Segregation model solution

Since the details of these equations are explained elsewhere, only key ideas are briefly described here. One of these is to classify the solute atom clusters into irradiation-induced clusters and irradiation-enhanced clusters. Irradiation-induced clusters correspond to solute atom clusters with or without Cu atoms, whose formation mechanism is assumed to be the segregation of solute atoms based on point defect cluster or matrix damage (heterogeneous nucleation). On the other hand, the irradiation-enhanced clusters correspond to so-called CRPs (Cu-rich precipitates) or CELs (Cu-enriched clusters), and the formation mechanism is the clustering of Cu atoms above the solubility limit enhanced by the excess vacancies introduced by irradiation. This model also assumes that the formation of solute atom clusters and matrix damage is not independent to each other, which is a very different model from the conventional two-feature models as described in the previous sections. Another key idea is the introduction of a concept of a thermal vacancy contribution in the diffusivity model. This idea is essentially identical to that shown in Rg. 11.11. This is a direct modeling of the results of atomic-level computer simulations. ... 

Equations 4.64 and 4.65 yield the final value of y, at 0 = 0, as the solution is obtained backwards, by starting with large values of 0 (in addition to the X values valid at these points) and ending up with 0 = 0. Similar to the segregation model, a precondition for the maximum mixedness model is that the residence time functions—or more precisely the intensity function—are available, either from theory or from experiments. 

Numerical solutions of the maximum mixedness and segregated flow equations for the Erlang model have been obtained by Novosad and Thyn (Coll Czech. Chem. Comm., 31,3,710-3,720 [1966]). A few comparisons are made in Fig. 23-14. In some ranges of the parameters n or fte ihe differences in conversion or reaclor sizes for the same conversions are substantial. On the basis of only an RTD for the flow pattern, perhaps only an average of the two calculated extreme performances is justifiable. 

The "Filtration Model" as proposed in this study represents the overall rate of the solute transfer (macro-segregation) in zone refining very well for a wide range of experimental conditions. 

Using the proposed model along with the two generalized correlations for the experimental transmission coefficients in the partially solidified zone, a fairly close prediction of the solute redistribution (macro-segregation) of eutectic-forming mixtures after a single zone pass can be made. 

Unfortunately, the dilute solution model is limited in its applicability to concentrated solutions. This causes problems for alloys such as Ni-based superalloys, high alloy steels, etc., and systems where elements partition strongly to the liquid and where solidification processes involve a high level of segregation. It is also not possible to combine dilute solution databases which have been assessed for different solvents. The solution to this problem is to use models which are applicable over the whole concentration range, some of which are described below. 

Statistical thermodynamic descriptions of these transitions in substitutional alloys have been developed for the cases of both binary and ternary alloys , using a simple nearest neighbor bond model of the surface segregation phenomenon (including strain energy effects). Results of the model have been evaluated here using model parameters appropriate for a Pb-5at%Bi-0.04at%Ni alloy for which experimental results will be provided below. However, the model can be applied in principle to the computation of equilibrium surface composition of any ternary solution. 

The finite volume methods have been used to discretised the partial differential equations of the model using the Simple method for pressure-velocity coupling and the second order upwind scheme to interpolate the variables on the surface of the control volume. The segregated solution algorithm was selected. The Reynolds stress turbulence model was used in this model due to the anisotropic nature of the turbulence in cyclones. Standard fluent wall functions were applied and high order discretisation schemes were also used. 

One limitation of the one-solubility parameter model is that it assumes that the solute can only interact with the organic matter through London forces. Although this assumption may be reasonable for SOM, DOM is typically more polar and can participate in other types of van der Waals interactions. These include permanent dipole-induced dipole (Debye) and permanent dipole-permanent dipole (Keesom) interactions in which the degree of binding that occurs depends on the polarizability of the DOM (Gauthier et al., 1987 Uhle et al., 1999). To account for these types of interactions Chin and Weber (1989) segregated the solubility parameter terms into three components to account for all these different types of molecular interactions to... 

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