang et al. [12] Over 14.03–116.064 MPa and 310.9–470 K
α1	−33.0606	−43.6755	−33.555	−37.8109
α2	−4.75512	−4.73015	−4.74684	−4.66981
R2 (coefficient of correlation)	0.9999	1.0000	0.9993	0.9997
RMSE (relative difference between the correlation and measured or simulation density values)	0.0032	0.0014	0.0061	0.0051
(ρData/ρEOS − 1)100 (percentage average relative absolute deviation of density correlation values from data)	0.15%	0.08%	0.47%	0.39%

. Figure 1 shows (a) the correlation values matching very closely to the hydrogen density experimental data of Weber [21] and Sakoda et al. [9] and (b) the comparison of the hydrogen density obtained by the correlation with the experimental data of Weber [21] and Sakoda et al. [9]. The new EOS is very accurate, as indicated by the coefficients of correlation almost equal to unity (R² = 0.9999) and the relative difference between the correlation and measured density values very close to zero (RMSE = 0.0032). The percentage average absolute value of the relative deviation (ρ_Data/ρ_EOS − 1)100 of the EOS correlation density values ρ_EOS from the experimental density values ρ_Data indicates 0.15% average uncertainty in the present case. Therefore, the correlation with the new simple two-parameter EOS is computationally more convenient and better than the virial EOS developed for hydrogen by Sakoda et al. [9] containing thirteen empirical parameters and having 0.22% average uncertainty.

Next, only the hydrogen gas density experimental data of Sakoda et al. [9] over 0–100 MPa pressures and 353–473 K temperatures were used for the solution of Equation (79) to obtain the optimum values of the α₁ and α₂ constants, as shown in Table 2. The 0.15% average uncertainty involved in the correlation of the combined data of Weber [21] and Sakoda et al. [9] dropped down to 0.08%, indicating that the quality of the correlation improved significantly when the hydrogen density experimental data of Weber [21] was discarded. The new EOS is very accurate, as indicated by the coefficients of correlation equal to unity (R² = 1.0000) and the relative difference between the correlation and measured density values very close to zero (RMSE = 0.0014). The percentage average absolute value of the relative deviation (ρ_Data/ρ_EOS − 1)100 of the EOS correlation density values ρ_EOS from the experimental density values ρ_Data indicates 0.08% average uncertainty in the present case. Figure 2 shows (a) the correlation values matching very closely to the hydrogen density experimental data of Sakoda et al. [9] and (b) the comparison of the hydrogen density obtained by the correlation of the experimental data of Sakoda et al. [9].

Similarly, for comparison purposes, only the hydrogen gas density experimental data of Weber [21] over 0–80 MPa pressures and 220–300 K temperatures were used for the solution of Equation (79) to obtain the optimum values of the α₁ and α₂ constants, as shown in Table 2. The 0.15% average uncertainty involved in the correlation of the combined data of Weber [21] and Sakoda et al. [9] increased to 0.47%, indicating that the quality of the correlation dropped when the hydrogen density experimental data of Sakoda et al. [9] were discarded. Thus, the correlation is less accurate, as indicated by the coefficients of correlation a little lower than unity (R² = 0.9993), and the relative difference between the correlation and measured density values is not very close to zero (RMSE = 0.0061). The percentage average absolute value of the relative deviation (ρ_Data/ρ_EOS − 1)100 of the EOS correlation density values ρ_EOS from the experimental density values ρ_Data indicates a higher 0.47% average uncertainty in the present case.

Figure 3 shows (a) the correlation values matching reasonably to the hydrogen density experimental data of Weber [21] and (b) the comparison of the hydrogen density obtained by the correlation of the experimental data of Weber [21]. This exercise indicates that the quality of the hydrogen density experimental data of Weber [21] is a little lower than the quality of the hydrogen density experimental data of Sakoda et al. [9].

Finally, the molecular simulation data of Zhang et al. [12] obtained over 14.03–116.064 MPa pressures and 310.9–470 K temperatures were used. The solution of Equation (79) provides the optimum values of the α₁ and α₂ constants, as shown in Table 2. Figure 4 shows (a) the correlation values matching very closely to the hydrogen density molecular simulation data of Zhang et al. [12] and (b) a comparison of the hydrogen density obtained by the correlation of the molecular simulation data of Zhang et al. [12]. The EOS is accurate, as indicated by the coefficients of correlation almost equal to unity (R² = 0.9997), and the relative difference between the correlation and measured density values is close to zero (RMSE = 0.0051). The percentage average absolute value of the relative deviation (ρ_Data/ρ_EOS − 1)100 of the EOS correlation density values ρ_EOS from the experimental density values ρ_Data indicates 0.39% average uncertainty in the present case.

The correlations of the experimental data of Weber [21] and Sakoda et al. [9] and the molecular simulation data of Zhang et al. [12] obtained using the new EOS expressed by Equation (73) are very accurate, as indicated by the coefficients of correlations (R²) almost equal to unity and the relative difference (RMSE) between the correlation and measured or simulation density values very close to zero, as summarized in Table 2.

