High Resolution Methods

The purpose of this section is to give an overview of the pertinent high resolution methods often referred in the literature on multiphase reactor modeling. These are The Maker and Cell (MAC) method [96], the Simplified MAC method [6], the volume of fluid (VOF) method[108], the level set (LS) front capturing method [214, 20, 186], and finally the front tracking method [227, 221[. 

In the first phase of the MAC algorithm the pressure at time level tn+i is calculated, solving a Poisson equation in which the source term is a function of the velocity values at time level only. Thereafter, the full Navier-Stokes equations are used to find the time level velocities. It is emphasized that 

In this sub-section the basis elements of the volume of fluid method are described. In general the VOF model is composed of a set of continuity and momentum equations, as well as a transport equation for the evolution of a 

A common feature for all the different formulations of the VOF model is that the location and orientation of the interface are defined through a volume fraction function. The evolution of this volume fraction function in time and space is determined by a scalar advection equation defined by  

The propagation procedure contains the calculation of the volume fraction, ai, that represents the fractional volume of the cell occupied by fluid f. A unit value of ai corresponds to a cell full of fiuid f, while a zero value indicates that the cell contains no fluid f. Cells with ai values between zero and one must then contain an interface. 

Using the volume averaged Eulerian-Eulerian and Eulerian-Lagrangian multiphase models the spatial resolution considered deriving the constitutive relations is generally coarse compared to the size of the particles. However, small-scale structures in the fluid and most likely also within the dispersed fluid particles may under specific 

In this section the basis elements of the volume of fluid (VOF) method are described. In general the VOF model is composed of a set of continuity and momentum equations, as well as a transport equation for the evolution of a phase indicator function which is used to determine the location and orientation of the interface. We distinguish between the jump condition—and the whole field formulations of the method, in which both forms are based on a macroscopic view defining the interface as a 2D surface. The jump condition form is especially convenient for free-surface flow simulations, whereas the whole field formulation is commonly used for interfacial flow calculations in which the internal flow of all the phases are of interest. 

In the jump-condition formulation the physical problem is generally decomposed into k bulk phase domains where the continuity and momentum equations for isothermal incompressible flows holds, and at the interface between these domains boundary conditions are specified using the interface jump conditions. That is, across the interface some quantities are required to be continuous, while others are required to have specific jumps. The discontinuous (singular) momentum jump condition can be derived by use of the surface divergence theorem (see e.g., [68] p. 51 [28]). A rigorous derivation of the jump balances for the multi-fluid model is given in Sect. 3.4. 

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Breitmaier E and Voelter W 1986 Carbon-13 NMR Spectroscopy High Resolution Methods and Applications in Organic Chemistry (New York VCH)... 

L. Hagel, "Gel Filtration" in J-C. Janson and L. Ryden, ed.. Protein Purification Principles, High Resolution Methods, and Applications, VCH Pubhshers,... 

Traditionally, the penetration of chlorides and sodium is measured destructively by grinding layers of concrete and chemically analyzing the powder samples. These data are used to calculate diffusion coefficients for the ions. This procedure is very slow, has low spatial resolution, and is destructive. The measured data are critically important for the development of service life models and therefore a rapid, high-resolution method to monitor the ingress of these ions is desirable. 

High-resolution methods of substanees investigations allow understand not only strueture of those or that material but may be used more mueh widely in application fields. One of them may be mineralogical-geological reconstructions which get possibility to discover new mineral deposits and new natural raw materials [1],... 

Another way of looking at the traditional double-crystal method is that it measures the difference in reflectivity between the specimen and a reference perfect crystal. This is the original high resolution method and is still the best choice when good reference crystals are available and the specimens to be... 

High-Resolution Methods and Applications in Organic Chemistry and Biochemistry... 

As in the second edition, the authors survey the large number of L C NMR applications to organic molecules and natural products in a representative and systematic rather than an exhaustive way. New sections about coupling constants. organophosphorus and organometallic compounds as well as synthetic polymers have been added. The scope remains limited to high-resolution methods and molecular systems. 

Carbon 13 NMR spectroscopy high resolution methods and applications in organic chemistry and biochemistry / Eberhard Breitmaier Wolfgang Voelter. -3., completely rev. cd.. reprint, corr. -Wcinheim New York VCH. 1990 ISBN 3-527-26466-3 (Weinheim. .. )... 

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