Simplified Modeling

If the innermost sphere can be treated independently of the second, it would simplify calculation. This can be tested by considering induced dipoles in the first shell independently of the second using Eq. (73), or via those in the second shell using two simultaneous equations based on 

The first term in this useful and easily-handled expression combines the ion-permanent dipole energy -ppAA and its repulsive interaction with all permanent dipoles +PpEpA/2 = +Pp.AA(l - l/sA)/2 (compare Ref. 16). 

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Parameters of the simplified model for calculating hydrate formation temperatures. 

In principle, the reaction cross section not only depends on the relative translational energy, but also on individual reactant and product quantum states. Its sole dependence on E in the simplified effective expression (equation (A3.4.82)) already implies unspecified averages over reactant states and sums over product states. For practical purposes it is therefore appropriate to consider simplified models for tire energy dependence of the effective reaction cross section. They often fonn the basis for the interpretation of the temperature dependence of thennal cross sections. Figure A3.4.5 illustrates several cross section models. 

Troe J 1995 Simplified models for anharmonic numbers and densities of vibrational states. I. Application to NO2 and Chem. Phys. 190 381-92... 

Much use has been, and continues to be, made of simplified model schemes representative of general classes of chemical or themial feedback. The oregonator and Lengyel-Epstein models for the BZ and CDIMA systems have been given earlier. Pre-eminent among the more abstracted caricature models is the bnisselator introduced by Prigogine and Lefever [47] which has the following fomi ... 

It is difficult to observe tliese surface processes directly in CVD and MOCVD apparatus because tliey operate at pressures incompatible witli most teclmiques for surface analysis. Consequently, most fundamental studies have selected one or more of tliese steps for examination by molecular beam scattering, or in simplified model reactors from which samples can be transferred into UHV surface spectrometers witliout air exposure. Reference [4] describes many such studies. Additional tliemes and examples, illustrating botli progress achieved and remaining questions, are presented in section C2.18.4. 

The visuahzation of hundreds or thousands of connected atoms, which are found in biological macromolecules, is no longer reasonable with the molecular models described above because too much detail would be shown. First of aU the models become vague if there are more than a few himdied atoms. This problem can be solved with some simplified models, which serve primarily to represent the secondary structure of the protein or nucleic acid backbone [201]. (Compare the balls and sticks model (Figure 2-124a) and the backbone representation (Figure 2-124b) of lysozyme.)... 

A very important issue - disregard of which is a big source of bad modeling studies - is the dear distinction of transport processes (toxicokinetics) and interactions with targets such as membranes, enzymes, or DNA (toxicodynamics). Figure 10.1-6 gives a rather simplified model of a fish to illustrate this distinction. 

Figure 10.1-6. Simplified model of a higher aquatic organism and of the toxicokinetic processes taking place.

Within this simplified model, the energy differenee between the daughter and the parent speeies ean be written as follows (cjik represents the partieular spin-orbital that is added or removed) ... 

Another simplified model is the freely jointed or random flight chain model. It assumes all bond and conformation angles can have any value with no energy penalty, and gives a simplified statistical description of elasticity and average end-to-end distance. 

In the higher pressure sub-region, which may be extended to relative pressure up to 01 to 0-2, the enhancement of the interaction energy and of the enthalpy of adsorption is relatively small, and the increased adsorption is now the result of a cooperative effect. The nature of this secondary process may be appreciated from the simplified model of a slit in Fig. 4.33. Once a monolayer has been formed on the walls, then if molecules (1) and (2) happen to condense opposite one another, the probability that (3) will condense is increased. The increased residence time of (1), (2) and (3) will promote the condensation of (4) and of still further molecules. Because of the cooperative nature of the mechanism, the separate stages occur in such rapid succession that in effect they constitute a single process. The model is necessarily very crude and the details for any particular pore will depend on the pore geometry. 

The obtained model (1.15)-(1.17) of a shallow shell can be simplified once more. The values of kiQi, kiQ2 are small enough very often, and they can be omitted. By doing so, we obtain the simplified model of a shallow shell consisting of the following equilibrium equations ... 

