Theoretical Assessment of Compressibility Factor of Gases by Using Second Virial Coefficient

1. Introduction

The thermodynamic properties of gases used in many industrial applications require accurate evaluation. From the many possible thermodynamic properties, compressibility factor (Z), speed of sound, heat capacity, entropy, enthalpy, and internal energy are considered here. It is shown here that the compressibility factor is an important thermodynamic property to account for the behaviour of the gases [1], [2], [3]. It is also significant in chemical engineering calculations and in petroleum industry [4], [5], [6], [7], [8]. Several thermodynamic properties of gases such as density, isothermal compressibility and viscosity are calculated using Z value [4] and, therefore, it is still one of the main topics of research in the field of engineering [9], [10], [11]. Estimation of the gas Z is important in the gas industry for determining newly explored gas reservoirs, evaluating initial supply and gas reserves, and forecasting future gas production. It is also important in petroleum engineering, where it is used in calculations of gas flow through porous media, reservoir simulations, gas pressure gradients in tubing and pipelines, gas metering and gas compression [11], [12], [13]. It is also possible to determine the increase in the CO₂ content of natural gas according to the calculation of the compression factor. It can be seen from literature that the compressibility factor of gases was examined using experimental methods, equations of state, empirical correlations and analytical techniques [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22]. When Z<1, this indicates that a real gas has a lower pressure than an ideal gas and thus those molecules are more influenced by the attractive part of the potential than its repulsive part, due to their finite molecular volume. If the temperature is higher, then Z is positive for all pressures and repulsion dominates over attraction.

The equations of state are applicable to all types of gas within a certain range of pressure and temperature. Certain equations of state, such as the virial equation of state and the van der Waals equation of state, can be derived from statistical mechanics theory [23], [24]. It should be noted that the virial equation of state is one of the primary equations defining the thermodynamic properties of real gases at low-to-moderate densities [25], [26], [27], [28]. Generating potential and corresponding state law is a powerful tool in the theory of Lennard–Jones fluid as well as in various implementations. There are useful equations of state for accurate assessment of various properties of Lennard–Jones fluid [29].

In this study, we present a new method for evaluating the Z value of gases using an analytical expression [30] for the second virial coefficient with a Lennard–Jones (12-6) potential. In this study [31], the authors have proposed for an alternative simplest analytical method for accurate calculation of the second virial coefficient. It is to be noted that the Lennard–Jones (12-6) potential has widespread application as a powerful pair potential for gases and liquids. The results demonstrate that the proposed analytical expression is applicable to gases.

2. Definitions and Formulae

The compressibility factor of gases can be written in terms of the virial coefficients in the following form [32], [33]

where P is the pressure, V is the molar volume, T is the temperature, and R is the universal gas constant. The second, third, and fourth virial coefficients are B(T), C(T), and D(T), respectively. All virial coefficients are directly related to the interactions between the molecules. The second virial coefficients are defined as:

where f_ij=[exp(−u_ij/k_BT)−1] is the Mayer function, k_B is the Boltzmann constant, and u_ij is the intermolecular potential between the i^th and j^th atoms or molecules [34], [35], [36].

In this study, u_ij is the Lennard–Jones (12-6) potential defined as:

where ε is the depth of the potential well with the minimum at r_min=2^1/6σ, σ is the finite distance at which the inter-particle potential is zero, and r is the distance between the particles [37]. An alternative expression for the second virial coefficient can be obtained by the application of the series expansion [38]

to (2). This yields the following formula [27]

where b₀=2πN_Aσ³/3, N_A is the Avogadro constant, N is the upper limit of the summations, and the quantities Γ(α) are well-known gamma functions defined by [38]

In this study, we use the first two terms of the virial equation to evaluate the compressibility factor as follows:

As can be seen from (7), the choice of a reliable formula for calculation of the second virial coefficient is of prime importance in the accurate calculation of the compressibility factor, which depends on the temperature, pressure, and composition of the gas [39]. It is to be noted that for Z=1 the gas behaves as an ideal gas. With increasing pressure, different gases deviate from ideal behavior in varying ways [40], [41].

3. Numerical Results and Discussion

We present here an analytical approach for calculating the compressibility factor of gases. The physical significance of the second virial coefficient is that it takes into account deviations from ideal gas behavior that result from interactions between two molecules. At low densities, the deviation from the ideal state is adequately explained by the second virial coefficient, but at higher densities, higher virial coefficients must be taken into account. The second virial coefficient was then used to predict the compressibility factors at low densities.

As examples, the analytical expressions obtained for the compressibility factor are evaluated for the gases H₂, N₂, He, CO₂, CH₄, and air. On the basis of the formula derived here, we have constructed a program to compute the compressibility factor of gases using Mathematica 7.0 mathematical software. The parameters of the Lennard–Jones (12-6) potential for the gases examined are given in Table 1 [30], [33], [37]. Table 2 displays the results obtained from (8) and the comparison with data from the literature [32], [42]. The accuracy of the proposed approach is good, proving that it can be applied to evaluate the compressibility factor of real gases. Tables 3–6 show the results of calculations of the compressibility factor for H₂, N₂, He, CO₂, CH₄ and air using (7). As can be seen from these tables, the results presented show good agreement with the data in the literature [4], [14], [20], [43], [44], [45], [46], [47], [48]. The results for molecular systems also show a good rate of convergence and numerical stability. The advantage of this study is that it provides an alternative fast and accurate analytical approach for obtaining the quantities B(T). As can be seen from our obtained results with different molecules, the proposed algorithms are general and efficient in the evaluation of various properties of gases for a wide range of values of thermodynamic parameters. All calculations were performed with N=50. As a tool, this calculation method is conceptually valid over wide temperature ranges, and it offers some advantages over currently available methods.

Figures 1–4 show the compressibility factor Z, plotted against pressure P, for the gases H₂, N₂, He, CO₂, CH₄, and air at various values of temperature. As can be seen from these figures, the value of Z approaches 1 as the gas pressure approaches zero. The formula presented for the second virial coefficient gives satisfactory CPU times. As an example, in the case of the second virial coefficient with upper limits N=50, the CPU times to use (5) and [32] at temperatures T=400 and 1000 K are about 0.921 and 1.872 ms, and 0.845 and 1.716 ms, respectively. The compressibility factors of several gases show a lot of similarities, but they differ from one another. Therefore, the proposed method for evaluation of the compressibility factor is necessarily very accurate in representing the P and T behaviour of gases. In this sense, as seen from the results, our proposed method provides high accuracy for separate assessments of the compressibility factor of gases. Especially, the accurate estimate of compressibility factor of gases allows us an evaluation of the mass flow of gas transferred through pipelines. Finally, the gases used in chemical engineering applications and petroleum industry can be analysed using the formula for the compressibility factor presented here.

Figure 1:

The Z plotted against pressure P for CH₄ and N₂.

Figure 2:

The Z plotted against pressure P for CO₂ and He.

Figure 3:

The Z plotted against pressure P for Air.

Figure 4:

The Z plotted against pressure P for H₂.