3.4 Conflict between phase equilibria and critical line description in binary mixtures

Reliable models for thermodynamic and phase behavior description of binary mixtures are facing problems of adequate estimation of the binary interaction parameters from the restricted set of VLE data. In spite of availability of strict

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

relationships to obtain from EoS model, the different derivatives of thermodynamic values, for instance, the prediction of critical lines from the EoS with parameters restored from VLE data, is questionable due to diverse sources of uncertainty in both the used models and experimental data.

For illustration, we consider thermodynamic and phase behavior of water-carbon dioxide system near the critical point of water. Experimental data on p-ρ-T-x properties have been taken from [18]; data on phase equilibria and critical curve have been extracted from [19, 20]. The critical lines have been calculated using algorithm developed in [21]. Phase equilibria calculations have been carried out with Michelsen and Mollerup method [22] for cubic EoS. We performed phase equilibria and critical line calculations for the RK EoS with binary interaction parameters k12 and l12 used in standard mixing rules

$$\begin{aligned} b &= b\_{11}\mathbf{x}^2 + l\_{12}\mathbf{x}(\mathbf{1}-\mathbf{x})\left(\frac{b\_{11}+b\_{22}}{2}\right) + b\_{22}(\mathbf{1}-\mathbf{x})^2\\ a &= a\_{11}\mathbf{x}^2 + k\_{12}\mathbf{x}(\mathbf{1}-\mathbf{x})\sqrt{a\_{11}a\_{22}} + a\_{22}(\mathbf{1}-\mathbf{x})^2 \end{aligned} \tag{14}$$

and restored from different crisp convolution schemes. For simplicity, the results of calculation for phase equilibria and critical line are presented in Figures 5 and 6 only for most selected compromise schemes. The challenge of conflict between parameters retrieved from different sections of thermodynamic surface remains important for practically all EoS including sophisticated multiparameter equation of

## Figure 5.

Critical line of Н2О▬СО<sup>2</sup> system [21]. ● Experimental data ▬ Best fit of VLE data °°°° von Neumann's coalition scheme [16] ─ ─ Best fit of critical curve.

where wi are the weight coefficients, K<sup>0</sup>

Distillation - Modelling, Simulation and Optimization

KCð Þ¼ X min ∑

KCð Þ¼ <sup>X</sup> minY<sup>n</sup>

is extenuation of uncertainty driving from vagueness.

find the values Х and λ that maximize λ that is subject to

i¼1

�

n i¼1

scalar nonlinear programming problem:

Neumann coalition scheme [16].

3.3 Fuzzy convolution scheme

ing of phase equilibria.

mixtures

78

is acceptable for decision-maker remaining in the coalition, and KC is a global tradeoff criterion. If it will be possible to come to an agreement about preference (weight) for each criterion, then the final decision can be found as a solution of

wi Kið Þ� <sup>X</sup> <sup>K</sup><sup>0</sup>

If no concordance among decision-makers is concerning the weight choice, then arbitration network is preferable. Classical arbitration scheme was derived mathematically rigorously by Nash but very often criticized from common sense [16]:

Kið Þ� <sup>X</sup> <sup>K</sup><sup>0</sup>

All crisp convolution schemes under discussion try to reduce an uncertainty deriving from conflict among different criteria in the Pareto domain. The next step

Zadeh [14] put the theory of fuzzy sets forward with explicit reference to the vagueness of natural language, when describing quantitative or qualitative goals of the system. Here we assume that local criteria as well as different constraints in the ill-structured situation can be represented by fuzzy sets. A final decision is defined by the Bellman and Zadeh model [17] as the intersection of all fuzzy criteria and constraints and is represented by its membership function μ(Х) as follows:

This problem is reduced to the standard nonlinear programming problems: to

λ≤ μKið Þ X , i ¼ 1, 2, …, n;

λ≤μCjð Þ X , j ¼ 1, 2, …, m

Here, by way of illustration, the new approach was introduced to estimate the Redlich-Kwong EoS parameters for simultaneous description of the phase equilibria and critical line data in binary mixtures, thermodynamically consistent description of the inhomogeneous data, and other inconsistency problems arising in the model-

3.4 Conflict between phase equilibria and critical line description in binary

Reliable models for thermodynamic and phase behavior description of binary mixtures are facing problems of adequate estimation of the binary interaction parameters from the restricted set of VLE data. In spite of availability of strict

� �

i

μð Þ¼ Х μКð Þ Х μСð Þ Х , X ∈ XP: (12)

In terms of game-theoretical approach, such coalition is defined as the von

� � � � �

i

<sup>i</sup> is an infimum of the desired result that

�, X ∈ XP: (10)

�, X ∈ XP: (11)

(13)

for imidazolium (IL)-based ionic liquids and industrial refrigerant mixtures. IL doping leads to the breaking of azeotrope in binary refrigerant mixtures that open the way for the azeotrope refrigerant mixture separation technologies in order to

