Discretization Approaches

In this section an outline of a selection of discretization methods frequently used in chemical reactor engineering is given. 

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The simplest discrete approach is the solvaton method 65) which calculates above all the electrostatic interaction between the molecule and the solvent. The solvent is represented by a Active molecule built up from so-called solvatones. The most sophisticated discrete model is the supermolecule approach 661 in which the solvent molecules are included in the quantum chemical calculation as individual molecules. Here, information about the structure of the solvent cage and about the specific interactions between solvent and solute can be obtained. But this approach is connected with a great effort, because a lot of optimizations of geometry with ab initio calculations should be completed 67). A very simple supermolecule (CH3+ + 2 solvent molecules) was calculated with a semiempirical method in Ref.15). 

This problem can be solved using a combined optimization and constraint model solution strategy (Muske and Edgar, 1998) by converting the differential equations to algebraic constraints using orthogonal collocation or some other model discretization approach. 

Computational methods to study solvent effects on NMR (Sadlej Pecul) and EPR (Barone, Cimino Pavone) parameters are presented and discussed within the PCM as well their generalizations to hybrid continuum/discrete approaches in which the presence of specific interactions (e.g. solute-solvents H-bonds) is explicitly taken into account by including some solvent molecules strongly interacting with the solute. 

Solvent effects on vibrational spectroscopies are analyzed by Cappelli using classical and quantum mechanical continuum models. In particular, PCM and combined PCM/discrete approaches are used to model reaction and local field effects. 

Lente proposed a discrete-state stochastic modeling approach in which chiral amplification could be described by a quadratic autocatalytic model without considering cross-inhibition [67,68]. However, the discrepancy between the usually employed deterministic kinetic approach, which reinforces the need for cross-inhibition, and the discrete-state stochastic approach is only apparent. The discrete approach considers the repetitive reproduction of single molecules which, in the case of a chiral system, obviously are individually all enantiomerically pure. Hence, basically no amplification of the ee occurs at all during the discrete scenario. It has been indicated that deter-... 

Calculating the dispersions of the random processes Aff s) and A/ (s), one has to take into account that relations (3.50) and (3.51) for the random forces are written for the continuous time, so that in the discrete approach one has... 

Garcia L., D Alessandro G., Bioulac B., Hammond C. High-frequency stimulation in Parkinson s disease more or less Trends in Neuroscience, 2005, 28, 209-216. Garenne A. and Chauvet G. A. A discrete approach for a model of temporal learning by the cerebellum in silico classical conditioning of the eyeblink reflex. / Integr Neurosci, 2004, 3,301-18. 

This PWE was used in [18] to obtain the numerical results. For the numerical implementation the B-spline approximation [21] was chosen that represents actually the refined version of the space discretization approach. In Table 1 the convergence of the PWE approach with the multicommutator expansion is presented for the lowest-order SE correction for the ground state of hydrogenlike ions with Z = 10. The minimal set of parameters for the numerical spline calcuations was chosen to be the number of grid points N = 20, the number of splines k = 9. This minimal set allowed to keep a controlled inaccuracy below 10%. What is most important for the further generalization of the PWE approach to the second-order SESE calculation is that with Zmax = 3 the inaccuracy is already below 10% (see Table 1). The same picture holds with even higher accuracy for larger Z values. The direct renormalization approach is not necessarily connected with the PWE. In [19] this approach in the form of the multicommutator expansion (Eq. (16)) was employed in combination with the Taylor expansion in powers of (Ea — En>)r 12 The numerical procedure with the use of B-splines and 3 terms of Taylor series yielded an accuracy comparable with the PWE-expansion with Zmax = 3. 

Figure 3. Interaction free energy in the discrete approach as a function of the separation distance between surfaces, calculated in the discrete model for various values of g (a, top) linear scale (which shows better the oscillatory behavior near the surface (b, bottom) logarithmic scale (which better reveals the behavior at large distances).

Figure 5. (a, top) Polarization as a function of the distance from one surface. The solution of the discrete approach (eq 40, circles) and its analytical interpolation (eq 41, line 1) are compared to the solution obtained via the continuous approximation (eq 31, line 2). (b, bottom) Interaction energy, as a function of separation distance, for the discrete approach and for the continuous approximation. 

