2. Phase diagram classes and global phase diagram

Mapping of the global equilibrium surface into the parameter space of the equation of state model provides the most comprehensive set of criteria for prediction phase behavior of the binary mixture.

The impact of critical parameters of components on phase topology is provided via global phase diagrams. The diagrams are depicted in the space of the equation of state parameters, e.g., van der Waals a and b parameters. The specific points and lines of global phase diagrams, including tricritical points, double critical endpoints, azeotropic lines, etc., create the boundaries at the diagram and divide the model parameter space into the regions corresponding to the various types of the phase behavior. Generally, global phase diagram is expressed in dimensionless parameters which depend on the equation of state model. The global phase diagrams of such different models as the one-fluid EoS of binary Lennard-Jones fluid [6, 7] and the Redlich-Kwong model [5, 8, 9] are almost identical, in particular for the case of equal-sized molecules.

We consider here the cubic-type models:

$$p = \frac{\text{RT}}{v - b} - \frac{a(T)}{v(v + b)}\tag{1}$$

where R is the universal gas constant and the EoS parameters a and b of mixture depend on the mole fractions xi and xj of the components i and j and on the corresponding parameters aij and bij for different pairs of interacting molecules:

$$a = \sum\_{i=1}^{2} \sum\_{j=1}^{2} \mathbb{x}\_{i} \mathbb{x}\_{j} a\_{\text{i}\dagger} (1 - k\_{\text{i}\dagger}),\\b = \sum\_{i=1}^{2} \sum\_{j=1}^{2} \mathbb{x}\_{i} \mathbb{x}\_{j} b\_{\text{i}\dagger} \tag{2}$$

The convenient set of dimensionless parameters for the Redlich-Kwong model is as follows [13]:

represent boundaries between types of phase behavior in a dimensionless space of

GPD also provides good visualization of the impact of model parameters of mixture components to the topology of phase behavior. The proposed types of phase behavior classification [1] are generally used to characterize the different

Varchenko [2] has provided a more rigorous classification for conventional features of equilibrium surface and phase diagrams for binary mixtures with strict

The various phase diagram classes and p–T projections of the main types of phase diagrams have been described in literature [1–3]. Global phase diagrams are a technique which can be used for the prediction of different phase behavior in the

Patel and Sunol [12] developed a robust automated routine for global phase diagram generation in binary systems. The approach uses any equation of state models, takes into account solid-phase existence, and provides type VI phase diagram generation. The generated data set includes calculations of the critical endpoints, quadruple points, critical azeotrope points, azeotrope endpoints, pure azeotrope points, critical line, liquid–liquid–vapor line, and azeotrope line. Azeotrope-breaking is important for the successful distilling of industrially important mixtures. To simulate the mixture phase behavior, models based on the equation of state (EoS) presentation for thermodynamic properties are more preferable. The conventional methods of parameter identification use the statistical paradigm, which is based on maximum likelihood or a posteriori probability criteria and does not take into account uncertainties of vague nature. Decision-making process under various uncertainties requires mathematical methods, which include uncertainty evaluation a priori. Statistical methods interpret all variety of uncertainty types in the framework of the randomness concept. Nevertheless, there are ill-structured situations, which have not any strictly defined boundaries and cannot

The main challenge is to deal with such kind of expressions like "neighborhood" or "best fit," which do not have strict boundaries, separating one class of objects from others. Generally, there are ambiguous verbal models, which can be treated as fuzzy formulated targets, depending on biased assessment of boundaries for approximations used. As a case study, we provide estimation of the optimal parameters of the Redlich-Kwong equation of state [13], retrieved from the different conflicting data sets resulting from the inconsistency problems arising in the modeling of phase equilibria. Such process reflects different types of uncertainties, including uncertainties of nonstatistical origin. The parameters of phase equilibria models are considered as alternatives, i.e., they allow meeting the targets and prescribed constraints. The parameter estimation problem of phase equilibrium modeling multicriteria approach is applied. To describe the uncertainties of such

type, the "fuzzy set" approach introduced by L.A. Zadeh [14] is used.

schemes to reduce a vector criterion into the scalar are provided.

The problem of optimum parameter estimation of thermodynamic and phase behavior under the uncertainty is a search of the Pareto set. The diverse computational methods of the Pareto-optimum parameter convolution crisp and fuzzy

Ionic liquids can be treated as "adjustable" working fluids given the fact that variations of different "R – " groups and cation/anion ratio selection ensure meeting preferred trade-off solution between density, viscosity, melting point,

Current topological analysis of equilibrium surfaces of binary fluid systems provides 26 singularities and 56 scenarios of phase behavior evolution depicted in p–T

determination of the eight topologically different rearrangements.

mixtures without vapor–liquid equilibrium calculations [4–11].

EOS parameters.

diagrams [3].

be accurately formulated.

72

types of phase behavior in binary mixtures.

