2.2 Calculation of the efficient fronts

To obtain the optimal parameters of the different sub-models that represent the data set iso-p VLE and h<sup>E</sup> , the multi-objective optimization (MOO) procedure is used, which is characterized by the vector

$$\mathbf{OP} = \left[ s(\mathbf{g}^{\mathrm{E}}/RT) \ s(h^{\mathrm{E}}) \right] \tag{6}$$

where s(y E ) is the root of the mean square error (or RMSE) calculated by the model when representing the generic property y E :

$$\log(p^{\mathrm{E}}) = \left[\sum \left(\mathbf{y}\_{\mathrm{exp}}^{\mathrm{E}} - \mathbf{y}\_{\mathrm{cal}}^{\mathrm{E}}\right)^{2}/m\right]^{0.5}; m = \text{number of experimental points} \tag{7}$$

The result of such a problem is a discrete set of values, taken from the efficient front, s(g E /RT) = f[s(h<sup>E</sup> )], where each one of these give rise to different estimates of the thermodynamic behavior. Here the ε-constraint algorithm [3] is used to solve the MOO optimization problem. This procedure converts the multi-objective nonlinear problem (MNLP) into multiple single-objective problem (constrained nonlinear problem (CNLP)), thence minimizing only the first objective in Eq. (6), that is,

$$\text{OF}\_{\text{e}} = \text{s} \, \text{(g}^{\text{E}} / \text{RT)} \, \tag{8}$$

using the other vector element to establish a constraint in the calculation, that is,

$$
\textit{constant} \rightarrow \textit{s} \left( h^{\mathrm{E}} \right) \leq \varepsilon \tag{9}
$$

positive, ruling out the use of these series for other subsequent tasks, such as

T = 397.67 K; Ref. [19], (□) T = 344.19 K, (+) T = 363.19 K, (◊) T = 358.18 K.

Plot of VLE data of the binary: acetone(1) + ethanol(2), iso-p ≈ 101.32 kPa. (a) T,x,y; (c) γ,x; (d) gE

() Ref. [10]; (∇) Ref. [11]; (○) Ref. [12]; (◊) Ref. [13]; (+) Ref. [14]; (△) Ref. [15]; (□) Ref. [16]; (b) p,x,y. iso-T: (▷) Ref. [14]; (○) Ref. [17]; Ref. [18] () T = 372.67 K; (▽) T = 422.56 K, (△)

A Practical Fitting Method Involving a Trade-Off Decision in the Parametrization Procedure…

DOI: http://dx.doi.org/10.5772/intechopen.85743

in Figure 2(a and b), where the different series show an acceptable coherence,

In studies related to the thermodynamics of solutions, the experimental information generated by the binary benzene + hexane, for different properties, has been, and still is, used by researchers in that area as a reference of their investigations; therefore, the choice of this system is justified. Twenty-three useful VLE data series were found in the bibliography [26–42], 16 iso-p and 7 iso-T. Figure 3 shows important errors in some of the series [29, 31, 37] at 101 kPa, with data missing the trend observed in the other series. These are propagated to the T vs x,y representations (Figure 4(a)). The random error for these series is serious as evidenced in

, six references were found [20–25]. All extracted values are shown

/RT,x.

modeling and, specially, simulation.

/dT)p,x > 0.

Regarding h<sup>E</sup>

3.2 Benzene + hexane

having (dh<sup>E</sup>

Figure 1.

Figure 4(b).

51

The value of ε is limited, from 0 to 500 J mol�<sup>1</sup> , since greater errors in the h<sup>E</sup> are not acceptable. The efficient front is then achieved by solving the CNLP (Eqs. (8) and (9)) for different values of ε in the indicated range. To do so, we used a hybrid evolutionary algorithm [3], coupled with Nelder-Mead algorithm [9], for local refinement.
