Introducing mechanistic kinetics to the Lagrangian Gibbs energy calculation

doi:10.1016/j.compchemeng.2006.03.001

Computers & Chemical Engineering

Volume 30, Issues 6–7, 15 May 2006, Pages 1189-1196

https://doi.org/10.1016/j.compchemeng.2006.03.001 Get rights and content

Abstract

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.

Introduction

Simulation of chemical processes and reactive flows by efficient computer program has become everyday practice in both research and industry. For those processes, which involve time-dependent changes both in terms of temperatures and chemical composition, most often models based on global reaction rates are used. The mechanistic approach, which involves elementary reaction kinetics is also frequently applied. The conventional method to simplify treatments of multi-step reaction mechanisms is also to suggest that the mechanism consists of fast and slow reactions, so that the former are assumed to reach equilibrium, and the latter then are regarded as the rate-limiting steps of the chemical change. The reduced mechanism thus gained is often more convenient to use in process modeling than the alternative of compiling the s.c. complete reaction mechanism in terms of their forward and reverse reaction rates. In all these models, thermodynamic quantities are generally used to support material and energy balances and to check final equilibrium conversions.

Development of robust and efficient methods for the computation of thermodynamic multi-phase systems has long been a challenge in both chemical engineering and materials science. A comprehensive classification of the various alternative approaches to solve the free energy minimization problem is given, e.g. by Smith and Missen (1991). A review on the methods used for multi-phase equilibria with chemical reactions is given, e.g. by Seider and Widagdo (1996), with emphasis on the stability analysis performed by the reaction tangent plane method developed extensively by Michelsen (1994). Harding and Floudas (2000) obtained a theoretical confirmation for the stability of the tangent plane optimization problem. Nichita, Gomez, & Luna (2000) presented the tunneling method for global optimization of the Gibbs energy problem, so as to avoid local minima and saddle points while searching the global solution. They also give a broad overview on the history of various algorithms and programs developed for the reactive multi-phase approach in chemical and petroleum engineering calculations. Quite recently, the obvious advantages of the multi-phase approach have yet gained increasing interest when approaching practical chemical engineering problems (Ballard & Sloan, 2004). Gibbs free energy minimization, performed typically with the Lagrange multiplier method and applied for thermochemical process models, has also been extensively used in high temperature and materials chemistry (Eriksson & Hack, 1990). In what follows, we apply the method of Lagrangian multipliers as described also by Walas (1985) and by, e.g. Smith and Missen (1991), 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.

As referred in an earlier paper (Koukkari, Pajarre, & Hack, 2001), there has been a search for the link between reaction kinetics and multi-component thermodynamic calculations for quite some time. For diffusional aspects of phase formation and transformation programs have been used extensively and successfully (Anderson, Höglund, Jönsson, & Agren, 1990; Krupp & Christ, 1999). For cases in which mainly materials transport plays the limiting role coupling with fluid dynamics or even with simple flowsheeting with the concept of local equilibrium has proven quite successful (Koukkari et al., 2001, Robertson, 1995). For simple cases such as linearly coupled equilibrium cells which describe the steady state of a silicon shaft furnace process it is even possible to use standard software (Eriksson & Hack, 1990). Some approaches have also been made to link equilibrium aspects of multi-component systems with kinetic inhibitions with single reaction rates (e.g. the work by Korousic and Stupnisek (1995) on the NH₃ kinetics in nitriding gases). An early approach in combining reaction rates with multi-phase calculations was the extension of the stoichiometric matrix of a Gibbsian multi-component system by the image component (Koukkari, 1993). The image method was proven successful in many cases, where the driving force of a single reaction is sufficient for reaching 100% conversion from reactants to products. The introduction of conserved groups as additional system components to the stoichiometric matrix in the Lagrangian method of Gibbs energy minimization was first used to preserve aromatic rings in a benzene flame model (Alberty, 1989). With a further matrix extension, this technique was shown to be applicable to kinetically conserved species (Pajarre, 2001) and could be applied to several related problems (Koukkari et al., 2001).

In the present paper, the method of the extended stoichiometric matrix in terms of the conserved (kinetically constrained) species is re-formulated by using the Lagrangian function. By using the partial derivatives of the Lagrangian the necessary conditions as regards to the equilibrium processes and the kinetically controlled species, respectively, are deduced. The method will also be applied to simple examples, by which we evaluate the congruousness of the method.

