Nonlinear diffusion model

Fig. 15. Theoretical breakthrough curves for a nonlinear (Langmuir) system showing the comparison between the linear driving force (—), pore diffusion (--------------------), and intracrystalline diffusion (-) models based on the Glueckauf approximation (eqs. 40—45). From Ref. 7.

The hydraulic permeation model predicts highly nonlinear water content profiles, with strong dehydration arising only in the interfacial regions close to the anode. Severe dehydration occurs only at current densities closely approaching/p,. The hydraulic permeation model is consistent with experimental data on water content profiles and differential membrane resistance, i i as corroborated in Eikerling et al. The bare diffusion models exhibit marked discrepancies in comparison with these data. 

Depending on the distribution chosen, as few as three fitting parameters may be required to define a distribution of diffusion rates. In some cases, a single distribution was used to describe both fast and slow rates of sorption and desorption, and in other cases fast and slow mass transfer were captured with separate distributions of diffusion rates. For example, Werth et al. [42] used the pore diffusion model with nonlinear sorption to predict fast desorption, and a gamma distribution of diffusion rate constants to describe slow desorption. 

Reaction-diffusion models have very successfully accounted for the interaction between normal and tumor cells. The diffusion terms in these models can be broadly divided into two categories, linear and nonlinear diffusion. In linear diffusion models, the flux of one cell type depends only on the concentration of cells of the same type. In nonlinear diffusion models, the presence of one cell type affects the diffusion of cells of a different type. Models with nonlinear diffusion have described the spatial dispersal and temporal development of tumor tissue, normal tissue, and excess H ion concentration [155]. They assume that transformation-induced reversion of neoplastic tissue creates a microenvironment around the tumor where tumor cells survive and proliferate, whereas normal cells do not remain viable. These conditions, favorable for tumor cells and unfavorable for normal cells, are due to... 

FIGURE 11.8 First- and second-order FRFs for micropore diffusion model. (From Petkovska, M. and Do, D.D., Nonlinear Dyn., 21, 353 376, 2000. With permission.)... 

The effect of laser phase fluctuations on PIER4 has been considered by Agarwal, and detailed line shapes were calculated. In a recent publication we have derived (within the phase diffusion model) equations of motion in the limit of short correlation times. The effect of the stochastic phase fluctuations was shown to be similar to T2 dephasing processes, and a procedure was given for the inclusion of this similarity in many nonlinear processes. In particular, two predictions were made ... 

Theoretical breakthrough curves for nonlinear systems may be calculated by numerical solution of the model equations using standard finite difference or collocation methods. Such solutions have been obtained by many authors and a brief summary is given in Table 8.4. In all cases plug flow was assumed and the equilibrium relationship was taken to be of cither Langmuir or Freundlich form. As linearity is approached ( ->1.0) the linearized rate models approach the Anzelius model (Table 8.1, model la) while the diffusion models approach the Rosen model (Table 8.1, model la). The conformity of the numerical solution to the exact analytic solution in the linear limit was confirmed by Garg and Ruthven. ... 

FIGURE 8.17. Theoretical breakthrough curves for nonlinear (Langmuir) systems showing comparison between linear rale, pore diffusion, and intracrystalline diffusion models. See Table 8.4, models 16, 2c, 36.)... 

The adequacy of the Glueckauf linear rate approximation may be judged by comparing the curves calculated from the diffusion models with the curves calculated fom the linear rate expression, which are also shown in Figure 8.17. The linearized rate approximation provides a good representation of the desorption curves over the entire range of nonlinearity. The approximation is... 

FIGURE 8.18. Theoretical breakthrough curves calculated for a nonlinear (Langmuir) system according to the pore diffusion model (model 3i, Table 8.4) showing effect of nonlinearity. In b) the desorption curves of (a) are shown plotted on an inverted concentration scale with... 

The breakdown of the linear rate approximation for nonlinear systems was noted by Vermuelen who developed modified lumped parameter approximations which represent the diffusion models more accurately than the simple linear rate expression. For solid diffusion (model 2a or 2b) a quadratic driving force approximation is recommended ... 

FIGURE 8.21. Comparison of dimensionless bed lengths required to approach within 5% of the constant-pattern solution, as measured by the 10-90% times, according to the linear rate (solid film), intracrystalline, and pore diffusion models (Table 8.6, models lb, 2b, and 36) showing effect of isotherm nonlinearity. 

Magyar E. Exact analytical solution of a nonlinear reaction— diffusion model in porous catalysts. Chemical Engineering Journal 2008 143 167-171. 

Rohrlich, B. Parisi, J. Peinke, J. Rossler, O. E. 1986. A Simple Morphogenetic Reaction-Diffusion Model Describing Nonlinear Transport Phenomena in Semiconductors, Z. Phys. B Cond. Matter 65, 259-266. 

For ionomeric systems in which the strong interactions between ionic sites and the penetrant (water) result in concentration dependent diffusion coefficients, the Fickian diffusion model, which assumes the solubility coefficient is independent of the concentration, is not valid. The commonly used Fickian diffusion constants actually contain mobility and solubility gradient contributions [19], In addition, the concentration dependent solubilities lead to nonlinear concentration profiles during steady state diffusion. Therefore, mobility measurements which generate average diffusion coefficients are generally not satisfactory. 

The lateral particle dispersion coefficient was also studied by Shi and Fan (1984) and Subbarao et al. (1985). A one-dimensional diffusion model was used by Shi and Fan (1984) to characterize lateral mixing of solids. Through dimensional analysis and nonlinear regression analysis of the literature data, they arrived at an equation for the lateral dispersion coefficient for general application. 

Lefebvre et developed a generalized Fickean diffusion model using the free-volume concept. A finite-element model that accounts for Schapery s nonlinear viscoelastic constitutive relation(25) and the nonlinear diffusion model of Lefebvre et was discussed by Roy and Reddy. ( 4,45)... 

The two-dimensional nonlinear Fickean diffusion model used in the present study is the one developed by Lefebvre et Fick s law for the two-dimensional diffusion of a penetrant within an isotropic material is given by... 

A review of the theoretical basis, finite-element model, and sample applications of the program NOVA are presented. The updated incremental Lagrangian formulation is used to account for geometric nonlinearity (i.e., small strains and moderately large rotations), the nonlinear viscoelastic model of Schapery is used to account for the nonlinear constitutive behavior of the adhesive, and the nonlinear Fickean diffusion model in which the diffusion coefficient is assumed to depend on the temperature, penetrant concentration, and dilational strain is used. Several geometrically nonlinear, linear and nonlinear viscoelastic and moisture... 

To conclude this section, systems of diffusively coupled nonlinear oscillators act as conceptual models that can explain the phenomenon of ERD. In general, the model discussed here not only applies to explaining ERD in the brain but can also explain similar occurrences of ERD in other systems such as Josephson junction arrays, Bose-Einstein condensates, system of coupled spin torque nano-oscillators, and the like. 

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