Dear Colleagues,

The generalized thermodynamic equilibrium problem (GTEP) applies to multicomponent heterogeneous systems of any number of phases (vapor, gas, liquid, solid) and any number of chemical species and equilibrium reactions, limited only by Gibbs' phase rule. The number of phases is intrinsic to the problem, which is also a major challenge in GTEP. Not only is the global optimal solution needed, the objective function itself is part of the optimization problem.

Mathematically, GTEP belongs to the class of non-linear programming where the objective function can be identified as a generic thermodynamic potential (valid for the phase assembly) and its derivatives, but without referring to domain-specific physical quantities like activity and/or fugacity coefficients.

In practice, global optimization is sought after using first- and second-order derivatives only, but current advances in automatic differentiation tools suggest that structured variable transformations or higher-order derivatives can also be of theoretical interest.

The phase stability analysis of GTEP with heterogeneous reactions is particularly difficult because several phases can enter or leave the phase assembly simultaneously. Heuristics is best avoided and mathematical analysis stands out as the preferred alternative. Additionally, since thermodynamics lacks a natural metric, the convergence criteria are always an issue.

Step length algorithms are arguably the most important resource for any GTEP solver due to convergence, and because they can help to avoid stepping into unphysical domains. This is a model-specific dilemma and the underlying equations of state must be given the last word, but not by lending access to the underlying physical database. The solver should interact with the models only via a generic query API. The same applies to initialization. One example is VLE calculations using Tc and Pc from the database, which has no meaning in condensed phase equilibrium problems.

This Special Issue on "Calculating Generalized Thermodynamic Equilibrium" aims to resolve a longstanding theoretical challenge in academia and process industry by looking into novel advances in non-linear optimization, but also to re-visit classic solution strategies, and not to forget by elevating the problem formulation from a mere Gibbs energy formalism to include any generic potential of thermodynamic origin (i.e., energy, entropy, volume etc.) in canonical state variables. Topics include, but are not limited to:

• Recent advances in global optimization applied to thermodynamic problems;
• Revisiting classic techniques with the same focus;
• Exhaustive searches for global equilibrium in two- and three-component systems;
• Thermodynamic metric and convergence criteria;
• Phase stability analysis of heterogeneous reactions;
• Generic query API for model-specific information useful for initialization and step length control.

Prof. Dr. Tore Haug-Warberg
Dr. Volker Siepmann
Dr. Olaf Trygve Berglihn
Guest Editors

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• functional analysis
• global optimization
• fast convergence
• thermodynamic modeling, calculation, and stability analysis
• process systems technology, chemical thermodynamics
• solution to heterogeneous equilibrium problems