# Thermodynamics Models

## Thermodynamic Models

The knowledge of the real behavior of mixtures is important for different applications. This is especially true for the synthesis, simulation and optimization of separation processes, where the phase equilibrium of the system to be separated has to be known. For the description of phase equilibria today modern thermodynamic models are available. Starting from the isofugacity criterion:

fiα = fiβ

the different phase equilibria can be described by using activity or fugacity coefficients to account for the real behavior of the different phases. For vapor-liquid equilibria it can be distinguished between two different methods. One method only requires fugacity coefficients φi for the liquid (L) and the vapor (V) phase:

xiφiL = yi φiV

whereby the required fugacity coefficients can be derived from the PvTx(y)-behavior e.g. by using cubic equations of state such as the Soave-Redlich-Kwong or Peng-Robinson equation of state and suitable mixing rules. Another method (ϒ - φ- approach) employs activity coefficients to describe the real behavior in the liquid and fugacity coefficients for the vapor phase, whereby for not strongly associating compounds at moderate pressures often the following simplified relation can be used:

xi ϒi Pis = yi P

For the calculation of activity coefficients in multicomponent systems different gE-models (e.g. Wilson, NRTL, UNIQUAC) are available. Both methods have different advantages and disadvantages. They have in common that phase equilibria of multicomponent mixtures can be calculated using binary data alone. This is most important since nearly no data are available for multicomponent systems. But for fitting the required binary parameters reliable phase equilibrium information for the whole concentration and a large temperature range is required. Using equations of state there is the great advantage that besides phase equilibria different pure component and mixture properties for the different phases can be calculated (densities, vapor pressures, heat of vaporization, thermodynamic properties such as enthalpies, entropies, ..) and that this method also can be used to handle supercritical compounds.

For actual problems often at least a part of the required binary data is missing. This means that in many cases the methods mentioned above cannot directly be applied. To overcome these problems the missing binary data have to be measured or ideal behavior has to be assumed. Since the measurement of phase equilibrium data is very time consuming and the assumption of ideal behavior can lead to very erroneous results it would be most desirable to apply reliable predictive methods. As already planned in 1973 the worldwide available phase equilibrium data and excess properties stored in the Dortmund Data Bank were used for the development or further development of different predictive methods for non-electrolyte and electrolyte systems, such as

• UNIFAC (in collaboration with Prof. Dr. Aa. Fredenslund (Lyngby, Denmark))
• mod. UNIFAC
• ASOG (in collaboration with Prof. Dr. K. Kojima and Prof. Dr. K. Tochigi (Tokyo, Japan)
• PSRK
• VTPR
• LIQUAC
• LIFAC

which are now used worldwide in chemical industry. The possibility to predict reliably the real behavior of fluid mixtures using gE-models (Wilson, NRTL, UNIQUAC), equations of state (SRK) and group contribution methods (UNIFAC, mod. UNIFAC, ASOG, PSRK, VTPR, LIQUAC, LIFAC) opens a wide field of different applications. A selection of important applications is given in Fig. 1.

Fig. 1 Different applications related to real mixture behavior

With the knowledge of the real behavior it is for example possible to calculate residual lines for ternary and higher systems. Fig. 2 shows these lines together with the boundary lines which separate the different distillation fields for different systems at a pressure of 1.013 bar. Whereas during the separation of binary systems by simple distillation the azeotropic point cannot be passed, in ternary systems exist boundary lines and in quaternary systems boundary surfaces, which likewise cannot be crossed.

Fig. 2 Residue curves and boundary lines for different systems

Using the information on the real behavior of mixtures, it is often possible to find simple solutions for separation problems. Fig. 3 shows the results of a column calculation, where pure water and methanol are obtained from a four component mixture in only one column, since the impurities (ethanol and isobutanol) can be removed by a small side-stream.

Fig. 3 Simulation of a rectification column using the Wilson model

(1 - methanol, 2 - water, 3 - ethanol, 4 -iso butanol)

(Stand: 20.04.2022)