Simplified single-particle model

Apart from the demands of the Pauli principle, the motion of electrons described by the wavefunction P° attached to the Hamiltonian H° is independent. This situation is called the independent particle or single-particle picture. Examples of single-particle wavefunctions are the hydrogenic functions (pfr,ms) introduced above, and also wavefunctions from a Hartree-Fock (HF) approach (see Section 7.3). HF wavefunctions follow from a self-consistent procedure, i.e., they are derived from an ab initio calculation without any adjustable parameters. Therefore, they represent the best wavefunctions within the independent particle model. As mentioned above, the description of the Z-electron system by independent particle functions then leads to the shell model. However, if the Coulomb interaction between the electrons is taken more accurately into account (not by a mean-field approach), this simplified picture changes and the electrons are subject to a correlated motion which is not described by the shell model. This correlated motion will be explained for the simplest correlated system, the ground state of helium. 

One of the simplified heat transfer models of two-phase flows is the pseudocontinuum one-phase flow model, in which it is assumed that (1) local thermal equilibrium between the two phases exists (2) particles are evenly distributed (3) flow is uniform and (4) heat conduction is dominant in the cross-stream direction. Therefore, the heat balance leads to a single-phase energy equation which is based on effective gas-solid properties and averaged temperatures and velocities. For an axisymmetric flow heated by a cylindrical heating surface at rw, the heat balance equation can be written as... 

When going over to a model without ordering of the particles, a state of the system is described by variables 9h 6tJ r) and the system (50) is simplified singling out of sites 8 and y is absent, and the migration contributions between different sublattices are compensated. The system of equations acquires the form... 

In order to predict emissions from AFBC s it is necessary to couple a model of the sulfur capture of individual particles into a system s model which takes into account the SO2 formation, removal, and transport. Because the single particle behavior is so complex, most such models (10, 20, 21, 22) use simplified, usually empirical, fits of single particle behavior determined, for example, from thermogravimetric analysis. 

Alternatively, if one argues that all particles are statistically identical, then Ajp )i and should not depend on n. Although this assumption is not necessary for deriving Eq. (4.39), it is usually invoked to simplify the physical modeling. In other words, if all particles are statistically identical, the mesoscale models can be validated by considering single-particle statistics from DNS for randomly selected particles. 

Simplified Model. The dissolution rate of a single particle of volume V is given by ... 

The total fraction remaining, F, calculated from the volume of HC1 added is given as a function of dimensionless time, t/t50, where t50 is the time required to dissolve 50% of the CaC03. Calculated curves are also given using the simplified mass transfer model with a single particle size (monodisperse) and with the actual polydisperse size distribution (Table 2). The polydisperse model fits the shape of the curve very well at all times. The monodisperse model is only satisfactory for t/t50 less than 1. 

The three-dimensional plate on coil problem was simplified by choosing a suitable two-dimensional unit cell (Figure 2.10). The unit cell consisted of a single coil cross-section and an appropriate section of the plate. The model was first configured by an empirical fit of the unknown current in the coil. This current was then applied for all other simulations. All other parameters, like material properties and experimental parameters, have been chosen according to appropriate literature or real test parameters. Various models have been established covering the range of matrix/particle models with various particle distributions e.g., random/pattern), particle sizes, the incorporation of polymer fibers, and the use of different material combinations. 

Since we are interested in cellular activities where several motors could be involved, a simplified mean-filed model emerge with the assumption that motors a) work independent of each other b) share the applied load equally. These equations are able to predict the transport properties of cargo particles such as their effective velocity and average run length. Two most important quantities that need to be mentioned here are the force-velocity relation and detachment rate of motor when a load is applied to the cargo. Several experiments have shown that velocity Vn(F) of a single motor in system of n bound motors decreases almost linearly with the force F applied against the motor movement. 

In the above discussion of the frequency dependent permittivity, the analysis has been based on either the single particle rotational diffusion model of Debye, or empirical extensions of this model. A more general approach can be developed in terms of time correlation functions [6], which in turn have to be interpreted in terms of a suitable molecular model. While using the correlation function approach does not simplify the analysis, it is useful, since experimental correlation functions can be compared with those deduced from approximate theories, and perhaps more usefully with the results of molecular dynamics simulations. Since the use of correlation functions will be mentioned in the context of liquid crystals, they will be briefly introduced here. The dipole-dipole time correlation function C(t) is related to the frequency dependent permittivity through a Laplace transform such that ... 

It seems probable that a fruitful approach to a simplified, general description of gas-liquid-particle operation can be based upon the film (or boundary-resistance) theory of transport processes in combination with theories of backmixing or axial diffusion. Most previously described models of gas-liquid-particle operation are of this type, and practically all experimental data reported in the literature are correlated in terms of such conventional chemical engineering concepts. In view of the so far rather limited success of more advanced concepts (such as those based on turbulence theory) for even the description of single-phase and two-phase chemical engineering systems, it appears unlikely that they should, in the near future, become of great practical importance in the description of the considerably more complex three-phase systems that are the subject of the present review. 

Therefore, in many fundamentally oriented studies the complex catalyst is replaced by a simplified model, which is better defined. Such models range from supported particles from which all promoters have been removed, via well-defined particles deposited on planar substrates, to single crystals (Fig. 4.1). With the latter we are in the domain of surface science, where a wealth of informative techniques is available that do not work on technical catalysts. 

In order to derive approximate laws for the growth of a two-dimensional layer, we consider a simplified model in which all isolated clusters, i.e. clusters that do not touch another cluster, axe circular. For the moment, consider a single such cluster of radius r(t). New particles can only be incorporated at its boundary. Assuming that this incorporation is the rate-determining step, the number N(t) of particles belonging to the cluster obeys the equation ... 

The foregoing treatment assumes that at least one of the reactants is a single crystal and the reactant/product geometry is well-defined. Many technologically important ceramic reactions, on the other hand, are usually carried out between polycrystalline powders. The reaction kinetics in these cases depend on several physical factors such as particle size, packing density, porosity, and so on. Jander (1927) and Carter (1961) have proposed models for powder reactions making several simplifying assumptions. 

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