SOLVATION MODELS FOR MOLECULAR PROPERTIES CONTINUUM VERSUS DISCRETE APPROACHES... 

The collective properties of bulk material typically reflect the behavior of tens of thousands of molecules in a volume of at least 106 A3 [123], The description of the electrostatic and response properties of such volumes is obviously beyond any discrete approach and one has to resort to experimental information, i.e., the dielectric constant. In the Introduction we argued that if sufficient solvation shells are included in a calculation, the effect of an enveloping continuum can be neglected. Nevertheless we give here, for completeness sake, an explicit formulation of the coupling between a set of point charges and polarizabilities and a dielectric continuum. 

We present the major results established in the description of crazing and the recent developments in this field. Crazing has been investigated within continuum or discrete approaches (e.g., spring networks or molecular dynamics calculations to model the craze fibrils), which have provided phenomenological or physically based descriptions. Both are included in the presentation of the crazing process, since they will provide the basis for the recent cohesive surface model used to represent crazing in a finite element analysis [20-22],... 

In this section, reference is made to discrete approaches for the modeling of gas/condensate flow through mesoporous structures. Capillary network models are developed and evaluated by comparison with experimental results from the literature. Finally, experimental results obtained in our laboratory are presented on two mesoporous membranes, made by compaction of alumina microspheres, with porosities 0.41 and 0.48, respectively. 

Figure 1 - Illustrative example of the discrete approach - initialization.

The combination of all the points reported above seems to indicate versatile and efficient ab initio procedures as the best choice. However, there are other considerations to be added. Both continuum and discrete approaches suffer from limitations due to the separation of the whole liquid system into two parts, i.e. the primary part, or solute, and the secondary larger part, the solvent. These limitations cannot be eliminated until more holistic methods will be fully developed. We have already discussed some problems related to the shape of the cavity, which is the key point of this separation in continuum methods. We would like to remark that discrete methods suffer from similar problems of definition a tiny change in the non-boded interaction parameters in the solute-solvent interaction potential corresponds to a not so small change in the cavity shape. 

Discrete approaches allow computing G directly from the distance map but generally give results that are prone to high digitization effects. Our approach takes... 

Discretization Approaches 999 Substitution of this choice of Wi(z) gives ... 

Discretization Approaches 1005 Thus, in the Galerkin method the weak form (12.31) becomes... 

In order to accelerate the process optimization, Kawajiri and Biegler (2006b) have developed an efficient full discretization approach combined with a large-scale nonlinear programming method for the optimization of SMBs. More recently, they have extended this approach to a superstructure SMB formulation and used the e-constraint method to solve the bi-objective problem, where throughput and desorbent consumption were optimized (Kawajiri and Biegler, 2006a). 

When designing sensors, an initial decision has to be made between the discrete approach, with separated sensing element and electronic signal evaluation circuit,... 

The drawbacks of discrete analyzers are their mechanical complexity and high cost of operation. Sample cups, disposable cuvettes, rotors, and prepacked reagents increase the cost of individual assays above the acceptable limit for the strained budgets of most clinical laboratories. In addition, these machines are seldom used outside the clinical laboratory, because they are designed to handle three dozen of the most frequently required clinical tests. The advantages of the discrete approach are the ability of some of these instruments to perform assays via random access—which allows sequential assay of diverse analytes at will—and the capability of stat operation, which yields the analytical readout within 5-10 min after the machine has been switched on and a sample has been inserted by a technician. 

In terms of numerical methods, the dominance by FEM with equivalent continuum approach might not be most suitable for sparsely or moderately fractured hard rocks and more advanced methods and codes using discrete approach are needed. The issue of applicability of the equivalent media approach, the associated scale effects, and uncertainty evaluations need to be fully explored. The processes are dominated by coupled stress-flow problems and effects of thermo-chemical effects need more attention. More works for soils, clays, sands and other similar media, which are equally, if not more, important in the fields of geo-engineering and environments, seem also needed. 

A DISCRETE APPROACH TO MODELLING HYDROMECHANICAL ROCK RESPONSE OF FEDEX TUNNEL EXCAVATION (GRIMSEL UNDERGROUND RESEARCH LABORATORY, SWITZERLAND)... 

Comparing (6) with (1), we find that the conditions of equivalence between the (one-dimensional) continuum approach and the discrete approach are ... 

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