Substituting the best estimate values of the α₁ and α₂ constants presented in Table 2 into Equation (72) yields the high-end limit density ρ_∞ vs. temperature correlations of the experimental data of Weber [21] and Sakoda et al. [9] combined and the molecular simulation data, respectively, as follows:

$${\displaystyle \frac{1}{{\rho }_{\infty }\left(T\right)}}=\exp \left({\displaystyle \frac{-33.0606}{T}}-4.75512\right) \mathrm{obtained\; using\; all\; experimental\; data}$$

(80)

and

$${\displaystyle \frac{1}{{\rho }_{\infty }\left(T\right)}}=\exp \left({\displaystyle \frac{-37.8109}{T}}-4.66981\right) obtained\; using\; all\; simulation\; data$$

(81)

The quality of these correlations is better than that of the previous correlations developed by Civan [19] using Equation (59), as indicated by the coefficients of correlations almost equal to R² = 1.0. Therefore, the following new and improved EOSs for hydrogen were obtained by substituting Equations (80) and (81), respectively, into Equation (71):

$$\rho \left(P,T\right)={\left[{\displaystyle \frac{RT}{MP}}+\exp \left({\displaystyle \frac{-33.0606}{T}}-4.75512\right)\right]}^{-1} \mathrm{obtained\; using\; all\; experimental\; data}$$

(82)

$$\rho \left(P,T\right)={\left[{\displaystyle \frac{RT}{MP}}+\exp \left({\displaystyle \frac{-37.8109}{T}}-4.66981\right)\right]}^{-1} \mathrm{obtained\; using\; all\; simulation\; data}$$

(83)

Note that the molecular weight M of hydrogen is included in Equations (82) and (83) because the units of ρ(P,T) and ρ_∞(T) are expressed as kg/m³ instead of kmol/m³.

Figure 5 shows a comparison of the hydrogen density obtained by (a) the correlation of the experimental data of Weber [21] and Sakoda et al. [9] and (b) the correlation of the molecular simulation data of Zhang et al. [13]. The correlations of the experimental data of Weber [21] and Sakoda et al. [9] and the molecular simulation data of Zhang et al. [12] were obtained very accurately.

5. Development of Hydrogen Compressibility Factor Chart

Li et al. [27] developed a compressibility factor chart of hydrogen gas theoretically over the pressure range of 0.1–100 MPa and the temperature range of 233–363 K using four different cubic EOSs by van der Waals (VDW) (Laghaei et al. [28]), Redlich–Kwong (RK) (Redlich and Kwong [13]), Soave–Redlich–Kwong (SRK) (Soave [14]), and Peng–Robinson (PR) (Peng and Robinson [15]).

In this section, an application of the new EOS developed in the previous section is presented for the theoretical development of a hydrogen compressibility factor chart. The compressibility factor Z is determined as follows by substituting Equation (82) into Equation (1) over 0–100 MPa and 220–473 K:

$$Z={\displaystyle \frac{MP}{\rho RT}}=1+{\displaystyle \frac{MP}{{\rho }_{\infty }RT}}=1+{\displaystyle \frac{MP}{RT}}\exp \left({\displaystyle \frac{-33.0606}{T}}-4.75512\right)$$

(84)

Note that the molecular weight M of hydrogen is included in Equation (84) because the units of ρ(P,T) and ρ_∞(T) are expressed as kg/m³ instead of kmol/m³.

Figure 6 shows the compressibility factor chart of hydrogen gas developed theoretically using Equation (84) based on the correlation of the combined experimental data of Weber [21] and Sakoda et al. [9].

6. Conclusions

In this paper, a new simple and improved two-parameter EOS was developed for hydrogen that can be used conveniently for storage conditions in underground tanks, salt caverns, and drained hydrocarbon reservoirs. The variation in hydrogen gas density with pressure was described by the modified power-law equation with the high-end limit density dependence on temperature described by an Arrhenius [20]-type asymptotic exponential function. This EOS is thermodynamically consistent because density and its derivative with respect to pressure satisfy the conditions necessary for the EOS to become an ideal gas EOS when pressure approaches zero. The new simple two-parameter EOS developed for hydrogen is much more convenient than other EOSs reported in the literature and the thirteen-parameter virial EOS developed by Sakoda et al. [9].

The correlations of the experimental data of Weber [21] and Sakoda et al. [9] combined and the molecular simulation data of Zhang et al. [12] obtained using this new EOS are very accurate, as indicated by the coefficients of correlations (R²) almost equal to unity and the relative difference (RMSE) between the correlation and measured density values close to zero. However, it was determined that the quality of the hydrogen density experimental data of Weber [21] is a little lower than the quality of the hydrogen density experimental data of Sakoda et al. [9]. Nevertheless, the quality of the correlation obtained by the new simple two-parameter EOS is better, and this EOS is computationally more convenient than the virial EOS containing thirteen empirical parameters developed for hydrogen by Sakoda et al. [9]. The correlations obtained using the experimental data of Weber [21] and Sakoda et al. [9] combined and the molecular simulation data of Zhang et al. [12] are very similar. Therefore, it is concluded that molecular simulation data may be used instead of experimental data for correlating the hydrogen EOS. However, further research may be required for a further investigation of this issue.

Table 2 presents the temperature and pressure ranges of the data used for the correlations of the EOS developed in this paper. For example, the correlation of the experimental data of Weber [21] and Sakoda et al. [9] combined is valid over 0–100 MPa and 220–473 K.

The EOS developed in this paper has a theoretical-based advantage because it is based on the modified power-law equation with the high-end limit density dependence on temperature described by an Arrhenius [20]-type asymptotic exponential function. This explicit two-parameter EOS is simple and requires much less computational effort than many implicit and computationally complicated EOSs proposed in the literature reviewed in Table 1. This EOS satisfies the end-limit density values ρ_o and ρ_∞(T) under the low-end and high-end limit pressure conditions denoted respectively by P_o = 0 and P_∞ → ∞, and it is thermodynamically consistent. As shown in Table 2, however, the parameter values of the Arrhenius [20]-type asymptotic exponential function depend on the range of data used for the correlation of this EOS.

In this paper, an application of the new EOS was successfully achieved in a simple practical manner for the theoretical development of a hydrogen compressibility factor chart over the pressure and temperature ranges of 0–100 MPa and 220–473 K.