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The theoretical background of

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GTT-Technologies The theoretical background of FactSage The following slides give an abridged overview of the major underlying principles of the calculational modules of FactSage.

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GTT-Technologies The Gibbs Energy Tree Mathematical methods are used to derive more information from the Gibbs energy ( of phase(s) or whole systems ) Gibbs Energy Minimisation Gibbs-Duhem Legendre Transform. Partial Derivatives with Respect to x, T or P Equilibria Phase Diagram Maxwell H, U, F i,c p(i), H (i),S (i),a i,v i Mathematical Method Calculational result derived from G

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GTT-Technologies Thermodynamic potentials and their natural variables Variables Gibbs energy:G=G(T,p, n i,...) Enthalpy:H=H(S,P, n i,...) Free energy:A=A(T,V, n i,...) Internal energy:U=U(S,V, n i,...) Interrelationships: A=U T S H=U P V G=H T S = U P V T S

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GTT-Technologies Maxwell-relations: Thermodynamic potentials and their natural variables P P T T SUV HA G SV and

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GTT-Technologies Thermodynamic potentials and their natural Equilibrium condition:

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GTT-Technologies Temperature Composition Use of model equations permits to start at either end! Gibbs-Duhem integrationPartial Operator Integral quantity: G, H, S, c p Partial quantity: µ i, h i, s i, c p(i) Thermodynamic properties from the Gibbs-energy

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GTT-Technologies With (G is an extensive property!) one obtains J.W. Gibbs defined the chemical potential of a component as: Thermodynamic properties from the Gibbs-energy

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GTT-Technologies Transformation to mole fractions : = partial operator Thermodynamic properties from the Gibbs-energy

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GTT-Technologies Gibbs energy function for a pure substance G(T) (i.e. neglecting pressure terms) is calculated from the enthalpy H(T) and the entropy S(T) using the well-known Gibbs-Helmholtz relation: In this H(T) is and S(T) is Thus for a given T-dependence of the c p -polynomial (for example after Meyer and Kelley) one obtains for G(T):

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GTT-Technologies Gibbs energy function for a solution reference term ideal termexcess termAs shown above G m (T,x) for a solution consists of three contributions: the reference term, the ideal term and the excess term. For a simple substitutional solution (only one lattice site with random occupation) one obtains using the well-known Redlich-Kister-Muggianu polynomial for the excess terms:

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GTT-Technologies Equilibrium condition: or Reaction : n A A + n B B +... = n S S + n T T +... Generally : dT = 0dp = 0 For constant T and p, i.e. dT = 0 and dp = 0, and no other work terms: Equilibrium considerations a) Stoichiometric reactions

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GTT-Technologies changes dn i change in extend of reaction d For a stoichiometric reaction the changes dn i are given by the stoichiometric coefficients n i and the change in extend of reaction d . Thus the problem becomes one-dimensional. One obtains: [see the following graph for an example of G = G(x) ] Equilibrium considerations a) Stoichiometric reactions

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GTT-Technologies Gibbs Energy as a function of extent of the reaction 2NH 3 N 2 + 3H 2 for various temperatures. It is assumed, that the changes of enthalpy and entropy are constant. Extent of Reaction Gibbs energy G 0.00.10.20.30.40.50.60.70.80.91.0 T = 400K T = 500K T = 550K Equilibrium considerations a) Stoichiometric reactions

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GTT-Technologies Separation of variables results in : Thus the equilibrium condition for a stoichiometric reaction is: Introduction of standard potentials i ° and activities a i yields: One obtains: Equilibrium considerations a) Stoichiometric reactions

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GTT-Technologies Law of Mass Action It follows the Law of Mass Action: where the product or is the well-known Equilibrium Constant. Equilibrium considerations a) Stoichiometric reactions The REACTION module permits a multitude of Law of Mass Action. calculations which are based on the Law of Mass Action.