Azeotrope-Breaking Potential of Binary Mixtures in Phase Equilibria Modeling

Applying the cubic model of the equation of state, only the critical properties and acentric factor of the individual components in mixtures are sufficient to define

The degenerated critical azeotrope point is a boundary state separating azeotrope and non-azeotrope and produces the limit of the critical azeotrope at xi ! 0 or at xi ! 1. The solution of the thermodynamic equation system for a degenerated

> 1 � Z<sup>4</sup> 1 � Z<sup>3</sup>

<sup>c</sup>2=Pc<sup>2</sup> � 2 1ð Þ � k<sup>12</sup>

T2

<sup>c</sup>2=Pc<sup>2</sup> <sup>þ</sup> <sup>T</sup><sup>2</sup>

where the upper signs + or � correspond to x2 = 0 and the lower signs - to x2 = 1,

To define the azeotropic states, the parameters Zi can be expressed via critical (pseudocritical) temperatures Tci and pressures Pci of pure components and empirical binary interaction parameters k12 and l12 (17). The unlike pair interaction parameters Z2 and Z4 (i.e., a12 and b12) can be estimated solving simultaneously the system of Eqs. (17) for the given Z1 and Z3 (or the set of pure component constants a11, b11, a22,

> <sup>Z</sup><sup>3</sup> <sup>¼</sup> Tc2=Pc<sup>2</sup> � Tc1=Pc<sup>1</sup> Tc2=Pc<sup>2</sup> þ Tc1=Pc<sup>1</sup>

<sup>Z</sup><sup>4</sup> <sup>¼</sup> Tc1=Pc<sup>1</sup> � ð Þ <sup>1</sup> � <sup>l</sup><sup>12</sup> <sup>ð</sup>Tc1=Pc<sup>1</sup> <sup>þ</sup> Tc2=Pc2Þ þ Tc2=Pc<sup>2</sup> Tc2=Pc<sup>2</sup> þ Tc1=Pc<sup>1</sup>

The boundary between azeotrope and zeotrope states in Z1–Z2 plane at fixed values of Z3 and Z4 is a straight line. The definition of azeotropic and zeotropic

Double azeotropic phenomena can be observed in the global phase diagram only in the vicinity of the crossing point of two straight azeotropic borders in (Z1, Z2)

<sup>2</sup> <sup>þ</sup> <sup>Ψ</sup> <sup>þ</sup> <sup>þ</sup> <sup>Ψ</sup> � , where <sup>Ψ</sup> � <sup>¼</sup> <sup>1</sup> � <sup>Z</sup><sup>4</sup>

<sup>1</sup> <sup>¼</sup> <sup>Ψ</sup> <sup>þ</sup> <sup>þ</sup> <sup>Ψ</sup> �

� 1

� �Λ, (16)

Tc1Tc<sup>2</sup> ffiffiffiffiffiffiffiffi pc1pc<sup>2</sup> <sup>p</sup> <sup>þ</sup> <sup>T</sup><sup>2</sup>

<sup>c</sup>1=Pc<sup>1</sup>

1 � Z<sup>3</sup>

� 1 � �Λ: (18)

<sup>c</sup>1=Pc<sup>1</sup>

, (17)

:

,

ð15Þ

remove the environmentally harmful substances.

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

the phase behavior in the interested sector of the parameters.

gives the following relationship for dimensionless parameters Zi:

b22) that are determined from the critical parameters of the components:

, Z<sup>2</sup> <sup>¼</sup> <sup>T</sup><sup>2</sup>

Z<sup>2</sup> ¼ ∓Z<sup>1</sup> � ð Þ 1 � Z<sup>1</sup>

4.1 Azeotrope-breaking criteria

critical azeotrope [5–7, 9]

<sup>Z</sup><sup>1</sup> <sup>¼</sup> <sup>T</sup><sup>2</sup>

plane (Figures 8 and 9):

<sup>Z</sup><sup>2</sup><sup>i</sup> <sup>¼</sup> <sup>Ψ</sup> <sup>þ</sup> � <sup>Ψ</sup> �

81

<sup>2</sup> <sup>þ</sup> <sup>Ψ</sup> <sup>þ</sup> <sup>þ</sup> <sup>Ψ</sup> � , Z<sup>i</sup>

T2

<sup>c</sup>2=Pc<sup>2</sup> � <sup>T</sup><sup>2</sup>

<sup>c</sup>2=Pc<sup>2</sup> <sup>þ</sup> <sup>T</sup><sup>2</sup>

states for refrigerants is depicted in Figure 7.

4.2 Double azeotrope-breaking criterion

<sup>c</sup>1=Pc<sup>1</sup>

<sup>c</sup>1=Pc<sup>1</sup>

respectively.
