Simplified Explicit Methods

Both the exact solution and the Toor, Stewart, and Prober methods discussed above require an iterative approach to the calculation of the fluxes. In addition, the calculations are somewhat time consuming, especially when done by hand. It would be nice to have a method of calculating the fluxes that involved no iterations and yet was sufficiently accurate (when compared to these more rigorous methods) to be useful in engineering calculations. 

There are two such methods one is due to Krishna (1979d, 1981b), the other to Burghardt and Krupiczka (1975) and its generalization by Taylor and Smith (1982). 

Our starting point for the development of these explicit methods is the ditfusion equation 

Equations 8.5.1.3 are quite general they involve no assumptions regarding the constancy of particular matrices they apply to mixtures with any number of components and for any relationship between the fluxes. It is at this point where any assumptions necessary to solve Eqs. (8.5.1-8.5.3) must be made. In the three methods to be discussed below we proceed in exactly the same way as we did when deriving the exact solution and the solution to the linearized equations first obtain the composition profiles, then differentiate to obtain the gradients at the film boundary, and combine the result with Eq. 8.5.3 to obtain the working flux equations. 

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For physically realistic and bounded results, it is necessary to ensure that all the coefficients of the discretization equation are positive. This requirement imposes restrictions on the time step that can be used with different values of 0. It can be seen that a fully implicit method with 0 equal to unity is unconditionally stable. Detailed stability analysis is rather complex when both convection and diffusion are present. In general, simplified criteria may be used when an explicit method is used in practical simulations ... 

In the case of the Euler methods, the problem can be simplified by first applying the explicit method to predict a value y. ... 

As indicated by its name, it involves assessing layers of protection other than just the instrument protective functions. For instance, a contribution toward risk reduction by independent protective layers (IPLs) such as alarms and operators or basic process control is explicitly defined as a risk reduction factor. The combination of the risk reduction factors for all IPLs provides the total risk reduction possible. It is fundamentally a simplified quantitative method that considers the risk reduction contributed from each IPL typically by order of magnitude risk reduction (i.e., say 0.1 for a DCS, or 0.01 for a relief valve, etc.). 

The Extended Iliickel method, for example, does not explicitly consider the elTects of electron-electron repulsions but incorporates rep 11 Ision s into a sin gle-clectron poten tial. Th is simplifies th c solution of the Schrbdinger equation and allows IlyperChem to compute the poten tial energy as the sum of the energies for each electron. 

Another way is to reduce the magnitude of the problem by eliminating the explicit solvent degrees of freedom from the calculation and representing them in another way. Methods of this nature, which retain the framework of molecular dynamics but replace the solvent by a variety of simplified models, are discussed in Chapters 7 and 19 of this book. An alternative approach is to move away from Newtonian molecular dynamics toward stochastic dynamics. 

Because of the success of the r12 method in the applications, one had almost universally in the literature adopted the idea of the necessity of introducing the interelectronic distances r j explicitly in the total wave function (see, e.g., Coulson 1938). It was there-fore essential for the development that Slater,39 Boys, and some other authors at about 1950 started emphasizing the fact that a wave function of any desired accuracy could be obtained by superposition of configurations, i.e., by summing a series of Slater determinants (Eq. 11.38) built up from a complete basic one-electron set. Numerical applications on atoms and molecules were started by means of the new modern electronic computers, and the results have been very encouraging. It is true that a wave function delivered by the machine may be the sum of a very large number of determinants, but the result may afterwards be mathematically simplified and physically interpreted by means of natural orbitals.22,17... 

Although complete, fully polarizable QM/MM schemes are computationally demanding, a simplified version of this formalism was arguably the first QM/MM approach to be described (Warshel and Levitt 1976), and the method still sees some use today. The simplification involves replacing explicit, polarizable MM molecules with a three-dimensional grid of fixed, polarizable dipoles - each a so-called Langevin dipole (LD) as it is required to obey... 

In some cases, the major disadvantage of second-order methods may be the programming effort required to derive explicit expressions for the Hessian elements, whose number increases as the square of the number of parameters. In Sect. 3.6, a simplified form of the Hessian matrix is derived by considering the particular form of the least-squares objective functions. 

Depending on the numerical techniques available for integration of the model equations, model reformulation or simplified version of the original model has always been the first step. In the Sixties and Seventies simplified models as sets of ordinary differential equations (ODEs) were developed. Explicit Euler method or explicit Runge-Kutta method (Huckaba and Danly, 1960 Domenech and Enjalbert, 1981 Coward, 1967 Robinson, 1969, 1970 etc) were used to integrate such model equations. The ODE models ignored column holdup and therefore non-stiff integration techniques were suitable for those models. 

The computational methods developed to deal with the various (explicitly correlated) resonance-theoretic models turn out to be most powerful for the more highly simplified models, of the preceding section 2. It is emphasized that these more highly simplified schemes need not necessarily entail significant approximation or loss of accuracy when properly parameterized. Moreover, these more simplified schemes often allow general conclusions for whole sequences or sets of molecules. For the higher-level models the manner of solution turns out to... 

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