**2. The PR and SRK cubic EoS models**

#### **2.1 Development of the cubic EoS models**

The ideal gas law *pvm* ¼ *RT*, where the gas constant *R* ¼ *kBNA* is defined as the product the Boltzmann constant and the Avogadro number, only considers the

*Perturbation Theory and Phase Behavior Calculations Using Equation of State Models DOI: http://dx.doi.org/10.5772/intechopen.93736*

elastic collision of molecules thus considering the thermodynamic behavior of the fluid as a purely kinetic process. As a result, it exhibits accurate predictions of the molar volume only when gases at pressures and temperatures close to the atmospheric ones are considered. On the other hand, the real gas law *pvm* ¼ *ZRT* can be used to describe accurately the properties of any fluid and at any conditions provided that the appropriate value of the compressibility factor *Z* (also known as deviation factor in the sense that it considers the deviation of the real gas law from the ideal gas one) can be computed. Clearly, the real gas law simplifies to the ideal one by simply setting *Z* ¼ 1.

Van der Waals was first to recognize the need to separately consider attractive and repulsive forces between the fluid molecules thus leading to the first cubic equation

$$p = RT/(v\_m - b) - a/v\_m^2. \tag{1}$$

Indeed, the *a* term in Eq. (1) can be thought of as a term accounting for the attractive forces between molecules as it reduces pressure. Parameter *b* accounts for the molecules volume which becomes significant at high pressures (i.e. liquid state) as lim *<sup>p</sup>*!<sup>∞</sup>*vm* ¼ *b*. Both parameters are functions of the properties of the component or mixture under consideration. Clearly, by setting both parameters to zero we revert back to the ideal gas law.

Ever since, various new cubic EoS models have been proposed with the Soave-Redlich-Kwong (SRK) and the Peng-Robinson (PR) ones [6] being by far the most commonly used ones in the chemical engineering industry. Both are pressure explicit and are defined by the following expression

$$p = \frac{RT}{v\_m - b} - \frac{a}{(v\_m + \delta\_1 b)(v\_m + \delta\_2 b)},\tag{2}$$

where the parameters values are given in **Table 1**. The temperature dependent term in that Table is given by

$$a(T) = \left(\mathbf{1} + m\left(\mathbf{1} - \sqrt{T/T\_\varepsilon}\right)\right)^2,\tag{3}$$

where *m* is a function of the component acentric factor *ω* defined by

$$m = \begin{cases} 0.48 + 1.574\omega - 0.176\omega^2 & \text{SRK} \\ 0.37464 + 1.54226\omega - 0.26992\omega^2 & \text{PR, } \omega \le 0.49 \\ 0.3796 + 1.485\omega - 0.1644\omega^2 + 0.01667\omega^3 & \text{PR, } \omega > 0.49 \end{cases} \tag{4}$$

The required properties of pure components can be found in any standard petroleum thermodynamics textbook [7]. When pseudo-components are used to describe the fluid composition, such as such as pseudo-C8 and pseudo-C11 in petroleum mixtures, average values can also be obtained from the literature. Custom


**Table 1.** *Cubic EoS models constants.*

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pseudo-components such as petroleum mixtures heavy end need to be treated by means of suitable correlations which utilize molar mass and density to provide estimates of the critical properties and the accentric factor or other required properties [8]. When it comes to mixtures, parameters mixing rules need to be utilized to estimate *a* and *b*. For a mixture of known composition *zi*, they are given by

$$\begin{aligned} a\_{\text{mix}} &= \sum\_{i=1}^{n} \sum\_{j=1}^{n} z\_i z\_j \sqrt{a\_i a\_j} (1 - k\_{ij}) \\ b\_{\text{mix}} &= \sum\_{i=1}^{n} z\_i b\_i. \end{aligned} \tag{5}$$

The Binary Interaction Parameters (BIP) *kij* account for the interaction between different constituents and are usually initialized either to zero or by the Prausnitz [9] rule

$$k\_{\vec{\eta}} = \mathbf{1} - \left(\frac{2\boldsymbol{v}\_{c\_i}^{1/6}\boldsymbol{v}\_{c\_j}^{1/6}}{\boldsymbol{v}\_{c\_i}^{1/3} + \boldsymbol{v}\_{c\_j}^{1/3}}\right)^{\theta},\tag{6}$$

where the critical molar volume is obtained by solving the EoS at critical conditions

$$
\sigma\_c = Z\_c \mathbf{R} T\_c / p\_c,\tag{7}
$$

and the critical value *Zc* of the compressibility factor for the PR EoS equals to 0.3074. Parameter *θ* is user dependent and is usually set to 1.2. Note Eq. (6) is only used to determine BIPs between hydrocarbon components. BIPs between nonhydrocarbons or between hydrocarbon and nonhydrocarbon components are taken from Tables [6].

