Segregation equation solutions

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. ... 

Equation 20 also holds for a strongly segregated polymer solution (but then / is of order unity). 

The basic expression describing segregation. Equation (5.9), with the appropriate boundary conditions, can be solved when the bulk velocity, the percolation velocity, and the diffusion coefficients are all determined a priori. The solution of the differential equation can be achieved by considering the problem in terms of several specific cases. 

The value of the concentration C satisfying this equation at the center of each box is computed numerically by iteration. This expression is the classical one for the segregation of solute atoms given by Cottrell and Jawson (1949)... 

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. 

Solution The limits you can calculate under part (a) correspond to the three apexes in Figure 15.14. The limits are 0.167 for a PFR (Equation (1.47)), 0.358 for a CSTR (Equation (1.52)), and 0.299 for a completely segregated stirred tank. The last limit was obtained by integrating Equation (15.48) in the form... 

Deactivating catalyst 319 Dead zones 159, 162, 163 Degree of segregation 471 Density influences 492 Desorption of solute 578, 579 Difference differential equation 579 Difference formulae for partial differential equations 268 Differential column 167... 

However, the solution for the CSTR obtained by the RTD equation is correct only for first-order kinetics. For other rate expressions the conversion predicted by the RTD is incorrect for a mixed reactor because molecules do not simply react for time t, after which they leave the reactor. Rather, the fluid is continuously mixed so that the history of the fluid is not describable in these terms. This expression for conversion in the CSTR is applicable for segregated flow, in which drops of fluid enter the reactor, swirl in the reactor, and exit after time t because then each drop behaves as a batch reactor with the RTD describing the probability distribution of the drops in the CSTR. 

Solution. As indicated in the text, Eq. 9.9 must have the form of Eq. 9.15 in order to satisfy the segregation condition k = c2 /c2L at the boundary slab. Equation 9.10 then becomes... 

A mean field approach was applied to determine homopolymer distributions in the lamellar phase of a blend of AB diblock and A homopolymer by Shull and Winey (1992). In the strong segregation limit, complete segregation of the A homopolymer into the A microdomain was predicted. Furthermore, in this limit, the diblocks were treated as brushes , wetted by homopolymer in the A domain. Composition profiles showing the distribution of homopolymer and copolymer were determined by numerical solution of the self-consistent field equations. 

In equations 5-8, the variables and symbols are defined as follows p0 is reference mass density, v is dimensional velocity field vector, p is dimensional pressure field vector, x is Newtonian viscosity of the melt, g is acceleration due to gravity, T is dimensional temperature, tT is the reference temperature, c is dimensional concentration, c0 is far-field level of concentration, e, is a unit vector in the direction of the z axis, Fb is a dimensional applied body force field, V is the gradient operator, v(x, t) is the velocity vector field, p(x, t) is the pressure field, jl is the fluid viscosity, am is the thermal diffiisivity of the melt, and D is the solute diffiisivity in the melt. The vector Fb is a body force imposed on the melt in addition to gravity. The body force caused by an imposed magnetic field B(x, t) is the Lorentz force, Fb = ac(v X v X B). The effect of this field on convection and segregation is discussed in a later section. 

Solute segregation with bulk convection is given rigorously by solving the two-dimensional solute balance equation [for a two-dimensional velocity... 

Fluent is a commercially available CFD code which utilises the finite volume formulation to carry out coupled or segregated calculations (with reference to the conservation of mass, momentum and energy equations). It is ideally suited for incompressible to mildly compressible flows. The conservation of mass, momentum and energy in fluid flows are expressed in terms of non-linear partial differential equations which defy solution by analytical means. The solution of these equations has been made possible by the advent of powerful workstations, opening avenues towards the calculation of complicated flow fields with relative ease. 

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. 

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