Predict VLE and LLE from TC lines.

Definition of the TC line
It is shown [refs. 1 and 2] that VLE data converge to a line, if the following Y and Z are plotted versus X defined as follows:
X=(ps1+ps2)/(Pc1+ps2)
Y=(G1infps1-ps2)/(Pc1-ps2)
Z=(ps1-G2infps2)/(Pc1-ps2)

where ps1 and ps2 are vapor pressures of components 1 and 2, respectively; Pc1 is the critical pressure of component 1. Furthermore, G1inf and G2inf are the infinite dilution activity coefficients (IDAC) of components 1 and 2, respectively.

Examples of the TC line
In Fig. 1, Y and Z are plotted versus X for the benzene (1) + ethanol (2) binary. Figure 1 shows that all of the experimental data can be satisfactorily represented by the lines passing through the origin. The following two equations are useful for representing the VLE data:
Y, Z =aXb
Y, Z =aX +bX2 +cln(1+X)
where a, b and c are constants. Figure 1 includes both of the constant-temperature data and constant-pressure data. It is important to use appropriate infinite dilution activity coefficient equations satisfactorily representing P, x data.

Fig. 1 Y and Z versus X for the benzene (1) + ethanol (2) binary, (solid line) XY TC line (, , x) Y from experimental data, (-----) XZ TC line and (, , +) Z from experimental data.

Examples of T, x, y prediction from the TC line
In Fig. 2, T/Tb2 and y1 are plotted versus x1 for the benzene (1) + ethanol (2) binary system at 101.3 kPa, where T and Tb2 denote the system temperature and the boiling point of component 2, respectively, and where x1 and y1 denote the mole fractions of component 1 in the liquid and vapor phases, respectively. As shown in Fig. 2, the experimental y1 data, denoted by the (+) symbols, are fluctuating. Unfortunately, there is no method for identifying accurate data, as noted at BeAware.htm. However, Fig. 2 clearly shows that the green solid line predicted using the TC lines in Fig. 1 at 101.3 kPa is accurate, as it represents the (+) data.

Fig. 2 T/Tb2 and y1 versus x1 for the benzene (1) + ethanol (2) binary at 101.3 kPa, (red solid line) T/Tb2
predicted from TC lines, (x) T/Tb2 from experiments, (green solid line) y1 predicted from TC lines and (+) y1 from experiments.

References
1) Satoru Kato(
Tokyo Metropolitan Univ.), Juergen P. Schmelzer (Univ. of Applied Sci., Germany), Peter Gostomski (Univ. of Canterbury, NZ), Kenneth N. Marsh (Univ. of Canterbury, NZ), J. Chem. Eng. Data, 56 (2011) 4927-4934.
2) Satoru Kato (Tokyo Metropolitan Univ.,
TC Lines), Peter Gostomski (Univ. of Canterbury, NZ), Akiyoshi Oda (Nippon Refine, Japan), Joerg Freitag (Nippon Refine, Japan), Seiya Hirohama (Invensys, Japan), David Bluck (Invensys, U.S.A.), Fluid Phase Equilibria, 357 (2013) 36-42.