Distillation - Modelling, Simulation and Optimization

Distillation - Modelling, Simulation and Optimization

$$\begin{aligned} Z\_1 &= \frac{d\_{22} - d\_{11}}{d\_{22} + d\_{11}}, \\ Z\_2 &= \frac{d\_{22} - 2d\_{12} + d\_{11}}{d\_{22} + d\_{11}}, \\\\ Z\_3 &= \frac{b\_{22} - b\_{11}}{b\_{22} + b\_{11}}, \\\\ Z\_4 &= \frac{b\_{22} - 2b\_{12} + b\_{11}}{b\_{22} + b\_{11}} \end{aligned} \tag{3}$$

where.

$$d\_{\vec{\mathbb{H}}} = \frac{T\_{\vec{\mathbb{H}}} b\_{\vec{\mathbb{H}}}}{b\_{\vec{\mathbb{H}}} b\_{\vec{\mathbb{H}}}}, \quad T\_{\vec{\mathbb{H}}} = \left(\frac{\Omega\_b a\_{\vec{\mathbb{H}}}}{R \Omega\_a b\_{\vec{\mathbb{H}}}}\right)^{23}, \ \ \ \Omega\_a = \left[\mathcal{G}\left(2^{13} - 1\right)\right]^{-1}, \ \ \ \ \Omega\_b = \frac{2^{13} - 1}{3}.$$

In the case of the Redlich-Kwong (RK) model [13], the value Λ is given by

$$A = \frac{\Theta\_b}{\Theta\_{\text{ab}} (\Theta\_b - 1)^2} + \frac{1}{\Theta\_b + 1} = 0.67312,\tag{4}$$

where <sup>Θ</sup>ab <sup>¼</sup> <sup>Ω</sup><sup>a</sup> Ωb , <sup>Θ</sup><sup>b</sup> <sup>¼</sup> zc Ωb . According to Eq. (4), the boundary between azeotropic and non-azeotropic states in Z1–Z2 plane at fixed values of Z3 and Z4 is a straight line. The values are equal to Ω<sup>a</sup> = 0.42747, Ω<sup>b</sup> = 0.08664, and Zc = 0.333.

The combining rules for the binary interaction parameters are

$$a\_{\vec{\mathbf{i}}\rangle} = \left(\mathbf{1} - k\_{\vec{\mathbf{i}}\rangle}\right) \sqrt{a\_{\vec{\mathbf{i}}\vec{a}} a\_{\vec{\mathbf{j}}\rangle} \qquad b\_{\vec{\mathbf{i}}\vec{\mathbf{j}}} = \left(\mathbf{1} - l\_{\vec{\mathbf{i}}\vec{\mathbf{j}}}\right) \frac{b\_{\vec{\mathbf{i}}\vec{\mathbf{i}}} + b\_{\vec{\mathbf{j}}\vec{\mathbf{j}}}}{2}.\tag{5}$$

where kij and lij are fitting coefficients in the Lorentz-Berthelot combining rule (kij = lij = 0).

The simplest boundary is a normal critical point when two fluid phases are becoming identical. Critical conditions are expressed in terms of the molar Gibbs energy derivatives in the following way:

$$\left(\frac{\partial^2 G}{\partial \mathbf{x}^2}\right)\_{p,T} = \left(\frac{\partial^3 G}{\partial \mathbf{x}^3}\right)\_{p,T} = \mathbf{0}.\tag{6}$$

Here C1 and C2 are critical points of components 1 and 2; Cm is hypothetic critical point beyond solidification line. The types of phase behavior indicated as

Accounting the effects of uncertainty in the fluid-phase equilibria, modeling became important in the recent decade. Generally, the uncertainty tried to be resolved via probability and random process theories. However, the probabilistic methods can lead to the unreliable estimation of parameters. There are three main

3. Uncertainties and conflicts in the parameter estimation

complimentary models of uncertainty described in literature:

Roman numbers are described in [1–3].

Global phase diagram of the RK model [13], Z3 = Z4 = 0.25.

Global phase diagram of the RK model [13], Z3 = Z4 = 0.

DOI: http://dx.doi.org/10.5772/intechopen.83769

Azeotrope-Breaking Potential of Binary Mixtures in Phase Equilibria Modeling

Figure 1.

Figure 2.

75

Corresponding critical conditions for the composition-temperature-volume variables are

$$\begin{aligned} A\_{\mathbf{xx}} - \mathbf{W} \mathbf{A}\_{\mathbf{x} \mathbf{V}} &= \mathbf{0}; \\ A\_{\mathbf{xxx}} - \mathbf{3} \mathbf{W} \mathbf{A}\_{\mathbf{x} \mathbf{V}} + \mathbf{3} \mathbf{W}^2 A\_{\mathbf{x} \mathbf{V}} - \mathbf{3} \mathbf{W}^3 A\_{\mathbf{V} \mathbf{V} \mathbf{V}} &= \mathbf{0}; \end{aligned} \tag{7}$$

where A is the molar Helmholtz energy and.

<sup>W</sup> <sup>¼</sup> <sup>A</sup>xx <sup>A</sup>VV and <sup>A</sup>mVnx <sup>¼</sup> <sup>∂</sup>nþmA ∂xn∂V<sup>m</sup> � � <sup>T</sup> are the contracted notations for differentiation operation which can be solved for VC and TC at a given concentration x.

