WS model

More recently, we have created full-bed periodic two-layer models for packings of cylindrical particles (Taskin et al., 2006). The geometry shown in Fig. 3b was created specifically for comparison of the WS model with cylindrical particles described in the following section, with the structures identical within a 120° segment of the bed. 

Fig. 4. The 120° wall-segment (WS) model top (spheres) and front (cylinders).

When simulations are done in a WS model, the results need to be validated against a full-bed model. The main reason for this is not only to see if the WS model results are representative for a full bed but also to check that the symmetry boundaries, which are relatively close to all parts of the segment model, do not influence the solution. 

A section in the full-bed models was isolated that was comparable to the WS model. The layout of these different sections was identical, except that the WS model had a two-layer periodicity and the full-bed models had a six-layer periodicity. To be able to make direct comparisons of velocity profiles, several sample-points needed to be defined. In the three different models seven tangential planes were defined and on each plane three axial positions were defined. This reduced the data to single radial velocity profiles at corresponding positions in all three models, as shown in Fig. 10, for the WS model. Identical planes were defined in the full-bed models. Some spheres and sample planes 4 and 5 are not displayed to improve the visibility of the sample planes and lines. In the right-hand part of the figure, plane 4 is shown with the axial positions at which data were taken and compared. 

The data from the WS model in some cases deviated slightly from the full-bed models. This could be explained by the slightly different layout of the WS model. Some spheres had to be relocated in the WS model to create a two-layer periodicity from the six-layer periodicity in the full-bed models. The differences in velocity magnitudes were mainly found in the transition area between the wall layers and the center layers. The effect of slightly larger gaps between spheres from the nine-sphere wall layers and the three-sphere central layers, due to the sphere relocations, had a noticeable effect on the velocity profile. Differences were also found in the central layer area where the sphere positions were not identical. 

Fourth, the significance of a proper hard-core term near the critical density remains subject to debate. Near the critical density, the MSA theory is more sensitive to the choice of the hard-core term than is the DH theory [198,199]. At high ion densities the need for an appropriate choice of the hard-core term is unquestionable—for example, to prevent the coexistence curve from reaching states beyond the close packing of the b.c.c. solid phase of the RPM (Pbcc — 1-3 [252]). Figure 9 compares the coexistence curves of the FL and WS models with the simulation results of Orkoulas and Panagiotopoulos [52],... 

To demonstrate the differences between the WS and the HVO models, the results of VLE predictions for the 2-propanol and water binary system at 353 K with the parameters obtained from the DECHEMA tables at 303 K are shown in Eigure 4.3.9 in which the solid line is the prediction with the WS mixing rule and the dashed line describes the results of the HVO model. The significant advantage of the WS model over the HVO model in predictions is clearly visible in this figure. 

Before proceeding, it is necessary to stress once again a characteristic difference between the WS model and the other five models used here. All the EOS-G models,... 

Figure 5.1.1. VLE prediction for the methanol and benzene binary system at 293 K by various methods. Circles represent experimental data, the solid line with crosses shows the UNIFAC predictions, and the smooth solid line denotes the results of the WS model. The large, medium, and short dashed lines are from the HVOS, HVO, and MHVl models, respectively the dotted line is from the MHV2 model and the dot-dash line reflects the results of the LCVM model. (Points are VLE data from the DECHEMA Chemistry Series, Gmehling and Onken 1977, Vol. 1, Pt, 2a, p. 220 the data file name on the accompanying disk for this system is MB20.DAT.)...

In Figure 5.1.3 the prediction of the VLB behavior of the various models for the methanol and benzene binary system at 453 K is shown. The direct use of the UNIFAC activity coefficient model in the y-4> model qualitatively behaves differently than the other models and performs relatively poorly. The various EOS-G models show similar behavior that is in qualitative agreement with the experimental data however, quantitatively the WS model is the most accurate. 

Among the mixing rules tested here in the predictive EOS-G formalism, only the WS model can be made to match the excess free energy from a conventional activity coefficient model closely by varying the model parameter kij. This flexibility can also be used to incorporate infinite dilution activity coefficient information into this model, as discussed in the next section. 

Figure 6.2.2. Excess enthalpy for the benzene and cyclohexane system at 293 K (dots) and at 393 K (triangles), Tlie lines denote correlations at 293 K and predictions at 393 K using various models. The solid line reflects predictions using the 2PVDW model, the dotted line represents the predictions using the van Laar activity coefficient mode , the short dashed lines signify predictions using the HVOS model, and the long dashed line denotes predictions made with the WS model. Data are from the DECHEMA Chemistry Series (Gmehling and Onken 1977, Vol. 3, Pt. 2, p. 992).

IN ADDITION Kl2 PARAMETER OF THE WS MODEL IS FIT (other parameters such as alpha of the NRTL model, or UNIQUAC pure component parameters must be supplied by user.) "... 

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