18th European Symposium on Computer Aided Process Engineering – ESCAPE 18 Bertrand Braunschweig and Xavier Joulia (Editors) © 2008 Elsevier B.V./Ltd. All rights reserved. Thermochemical Multi-phase Models Applying the Constrained Gibbs Energy Method Risto Pajarre, Peter Blomberg, Pertti Koukkari VTT, Process Chemistry, P.O. Box 1000 FIN-02044 VTT, Finland; risto.pajarre@vtt.fi 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 undetermined 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. Keywords: Gibbs energy calculation, multi-phase chemistry, surface tension, ion exchange, reaction rate 1. Theory When additional energy or work terms affect the Gibbs free energy, [G = G(T,p,nk)] it is customary to transform the total differential of the Gibbs function to read as follows (Alberty 2001): N dG SdT Vdp k dnk k 1 F dx i i i (1) R. Pajarre et al. 2 where S denotes the entropy and V the volume of the system, μk = μk(T, p, nk) is the chemical potential of the species (k), T is temperature, p is pressure and nk refers to mole amounts of chemical substances. The last term (Fidxi) refers to additional generalised work effects. The Gibbs energy minimisation requires the use of the chemical potentials μk (partial molar Gibbs energies) of the constituents of the system. If additional work terms are not included, the multi-phase chemical equilibrium is solved with the mass balance constraints as the necessary subsidiary conditions. Yet, in many physical circumstances the potentials appear constrained also by other typically work-related factors. Then, immaterial constraints for chemical potentials can be applied together with the mass balance relations in a multi-phase Gibbs energy minimisation software. The additional conditions arise from a work or energy effect or they are due to a pre-defined affinity or extent of reaction. In the Gibbs energy minimisation performed by the Lagrange method the new constraints are set with additional Lagrangian multipliers (Koukkari, Pajarre 2006a). They assume the same mathematical form as the mass balance conditions, but with a zero molar mass for the new components. The equations governing the chemical potentials and their respective constraints in the multi-phase system become as follows: k N a k 1 kj l a j (k 1,2,...N ) (2) nk j 0 ( j 1,2, l ) (3) j 1 kj where N is number of different chemical constituents as formed from l system components by means of the conservation matrix akj. The πj’s represent the chemical potentials of the components in the system (Koukkari, Pajarre 2006a; Perez-Cisneros et al. 2007) and equation (3) includes in addition to the mass balance constraints the necessary immaterial conditions, such as electroneutrality, conservation of surface area, metastability of a given constituent etc.. Equations (2) and (3) can then be used to solve for the unknowns (nk, πj). For equilibrium problems, the Gibbs energy minimisation with given input amounts and the system temperature and pressure is sufficient to solve for the constraint potential and system composition. In non-equilibrium systems, an external algorithm of e.g. reaction kinetics can furthermore be necessary to set the extent of change (Koukkari, Pajarre 2006b). In Table 1 some examples of the constraints and their respective conjugate potentials have been listed. Thermochemical Multi-phase Models Applying the Constrained Gibbs Energy Method Table 1. Examples of constraint potentials in the Gibbs free energy method System Constraint Conjugate potential Examples Chemical equilibrium systems Mass balance Chemical potential of system components Multi-phase chemical equilibria N a k 1 Systems with surface/interface equilibria kj nk b j Surface area Ns Ak nks A j G k b j a kj Surface energy per unit area (A0) area k 1 Electrochemical multi-phase systems Charge neutrality N z k nk 0 Electric potential energy per mole of charge q k 1 Rate-controlled complex systems Reacted amount from the extent of reaction N a k 1 kr n k bri G A0 barea G F bq Affinity of chemical reactions r i G G Affi bri i Phase diagrams Interface compositions Nanoparticle phase diagrams Membrane systems fibre and pulp suspensions Metastable phases or constituents; Non-equilibrium chemical reactors akj = element of conservation matrix ; N = nr. of constituents. ; bj = molar amount of component jth component, b(X) = molar amount of component ‘X’ (b(area) = A/A0) πi = Chemical potential of ith component ; π(X) = Chemical potential of component ‘X’ A = total surface area; Ak = molar surface area ; = surface tension; A0 = constant with dimensions of m2/mol (numerical value arbitrary) zk = charge number of species ; F = Faraday const. ; = electric potential difference between phases Affi = Affinity of ith reaction 2. Example cases 2.1. Surface energy of mixtures Surface energy can appear as an additional term in the Gibbs energy function of a multicomponent system. Assuming a flat, monomolecular surface layer where each constituent of the surface phase occupies a characteristic molar surface area Ak, the Gibbs energy can be written as: 3 R. Pajarre et al. 4 G kb nkb ks nks Ak nks k k (4) k where supersrcipts b and s refer to bulk and surface respectively. At equilibrium, the total surface area (A) is constant and the immaterial constraint corresponding equation (3) can be written as presented in Table 1. (Pajarre et al. 2006). With this and material balance constraints formulated in such a way, that a minor amount of the surface phase will appear, the surface tension can be solved by min(G) calculation as a conjugate potential, together with the chemical potentials of the bulk and surface phases.. The molar surface areas and surface energies of pure substances are needed for the standard chemical potentials of the surface constituents: k0, s k0 Ak k (5) It follows that the composition and surface tension of a mixture can be solved by Gibbs energy minimisation when the initial amounts, the molar surface areas and surface tensions of pure substances and the excess energy functions are given as input data. In Fig. 1 the method has been applied to a binary water-ethanol system. Surface tension (N/m) 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0 0.2 0.4 0.6 0.8 1 x(C2H5OH) bulk Figure 1. Surface tension of a water-ethanol system. 