Composition gradient model

The starting point for developing the model is the set of diffusion equations for a gas mixture in the presence of temperature, pressure and composition gradients, and under the influence of external forces." These take the following form... 

The complete problem with composition gradients as well as a pressure gradient, may be regarded as a "generalized Poiseuille problem", and its Solution would be valuable for comparison with the limiting form of the dusty gas model for small dust concentrations. Indeed, it is the "large diameter" counterpart of the Knudsen solution in tubes of small diameter. 

For the WC phase, the milder composition gradients, when revealed at the surface, make smoother transitions. This produces on the average lower C/He ratios, in very good agreement with observations [1]. The abundances in WC stars are not equilibrium values, but are products of the partial He-burning, thus they are model dependent and offer a most interesting test on the quality of stellar models. 

All the experimental data in Table 6.1 refer to pure gases. Separation experiments, in which surface diffusion is the separation mechanism, are scarcely reported. Feng and Stewart (1973) and Feng, Kostrov and Stewart (1974) report multicomponent diffusion experiments for the system He-Nj-CH in a y-alumina pellet over a wide range of pressures (1-70 bar), temperatures (300-390 K) and composition gradients. A small contribution of surface diffusion (5% of total flow) to total transport could be detected, although it is not clear, which of the gases exhibits surface difiusion. The data could be fitted with the mass-flux model of Mason, Malinauskas and Evans (1967), extended to include surface diffusion. 

The general approach of graded radiation exposure can also be used to examine light driven processes such as photopolymerization [19]. For example, Lin-Gibson and coworkers used this library technique to examine structure-property relationships in photopolymerized dimethacrylate networks [38] and to screen the mechanical and biocompatibility performance of photopolymerized dental resins [39]. In another set of recent studies, Johnson and coworkers combined graded light exposure with temperature and composition gradients to map and model the photopolymerization kinetics of acrylates, thiolenes and a series of co-monomer systems [40 2]. 

In the two-film model (Figure 13), it is assumed that all of the resistance to mass transfer is concentrated in thin stagnant films adjacent to the phase interface and that transfer occurs within these films by steady-state molecular diffusion alone. Outside the films, in the bulk fluid phases, the level of mixing is so high that there is no composition gradient at all. This means that in the film region, only one-dimensional diffusion transport normal to the interface takes place. 

The reaction considered is the gas-phase, irreversible, exothermic reaction A + B — C occurring in a packed tubular reactor. The reactor and the heat exchanger are both distributed systems, which are rigorously modeled by partial differential equations. Lumped-model approximations are used in this study, which capture the important dynamics with a minimum of programming complexity. There are no sharp temperature or composition gradients in the reactor because of the low per-pass conversion and high recycle flowrate. 

In flow the challenge has been to write convincing equations that couple concentration and composition gradients to elastic stresses and the bulk flow field. When done within a two-fluid model for polymer solutions transitions in light-scattering patterns seen in experiment may be explained. Extensions to polymer blends are potential candidates as explanations of shear-induced shifts of the spinodal and biphasic islands seen experimentally. - ... 

Figure 8.1. Film model for transfer in phase x. Turbulent eddies wipe out composition gradients in the bulk fluid phase. Composition variations are restricted to a layer (film) of thickness f adjacent to the interface. Model due to Lewis and Whitman.

The equations derived in the previous section represent penneation rates and flux, and Examples 18.1 and 18.2 are apphcations to a speciflc model. The model assumes the fluid on each side of the membrane to have a constant composition parallel to the membrane. The compositions normal to the membrane are also assumed constant, with the possible exception of composition gradients in the films adjacent to the membrane. The bulk phase compositions on both sides of the membrane had to be given since no material balances were considered. The flow pattern implied in this model is that of perfect mixing (if the film resistances next to the membrane are neglected). Other flow patterns include cross flow, countercurrent flow, and cocurrent flow. 

Desre et al. [2.88] have proposed a mechanism for the suppression of nucleation of intermetallics in the case that an amorphous layer has already formed. In this model, nucleation of the intermetallic is impeded by the composition gradient in the growing amorphous interlayer. According to Figs. 2.18, 20, this composition gradient is given by... 

In many practical instances involving multtcompouant diffusion the diffusive flux is simply modeled in terms of an effective diffiisivity of i in Ihe mixture (/>, ) which relates the flux of i only to its owa composition gradient. 

However, current thermodynamic theories of compositional equilibrium under the combined influence of gravity and temperature fields do not adequately explain the large compositional gradients that are often encountered, except at conditions close to critical (Schulte 1980 Holt et al. 1983 Creek Schrader 1985 England et al. 1987 Nutakki et al. 1996). It is now quite common for the phenomenon of strong compositional grading to be associated with near-critical fluids, but the definition of near-critical fluids is rather broad and hazy Another problem with these theories is that they often do not predict the shape of these compositional depth trends at all well. In fact, Hoier Whitson (2001) doubt that most petroleum fields satisfy the fundamental assumptions in these models, especially that of zero mass flux (i.e. stationary state equilibrium). 

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