Global phase diagrams generated for Redlich-Kwong model is presented in Figures 1 and 2.

Azeotrope-Breaking Potential of Binary Mixtures in Phase Equilibria Modeling DOI: http://dx.doi.org/10.5772/intechopen.83769

Figure 1.

<sup>Z</sup><sup>1</sup> <sup>¼</sup> <sup>d</sup><sup>22</sup> � <sup>d</sup><sup>11</sup> d<sup>22</sup> þ d<sup>11</sup>

<sup>Z</sup><sup>3</sup> <sup>¼</sup> <sup>b</sup><sup>22</sup> � <sup>b</sup><sup>11</sup> b<sup>22</sup> þ b<sup>11</sup>

In the case of the Redlich-Kwong (RK) model [13], the value Λ is given by

and non-azeotropic states in Z1–Z2 plane at fixed values of Z3 and Z4 is a straight

<sup>p</sup> , bij <sup>¼</sup> <sup>1</sup> � <sup>l</sup>ij

where kij and lij are fitting coefficients in the Lorentz-Berthelot combining rule

The simplest boundary is a normal critical point when two fluid phases are becoming identical. Critical conditions are expressed in terms of the molar Gibbs

> <sup>¼</sup> <sup>∂</sup><sup>3</sup> G ∂x<sup>3</sup> � �

Corresponding critical conditions for the composition-temperature-volume

p,T

<sup>A</sup>xVV � <sup>3</sup>W<sup>3</sup>

Axx � WAxV ¼ 0;

AVVV ¼ 0;

<sup>T</sup> are the contracted notations for differentiation

<sup>Θ</sup>abð Þ <sup>Θ</sup><sup>b</sup> � <sup>1</sup> <sup>2</sup> <sup>þ</sup>

line. The values are equal to Ω<sup>a</sup> = 0.42747, Ω<sup>b</sup> = 0.08664, and Zc = 0.333. The combining rules for the binary interaction parameters are

aiiajj

p,T

operation which can be solved for VC and TC at a given concentration x.

Global phase diagrams generated for Redlich-Kwong model is presented in

where.

<sup>d</sup>ij <sup>¼</sup> <sup>T</sup>ijbij biibjj

where <sup>Θ</sup>ab <sup>¼</sup> <sup>Ω</sup><sup>a</sup>

(kij = lij = 0).

variables are

<sup>W</sup> <sup>¼</sup> <sup>A</sup>xx

Figures 1 and 2.

74

Ωb

, <sup>Θ</sup><sup>b</sup> <sup>¼</sup> zc Ωb

aij ¼ 1 � kij

energy derivatives in the following way:

� � ffiffiffiffiffiffiffiffiffi

∂2 G ∂x<sup>2</sup> � �

<sup>A</sup>xxx � 3WAxxV <sup>þ</sup> <sup>3</sup>W<sup>2</sup>

∂xn∂V<sup>m</sup> � �

where A is the molar Helmholtz energy and.

<sup>A</sup>VV and <sup>A</sup>mVnx <sup>¼</sup> <sup>∂</sup>nþmA

, Tij <sup>¼</sup> <sup>Ω</sup>baij

Distillation - Modelling, Simulation and Optimization

RΩabij � �<sup>23</sup>

<sup>Λ</sup> <sup>¼</sup> <sup>Θ</sup><sup>b</sup>

<sup>Z</sup><sup>2</sup> <sup>¼</sup> <sup>d</sup><sup>22</sup> � <sup>2</sup>d<sup>12</sup> <sup>þ</sup> <sup>d</sup><sup>11</sup> d<sup>22</sup> þ d<sup>11</sup>

<sup>Z</sup><sup>4</sup> <sup>¼</sup> <sup>b</sup><sup>22</sup> � <sup>2</sup>b<sup>12</sup> <sup>þ</sup> <sup>b</sup><sup>11</sup> b<sup>22</sup> þ b<sup>11</sup>

,

,

, <sup>Ω</sup><sup>a</sup> <sup>¼</sup> 9 213 � <sup>1</sup> � � � � �<sup>1</sup>

1

,

, <sup>Ω</sup><sup>b</sup> <sup>¼</sup> <sup>2</sup><sup>13</sup> � <sup>1</sup>

<sup>Θ</sup><sup>b</sup> <sup>þ</sup> <sup>1</sup> <sup>¼</sup> <sup>0</sup>:67312, (4)

. According to Eq. (4), the boundary between azeotropic

� � <sup>b</sup>ii <sup>þ</sup> <sup>b</sup>jj

<sup>3</sup> :

<sup>2</sup> : (5)

¼ 0: (6)

(7)

(3)

Global phase diagram of the RK model [13], Z3 = Z4 = 0.

Figure 2. Global phase diagram of the RK model [13], Z3 = Z4 = 0.25.

Here C1 and C2 are critical points of components 1 and 2; Cm is hypothetic critical point beyond solidification line. The types of phase behavior indicated as Roman numbers are described in [1–3].