2.2. Donnan potential and salt solubilities in pulp suspensions The concept of ion-exchanging (Donnan) equilibrium has been conventionally used to characterize electrolyte interactions with cellulose fibres (Towers, Scallan 1996). Because of its simplicity it is an attractive approach to modelling of fibre-ion interactions. The aqueous solvent in the pulp system is divided between two subvolumes: The volume of water surrounding the fibres and the volume of water inside the fibre walls. The amount of water enclosed by the fibre walls depends on the type of pulp and to some extent the chemical composition of the suspension liquid, typical values being around 1-1.5 l/kg of dry weight. This value can be measured and is then regarded as constant. Pulp contains fiber-bound acidic groups that can dissociate giving the fibre an anionic charge. The amount of these groups, as well as their dissociation constants can be quantified with standard potentiometric methods, and thus, the respective G-values for Gibbs energy calculations are accessible. A basic Donnan model however, is insufficient to deal several practical problems that deal with dissolution and precipitation equilibria. These may involve carbonaceous and other salts as well as the effect of gaseous pH-agents, such as carbon dioxide and chlorine dioxide. Thence, a multi-phase approach based on the Gibbs free energy Thermochemical Multi-phase Models Applying the Constrained Gibbs Energy Method minimisation was developed, where the Donnan theory for ion exchange between two aqueous phases was incorporated to a multi-phase Gibbs’ian system. (Koukkari et al. 2002). The necessary constraint is electroneutrality in both aqueous phases, and the electrochemical potential difference becomes solved formally as another Lagrange coefficient as presented in Table 1. This approach also enables the use of a more elaborate ionic interaction models, such as that of Pitzer, instead of the ideal solution assumption. Model curve in Fig. 2 has been calculated assuming just one acidic group (pKa=4) as reported by Towers and Scallan. Donnan potential, V . 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 -0.01 0 2 4 6 8 10 12 pH Ca Mg Mn model Figure 2. The Donnan potential has been calculated as a function of pH and compared to measured values of the concentration ratios of cationic species Ca2+, Mg2+ and Mn2+ in a pulp suspension. The open diamond at highest pH value is for Manganese corrected for calculated precipitation of solid hydroxide. 2.3. Affinity of a rate constrained reaction Analogously with the equilibrium work terms it is possible to constrain the chemical affinity of selected reactions of a multicomponent system. Thus the Gibbs energy minimisation can be done for systems where one encounters metastable species or nonequilibrium constituents whose molar amounts are controlled by reaction rates. The constraining by reaction rates is most practically performed by determining the extent of a given reaction (r) from the experimental rate data (Koukkari and Niemelä 1997; Perez-Cisneros et al. 1997). As this can be done incrementally for any reaction with known reaction kinetics, the extent of reaction can be ‘frozen’ for any instant of time while the min(G) procedure is performed for the whole system, as presented in Table 1. The amount of the constraining component remains conserved during the minimisation, while otherwise the system may reach equilibrium. In terms of affinities of the reactions one may conclude that the affinity of fast (equilibrium) reactions becomes zero, while the affinity of the constrained reaction results with a positive (non-zero) value until the global equilibrium is reached (Fig. 3). Performing these calculations assumes local chemical equilibrium to be valid for the homogeneous (well-stirred) system (or in system that can be divided to such parts) and that the chemical relaxation rate is slow compared with other relaxation times, including local heat exchange. The model results can be validated by measured laboratory or process data. 5 R. Pajarre et al. 6 Gibbs free energy [kJ] -1930 -1940 -1950 -1960 140 1 0.8 0.6 0.4 0.2 0 120 100 0 5 -1970 10 15 20 25 Time [ms] Gibbs energy -1980 Affinity 80 60 40 Affinity [kJ/mol] Extent of reaction -1920 -1990 20 -2000 -2010 0 0 5 10 15 20 25 Time [ms] Figure 3. The reaction rate controlled oxidation reaction of TiCl 4 in the Ti-O2-Cl2 -system at 1360 C, showing the monotonically decreasing total Gibbs free energy and the affinity of the oxidation reaction on the same time scale. 3. Conclusion The constrained Gibbs energy applies to modeling of such systems as the electrochemical Donnan equilibria in aqueous systems containing semi-permeable interfaces, phase composition in surface-energy controlled systems and in systems with affinities controlled by chemical reaction kinetics. The new method has been successfully applied in calculating distribution coefficients for metal cations together with pH-values in pulp suspensions, in calculation of surface tension of various mixtures and in thermochemical process modelling involving chemical reaction rates. It can be used in assessment of thermodynamic properties, service and design applications, process scale-up, study of new chemical concepts and search for new, more economic process alternatives. References R. A. Alberty, 2001, Use of Legendre transforms in chemical thermodynamics, Pure Appl. Chem., 73(8), 1349-1380. P. Koukkari, J. Niemelä, J, 1997, Time-dependent reactor simulation by stationary state thermochemistry, Comput. Chem. Eng., 21(3), 245 – 253 P. Koukkari, R. Pajarre, 2006, Calculation of constrained equilibria by Gibbs energy minimisation, Calphad, 30, 18-26. P. Koukkari R. Pajarre, 2006, Introducing mechanistic kinetics to the Lagrangian Gibbs energy calculation, Comput. Chem. 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