FAST 1.0 model

As shown in 4.4.1., two stages usually occur in surface nucleation, nucleation and then nuclei growth. If surface nucleation is fast (Model C) it is likely due to reaction of a gas with the solid particle. The reaction of a liquid is the other possibility, i.e.-... 

These observations imply that there exists a process variable - namely the total impurity holdup - whose dynamics are slow (in the sense defined in Chapter 2) and whose evolution is thus not captured by the fast model (4.20). As a consequence, one of the differential equations in (4.20) is redundant (i.e., these equations are not linearly independent). The steady-state conditions corresponding to the fast dynamics,... 

The fast model (6.12) describes the evolution of the enthalpies/temperatures of the individual units. Thus, control objectives related to the individual units (e.g., reactor temperature control) should be addressed in this time scale. The significant energy flows w1 associated with the internal streams are available as manipulated inputs to this end. Note that it is often practical to vary w1 by modifying a material flow rate, rather than by varying a stream s enthalpy/ temperature. 

Engelen, S. and Hubert, M., Fast model selection for robust calibration methods, Analytica Chimica Acta, 544, 219-228, 2005. 

Chipman, H. A. (1998). Fast model search for designed experiments. In Quality Improvement Through Statistical Methods. Editor B. Abraham, pages 205-220. Birkhauser, Boston. 

The process concept can be described by the advanced numerical model as detailed in the previous section. However, by assuming that the fronts that are formed during the different process steps are perfectly well defined (sharp), a simplified and relatively easy-to-solve and fast model can be developed, referred to as the sharp front approach . In the first place, this approach is a very useful tool to quickly investigate the influence of process parameters on process behaviour. Furthermore, it can be used in conceptual design studies. This section describes this sharp front approach and compares the outcomes with the more advanced numerical model. Finally, the influences of several process parameters are studied. 

Perhaps the earliest, and most well-known technique, is that developed by Smith (Reference 1). It still employs a PID controller but in addition includes a. fast model that predicts how the process will behave in the future. The controller uses the output of this model, rather than the actual PV, and can therefore be tuned as if there is no deadtime. A plant model is also included. Its purpose is to check whether the actual PV eventually matches the prediction and, if not, generate a correction term. Figure 7.1 shows the configuration. 

This is in stark contrast to the current model building workflow, where the user spends a lot of time doing tedious tasks that does not really add value. Moreover, as more information is gained over the lifecycle of the reservoir, the models are not necessarily updated due to the amount of model reworking that needs to be done. Using the framework presented here, new information is visible across all scales and is directly ready for utilization in all models once it becomes available. As such, fast model updates are now possible. 

We will describe integral equation approximations for the two-particle correlation fiinctions. There is no single approximation that is equally good for all interatomic potentials in the 3D world, but the solutions for a few important models can be obtained analytically. These include the Percus-Yevick (PY) approximation [27, 28] for hard spheres and the mean spherical (MS) approximation for charged hard spheres, for hard spheres with point dipoles and for atoms interacting with a Yukawa potential. Numerical solutions for other approximations, such as the hypemetted chain (EfNC) approximation for charged systems, are readily obtained by fast Fourier transfonn methods... 

For very fast reactions, as they are accessible to investigation by pico- and femtosecond laser spectroscopy, the separation of time scales into slow motion along the reaction path and fast relaxation of other degrees of freedom in most cases is no longer possible and it is necessary to consider dynamical models, which are not the topic of this section. But often the temperature, solvent or pressure dependence of reaction rate... 

Figure A3.9.3. Time-of-flight spectra for Ar scattered from Pt(l 11) at a surface temperature of 100 K [10], Points in the upper plot are actual experimental data. Curve tinough points is a fit to a model in which the bimodal distribution is composed of a sharp, fast moving (lienee short flight time), direct-inelastic (DI) component and a broad, slower moving, trapping-desorption (TD) component. These components are shown...

The approach is ideally suited to the study of IVR on fast timescales, which is the most important primary process in imimolecular reactions. The application of high-resolution rovibrational overtone spectroscopy to this problem has been extensively demonstrated. Effective Hamiltonian analyses alone are insufficient, as has been demonstrated by explicit quantum dynamical models based on ab initio theory [95]. The fast IVR characteristic of the CH cliromophore in various molecular environments is probably the most comprehensively studied example of the kind [96] (see chapter A3.13). The importance of this question to chemical kinetics can perhaps best be illustrated with the following examples. The atom recombination reaction... 

Fast P L and Truhlar D G 1998 Variational reaction path algorithm J. Chem. Phys. 109 3721 Billing G D 1992 Quantum classical reaction-path model for chemical reactions Chem. Phys. 161 245... 

Figure C3.2.12. Experimentally observed electron transfer time in psec (squares) and theoretical electron transfer times (survival times, Tau a and Tau b) predicted by an extended Sumi-Marcus model. For fast solvents tire survival times are a strong Emction of tire characteristic solvent relaxation dynamics. For slower solvents tire electron transfer occurs tlirough tire motion of intramolecular degrees of freedom. From [451.

Excitable media are some of tire most commonly observed reaction-diffusion systems in nature. An excitable system possesses a stable fixed point which responds to perturbations in a characteristic way small perturbations return quickly to tire fixed point, while larger perturbations tliat exceed a certain tlireshold value make a long excursion in concentration phase space before tire system returns to tire stable state. In many physical systems tliis behaviour is captured by tire dynamics of two concentration fields, a fast activator variable u witli cubic nullcline and a slow inhibitor variable u witli linear nullcline [31]. The FitzHugh-Nagumo equation [34], derived as a simple model for nerve impulse propagation but which can also apply to a chemical reaction scheme [35], is one of tire best known equations witli such activator-inlribitor kinetics ... 

Conical intersections can be broadly classified in two topological types peaked and sloped [189]. These are sketched in Figure 6. The peaked case is the classical theoretical model from Jahn-Teller and other systems where the minima in the lower surface are either side of the intersection point. As indicated, the dynamics of a system through such an intersection would be expected to move fast from the upper to lower adiabatic surfaces, and not return. In contrast, the sloped form occurs when both states have minima that lie on the same side of the intersection. Here, after crossing from the upper to lower surfaces, recrossing is very likely before relaxation to the ground-state minimum can occur. 

The first requirement is the definition of a low-dimensional space of reaction coordinates that still captures the essential dynamics of the processes we consider. Motions in the perpendicular null space should have irrelevant detail and equilibrate fast, preferably on a time scale that is separated from the time scale of the essential motions. Motions in the two spaces are separated much like is done in the Born-Oppenheimer approximation. The average influence of the fast motions on the essential degrees of freedom must be taken into account this concerns (i) correlations with positions expressed in a potential of mean force, (ii) correlations with velocities expressed in frictional terms, and iit) an uncorrelated remainder that can be modeled by stochastic terms. Of course, this scheme is the general idea behind the well-known Langevin and Brownian dynamics. 

Abstract. Molecular dynamics (MD) simulations of proteins provide descriptions of atomic motions, which allow to relate observable properties of proteins to microscopic processes. Unfortunately, such MD simulations require an enormous amount of computer time and, therefore, are limited to time scales of nanoseconds. We describe first a fast multiple time step structure adapted multipole method (FA-MUSAMM) to speed up the evaluation of the computationally most demanding Coulomb interactions in solvated protein models, secondly an application of this method aiming at a microscopic understanding of single molecule atomic force microscopy experiments, and, thirdly, a new method to predict slow conformational motions at microsecond time scales. 

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