Pseudo-continuum approach

Section 4.1 via Section 4.1.2 formally illustrates vapor-Uquid equilibria vis-a-vis distillation in a closed vessel along with bubble-point and dew-point calculations for multicomponent systems. How vapor-liquid equilibrium is influenced by chemical reactions in the liquid phase is treated in Section 5.2.1.2, where two subsections, 5.2.1.2.1 and 5.2.1.2.2, deal with reactions influencing vapor-Uquid equilibria in isotopic systems. We next encounter open systems in Chapter 6. The equations of change for any two-phase system (e.g. a vapor-Uquid system) are provided in Section 6.2.1.1 based on the pseudo-continuum approach for the dependences of species concentrations... 

The solution of the problem of flxed-bed adsorption of a strongly adsorbed species i present in an inert mobile feed liquid is at hand when the concentration Ca of the desired species in the liquid phase (moles of species i per unit volume of the liquid phase) is obtained as a function of time t and spatial coordinates x, y, z. Complexity of the fluid dynamics in a packed bed and the presence of convective dispersion require, however, considerable reduction in this goal for practical purposes. In the "pseudo-continuum approach (Lee et al. (1977a) see Section 6.2.1.1), only the axial (mean flow direction, z-coordinate) variation is retained, i.e. we look only for a... 

Figure 7.1.1. (a) Movement of the color front in the packed adsorbent bed with time and the corresponding concentration of the coloring species at the bed outlet (b) packed-bed schematic and its idealization in pseudo-continuum approach for adsorption from a liquid. (After Lightfbot et aL, 1962.)... 

An alternative and complementary use of CFD in fixed bed simulation has been to solve the actual flow field between the particles (Fig. lb). This approach does not simplify the geometrical complexities of the packing, or replace them by the pseudo-continuum that is used in the first approach. The governing equations for the interstitial fluid flow itself are solved directly. The contrast is thus between the interstitial flow field type of simulation and the superficial flow... 

Fio. 1. Comparison of (a) pseudo-continuum and (b) interstitial CFD approaches to packed-tube simulation. 

In practical implementations of complex scaling, the Hamiltonian is regularly discretized in finite space, for example, in a box of radius R. This yields a discrete pseudo-continuum with energies that fulfill Ek e w for Z = 0 and approaches it with increasing k and R for Z 0. If exterior complex scaling is made in such a finite box, Eq. (15) is adjusted to... 

Figure 5.9 Energy eigenvalues (dots) to the discretized and complex rotated one-particle Dirac Hamiltonian with s-symmetry. The high-energy pseudo-continuum states are rotated with an angle 9 (indicated with the dashed line) down from the real axis, where 6 is the complex rotation angle. For low energies, the line of dots bends and approaches the solid line, which indicates the expected rotation of 29 in the nonrelativistic domain.

Koelman and Hoogerbrugge (1993) have developed a particle-based method that combines features from molecular dynamics (MD) and lattice-gas automata (LGA) to simulate the dynamics of hard sphere suspensions. A similar approach has been followed by Ge and Li (1996) who used a pseudo-particle approach to study the hydrodynamics of gas-solid two-phase flow. In both studies, instead of the Navier-Stokes equations, fictitious gas particles were used to represent and model the flow behavior of the interstial fluid while collisional particle-particle interactions were also accounted for. The power of these approaches is given by the fact that both particle-particle interactions (i.e., collisions) and hydrodynamic interactions in the particle assembly are taken into account. Moreover, these modeling approaches do not require the specification of closure laws for the interphase momentum transfer between the particles and the interstitial fluid. Although these types of models cannot yet be applied to macroscopic systems of interest to the chemical engineer they can provide detailed information which can subsequently be used in (continuum) models which are suited for simulation of macroscopic systems. In this context improved rheological models and boundary condition descriptions can be mentioned as examples. 

Analysis. To answer the question posed above, we treat the fuel cloud as a pseudo homogeneous or continuum phase (6, 7) with the individually burning particles treated as point sinks of oxidizer. Such an approach is rigorously defensible when the cloud is suflBciently dilute that ... 

Approach. The continuum total group combustion criterion is established by asking when will the cloud bum as a large pseudo-droplet with the flame located just outside of the cloud droplet region That is, when will the evaporating particles inside the cloud provide sufficient vapor so that fuel and oxidizer mix in stoichiometric amounts at the cloud boundary ... 

The state of the art in understanding and describing droplet formation processes is divided into two different perspectives flie cmitinuum mechanics or fluid dynamics approach and the particle-based approach. Whilst the cmitinuum mechanics approach is based usually on the Navier-Stokes equation, the continuum equation and the energy equation, the particle-based approach is based on the interconnecting forces between neighboring particles like molecules or pseudo particles. 

It is almost impossible to cover the entire range of models in Figure 25.1, and in this chapter we will limit ourselves to the different modeling approaches at the continuum level (micro-macroscopic and system-level simulations). In summary, there are computational models that are developed primarily for the lower-length scales (atomistic and mesoscopic) which do not scale to the system-level. The existing models at the macroscopic or system-level are primarily based on electrical circuit models or simple lD/pseudo-2D models [17-24]. The ID models are limited in their ability to capture spatial variations in permeability or conductivity or to handle the multidimensional structure of recent electrode and solid electrolyte materials. There have been some recent extensions to 2D [29-31], and this is still an active area of development As mentioned in a recent Materials Research Society (MRS) bulletin [6], errors arising from over-simplified macroscopic models are corrected for when the parameters in the model are fitted to real experimental data, and these models have to be improved if they are to be integrated with atomistic... 

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