Combining reaction kinetics to the multi-phase Gibbs energy calculation

Transcription

1 7 th European Symposium on Computer Aided Process Engineering ESCAPE7 V. Pesu and P.S. Agachi (Editors) 2007 Esevier B.V. A rights reserved. Combining reaction inetics to the muti-phase Gibbs energy cacuation Pertti Kouari, Risto Paarre VTT Process Chemistry, P.O. Box 000 FIN-02044, VTT FINLAND pertti.ouari@vtt.fi;risto.paarre@vtt.fi Abstract Deveopment of robust and efficient methods for the computation of mutiphase systems has ong been a chaenge in both chemica and petroeum engineering as we as in materias science. Severa techniques have been deveoped, particuary those which appy the Gibbs free energy minimization. In addition to cacuation of goba equiibrium probems, practica process simuation woud benefit from agorithms, where reaction rates coud be taen into account. In the present wor, the method of Lagrange mutipiers has been used to incorporate such additiona constraints to the minimization probem, which aow a mechanistic reaction rate mode to be incuded in the Gibbsian muti-component cacuation. The method can be used to cacuate the thermodynamic properties of a mutiphase system during a chemica change. The appications incude computationa materias science, industria process modeing with nown reaction rates combined with compex heat and mass transfer effects and studies of other nonequiibrium systems. Keywords Gibbs energy minimization, reaction rate constraint, modeing of muti-phase processes

2 2 P. Kouari et a.. Introduction Severa industria processes in the fieds of chemistry, petroeum engineering, metaurgy, stee-maing or e.g. in pup and paper manufacturing incude production stages, where chemica changes occur in true muti-phase fashion. The quantitative treatment of such processes is best performed with thermodynamic muti-phase methods. In a compex reactive system, where severa phases can form, the minimum of Gibbs energy provides a genera criterion for equiibrium cacuations. Yet, many of the probems invove nonequiibrium reactions and processes, for which the Gibbs ian approach as such is not appicabe. Consequenty, there has been a search for ways to in reaction inetics and muti-component thermodynamic cacuations. Whie performing the Gibbs energy minimization the cacuation is usuay constrained by the mass baance of the cosed system. As a resut, the goba equiibrium composition at a given temperature and pressure is reached. In what foows, a Gibbs free energy technique based on the extension of the (stoichiometric) conservation matrix of the Lagrange method is presented. The extension of the conservation matrix can be used to incude reaction inetic restrictions to contro the extent of seected chemica reactions in terms of their reaction rates. 2. Setting rate constraints into the Gibbs energy cacuation 2.. Gibbs free energy minimization by the Lagrange method The Gibbs free energy of the muticomponent system is written in terms of the chemica potentias as foows G = n μ () where μ is the chemica potentia of the species () in the respective phase and n is its moar amount. The respective chemica muti-component system with N constituents, which are formed of components can be described with a (N ) conservation matrix A, the eement of which is a, defining the stoichiometry of component in the respective constituent. The minimization of the Gibbs free energy can be performed by using severa aternative techniques [-2]. Perhaps the most widespread mathematica routine is that of Lagrangian mutipiers, where a new function (L) is defined in terms of the mass baance reations of the system components:

3 Combining reaction inetics to the muti-component Gibbs Energy cacuation 3 L = G λ φ (2) where λ s are the undetermined mutipiers of the Lagrange method and represent the mass baances of the system components. Using these and the partia derivatives of the Lagrange function, the Gibbs energy minimum can be soved in terms of the equiibrium moar amounts at a given temperature and pressure. The Lagrange mutipiers as we become soved representing the equiibrium chemica potentias of the system components [,3,4]. Thus, for the chemica potentias of any constituent there is: μ = aλ ( =,2,... N) (3) = Equation (3) gives the chemica potentia of any constituent as a inear combination of the respective potentias of the system components. The muti-phase system may aso be reaction rate controed or infuenced by other stress- or fied- reated factors, in which case additiona constraining equations are needed. It is often possibe to find such extension of the conservation matrix by which a new Lagrangian mutipier can be used to sove the constraint potentia in the muti-phase system [3]. The conservation matrix A of the additionay constrained system is made up of the coefficients of a conservation equations vaid in the system. The rows of the matrix represent again its atogether N constituents. When merey chemica reaction equiibria are considered, the conventiona stoichiometric reations determine the conservation of atoms of eements and eectric charge and there are respective coumns for these. When additiona conservation equations are required, the new constraint appears as an additiona coumn (+) in the conservation matrix: a, a, a, + () () () ' an, a N, a N, + A = (4) ( 2) ( 2) ( 2) a N +, a N+, an +, + ( ) ( ) ( ) Ψ Ψ Ψ an, an, an, + Here the matrix eements for the stoichiometric phase constituents are presented in the firs coumns and the additiona coumn (+) represents the new conservation equation. Thus, the matrix eement a, + = 0 for a those φ

