Fixed phase algorithm

A second drawback of the fixed-time approach is that it is mathematically not exact. The accuracy of the simulation is governed by the choice of the time step used. Only in the limit of an infinitesimal time step does the method becomes mathematically exact. For simulations performed at very small time steps, accuracy is not an issue. Although the fixed time algorithm has proven to be fairly effective for certain gas-phase reaction systemsl " , nearly all of the published studies on surfaces use what is known as the variable time-step approachl . 

The KTTS depends upon an absolute 2ero and one fixed point through which a straight line is projected. Because they are not ideally linear, practicable interpolation thermometers require additional fixed points to describe their individual characteristics. Thus a suitable number of fixed points, ie, temperatures at which pure substances in nature can exist in two- or three-phase equiUbrium, together with specification of an interpolation instmment and appropriate algorithms, define a temperature scale. The temperature values of the fixed points are assigned values based on adjustments of data obtained by thermodynamic measurements such as gas thermometry. 

When the kinetics are unknown, still-useful information can be obtained by finding equilibrium compositions at fixed temperature or adiabatically, or at some specified approach to the adiabatic temperature, say within 25°C (45°F) of it. Such calculations require only an input of the components of the feed and produc ts and their thermodynamic properties, not their stoichiometric relations, and are based on Gibbs energy minimization. Computer programs appear, for instance, in Smith and Missen Chemical Reaction Equilibrium Analysis Theory and Algorithms, Wiley, 1982), but the problem often is laborious enough to warrant use of one of the several available commercial services and their data banks. Several simpler cases with specified stoichiometries are solved by Walas Phase Equilibiia in Chemical Engineering, Butterworths, 1985). 

For planar particles covered by an ion-penetrable membrane an efficient numerical algorithm is available for the resolution of (1). The distribution of fixed charge in the membrane phase can be nonuniform, and the electrolyte in the liquid phase can be either a b or mixed (a b) + (c d) [35]. For two identical, planar parallel particles, this algorithm can be applied to estimate... 

The computer algorithm for determining the compositions of three and four equilibrium phases at a fixed temperature and pressure is based on the successive substitution algorithm described by Heidemann (10). An initial estimate of the phase compositions was obtained from either the measured phase compositions at the temperature and pressure of interest, or from previous calculations at a nearby temperature and pressure. Trivial solutions giving two phases of identical composition would often result when the initial compositions were not close to the actual equilibrium compositions. 

The fixed step-size parameter, fi, used in the TAG algorithm was also calculated during the tuning phase. From the theory of the LMS algorithm it can be shown that for stability the step size must satisfy the condition [7]... 

We observed in Chapter 3 that a chemical system is at thermodynamic equilibrium when the Gibbs free energy of the system is at a minimum. For a given pressure, temperature, and bulk composition, at equilibrium there will be one or more phases in which the concentrations of all species are fixed. Many different methods have been developed to compute the equilibrium state, as outlined by Van Zeggeren and Storey (1970) and Smith and Missen (1982). As an illustration, we will summarize an algorithm derived by Harvie et al. (1987), which has been applied with great success to geochemical systems. 

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