Continuous-solid model

Continuous-Solid Model. In this model, the axial conduction, in both phases, is included through the use of effective thermal conductivities kfe and kse. This gives... 

Dispersion-Particle-Based Model. This is an improvement over the continuous-solid model and allows for dispersion. The results are... 

Figure 9.7 The continuous melting model for Dt=0.001 and diverse values of the residual porosity

Limitations of the Shrinking Core Model. The assumptions of this model may not match reality precisely. For example, reaction may occur along a diffuse front rather than along a sharp interface between ash and fresh solid, thus giving behavior intermediate between the shrinking core and the continuous reaction models. This problem is considered by Wen (1968), and Ishida and Wen (1971). 

There are, however, two broad classes of exceptions to this conclusion. The first comes with the slow reaction of a gas with a very porous solid. Here reaction can occur throughout the solid, in which situation the continuous reaction model may be expected to better fit reality. An example of this is the slow poisoning of a catalyst pellet, a situation treated in Chapter 21. 

For the coke that burns in the regenerator, the continuous reaction model for solids of unchanging size is used because of its simplicity. 

This applies, however, only within limits. Many solids are somewhat mobile and can flow very slowly. In that case methods and models of capillarity can be applied. One case where capillarity plays an important role is sintering. In sintering a powder is heated. At a temperature of roughly 2/3 of the melting point of the material the surface molecules become mobile and can diffuse laterally. Thereby the contact areas of neighbouring particles melt and menisci are formed. When cooling, the material solidifies in this new shape and forms a continuous solid. 

Therefore under a constant stress, the modeled material will instantaneously deform to some strain, which is the elastic portion of the strain, and after that it will continue to deform and asynptotically approach a steady-state strain. This last portion is the viscous part of the strain. Although the Standard Linear Solid Model is more accurate than the Maxwell and Kelvin-Voigt models in predicting material responses, mathematically it returns inaccurate results for strain under specific loading conditions and is rather difficult to calculate. 

We close this introduction with a final remark about the modelling of the failure. In a real situation, failure takes place in solid samples which are, by nature, continuous in space. However, many studies (numerical and experimental) have been made on lattices. In all these studies, it is an implicit assumption that one can replace a continuous solid by a lattice. For example, a conducting solid can be described by a lattice in which the bonds between sites are identical resistors. It is a very common practice in percolation type models of disordered solids. We stress that this transformation (continuous solid to lattice) defines a particular length scale the length of the unit cell of the lattice. This implies that defects appear by discrete steps and this does not correspond always to real situations. We shall see later how to remove this limitation. 

The first two illustrations are of typical pilot plant-size centrifuges. Figure I is a continuous, solid bowl centrifuge especially suitable for studying the recovery and concentration of polymer solids from hydrocarbon mother liquor. It is of the standard model that has been widely used for this purpose in the past. A similar size centrifuge is also available for operation under pressures up to 1 atm. gage and higher if required, for continuous operation in a pressurized system. 

It is well known that the energy of interaction of an atom with the continuous solid is 2-3 times less than with the discrete (atomic) model (cf., e.g., Ref. [38], Figs. 2.2-2.4). Thus, to obtain the same Henry s Law constants with the two models, one has to increase e for the continuous model. This, however, does not discredit the continuous model which is frequently used in adsorption calculations. In particular, we can use the above mentioned results of Ref. [37] to predict the value of e for Ar which would have been obtained if one had carried out Henry s Law constant calculations for Ar in the AO model of Ref. [17] and compared them with experiment. One can multiply the value of e for CH4 obtained from AO model by the ratio of e values for Ar and CH4 in the CM model [36] to obtain tjk = 165A for Ar in the AO model. This is very close to the value of 160 K obtained in Ref. [21, 28] by an independent method in which the value of the LJ parameter e for the Ar - oxide ion interaction was chosen to match the results of computer simulation of the adsorption isotherm on the nonporous heterogeneons surface of Ti02. Considering the independence of the calculations and the different character of the adsorbents (porous and nonporous), the closeness of the values of is remarkable (if it is not accidental). The result seems even more remarkable in the light of discussion presented in Ref. [28]. Another line of research has dealt with the influence of porous structure of the silica gel upon the temperature dependence of the Henry constants [36]. 