Heat Transfer in Rotary Kilns. Heat transfer in rotary kilns occurs by conduction, convection, and radiation. In a highly simplified model, the treatment of radiation can be explained by applying a one-dimensional furnace approximation (19). The gas is assumed to be in plug flow the absorptivity, a, and emissivity, S, of the gas are assumed equal (a = e ) and the presence of water in the soHds is taken into account. Energy balances are performed on both the gas and soHd streams. Parallel or countercurrent kilns can be specified. 

Simplified models for proteins are being used to predict their stmcture and the folding process. One is the lattice model where proteins are represented as self-avoiding flexible chains on lattices, and the lattice sites are occupied by the different residues (29). When only hydrophobic interactions are considered and the residues are either hydrophobic or hydrophilic, simulations have shown that, as in proteins, the stmctures with optimum energy are compact and few in number. An additional component, hydrogen bonding, has to be invoked to obtain stmctures similar to the secondary stmctures observed in nature (30). 

A simplified model usiag a stagnant boundary layer assumption and the one-dimension diffusion—convection equation has been used to calculate wall concentration ia an RO module. The iategrated form of this equation, the widely appHed film theory (41), is given ia equation 8. 

Using this simplified model, CP simulations can be performed easily as a function of solution and such operating variables as pressure, temperature, and flow rate, usiag software packages such as Mathcad. Solution of the CP equation (eq. 8) along with the solution—diffusion transport equations (eqs. 5 and 6) allow the prediction of CP, rejection, and permeate flux as a function of the Reynolds number, Ke. To faciUtate these calculations, the foUowiag data and correlations can be used (/) for mass-transfer correlation, the Sherwood number, Sb, is defined as Sh = 0.04 S c , where Sc is the Schmidt... 

Kinetic Models Used for Designs. Numerous free-radical reactions occur during cracking therefore, many simplified models have been used. For example, the reaction order for overall feed decomposition based on simple reactions for alkanes has been generalized (37). 

Since biological systems can reasonably cope with some of these problems, the intuition behind neural nets is that computing systems based on the architecture of the brain can better emulate human cognitive behavior than systems based on symbol manipulation. Unfortunately, the processing characteristics of the brain are as yet incompletely understood. Consequendy, computational systems based on brain architecture are highly simplified models of thek biological analogues. To make this distinction clear, neural nets are often referred to as artificial neural networks. 

For gases, Ctf/D is usually close to 1, since the same basic transfer mechanism exists. For liquids, Ctf/D is invariably much greater than 1. A simplified model yields the relation... 

A simplified model of PC combustion includes the following sequence of events (I) on entering the furnace, a PC particle is heated rapidly, driving off the volatile components and leaving a char particle (2) the volatile components burn independently of the coal particle and (3) on completion of volatiles combustion, the remaining char particle burns. Whue this simple sequence may be generally correct, PC combustion is an extremely complex process involving many interrelated physical and chemical processes. 

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. 

The model that best describes the mechanism is usually very complicated. For dynamic studies that require much more computation (and on a more limited domain) a simplified model may give enough information as long as the formalities of the Arrhenius expression and power law kinetics are incorporated. To study the dynamic behavior of the ethylene oxide reactor. 

The regenerator (Figure 4-80) is represented by a simplified model that ineludes the total volume and mass balanee ealeulation. The regenerator exit temperature is assumed eonstant for the duration of the transient. The third-stage separator is handled as a fixed volume and assoeiated pressure drop. Blow-down (bypass) flow is subtraeted from the input flow. 

Chapter 5 describes simplified methods of estimating airborne pollutant concentration distributions associated with stationary emission sources. There are sophisticated models available to predict and to assist in evaluating the impact of pollutants on the environment and to sensitive receptors such as populated areas. In this chapter we will explore the basic principles behind dispersion models and then apply a simplified model that has been developed by EPA to analyzing air dispersion problems. There are practice and study problems at the end of this chapter. A screening model for air dispersion impact assessments called SCREEN, developed by USEPA is highlighted in this chapter, and the reader is provided with details on how to download the software and apply it. 

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