Algorithm for calculating phase diagrams at set initial global composition involving fluid and solid-solution phases for multicomponent chemically reactive systems

Abstract

The goal of this work was to develop an algorithm for computing, over wide ranges of conditions, phase envelopes and three-phase envelopes for reactive systems including the possibility of presence, at equilibrium, of solid phases of the solid-solution type. To our knowledge, algorithms of such sort are not available in the literature. The familiar constant global composition (z) phase envelope (PE) of a multicomponent non-reactive system shows, in the pressure (P) versus temperature (T) plane, a number of curves, such as bubble point, dew point, liquid-liquid and/or solid (S) - fluid (F) (S+F) curves (z is a vector of global component mole fractions). In a given point of the PE (which often includes critical points), a phase of finite size (major phase) of composition z is at equilibrium with a phase of differential size (incipient phase) of composition generally different from z. The PE is the boundary between the homogeneous region and the two-phase (heterogeneous) region. The information on the phase behavior of the multicomponent system of interest becomes more complete if some additional types of lines are plotted within the heterogeneous region, e.g., the three-phase envelopes (3PEs) or three-phase lines (3PLs, for the case of binary systems). A 3PE is the boundary between a two-phase region and a three-phase region. In a point of a 3PE, two phases of finite size are at equilibrium with an incipient phase (still being the global composition equal to z). The set of all PE segments plus all the auxiliary lines (e.g., 3PEs or 3PLs) constitute a quite complete constant z diagram which is named ?isopleth? (IP). For reactive systems, the global composition z changes along the reactive PE (R-PE) and also along the reactive 3PE (R-3PE) or reactive 3PL (R-3PL) (depending on the degrees of freedom (DsOF) of the reactive three-phase system). Each point of any of these curves corresponds to the simultaneous phase and chemical equilibria. Both, in IPs and in reactive IPs (R-IPs) the global mole fractions of the atoms indeed remain constant, while the global mole fractions of the components are constant only for the non-reactive IPs. In a previous work [1], calculation algorithms were developed for all types of lines present in R-IPs involving only fluid phases. In the present work, the possibility of precipitation of solid solutions has been incorporated to the computed R-IPs. The modelling approach proposed by Porras et al. [2] was used to represent the thermodynamic properties of multi-component solid phases (Solid Solution Approach (SSA)). As case study, we have chosen the carbon dioxide (CO2) + 1,2-propylene oxide (PO) + propylene carbonate (PC) system, in which chemical bonds are broken or formed as prescribed by the chemical reaction CO2 + 1,2-propylene oxide ⟷ propylene carbonate. The fluid-state volumetric and phase behavior of this system was modelled through the well-known SRK-EoS coupled to quadratic mixing rules (QMRs). The EoS pure component parameter values, and the values for the binary interaction parameters (considered equal for both the fluid and solid phases), used in the calculations, are given in ref [3], together with the required ?standard state?-related parameters. Other pure component parameters appearing in the solid-solution global fugacity expression were obtained, either from databases, or from reproducing the experimental pure component solid-liquid equilibrium curves. For the chosen reactive system, complete reactive three-phase line segments and reactive phase envelope segments, including the possibility of presence of solid phases that are solid solutions, were computed over wide ranges of conditions using numerical continuation methods. The results are illustrated through the phase diagram, computed for the initial global composition zCO20=0.75, zPO0=0.25 and zPC0=0.0, which is shown in the Graphical Abstract (GA). The obtained results imply the existence of complex patterns of behavior for the computed R-IPs (see insert in GA). The circular marker in the GA is a reactive critical point. Furthermore, a predicted homogeneous solid-solution region is shown in the pressure-temperature plane in the GA. The presence of homogeneous-solid regions in mixture isopleths is of impossible prediction by models that allow precipitation only of solid phases made of pure components (i.e., models that exclude the possibility of solid solutions). The proposed algorithms have been found to be robust and this is ascribed to the applied numerical continuation method.