Once all parameters have been estimated for a mixture of known composition at fixed pressure and temperature, the EoS can be solved for volume. Usually a dimensionless form that can be solved for *Z* ¼ *pvm=RT* rather than for *vm* is preferred

$$\begin{array}{l} Z^3 + ( (\delta\_1 + \delta\_2 - 1)B - 1)Z^2 + (A + \delta\_1 \delta\_2 B - (\delta\_1 + \delta\_2)B(B+1))Z \\ - \left( AB + \delta\_1 \delta\_2 B^2(B+1) \right) \\ = 0, \end{array} \tag{8}$$

where the dimensionless EoS constants are given by

$$A = a\_{\rm mix} p / (RT)^2, \quad B = b\_{\rm mix} p / (RT). \tag{9}$$

## **2.2 Use of the cubic EoS models**

As soon as the EoS constants have been defined, the compressibility factor *Z* can be obtained by solving the cubic polynomial Eq. (8) [10]. When more than one real positive roots are obtained, the smallest one is selected when the fluid is a liquid whereas the largest one is used for a gas. Molar volume and density can be easily computed by

$$v\_m = \text{ZRT}/p,\ \rho = \text{pM}/\text{ZRT},\tag{10}$$

where *M* denotes the fluid molar mass. Components fugacity coefficients *φi*, hence fugacity *fi* ¼ *φizip*, can be computed by the following expressions

*Perturbation Theory and Phase Behavior Calculations Using Equation of State Models DOI: http://dx.doi.org/10.5772/intechopen.93736*

$$\begin{aligned} \ln \phi\_i &= B\_i / B(Z-1) - \ln \left( Z - B \right) + A / ((\delta\_1 - \delta\_2)B) \left( \mathbf{1} / A \{ \phi \mathbf{A} / \delta \mathbf{z} \}\_i - B\_i / B \right) \times \\ &\times \ln \left( Z + \delta\_1 B \right) / (Z + \delta\_2 B), \end{aligned} \tag{11}$$

where *Ai* <sup>¼</sup> *aip=*ð Þ *RT* <sup>2</sup> , *Bi* <sup>¼</sup> *bip=*ð Þ *RT* and f g *<sup>∂</sup>A=∂<sup>z</sup> <sup>i</sup>* <sup>¼</sup> <sup>P</sup>*<sup>n</sup> <sup>j</sup>*¼1*<sup>z</sup> <sup>j</sup>* <sup>1</sup> � *kij* � � ffiffiffiffiffiffiffiffiffiffi *AiA <sup>j</sup>* p . Derivative properties such as the Joule-Thomson coefficient *μJT* can be computed by differentiating the EoS and incorporating the derivatives in the rigorous thermodynamic definitions of the properties. For example,

$$
\mu\_{fT} = \frac{v\_m}{c\_p} \left( \frac{T}{v\_m} \frac{\partial v\_m}{\partial T} \Big|\_p - \mathbf{1} \right). \tag{12}
$$

### **2.3 Volume translation**

Cubic EoS models are notoriously known for their deficiency in estimating liquid density. A simple modification, known as volume shifting or volume translation, originally proposed by Peneloux [11], can greatly improve the capabilities of cubic EoS. The idea lies in "shifting" the predicted phase molar volumes *vEoS <sup>m</sup>* by some amount that depends on the fluid composition and its components properties. More specifically, the shifted volume is given by

$$
v\_{m} = v\_{m}^{EoS} - \sum\_{i=1}^{n} z\_{i}c\_{i}.\tag{13}$$

Parameters *ci* are component specific and they are usually given as functions of the covolume parameters *bi*, that is

$$s\_i = c\_i b\_i,\tag{14}$$

where values of *si* for common pure components are available in Tables [6].

It should be noted that "shifting" (or "translating") the volume also affects the Z factor which needs to be updated to ensure calculations consistency

$$Z = Z^{\mathrm{EoS}} - p/RT \sum\_{i=1}^{n} z\_i c\_i. \tag{15}$$

It can be shown that when applying volume translation to two phases that equilibrate, the fugacities of the components do change but they do in the same amount so that they remain equal, thus not disturbing the equilibrium. As a result, volume translation does not affect phase compositions in flash calculations or saturation conditions but only phase density.
