1. Ideal-gas law

The ideal-gas law is the simplest EOS. It is based on two basic assumptions: (1) gas molecules do not occupy volume and are elastic molecules and (2) no interaction force between molecules. The ideal gas law can be written as Eq. (3), and the pvT relationship is obtained by molecular motion theory and statistical methods.

$$p = \frac{RT}{v} \tag{3}$$

where p, T, and *v* are pressure, temperature, and molar volume, respectively, and R is the gas constant.

In the low-density limit, all kinds of EOS reduce to the ideal gas law. Ideal gas law cannot be applied with liquid.

#### 2.Virial equation of state

The virial equation of state (EOS) is a polynomial series in pressure or in inverse volume whose coefficients are functions only of T for a pure fluid.

$$Z = \mathbf{1} + B\left(\frac{P}{RT}\right) + \left(\mathbf{C} - B^2\right)\left(\frac{P}{RT}\right)^2 + \dotsb \tag{4}$$

where the coefficients B and C are the second and third virial coefficients, respectively.

The expressions for the second virial coefficient can be obtained from the Tsonopoulos model [27], which is only applied to the saturated vapor because it cannot be used for liquids or at high pressures.

The Benedict-Webb-Rubin (BWR) EOS is an empirical extension of the virial EOS. The BWR expressions are based on the pioneering work of Benedict, Webb, and Rubin (1940 and 1942) who combined polynomials in temperature with power series and exponentials of density into an eight-parameter form [28]. Additional terms and parameters were later introduced by others to formulate modified Benedict-Webb-Rubin (MBWR) EOS.

The general form of BWR/MBWR correlations is

$$Z = \mathbf{1} + f\_1(T)/\boldsymbol{\nu} + f\_2(T)/\boldsymbol{\nu}^2 + f\_3(T)/\boldsymbol{\nu}^\pi + f\_4(T) \left[ \left( a + \boldsymbol{\gamma}/\boldsymbol{\nu}^2 \right) / \mathbf{V}^\pi \right] \exp(-\boldsymbol{\gamma}/\mathbf{v}) \tag{5}$$

#### 3.Cubic equation of state

The first practical cubic equation of state was proposed by van der Waals in 1873 [29]; the development of the modern cubic equation of state started from the Redlich Kwong (RK) equation published in 1949 [30], which is improvements to the van der

*Utilizing Computational Methods to Identify Low GWP Working Fluids for ORC Systems DOI: http://dx.doi.org/10.5772/intechopen.1003740*

Waals equation. Based on the previous work, Peng and Robinson [31] proposed the Peng Robinson (PR) state equation in 1976, and its form is as follows:

$$p = \frac{RT}{v - b} - \frac{a(T)}{v(v + b) + b(v - b)}\tag{6}$$

The parameters a and b are related to the critical pressure, critical temperature, and acentric factor as shown in Eqs. (6) and (7).

$$a(T) = 0.457235 \frac{\text{R}^2 T\_c^2 a(T)}{p\_c} \tag{7}$$

$$b = 0.077796 \frac{\text{RT}\_c}{p\_c} \tag{8}$$

$$a(T) = \left[1 + \left(0.37464 + 1.54226\alpha - 0.26992\alpha^2\right)\left(1 - T\_{\rm r}^{0.5}\right)\right]^2\tag{9}$$

where ω is the acentric factor, *T*<sup>r</sup> = *T*/*T*c,*T* and *T*<sup>c</sup> represent reduced temperature and critical temperature.

In 1982, Patel and Teja [14] corrected the compressibility factor of the state equation to a parameter related to the substance and obtained the Patel Teja equation of state (PT EOS). The fluid properties of a certain component near the vapor-liquid critical point are difficult to obtain from cubic EOS. Modification of the Patel Teja EOS (mPT EOS) as proposed by Coquelet et al. [32] permits to better estimating the thermodynamic properties close to the critical point.

The RK, PR, PT, and mPT equations are representative of cubic EOS. This type of EOS is characterized by fewer parameters, simple form, and fast solution speed, and is widely used in industry. They differ in their accuracy to estimate the densities, particularly the liquid and supercritical fluid densities.

#### 4.Fundamental equation of state based on Helmholtz energy

The Helmholtz energy-based equations are the most accurate EOS at a time when they were just coming into widespread use. The Helmholtz energy model was developed by NIST and included in the REFPROP software. Helmholtz-type equations of state (also called fundamental equations of state) are explained in terms of reduced molar Helmholtz free energy (see Eq. 10) wherein superscript ig concerns ideal gas contribution and superscript *x* concerns residual contribution. This equation contains several adjustable parameters α<sup>i</sup> and αk, *t*k, *d*k, and *l*k. Experimental data are required to adjust these parameters. Therefore, numerous and accurate experimental data (phase equilibria and volumetric properties essentially) are required. In comparison to the previous cited equations, the fundamental equation of state guarantees a very accurate prediction of the thermodynamic properties at the condition of the availability and the quality of the experimental data.

$$a(\delta, \tau) = \frac{A}{RT} = a^{\mathrm{id}} + a^r + \sum N\_k \delta^{\mathrm{d}\_k} \tau^{\mathrm{h}} + \sum N\_k \delta^{\mathrm{d}\_k} \tau^{\mathrm{h}} \exp\left(-\delta^{\mathrm{l}\_k}\right) \tag{10}$$

Temperature and density are expressed in dimensionless variables *<sup>δ</sup>* <sup>¼</sup> *<sup>ρ</sup> ρC* and *<sup>τ</sup>* <sup>¼</sup> *<sup>T</sup> TC* . ρ<sup>C</sup> and *T*<sup>C</sup> are the critical density and temperature, respectively. This equation of state is parametrized for numerous working fluids that can be potentially used for ORC systems. These fluids can be pure components or mixtures. NIST REFPROP software has established a very accurate thermodynamics database for all the most commonly used working fluids based on the Helmholtz EOS. In addition, this model can be applied for the mixture. Mixing rules based on multifluid approximation are considered for the computation of the new critical properties (ρ<sup>C</sup> and *T*C) corresponding to the mixture. Binary interaction parameters have to be determined on the experimental base (phase equilibria and volumetric properties).
