Coupled-Phase Model

The coupled-phase c-p model is based on analysis of a two-phase fluid by solving four differential equations that govern the motion of the two-phase mixture. For the effective wave number k, the solution gives 

From the effective wave number, one obtains the effective wave velocity V = Re( oo/k) and attenuation a = Im(k). 

Variation of phase velocity over a range of solid-volume percentages are calculated for the above models for two types of particles, i.e., glass beads and kaolins (with acoustic impedances of 21.12 x 10s and 10.66 x 105 g/cm2-s, respectively). Calculated results are shown in Fig. 5-27 for glass beads and in Fig. 5-28 for kaolins. All models, except Biot-2, show decreasing phase velocity at lower volume fractions, then increasing phase velocity at higher volume fractions. 

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The originators of the Penn Kem system claim that it is capable of measuring particle size distributions in the size range 0.01 to 100 pm for slurry concentrations at volume concentrations as high as 50%. They report experimental work with an on-line system using titanium dioxide at volume concentrations from 3.5% to 42.3%. Quantitative comparison of data was carried out at eighteen frequencies and eleven concentrations by volume [248,249]. Theoretical work resulted in the development of a unified coupled phase model which successfully predicted the experimental data for suspensions, emulsions and aerosols [250]. 

The long-wave requirement provides a sufficient simplification of the theory for implementing particle-particle interactions. It has been done in the work in Ref 20 on the basis of the coupled phase model (18, 19). This theory (19) works up to 40% volume even for heavy materials including rutile. 

The Eulerian multiphase model is used to predict the dispersed gas-liquid flow in the airlift loop reactor. It involves a set of momentum and continuity equations for each phase. Model equation coupling is achieved through the pressure and interphase exchange coefBcdents [5],... 

A much more detailed and time-dependent study of complex hydrocarbon and carbon cluster formation has been prepared by Bettens and Herbst,83 84 who considered the detailed growth of unsaturated hydrocarbons and clusters via ion-molecule and neutral-neutral processes under the conditions of both dense and diffuse interstellar clouds. In order to include molecules up to 64 carbon atoms in size, these authors increased the size of their gas-phase model to include approximately 10,000reactions. The products of many of the unstudied reactions have been estimated via simplified statistical (RRKM) calculations coupled with ab initio and semiempirical energy calculations. The simplified RRKM approach posits a transition state between complex and products even when no obvious potential barrier... 

This relation corresponds to exchange-coupled crystallites diluted in an ideally soft magnetic matrix. The only modification made over the original single phase model is the inclusion of the crystalline volume fraction xcr. It should be noted that is scaling with the volume fraction in the same way as with the crystalline volume 1f. 

Several hybrid simulations on crystal growth can be found in recent literature. Examples include dendritic solidification by coupling finite-different discretization of a phase field model to a MC simulation (Plapp and Karma, 2000), coupling a finite difference for the melt with a cellular automata for the solidification (Grujicic et al., 2001), a DSMC model for the fluid phase with a Metropolis-based MC for the surface to address cluster deposition onto substrates (Hongo et al., 2002 Mizuseki et al., 2002), a step model for the surface processes coupled with a CFD simulation of flow (Kwon and Derby, 2001) (two continuum but different feature scale models), an adaptive FEM CVD model coupled with a feature scale model (Merchant et al., 2000), and one-way coupled growth models in plasma systems (Hoekstra et al., 1997). Some specific applications are discussed in more detail below. 

Two models are available for interpreting attenuation spectra as a PSD in suspensions with chemically distinct, dispersed phases using the extended coupled phase theory.68 Both models assume that the attenuation spectrum of a mixture is composed of a superposition of component spectra. In the multiphase model, the PSD is represented as the sum of two log-normal distributions with the same standard deviation, that is, a bimodal distribution. The appearance of multiple solutions is avoided by setting a common standard deviation to the mean size of each distribution. This may be a poor assumption for the PSD (see section 11.3.2). The effective medium model assumes that only one target phase of a multidisperse system needs to be determined, while all other phases contribute to a homogeneous system, the so-called effective medium. Although not complicated by the possibility of multiple solutions, this model requires additional measurements to determine the density, viscosity, and acoustic attenuation of the effective medium. The attenuation spectrum of the effective medium is modeled via a polynomial fit, while the target phase is assumed to have a log-normal PSD.68 This model allows the PSD for mixtures of more than two phases to be determined. 

All polarizable models share the ability to polarize, by varying their charge distribution in response to their environment. In addition, shell models and EE models with charge-dependent radii have the ability to modify their polarizability—the magnitude of this polarization response—in response to their local environment. Consequently, it is reasonable to expect that shell models and mechanically coupled EE models may be slightly more transferable to different environments than more standard PPD and EE models. To date, it is not clear whether this expectation has been fully achieved. Although some shell-based models for both ionic and molecular compounds have been demonstrated to be transferable across several phases and wide ranges of phase points, " it is not clear that the transferability displayed by these models is better than that demonstrated in PPD- or EE-based models. And even with an environment-dependent polarizability, it has been demonstrated that the basic shell model cannot fully capture all of the variations in ionic polarizabilities in different crystal environments. ... 

With any type of molecular modeling, there is generally a tradeoff between cost and reliability, and one typically shuns models that cost more without increasing reliability. In practice, this cost is usually expressed as computational effort, or computer time. In gas phase modeling, one typically finds molecular mechanics and semiempirical molecular orbital theory at the low-cost end and multireference configuration interaction or coupled-cluster theory at the other, with the choice dictated by the size of the system. System size also influences the choice of solvation model. We consider first the least expensive models, those that take no account of the quantum mechanical nature of the solute. 

In the geometrically coupled cell model, it is assumed that a fluid cell has contact with the front half of the particle in front of it. This model has been studied by Vanderveen el ai, (1968), and has been recently modified by Rhee et al. (1973) to account for thermal conductivity in the solid phase. 

Very interesting alkane functionalizations in the presence of metal ions in the gas phase have also been reported. Schwarz et al. have described recently [11a] a gas-phase model for the platinum-catalyzed coupling of methane and ammonia. Endothermic by 61 kcal mol" reaction... 

Miedema, A. R., de Boer, F. R., Boom, R., Model predictions for the enthalpy of formation of transition metal alloys, CALPHAD Comput. Coupling Phase Diagrams Thermochem., 1, (1977), 341-349. Cited on page 334. 

The application of the hydro-mechanically coupled three phase models to real scale experiments is however still a challenge. This paper presents a successful application of such a model to a complex real scale experiment involving numerous materials with very different properties. 

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