Modified cell model

Dee, G. T., and Walsh, D. J., A modified cell model equation of state for polymer liquids,... 

Rudolf et al. (1998) used the modified cell model of Dee and Walsh and the Simha-Somcynsky theory to investigate the phase behavior, excess volumes, the influence of pressure on miscibility, and the causes of miscibility. It was found that the theory of Dee and Walsh yields results similar to the previously investigated theories, whereas the Simha-Somcynsky theory does not. A modification of the latter theory for mixtures again resulted in predictions similar to that of Dee and Walsh and the earlier investigated theories. 

S.C. Davis, U.S. Patents 5,372,980, 13 Dec 1994, Appl. 03 June 1993, to Polysar G.T. Dee, D.J. Walsh, Equations of state for polymer liquids. Macromolecules 21, 811-815 (1988) A modified cell model equation of state for polymer liquids. Macromolecules... 

Two forms of the cell model (CM) are then developed harmonic oscillator approximation and square-well approximation. Both forms assnme hexagonal closed packing (HCP) lattice structure for the cell geometry. The model developed by Paul and Di Benedetto [13] assumes that the chain segments interact with a cylindrical symmetric square-well potential. The FOV model discnssed in the earlier section uses a hard-sphere type repulsive potential along with a simple cubic (SC) lattice structure. The square-well cell model by Prigogine was modified by Dee and Walsh [14]. They introduced a numerical factor to decouple the potential from the choice of lattice strncture. A universal constant for several polymers was added and the modified cell model (MCM) was a three-parameter model. The Prigogine cell EOS model can be written as follows. 

What is the significance of each parameter in the Dee and Walsh modified cell model, MCM ... 

Additional examples of equation of state models include the lattice gas model (Kleintjens et al, [33,34], Simha-Somcynsky hole theory [35], Patterson [36], the cell-hole theory (Jain and Simha [37-39], the perturbed hard-sphere-chain equation of state [40,41] and the modified cell model (Dee and Walsh) [42]. A comparison of various models showed similar predictions of the phase behavior of polymer blends for the Patterson equation of state, the Dee and Walsh modified cell model and the Sanchez-Lacombe equation of state, but differences with the Simha-Somcynsky theory [43]. The measurement and tabulation of PVT data for polymers can be found in [44]. 

To account for the variation of the dynamics with pressure, the free volume is allowed to compress with P, but differently than the total compressibility of the material [22]. One consequent problem is that fitting data can lead to the unphysical result that the free volume is less compressible than the occupied volume [42]. The CG model has been modified with an additional parameter to describe t(P) [34,35] however, the resulting expression does not accurately fit data obtained at high pressure [41,43,44]. Beyond describing experimental results, the CG fit parameters yield free volumes that are inconsistent with the unoccupied volume deduced from cell models [41]. More generally, a free-volume approach to dynamics is at odds with the experimental result that relaxation in polymers is to a significant degree a thermally activated process [14,15,45]. 

As noted in the Introduction, one of the defining characteristics of any fuel-cell model is how it treats transport. Thus, these equations vary depending on the model and are discussed in the appropriate subsections below. Similarly, the auxiliary equations and equilibrium relationships depend on the modeling approach and equations and are introduced and discussed where appropriate. The reactions for a fuel cell are well-known and were introduced in section 3.2.2. Of course, models modify the reaction expressions by including such effects as mass transfer and porous electrodes, as discussed later. Finally, unlike the other equations, the conservation equations are uniformly valid for all models. These equations are summarized below and not really discussed further. 

Appendix Mathematical models. The mathematical model for Figures 1,2 and 7 is a modified Deans-Lapidus cell model (3), similar to that used by Jaffe (2) except that a different flow scheme has been used. The following assumptions have been made throughout ... 

If an extra element is added to a Randles cell, a modified Randles model can be formed. 

There is an obvious similarity between the equations of the crossflow model and those of the modified mixing-cell model. With suitable redefinition of parameters, it can easily be shown that these partial differential equations are mathematically identical. Thus, the solution for the modified mixing-cell model is identical to the solution for the crossflow model, for the same set of boundary conditions. 

Three-parameter PDE model (Van Swaaij et aL106) This model is largely used to correlate the RTD curves from a trickle-bed reactor. The model is based on the same concept as the crossflow or modified mixing-cell model, except that axial dispersion in the mobile phase is also considered. The model, therefore, contains three arbitrary parameters, two of which are the same as those used in the cross-flow model and the third one is the axial dispersion coefficient (or the Peclet number in dimensionless form) in the mobile phase (see Fig. 3-11). 

Later, Polyakov and Tarakanov modified the model by presenting if of a hexahedral cell having an initial curvature (eccentricity) near the rods disposed in two perpendicular directions. This model, which makes allowance for the initial anisotropy of the plastic foam, satisfactorily describes the elastic properties of flexible foams under considerable strains, but generally predicts unduly high values of the elastic properties of foamed plastics. 

Das, T., Bratko, D., Bhuiyan, L.B., and Outhwaite, C.W. Polyelectrolyte solutions containing mixed valency ions in the cell model A simulation and modified Poisson-Boltzmann study. Journal of Chemical Physics, 1997,107, No. 21, p. 9197-9207. 

FIGURE 21.15 Example profile of modified cell transit model with disease modifying drug action. 

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