Results of recent theoretical and computer simulation studies of phase transitions in monolayer films of Lennard-Jones particles deposited on crystalline solids are discussed. DiflFerent approaches based on lattice gas and continuous space models of adsorbed films are considered. Some new results of Monte Carlo simulation study for melting and ordering in monolayer films formed on the (100) face of an fee crystal are presented and confronted with theoretical predictions. In particular, it is demonstrated that the inner structure of solid films and the mechanism of melting transition depend strongly on the effects due to the periodic variation of the gas - solid potential. 

Spatially periodic porous media are made up of structural elements whose arrangement in space is completely described by a single unit cell (similar to the representative elementary volume concept of Bear, 1969), that is then repeated ad infinitum (Adler, 1992). The structural elements can be discrete voids in a continuous solid phase or vice versa. The simplest spatially periodic models are comprised... 

According to the basic statement of the models we are going to summarize, the metal is conceived as a network of cations immersed in a cloud of free electrons in a crystalline structure. The transport of the ions controls the growth of the new phase (Figure 8.2). The ionic transport will depend on the nature of the system and on experimental conditions, such as temperature, local electric field, local concentration excess, etc. To better understand the continuous-film models, the main ionic transport mechanisms in crystalline solids are presented [1] as follows. 

Proposed mechanisms of solids production from unconsolidated sand reservoirs have been discussed (102,103). Dusseault and Santarelli (104) proposed a mechanism for massive solids production from poorly consolidated sandstones that was based on a general plastic yield of the reservoir brought about by a high pressure drawdown in the yielded region. The vertical stress that the reservoir experiences was also a contributing factor. Subsequently, Geilikman et al. (105-108) developed a model for continuous solids production from unconsolidated heavy oil reservoirs as a yield front propagation. This is different from predictive models previously discussed (45, 46), which dealt with transient and catastrophic production but which did not discuss continuous production explicitly. 

A simple case of a batch reactor will be explained briefly. If the powder is mixed well in a container or pasted on a wall (case a), the reaction proceeds continuously throughout the solid particle. If most of the primary particles have similar diameter, the reaction proceeds at the same rate for all the particles. Under such conditions, the reaction rate analysis is rather simple, and two idealized models have been presented continuous reaction model and unreacted core model. For the former case. 

Construction of a contour for a form feature in a part model and its validation by use of checks is explained in Figure 8-9. Contour C is sketched for form feature FFj. Validation of contour C revealed three errors and communicated them to the engineer. Contour C contained a break point This is not allowed for the selected type of form feature solid modeling requires a closed contour. Contour C was found to be open at point P, i.e., the gap is larger than the specified tolerance. Note that certain form features such as solid sweeps allow an open contour. Finally, the upper limit specified for the length L was exceeded. After correction, the modified contour was applied at the creation of form feature FFj. Construction of the part model can be continued by form feature FF2 because form feature FFj was proved correct during its repeated validation. 

For the heterogeneous catalytic process to be effective, the reactants present in the surrounding fluid phase must be transported to the surfece of the solid catalyst, and after the reaction, the products formed must be carried back from the surface to the bulk fluid. The path of the physical rate processes at the particle scale is divided into two parts, as depicted in the 7-step sequence of the continuous reaction model used in microkinetic analysis ... 

Ai-feiastic jg glastic enthalpy of the solid solution calculated based on the continuous elastic model proposed by Friedel (Friedel, 1954) and Eshelby (Eshelby, 1954 1956), Al-fstructure jg the structure enthalpy induced by the structural changes, and jg chemical... 

While Fig. 17 is patterned after actual data on PIB, other polymers could have been used as a model with similar results. In view of extensive studies on af-PS, cited herein, we reproduce in Figure 18 a plot of actual data assembled from a variety of sources by McCrum, Read, and Williams in their Fig. 10.35. Fig. 18 differs from the original in that not all data points are shown and also in our dashed extrapolations of the Tg and Tp loci. 7 431 K which is in excellent agreement with the asymptotic Tn limit in Fig. 3. The continuous solid lines for Tg and Tpv/CK drawn by McCmm et al.l as representing trends for the two processes. This figure is one more illustration of the Lobanov-Frenkel technique. ... 

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