4 4 P. Kouari et a. constituents, which are not affected by the additiona constraint, whereas a, + is not zero for those constituents, which are affected by the said constraint. The mass baance of the tota system is not affected, if the moecuar mass of the component M + is chosen to be zero. When using A, the soution of min(g) again resuts with equation (3), that is the chemica potentias of the constituents become soved in terms of the eements of the extended matrix and the respective Lagrange mutipiers. There wi be one new mutipier for each new constraint equation, and they then represent either physica factors such as the eectrochemica part of a chemica potentia or constrained affinities [3] Systems with ineticay constrained reactions The above method can be used to conserve the amount of a species at its input vaue in a Gibbs ian cacuation. This vaue can then be agorithmicay connected to a reaction rate, and a series of Gibbs energy cacuations be performed where one or severa species ony change with a given reaction rate. If severa conditions for ineticay constrained species are used, the (non-zero) affinities of the chemica reactions are derived by using equation (3) and the stoichiometric coefficients of reactants and products in the constrained system [5]. The resut can be summarized in the foowing two equations: = ν μ ν a λ = 0 (equiibrium reactions) (5) = ν μ = ν a λ 0 (ineticay constrained reactions) (6) ' = + where ν is the stoichiometric coefficient (positive for a product, negative for a reactant) for the species in the reaction in question; is the number components in the system incuding those defined for inetic restrictions Cacuation of the rutiisation system with the Gibbsian method To avai the use of the method in a wide-spread thermochemica patform which utiizes the Lagrange mutipier technique, a further practica tric has been used. When e.g the program ChemApp [6] is appied as the free energy minimiser, the conservation matrix is usuay given with constant (stoichiometric) coefficients. To enabe constraining of forward and reverse reactions, the matrix is further extended by introducing a new row to contro the desired reactions. For a singe reaction, the rows can be denoted with R+ and R-, as they are connected to the constraint component R with non-zero a -vaues.

5 Combining reaction inetics to the muti-component Gibbs Energy cacuation 5 Both of them are defined as stoichiometric phases for system input definitions, but are not aowed to present in the cacuated state of Gibbs energy minimum. For the formation reaction R+, a = and for the decomposing R-, a =-. The respective connection to the actua species, which are ineticay conserved, must aso have non-zero stoichiometric matrix eements. The standard chemica potentias, which appear as input data in the cacuation, are set to zero for the phases R+ and R-. More than one species can again be invoved and the corresponding a -vaues become defined by the actua reaction stoichiometry. Tabe. Data for the TiO(OH) 2 cacination reactions (T = 000 C) Reaction ΔH J. mo - ΔG J. mo - Reaction rate equation TiO(OH) 2 TiO 2 (An) + H 2 O assumed equiibrium TiO 2 (An) TiO 2 (Ru) (ineticay constrained reaction) E a J. mo - A h ξ = -(-. t) E+7 In Tabe., a simpe system representing Titanium oxyhydrate [TiO(OH) 2 ] cacination is described. From experience it is nown that the H 2 O reease from the oxyhydrate is a fast reaction at ca. 80 C. The anatase-rutie transformation is a sow, yet spontaneous, soid state reaction, for which the rate constant () is nown in terms of the activation energy (E a ) and frequency factor (A). In tabe 2, the conservation matrix and its extension to incude one reactive component R is presented. The shaded area shows the non-restricted equiibrium system. Tabe 2. Stoichiometric matrix for the ineticay constrained rutiisation system Species H 2 O TiO 2 R H 2 O-gas 0 0 TiO(OH) 2 0 TiO 2 (An) 0 0 TiO 2 (Ru) 0 R+ 0 0 R In Figure, the time-dependent rutie [TiO 2 (Ru)] fraction ξ=ξ(t) is soved by using the constrained Gibbs energy minimization method. At each constant temperature the Gibbs energy appears as a monotonicay descending curve (ony the curve for 950 C is shown).

6 6 P. Kouari et a Rutie fraction Reaction rate (mode) meas. 995 C meas. 020 C meas. 045 C G J (mode, 995 C) Time / min Gibbs energy / J Rutiisation exotherm C J/(mo. min) C 045 C Time / min Figure. Use of a reaction rate constraint in a thermodynamic Gibbs energy system. The couped mode is aso used to cacuate the time-dependent exotherm of the rutiisation reaction. Reaction rate data is given in tabe. 3. Concusions The presented method can aso be used to introduce more compex mechanisms into a muti-component cacuation. The matrix A must be extended with one coumn and two rows for each ineticay constrained species [6]. Though the method was deveoped by using Lagrangian undetermined mutipiers, the fundamenta equations (3)-(6) derived from the extremum condition are stricty thermodynamic and do not depend on the mathematica method. Thus, a simiar approach can be pursued for when appying other minimization procedures. The combination of reaction rates with the muti-component method enabes direct cacuation of the thermodynamic state properties during a chemica change. The obvious advantage of such modes is the simutaneous and interdependent cacuation of a thermochemica changes in the process, which opens new possibiities for advanced muti-phase simuation. Acnowedgements TEKES, The Finnish Funding Agency for Technoogy and Innovation. References. W.R. Smith and R.W. Missen, Chemica Reaction Equiibrium Anaysis: Theory and Agorithms, Krieger Pubishing, Maabar, FLA, W.D. Seider and S. Widagdo, Fuid Phase Eq., 23 (996) P. Kouari and R. Paarre, Caphad, 30 (2006) V. Pesu and P.S, Agachi, (eds.), Proceedings of the 6 th European Symposium on Computer-Aided Process Engineering, (2006) P. Kouari and R. Paarre, Comp.Chem.Eng., 30 (2006)