Pertti Koukkari

Introducing mechanistic kinetics to the Lagrangian Gibbs energy calculation

The Gibbs free energy minimum is usually calculated with the method of Lagrangian multipliers with the mass balance conditions as the necessary subsidiary conditions. Solution of the partial derivatives of the Lagrangian function provides the equilibrium condition of zero affinity for all stoichiometric equilibrium reactions in the multi-phase system. By extension of the stoichiometric matrix, reaction rate constraints can be included in the Gibbsian calculation. Zero affinity remains as the condition for equilibrium reactions, while kinetic reactions receive a non-zero affinity value, defined by an additional Lagrange multiplier. This can be algorithmically connected to a known reaction rate for each kinetically constrained species in the system. The presented method allows for several kinetically controlled reactions in the multi-phase Gibbs energy calculation.

A Gibbs energy minimization method for constrained and partial equilibria

The conventional Gibbs energy minimization methods apply elemental amounts of system components as conservation constraints in the form of a stoichiometric conservation matrix. The linear constraints designate the limitations set on the components described by the system constituents. The equilibrium chemical potentials of the constituents are obtained as a linear combination of the component-specific contributions, which are solved with the Lagrange method of undetermined multipliers. When the Gibbs energy of a multiphase system is also affected by conditions due to immaterial properties, the constraints must be adjusted by the respective entities. The constrained free energy (CFE) minimization method includes such conditions and incorporates every immaterial constraint accompanied with its conjugate potential. The respective work or affinity-related condition is introduced to the Gibbs energy calculation as an additional Lagrange multiplier. Thus, the minimization procedure can include systemic or external potential variables with their conjugate coefficients as well as non-equilibrium affinities. Their implementation extends the scope of Gibbs energy calculations to a number of new fields, including surface and interface systems, multi-phase fiber suspensions with Donnan partitioning, kinetically controlled partial equilibria, and pathway analysis of reaction networks.

Combination of overall reaction rate with Gibbs energy minimization

A method to calculate multi-component chemical reaction mixtures as a sequence of time-dependent, intermediate thermochemical states is presented. The method combines the overall reaction kinetics with thermodynamic Gibbs energy minimization. The overall reaction is assumed to proceed according to the Arrhenius rate law. During the time-course of the reaction, the temperature and composition of the reaction mixture are calculated by a thermodynamic subroutine, which minimizes the Gibbs energy of the system. The extent of the overall reaction is algorithmically constrained in the Gibbs energy minimization procedure. During the sequential calculation, the kinetic condition is removed by finite differences. The temperature of each intermediate state is reached by an iterative procedure, which takes into account the heat transfer between the system and its surroundings and the enthalpy changes due to the chemical reactions. Thus, the method allows for the effect of temperature on the reaction kinetics as the reaction evolves. The chemical species present in each intermediate state are virtually independent and there is a chemical potential assigned to each of these species. The gradual chemical change in the thermodynamic system proceeds from the initial state of mixed reactants to the final state of product mixture. Both stationary and transient phenomena may be calculated. The method has been applied to some well-known industrial multi-component reaction systems and a fair agreement between the calculated and measured values has been obtained. The application of the thermochemical algorithm in reaction calorimetry is discussed.

Reaction rates as virtual constraints in Gibbs energy minimization

The constrained Gibbs energy method has been developed for the use of immaterial entities in the formula conservation matrix of the Gibbs energy minimization problem. The new method enables the association of the conservation matrix with structural, physical, chemical, and energetic properties, and thus the scope of free energy calculations can be extended beyond the conventional studies of global chemical equilibria and phase diagrams. The use of immaterial constraints enables thermochemical calculations in partial equilibrium systems as well as in systems controlled by work factors. In addition, they allow the introduction of mechanistic reaction kinetics to the Gibbsian multiphase analysis. The constrained advancements of reactions are incorporated into the Gibbs energy calculation by using additional virtual phases in the conservation matrix. The virtual components are then utilized to meet the incremental consumption of reactants or the formation of products in the kinetically slow reactions. The respective thermodynamic properties for the intermediate states can be used in reaction rate formulations, e.g., by applying the reaction quotients.

A systematic method to create reaction constraints for stoichiometric matrices

Abstract Modeling rate-controlled chemically reactive systems in biocatalysis, fuel combustion, material science, and chemical process engineering involves the quantification and exploitation of interactions between many chemical species. These dynamic chemical systems, having relatively few limiting reactions, can be conceived as a series of snapshots where reactions have fixed extents but otherwise idle. Since the reactions affect the stoichiometric matrix of the internal constraints, such constrained equilibrium states cannot be defined in terms of conventional atomic mass balances. A systematic method for obtaining generalized equilibrium constraints for reaction mechanisms of arbitrary complexity is presented. Reaction matrices are converted into entity conservation matrices using row operations. The simultaneously introduced virtual components enable Gibbs energy calculations for complex reaction schemes including organic systems and enzyme-catalyzed biochemical transformations having multiple limiting reactions. Classical Gibbs energy minimization, which would otherwise readily model phase transformations and solvent interactions, is thereby made accessible to these emerging application fields.

