Global thermodynamic behaviour of supercritical fluids : example of methane and ethane

Do to the fluctuations associated with the critical region of fluids. The behavior of thermodynamic properties these can not be predicted by mean field theories. To do so, a global equation of state based on the crossover model has been used. This equation of state is formulated on the basis of comparison of selected measurements of pressure-density-temperature data, isochoric and isobaric heat capacity of fluids.The model can be applied in a wide range of temperatures and densities around the critical point for ethane and methane. It is found that the developed model represents most of the reliable experimental data accurately.


Introduction
Methane and ethane are important fluids which are present in natural gas and Petroleum.Many efforts are diploid to formulate an equation of state for describing the thermodynamic properties of these two systems.
In fact, the work described in this paper is part of a research effort to develop a comprehensive but preliminary fundamental equation for the thermodynamic properties of ethane and methane in the critical region that extends to the classical region.The formulated equation of state covers the entire range of temperatures and densities around the critical region and also can describe the behavior of the thermodynamic properties of ethane and methane in the classical region far away from the critical region.Several analytic equations of state as well as non-analytical equations of state were proposed earlier [1].
Accurate information on the thermodynamic properties of fluids is highly sought for the chemical technology.The thermodynamic properties of fluids near the critical region are strongly affected by the presence of fluctuations and therefore, cannot be described by conventional equation.We have investigated an interim formulation for the behavior of the thermodynamic properties of methane and ethane in the vicinity of the critical region.For this reason we have used the so-called "Crossover Model" to describe the thermodynamic properties of methane and ethane in a wide range of temperatures and densities [2,3].

Crossover model
The description of the thermodynamic properties in the neighborhood of the critical region, as well as in the classical region by using two different formalisms, will introduce discontinuities of the caloric properties; such as Cv, Cp and Cs.Therefore, we introduce another alternative: a unified function that describes these properties with a smooth transition from the critical region to the classic region (without a jump).
Let ρ be the density, T the temperature, P the pressure, µ the chemical potential and A/V the Helmholtz free energy per unit volume.We make these properties dimensionless with the aid of the critical parameters [4]: in addition we define are analytic background functions of T subject to the conditions that at the critical temperature 0 . In order to obtain a fundamental equation that can be applied in a large range of densities and temperatures around the critical point we retain six terms in the classical Landau expansion [5] for ∆Acl: As shown by Abbaci (1991) [4] the theoretically predicted asymptotic behavior can be recovered from this expansion by the following transformation Where the functions and In terms of a crossover function Y to be determined from Fluid system do not exhibit the symmetry in coexistence curve as magnetic systems, also colled mixing of the field variable is needed [6,7] such as The coefficients c, c t , c ρ and d 1 are system-dependent constants.Finally, the total Helmholtz free-energy density is obtained ( )

Application to methane and ethane
The crossover model as applied to these two fluids contains the following system-dependent parameters: The critical parameters T c , ρ c , and P c to be deduced either from an asymptotic analysis of the thermodynamicproperty data near the critical point or reported by several experiments.The crossover parameters u and Λ, the scaling-field parameters c, c t, c ρ and d 1 , the classical parameters 05 a , 06 a , 14 a 22 a and the background parameters Ã j which can be determined by fitting the crossover model to the P-ρ-T data of Douslin [8] for ethane, with σ p = 0.007%, σ T = 0.001K, and σ ρ = 0.02% as an estimated errors in pressure, temperature and density.For methane, we used P-ρ-T data of Wagner and CO [9], with σ p = 0.0001MP, σ T = 0.005K, and σ ρ = 0.01%.Finally the caloric background i µ ~ which can be determined from experimental isochoric specific heat capacity data reported by Shmakov [10] for ethane and Abdulagatov data [11] for methane.
The values of the critical parameters for C 2 H 6 are those used by Douslin [8], which are as follows Tc= 305.322K, ρc= 206.18 Kg.m-3, Pc= 4.8722MPa (9) The values of the critical parameters for CH 4 used by Wagner [9] are The values of the system-dependent parameters adopted for C 2 H 6 in this work are presented in Table 2.The equation of state of ethane is valid in the range of temperature, density and susceptibility correspond to The depended system parameters of CH 4 are represented in Table 3.The equation of state of methane is valid in the range of temperature, density and susceptibility correspond to

Table 2 . 1 Table 3 .Figure1.
Figure1.Percentage deviation of the experimental pressure data of methane obtained by Wagner and CO.[9] from calculated value.

Figure2.Figure3.
Figure2.Isochoric specific heat Cv of methane.The data points indicate the experimental data obtained by Abdulagatov[11] and the values calculated by the crossover model.