Dispersion rigorous

Axial Dispersion. Rigorous models for residence time distributions require use of the convective diffusion equation. Equation (14.19). Such solutions, either analytical or numerical, are rather difficult. Example 15.4 solved the simplest possible version of the convective diffusion equation to determine the residence time distribution of a piston flow reactor. The derivation of W t) for parabolic flow was actually equivalent to solving... 

Rapid Approximate Design Procedure. Several simplified approximations to the rigorous solutions have been developed over the years (57—60), but they aU. remain too compHcated for practical use. A simple method proposed in 1989 (61,62) uses a correction factor accounting for the effect of axial dispersion, which is defined as (57)... 

Fig. 19. Correction factor for axial dispersion as a function of NTU. SoHd lines are rigorous calculations broken lines, approximate formulas according to hterature (61). (a) Numbers on lines represent Pe values Pe = 20 /Lj = 0.8. (b) For design calculations. Numbers on lines represent Pep u ...

A classification by chemical type is given ia Table 1. It does not attempt to be either rigorous or complete. Clearly, some materials could appear ia more than one of these classifications, eg, polyethylene waxes [9002-88 ] can be classified ia both synthetic waxes and polyolefins, and fiuorosihcones ia sihcones and fiuoropolymers. The broad classes of release materials available are given ia the chemical class column, the principal types ia the chemical subdivision column, and one or two important selections ia the specific examples column. Many commercial products are difficult to place ia any classification scheme. Some are of proprietary composition and many are mixtures. For example, metallic soaps are often used ia combination with hydrocarbon waxes to produce finely dispersed suspensions. Many products also contain formulating aids such as solvents, emulsifiers, and biocides. 

Eor printing on polyester, the fixation conditions are more rigorous than on other disperse dyeable fibers, owing to the slower diffusion of disperse dyes in polyester. Eor continuous fixation the prints are exposed at atmospheric pressure to superheated steam of 170—180°C for 6—8 min. A carrier may be added to the print paste for accelerated and fliU fixation. Dry-heat fixation conditions of 170—215°C for 1—8 min are less popular for printed fabrics, but are sometimes employed because of lack of other equipment. 

The problem is made more difficult because these different dispersion processes are interactive and the extent to which one process affects the peak shape is modified by the presence of another. It follows if the processes that causes dispersion in mass overload are not random, but interactive, the normal procedures for mathematically analyzing peak dispersion can not be applied. These complex interacting effects can, however, be demonstrated experimentally, if not by rigorous theoretical treatment, and examples of mass overload were included in the work of Scott and Kucera [1]. The authors employed the same chromatographic system that they used to examine volume overload, but they employed two mobile phases of different polarity. In the first experiments, the mobile phase n-heptane was used and the sample volume was kept constant at 200 pi. The masses of naphthalene and anthracene were kept... 

HGSystem offers the most rigorous treatments of HF source-term and dispersion analysis a ailable for a public domain code. It provides modeling capabilities to other chemical species with complex thermodynamic behavior. It treats aerosols and multi-component mixtures, spillage of a liquid non-reactive compound from a pressurized vessel, efficient simulations of time-dependent... 

A rigorous treatment of dispersion in soils is beyond the scope of this book. However, some qualitative discussion is warranted because of the potential and existing problems already described. Two ntain problems arise bceause dispersion in soil (or land) is anisotropic (i.e., it varies with direction) and the penneability is not only a variable but also att unknown. 

Stokes law is rigorously applicable only for the ideal situation in which uniform and perfectly spherical particles in a very dilute suspension settle without turbulence, interparticle collisions, and without che-mical/physical attraction or affinity for the dispersion medium [79]. Obviously, the equation does not apply precisely to common pharmaceutical suspensions in which the above-mentioned assumptions are most often not completely fulfilled. However, the basic concept of the equation does provide a valid indication of the many important factors controlling the rate of particle sedimentation and, therefore, a guideline for possible adjustments that can be made to a suspension formulation. 

Some workers have attempted to treat particular effects more rigorously, e g., by scaled-particle theory [142] or by extending [95, 103] Linder s theory [143] of dispersion interactions to the case of an SCRF treatment of solute-solvent interactions. We will not review these approaches here. 

Rigorous treatment of the self-action problem needs the transformation of Eq.(2.1), (2.5) into a system of integro-differential equations. However, if just some orders of group velocity dispersion and nonlinearity are taken into account, an approximate approach can be used based on differential equations solution. When dealing with the ID-i-T problem of optical pulse propagation in a dielectric waveguide, one comes to the wave equation with up to the third order GVD terms taken into account ... 

Authors efforts in this part of the work have been concentrated on developing turbulence closures for the statistical description of two-phase turbulent flows. The primary emphasis is on development of models which are more rigorous, but can be more easily employed. The main subjects of the modeling are the Reynolds stresses (in both phases), the cross-correlation between the velocities of the two phases, and the turbulent fluxes of the void fraction. Transport of an incompressible fluid (the carrier gas) laden with monosize particles (the dispersed phase) is considered. The Stokes drag relation is used for phase interactions and there is no mass transfer between the two phases. The particle-particle interactions are neglected the dispersed phase viscosity and pressure do not appear in the particle momentum equation. 

Further details may be found in the above quoted references. In particular, de Josselin de Jong (D14) and Saffman (SI, S2, S3) give relatively rigorous developments that take into account the anistropy caused by the flow. Thus different estimates are obtained for the dispersion coefficients depending on whether or not the direction considered is perpendicular or parallel to the mean flow. 

Even with this enormous number of scientific papers on the subject, mathematically rigorous results on the subject are rare. Let us mention just ones aiming toward a rigorous justification of Taylor s dispersion model and its generalization to reactive flows. We could distinguish them by their approach... 

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