Time-dependent reactor simulation by stationary state thermochemistry

Abstract A physico-chemical algorithm to calculate the thermochemical properties of a stationary chemical reactor is presented. The reactor is considered either as a set of closed systems or as elements of a continuous system. The reactor conditions then are simulated by the timely invariant steady state properties of the volume elements. The extent of the global chemical reaction is followed by the overall reaction rate. Other properties for the reactor volume elements are calculated by a thermodynamic subroutine, which minimizes the Gibbs energy of the kinetically constrained multi-component system and takes into account the heat exchange between the reaction system and its surroundings. The stationary state within each volume element requires constant heat flow at the fixed reactor position and constant values of the thermodynamic quantities such as Entropy and Gibbs free energy for the element itself. The proposed algorithm pursues to satisfy both the overall reaction rate and the thermodynamic requirements of the stationary state, resulting with a sequence of calculated, intermediate reactor ‘states’. The method has been used to compute temperature, velocity and composition profiles in a tubular high-temperature flame reactor, which is used in titanium dioxide production.

Setting kinetic controls for complex equilibrium calculations

Publisher Summary This chapter discusses setting kinetic controls for complex equilibrium calculations. Many methods exist that cover only the kinetic aspects of a stoichiometric reaction and how it proceeds in time. Only a few attempts have been made so far to link equilibrium aspects of multi-component systems with kinetic inhibitions or even single reaction rates. None of these has led to a generally applicable link between the terms used in reaction kinetic equations and the Gibbs energy minimization method available for general equilibrium calculations. Mostly, dedicated solutions for special cases have been established. The image component method, although practical, is not fully consistent thermodynamically when used for solution phases. The chapter describes a method that combines multi-component multi-phase equilibrium thermodynamics with reaction kinetics.

Thermochemical multi-phase models applying the constrained gibbs energy method

Abstract Computation of chemical equilibria in multiphase systems by Gibbs free energy minimization under constraints set by the material balance has increasing interest in many application fields, including materials technology, metallurgy and chemical engineering. The results are utilised in multi-phase equilibrium studies or as parts of equilibrium-based process simulation. Yet, there exist a number of practical problems where the chemical system is influenced by other constraining factors such as surface energy or electrochemical charge transport. For such systems, an extended Gibbs energy method has been applied. In the new method, the potential energy is introduced to the Gibbs energy calculation as a Legendre transformed work term divided into substance specific contributions. The additional constraint potential is then represented by a supplementary undertermined Lagrange multiplier. In addition, upper bounds on the amounts of products can be set, which then limit the maximum extents of selected spontaneous chemical reactions in terms of affinity. The range of Gibbs energy calculations can then be extended to new intricate systems. Example models based on free energy minimisation have been made e.g. for surface and interfacial systems, where the surface, interfacial or adsorbed atomic or molecular layers are modeled as separate phases. In an analogous fashion the partitioning effect of a semi-permeable membrane in a two-compartment aqueous system can be modeled. In such system the large ions, not permeable through the membrane, cause an uneven charge distribution of ionic species between the two compartments. In this case, the electrochemical potential difference between the two aqueous phases becomes calculated for the multi-component system. The calculated results are consistent with the Donnan equilibrium theory; however the multi-phase system may also include the gas phase and several precipitating phases, which extends the applicability of the new method. Finally, similar constraints can also be set to extents of reaction advancements, allowing usage of Gibbs energy calculations in dynamic reaction rate controlled systems.

Surface tension of a liquid metal–oxygen system using a multilayer free energy model

A constrained free energy model for an interfacial system of interacting layers is derived in the regular solution and Redlich–Kister formalisms. Composition and interfacial energies are solved using a model based on the minimisation of the total free energy of the system with a fixed interfacial area. As example cases iron–oxygen and copper–oxygen surfaces and liquid–liquid metal alloy interfaces are studied.

The combination of transformed and constrained Gibbs energies.

Gibbs free energy is the thermodynamic potential representing the fundamental equation at constant temperature, pressure, and molar amounts. Transformed Gibbs energies are important for biochemical systems because the local concentrations within cell compartments cannot yet be determined accurately. The method of Constrained Gibbs Energies adds kinetic reaction extent limitations to the internal constraints of the system thus extending the range of applicability of equilibrium thermodynamics from predefined constraints to dynamic constraints, e.g., adding time-dependent constraints of irreversible chemical change. In this article, the implementation and use of Transformed Gibbs Energies in the Gibbs energy minimization framework is demonstrated with educational examples. The combined method has the advantage of being able to calculate transient thermodynamic properties during dynamic simulation.