Section snippets

Overview of the Lagrangian method

Among the most frequently used methods for Gibbs energy minimization is the one based on Lagrangian multipliers. The Lagrangian function (L) is formed from the Gibbs energy (G) by using the mass balances of the system components as the necessary subsidiary constraints: $L = G - λ ψ = G - \sum_{j = 1}^{l} λ_{j} (\sum_{k = 1}^{N} a_{k j} n_{k} - b_{j})$ In Eq. (1), λ = λ(λ₁,λ₂, …, λ_l) is the set of the Lagrangian undetermined multipliers, as connected with the mass balances of the system components ψ. In the right hand part of the equation the explicit

Extension of the Lagrangian method for conserved and reactive species

It was shown earlier that conservation of the molar amount of a species at its input value can be introduced to an equilibrium calculation by including an additional system component R, for which the matrix element a_kj ≠ 0 for the conserved constituent(s), and a_kj = 0 for all other species (Koukkari et al., 2001, Pajarre, 2001). Originally, this technique was used by Alberty to show the possibility to conserve the aromaticity when calculating partial equilibria in a benzene flame (Alberty, 1989).

Example 1: anatase–rutile transformation

As a simple example, we consider the formation of titanium dioxide in a calciner. The feed consists of wet titanium oxyhydrate slurry. The chemical formula of the oxyhydrate can be written as TiO(OH)₂·nH₂O, with n ≥ 0. During calcination, the slurry is dried and finally the hydrate decomposes, leaving the product titanium dioxide into the bed. From the oxyhydrate, at relatively low temperatures (ca. 200 °C) the crystalline form anatase, TiO₂(An), is formed first, and only in the high temperature

Example 2: several kinetically constrained reactions

The presented method can also be used to introduce more complex mechanisms into a multi-component calculation. The mass conservation (stoichiometric) matrix A must be extended with one column and two rows for each kinetically constrained species. In Table 4, an example is presented for mercury chlorination at temperatures around 900 °C. At such high temperatures, in the Hg–chlorine–argon-system, elemental mercury should be the thermodynamically dominant gaseous species. However, experimental

Discussion

The method of the extended matrix is directly applicable in the commonly used Gibbs energy routines, such as Solgasmix or ChemApp (Eriksson, Hack, & Petersen, 1997). At present, the method complies without difficulty for problems with reduced number of constrained reactions within the Gibbsian system. The extension of the stoichiometric matrix with new components increases the computing time when compared with the solution of the respective equilibrium system. With simple condensed-gas-systems

References (27)

A.L. Ballard et al.
The next generation of hydrate prediction. Part III. Gibbs energy minimization formalism
Fluid Phase Equilibria
(2004)
P. Koukkari
A physico-chemical method to calculate time-dependent reaction mixtures.
Computers and Chemical Engineering
(1993)
M.L. Michelsen
Calculation of multiphase equilibrium
Computers and Chemical Engineering
(1994)
D.V. Nichita et al.
Multiphase equilibria calculation by direct minimization of Gibbs free energy with a global optimization method
Computers and Chemical Engineering
(2002)
W.D. Seider et al.
Multiphase equilibria of reactive systems
Fluid Phase Equilibria
(1996)
R.N. Sliger et al.
Towards the development of a chemical kinetic model for the homogenous oxidation of mercury by chlorine species
Fuel Processing Technology
(2000)
R.A. Alberty
Thermodynamics of the formation of benzene series polycyclic aromatic hydrocarbons in a benzene flame
Journal of Physical Chemistry
(1989)
J.-O. Anderson et al.
G. Eriksson
Thermodynamic studies of high temperature equilibria. III
Acta Chemica Scandinavica
(1971)
G. Eriksson
Thermodynamic studies of high temperature equilibria XII
Chemica Scripta
(1975)

G. Eriksson et al.

ChemSage—A computer program for the calculation of complex chemical equilibria.

Metallurgical Transactions B

(1990)

Eriksson, G., Hack, K., Petersen, S. (1997). In Werkstoffwoche ’96, Symposium 8: Simulation, Modellierung,...

FACT Database Edition 2001. (2001). FactSage 5.0, CRCT. Montreal, Canada and GTT Technologies, Herzogenrath,...

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Introducing mechanistic kinetics to the Lagrangian Gibbs energy calculation

Abstract

Introduction

Section snippets

Overview of the Lagrangian method

Extension of the Lagrangian method for conserved and reactive species

Example 1: anatase–rutile transformation

Example 2: several kinetically constrained reactions

Discussion

Fluid Phase Equilibria

Computers and Chemical Engineering

Computers and Chemical Engineering

Computers and Chemical Engineering

Fluid Phase Equilibria

Fuel Processing Technology

Thermodynamics of the formation of benzene series polycyclic aromatic hydrocarbons in a benzene flame

Journal of Physical Chemistry

Thermodynamic studies of high temperature equilibria. III

Acta Chemica Scandinavica

Thermodynamic studies of high temperature equilibria XII

Chemica Scripta

ChemSage—A computer program for the calculation of complex chemical equilibria.

Metallurgical Transactions B