Model ignorance

By expressing the mean-field interaction of an electron at r with the N- 1 other electrons in temis of a probability density pyy r ) that is independent of the fact that another electron resides at r, the mean-field models ignore spatial correlations among the electrons. In reality, as shown in figure B3.T5 the conditional probability density for finding one ofA - 1 electrons at r, given that one electron is at r depends on r. The absence of a spatial correlation is a direct consequence of the spin-orbital product nature of the mean-field wavefiinctions... 

Although experimental results could be fitted well with irreversible rate models, ignoring thermodynamic facts could be disastrous. Although reversibility moderated the maximum temperature at runaway, it was not the most important qualitative result. In fact, the one dimensional (directional, or irreversible, correctly) model was not realistic at these conditions. For the prediction of incipient runaway and the AT ax permissible before runaway, the reversibility was obviously important. 

The simplest model ignores tortuosity and assumes the bed equivalent hydro-dynamieally to a matrix of straight tubes - like a bundle of drinking straws e.g. as in Figure 2.10. 

Point defects in solids make it possible for ions to move through the structure. Ionic conductivity represents ion transport under the influence of an external electric field. The movement of ions through a lattice can be explained by two possible mechanisms. Figure 25.3 shows their schematic representation. The first, called the vacancy mechanism, represents an ion that hops or jumps from its normal position on the lattice to a neighboring equivalent but vacant site or the movement of a vacancy in the opposite direction. The second one is an interstitial mechanism where an interstitial ion jumps or hops to an adjacent equivalent site. These simple pictures of movement in an ionic lattice, known as the hopping model, ignore more complicated cooperative motions. 

Such a model ignores restrictions on the bond angles and dihedral angles imposed by the local chemical structure. This may be taken care of empirically by defining a characteristic ratio, Cm, where... 

A number of the assumptions used in the BET theory have been questioned for real samples [6]. One assumption states that all adsorption sites are energetically equivalent, which is not the case for normal samples. The BET model ignores lateral adsorbate interactions on the surface, and it also assumes that the heat of adsorption for the second layer and above is equal to the heat of liquefaction. This assumption is not valid at high pressures and is the reason for using adsorbate pressures less than 0.35. In spite of these concerns, the BET method has proven to be an accurate representation of surface area for the majority of samples [9,10]. 

While the collision theory of reactions is intuitive, and the calculation of encounter rates is relatively straightforward, the calculation of the cross-sections, especially the steric requirements, from such a dynamic model is difficult. A very different and less detailed approach was begun in the 1930s that sidesteps some of the difficulties. Variously known as absolute rate theory, activated complex theory, and transition state theory (TST), this class of model ignores the rates at which molecules encounter each other, and instead lets thermodynamic/statistical considerations predict how many combinations of reactants are in the transition-state configuration under reaction conditions. 

The PFR model ignores mixing between fluid elements at different axial locations. It can thus be rewritten in a Lagrangian framework by substituting a = Tpfrz, where a denotes the elapsed time (or age) that the fluid element has spent in the reactor. At the end of the PFR, all fluid elements have the same age, i.e., a = rpfr. Moreover, at every point in the PFR, the species concentrations are uniquely determined by the age of the fluid particles at that point through the solution to (1.2). 

An alternative method to RTD theory for treating non-ideal reactors is the use of zone models. In this approach, the reactor volume is broken down into well mixed zones (see the example in Fig. 1.5). Unlike RTD theory, zone models employ an Eulerian framework that ignores the age distribution of fluid elements inside each zone. Thus, zone models ignore micromixing, but provide a model for macromixing or large-scale inhomogeneity inside the reactor. 

This is rarely done in practice. For example, all commonly used models ignore possible effects of chemical reactions on the scalar-mixing process. Compare this with the flamelet model, where mixing and reactions are tightly coupled. 

As discussed in Chapter 3, at very high Reynolds numbers, turbulent mixing theory predicts that the scalar dissipation rate will be independent of Re and Sc. Thus, most molecular models ignore all dependencies on these parameters, even at moderate Reynolds numbers. In general, the inclusion of dependencies on Re, Sc, or Da is difficult and, most likely, will have to be done on a case-by-case basis. 

In spite of the success of the BET theory, some of the assumptions upon which it is founded are not above criticism. One questionable assumption is that of an energetically homogeneous surface, that is, all the adsorption sites are energetically identical. Further, the BET model ignores the influence of lateral adsorbate interactions. 

Intrusion-extrusion hysteresis has been attributed to ink-bottle shaped pores. In pores of this type intrusion cannot occur until sufficient pressure is attained to force mercury into the narrow neck, whereupon the entire pore will fill. However, on depressurization the wide-pore body cannot empty until a lower pressure is reached, leaving entrapped mercury in the wide inner portion. The ink-bottle model ignores several factors which may reduce it to an untenable concept. These include the following ... 

The ink-bottle model ignores the question of the energy required to break the mercury column in the pore in order for the narrow entrance to empty while the inner cavity remains filled. 

The DTO model ignores the overall translations and rotations of the molecule as a whole and refers only to internal vibrational modes. It is therefore incapable of explaining on its own the viscosity of dilute polymer solutions. The enhanced viscosity of dilute polymer solutions is undoubtedly due to a hydrodynamic damping of the polymer as a whole as it translates and rotates in the shear field. This was very well described by Debye (21). We should point out that the Debye viscosity is alternatively derivable from the RB theory. 

Here, Iq is the electron affinity of C60 (Iq = 2.65 eV [50]). Thus, the 5-potential model ignores the finite thickness nature of the carbon cage within the model, A = 0. Furthermore, in the framework of this model, the size of the embedded atom ra is considered to be so small, compared to the size of C60, that the ground state electronic wavefunctions of the embedded atom coincide exactly with those for a free atom. In other words, the model assumes no interaction between the ground state encaged atom and the carbon cage at all. Therefore, the model is applicable only to the deep inner subshells of the encaged atom. As for the carbon atoms from... 

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. 

Comparison of the results in Table 7.9 with those in Table 7.7 (and Table 7.8) and Table 7.4 show significant differences in the design, operating policies, optimal recoveries of products, number of batches to be processed for each duty and total yearly profit. This clearly shows the importance of including allocation time and set up time between batches in the objective function. It is to be noted that in all cases a simple model but detailed plate-to-plate calculations (Type HI) with reasonable column holdup is used (unlike a short-cut model ignoring column holdup as in Logsdon et al.). 

So how can one handle situations where the steps are far apart One answer lies in the pioneering work of Burton et al. (1951), also known as the BCF model. The BCF model is a continuum PDE that describes adsorption of atoms to and desorption from terraces along with surface diffusion on terraces [see Eq. (2) below for a simplified version of the BCF model]. When the concentration of adatoms is relatively large, nucleation between distant steps is most likely to occur, because the probability of a diffusing adatom to reach steps before encountering another adatom is low. Under these conditions, the BCF model is inadequate since it does not account for nucleation. Furthermore, the boundary conditions in the BCF model ignore the discrete nature of steps and treat them... 

It is important to be aware of clustering caused by electrostriction in order to understand reactions of ions in supercritical rare gases. If a classical continuum model is used to calculate clustering, the magnitude of the volume change due to electrostriction would be overestimated because such a model ignores the density build-up around the ion. Because of this density augmentation, the compressibility of the fluid near the ion is less and, since electrostriction is proportional to compressibility, the actual electrostriction will be less... 

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