3.1 Acetone + ethanol

There are 10 useful references in literature containing VLE data for the acetone(1) + ethanol(2) system [10–19] showing a total of 15 experimental series. In Figure 1, seven sets are iso-p (Figure 1(a)), and eight sets are iso-T (Figure 1(b)).

For use we refer to data series with information for the two phases. This is a requirement to check the thermodynamic consistency. Figure 1(b) shows an azeotrope at high pressures, which moves from the acetone-rich region toward pure ethanol as pressure increases; the separation between the compositions in the two phases is reduced. Figure 1(c) and (d) contains, respectively, the variation of γ<sup>i</sup> and g E /RT with the liquid composition for the system at 101 kPa. Ethanol's activity coefficient shows some errors that surely affect the global consistency of the data series. High slopes are observed as x<sup>1</sup> ! 0 along with negative values of the natural logarithm of γ<sup>1</sup> for data series other than those referenced [14–16]. This is a clear sign of inconsistency.

Table 1 shows the results obtained in the application of different consistency methods. The observation of the said table produces some comments which are interesting in the work development. The Wisniak test and the direct van Ness test accept most of the experimental series. The first of the methods rejects two (noted as n° 10,12) while the second fails to validate only one, n° 8. The Kojima test rejects five data series, with the error observed in the system n° 12 being critical. A new methodology recently proposed by us [6, 7] was also applied, which rejects nine systems: three by the integral form (n° 9, 11, 12) and six by the differential form (n° 1, 3, 4, 8, 9, 12). These systems present obvious signs of inconsistency that are also observed by at least one of the other methods. Therefore, the overall assessment of these series (in other words, the quality of their experimental data) is not

A Practical Fitting Method Involving a Trade-Off Decision in the Parametrization Procedure… DOI: http://dx.doi.org/10.5772/intechopen.85743

Figure 1.

s y<sup>E</sup> <sup>¼</sup> <sup>∑</sup> <sup>y</sup><sup>E</sup>

/RT) = f[s(h<sup>E</sup>

E

front, s(g

that is,

refinement.

g E

50

3.1 Acetone + ethanol

sign of inconsistency.

exp � <sup>y</sup><sup>E</sup> cal <sup>2</sup>

0:<sup>5</sup>

Distillation - Modelling, Simulation and Optimization

The value of ε is limited, from 0 to 500 J mol�<sup>1</sup>

3. Verification and selection of data series

=m

The result of such a problem is a discrete set of values, taken from the efficient

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

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

There are 10 useful references in literature containing VLE data for the acetone(1) + ethanol(2) system [10–19] showing a total of 15 experimental series. In Figure 1, seven sets are iso-p (Figure 1(a)), and eight sets are iso-T (Figure 1(b)). For use we refer to data series with information for the two phases. This is a requirement to check the thermodynamic consistency. Figure 1(b) shows an azeotrope at high pressures, which moves from the acetone-rich region toward pure ethanol as pressure increases; the separation between the compositions in the two phases is reduced. Figure 1(c) and (d) contains, respectively, the variation of γ<sup>i</sup> and

/RT with the liquid composition for the system at 101 kPa. Ethanol's activity coefficient shows some errors that surely affect the global consistency of the data series. High slopes are observed as x<sup>1</sup> ! 0 along with negative values of the natural logarithm of γ<sup>1</sup> for data series other than those referenced [14–16]. This is a clear

Table 1 shows the results obtained in the application of different consistency methods. The observation of the said table produces some comments which are interesting in the work development. The Wisniak test and the direct van Ness test accept most of the experimental series. The first of the methods rejects two (noted as n° 10,12) while the second fails to validate only one, n° 8. The Kojima test rejects five data series, with the error observed in the system n° 12 being critical. A new methodology recently proposed by us [6, 7] was also applied, which rejects nine systems: three by the integral form (n° 9, 11, 12) and six by the differential form (n° 1, 3, 4, 8, 9, 12). These systems present obvious signs of inconsistency that are also observed by at least one of the other methods. Therefore, the overall assessment of these series (in other words, the quality of their experimental data) is not

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),

; m ¼ number of experimental points (7)

OF<sup>ε</sup> <sup>¼</sup> s gE=RT (8)

constraint ! s h<sup>E</sup> , <sup>ε</sup> (9)

, since greater errors in the h<sup>E</sup> are

)], where each one of these give rise to different estimates of

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

positive, ruling out the use of these series for other subsequent tasks, such as modeling and, specially, simulation.

Regarding h<sup>E</sup> , six references were found [20–25]. All extracted values are shown in Figure 2(a and b), where the different series show an acceptable coherence, having (dh<sup>E</sup> /dT)p,x > 0.
