Computational methods Gibbs

Most of this book concerns the development and application of theoretical thermodynamic models, as these are the basis of the CALPHAD method. However, none of this would be possible without the existence of the computational methods and software which allow these models to be applied in practice. In essence, the issues involved in computational methods are less diverse and mainly revolve around Gibbs energy minimisation. In addition, there are optimiser codes which are used for the thermodynamic assessment of phase equilibria. The essential aim of these codes is to reduce the statistical error between calculated phase equilibria, thermodynamic properties and the equivalent experimentally measured quantities. 

The local composition model (LCM) is an excess Gibbs energy model for electrolyte systems from which activity coefficients can be derived. Chen and co-workers (17-19) presented the original LCM activity coefficient equations for binary and multicomponent systems. The LCM equations were subsequently modified (1, 2) and used in the ASPEN process simulator (Aspen Technology Inc.) as a means of handling chemical processes with electrolytes. The LCM activity coefficient equations are explicit functions, and require computational methods. Due to length and complexity, only the salient features of the LCM equations will be reviewed in this paper. The Aspen Plus Electrolyte Manual (1) and Taylor (21) present the final form of the LCM binary and multicomponent equations used in this work. 

The definition of the system also specifies a list of the independent reactions taken into account, including inter-phase reactions. It is not necessary for those reactions to actually take place, because we are dealing here with equilibrium calculations. Yet it must be borne in mind that such equilibria impose specific relations between the molar Gibbs energies of their components. Certain automated computation methods, which are specialized for certain t5q3es of systems, reveal only those equilibria. 

Brinkley (1947) published the first algorithm to solve numerically for the equilibrium state of a multicomponent system. His method, intended for a desk calculator, was soon applied on digital computers. The method was based on evaluating equations for equilibrium constants, which, of course, are the mathematical expression of the minimum point in Gibbs free energy for a reaction. 

As stated, the most commonly used procedure for temperature and composition calculations is the versatile computer program of Gordon and McBride [4], who use the minimization of the Gibbs free energy technique and a descent Newton-Raphson method to solve the equations iteratively. A similar method for solving the equations when equilibrium constants are used is shown in Ref. [7],... 

The phase equilibrium between a liquid and a gas can be computed by the Gibbs ensemble Monte Carlo method. We create two boxes, where the first box represents the dense phase and the second one represents the dilute phase. Each particle in the boxes experiences a Lennard-Jones potential from all the other particles. Three types of motion will be conducted at random the first one is particle translational movement in each box, the second one is moving a small volume from one box and adding to the other box, the third one is removing a particle from one box and inserting in the other box. After many such moves, the two boxes reach equilibrium with one another, with the same temperature and pressure, and we can compute their densities. 

Gibbs free-energy minimization offers an elegant computational manner without the need to specify the stoichiometry. In addition, phase equilibrium is accounted for. Since occasionally the method fails, considering explicit reactions is safer. 

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