Risto Pajarre

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.

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.

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.

Industrial applications of multi-phase thermochemical simulation

Abstract The multi-component Gibbs energy simulation provides an efficient tool for quantifying measured data in complex industrial systems. The advantages of the multi-phase methods have been recognized and they are becoming widely accepted in different applications, ranging from metallurgy and mate rials processing to chemistry and energy and environmental technologies. The thermodynamic multi-phase theory provides an effective method to support industrial process development by quantitative process analysis. By using the Gibb senergy, based calculations the chemical equilibria in complex multi-phase systems are determined and quantitative results are received for all that can be defined with the thermody namic properties of their constituents. The multiphase Gibbs energy method enables simult aneous computation of chemical and energy changes. Further, the thermochemical approach provides a method to solve the bulk chemistry together with the speciation of man y minor constituents, such as harmful trace compounds. Recent development has brought new phenomena, such as ion exchange and surface properties to the Gibbsian multi-phase domain. The thermochemical calculations can be applied e.g. on aqueous pulp suspensions to follow the chemical composition and alkalinity of the pulp and paper-making processes. New advanced algorithms also allow for the use of reaction rate constraints in multi-phase Gibbsian simulations. Combination of reaction rates with the thermochemical, method brings about simultaneous and interdependent models for chemical and energy changes in such reactors that involve endo-and exothermic reactions together with operator controlled heat and mass transfer effects. The practical applications include process internsification, optimization, troubleshooting, scale-up, study of new chemical concepts and search for new, more economic process alternatives.

Combining reaction kinetics to the multi-phase Gibbs energy calculation

Abstract Development of robust and efficient methods for the computation of multiphase systems has long been a challenge in both chemical and petroleum engineering as well as in materials science. Several techniques have been developed, particularly those which apply the Gibbs free energy minimization. In addition to calculation of global equilibrium problems, practical process simulation would benefit from algorithms, where reaction rates could be taken into account. In the present work, the method of Lagrange multipliers has been used to incorporate such additional constraints to the minimization problem, which allow a mechanistic reaction rate model to be included in the Gibbsian multi-component calculation. The method can be used to calculate the thermodynamic properties of a multiphase system during a chemical change. The applications include computational materials science, industrial process modeling with known reaction rates combined with complex heat and mass transfer effects and studies of other nonequilibrium systems.

Thermodynamic affinity in constrained free-energy systems

Affinity is the generic measure of the deviation of a state from stable equilibrium. Affinity, as introduced by de Donder, is a thermodynamic state property defined in terms of p, T, and system composition during the course of a chemical change. When incorporating reaction kinetic constraints to minimization of Gibbs energy of a multiphase system, affinity can be followed in terms of the extents of the constrained reactions. This property then becomes calculated in terms of the constraint potentials received as additional Lagrange multipliers in the minimization routine. Thus, received affinities are consistent with the respective values calculated from the chemical potentials of the reactants and products of the constrained reactions and their limiting behaviour corresponds to that defined for both stationary and stable equilibrium states. The intermediate affinities can be used in the respective reaction rate calculations, or as input parameters, to define the local chemical equilibrium set by known reaction kinetic constraints. Thus, they become a useful concept in modelling reactive processes.Graphical abstract

IV.9 – Modelling TiO2 production by explicit use of reaction kinetics

Publisher Summary This chapter discusses modeling TiO 2 production by explicit use of reaction kinetics. Titanium dioxide (TiO 2 ) is a bulk commodity, which is used as a white pigment, for paints, plastics, paper, and rubber. TiO 2 pigment is produced by two major industrial routes from its ore, which is either ilmenite or either natural or synthetic rutile. Ilmenite-based raw materials are used for the wet sulfuric acid process, while rutile is the major input for the high-temperature chloride process. After years of development, the manufacturing processes remain in extensive use and represent both economically effective and environmentally sound industrial practice. The chapter presents the examples where the combined reaction rate–multi-phase calculation has been applied to two characteristic unit processes of TiO 2 production. As the Gibbs energy model inherently calculates other thermodynamic properties, such as heat capacities, enthalpies, and entropies, it is usually beneficial to use Gibbs energy-based modeling in process calculations.