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GTT-Technologies Complex Equilibria Many components, many phases (solution phases), constant T and p : with or Equilibrium considerations b) Multi-component multi-phase approach

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GTT-Technologies Massbalance constraint j = 1,..., n of components b Lagrangeian Multipliers M j turn out to be the chemical potentials of the system components at equilibrium: Equilibrium considerations b) Multi-component multi-phase approach

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GTT-Technologies Example of a stoichiometric matrix for the gas-metal-slag system Fe-N-O-C-Ca-Si-Mg a ij j i Equilibrium considerations b) Multi-component multi-phase approach

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GTT-Technologies Modelling of Gibbs energy of (solution) phases Pure Substance(stoichiometric) Solution phase Equilibrium considerations b) Multi-component multi-phase approach Choose appropriate reference state and ideal term, then check for deviations from ideality. See Page 11 for the simple substitutional case.

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GTT-Technologies Use the EQUILIB module to execute a multitude of calculations based on the complex equilibrium approach outlined above, e.g. for combustion of carbon or gases, aqueous solutions, metal inclusions, gas-metal-slag cases, and many others. constraints in such calculations NOTE: The use of constraints in such calculations (such as fixed heat balances, or the occurrence of a predefined phase) makes this module even more versatile. Equilibrium considerations Multi-component multi-phase approach

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GTT-Technologies Phase diagrams as projections of Gibbs energy plots Hillert has pointed out, that what is called a phase diagram is derivable from a projection of a so-called property diagram. The Gibbs energy as the property is plotted along the z-axis as a function of two other variables x and y. From the minimum condition for the equilibrium the phase diagram can be derived as a projection onto the x-y-plane. (See the following graphs for illustrations of this principle.)

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GTT-Technologies P T P T Unary system: projection from -T-p diagram Phase diagrams as projections of Gibbs energy plots

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GTT-Technologies Binary system: projection from G-T-x diagram, p = const. 300 400 500 600 700 1.0 0.5 0.0 -0.5 1.0 0.8 0.6 0.4 0.2 0.0 T Cu x Ni Ni G Phase diagrams as projections of Gibbs energy plots

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GTT-Technologies Ternary system: projection from G-x 1 -x 2 diagram, T = const and p = const Phase diagrams as projections of Gibbs energy plots

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GTT-Technologies Use the PHASE DIAGRAM module to generate a multitude of phase diagrams for unary, binary, ternary or even higher order systems. NOTE:The PHASE DIAGRAM module permits the choice of T, P, m (as RT ln a), a (as ln a), mol (x) or weight (w) fraction as axis variables. Multi-component phase diagrams require the use of an appropriate number of constants, e.g. in a ternary isopleth diagram T vs x one molar ratio has to be kept constant. Phase diagrams generated with FactSage

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GTT-Technologies Gibbs-Duhem: N-Component System (A-B-C-…-N) SVnAnBnNSVnAnBnN T -P µ A µ B µ N Extensive variables Corresponding potentials

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GTT-Technologies N-component system (1)Choose n potentials: 1, 2, …, n (2)From the non-corresponding extensive variables (q n+1, q n+2, … ), form (N+1-n) independent ratios (Q n+1, Q n+2, …, Q N+1 ). Example: Choice of Variables which always give a True Phase Diagram [ 1, 2, …, n ; Q n+1, Q n+2, …, Q N+1 ] are then the (N+1) variables of which 2 are chosen as axes and the remainder are held constant.

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GTT-Technologies MgO-CaO Binary System 1 = Tfor y-axis 2 = -Pconstant for x-axis S T V -P n MgO µ MgO n CaO µ CaO Extensive variables and corresponding potentials Chosen axes variables and constants

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GTT-Technologies S T V -P n Fe m Fe n Cr m Cr f 1 = T (constant) f 2 = -P (constant) x-axis (constant) Fe - Cr - S - O System

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GTT-Technologies Fe - Cr - C System - improper choice of axes variables S T V -P n C m C n Fe m Fe n Cr m Cr f 1 = T (constant) f 2 = -P (constant) f 3 = m C a C for x-axis and Q 4 for y-axis (NOT OK) (OK) Requirement:

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GTT-Technologies This is NOT a true phase diagram. Reason: n C must NOT be used in formula for mole fraction when a C is an axis variable. NOTE: FactSage users are safe since they are not given this particular choice of axes variables. M 23 C 6 M7C3M7C3 bcc fcc cementite log(a c ) Mole fraction of Cr 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 -3-2012 Fe - Cr - C System - improper choice of axes variables

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