Other Diffusion Models

Whereas the thermodynamic properties of the resin and environment will determine the maximum extent of moisture absorption, diffusion is a kinetic property which is temperature dependent. Normally, the latter is considered to be Fickian in nature. For highly polar polymers, other diffusion models need to be considered, but for most structural composite materials, the Fickian laws are applicable. 

The Life-365 software predicts the initiation period assuming ionic diffusion to be the dominant mechanism. This software differs from other diffusion models in that it accounts for the variability of the diffusion coefficient with age and with temperature. It also attempts to model the impact of various additives. For additives such as silica fume and fly ash it reduces the diffusion coefficient to reflect the lower permeability and for corrosion inhibitors it raises the chloride threshold required to initiate corrosion. To include the impact of sealers and membranes it reduces the rate of accumulation of the surface chloride concentration. The rate of accumulation and the maximum accumulation of surface chloride in this program are based on the type of structure, geographic location and exposure. ACI Committee 365 has also published a state-of-the-art report on service life prediction which is in the process of being updated (ACI 365.1R-00 (2000)). 

Of particular importance is the timescale over which diffusion occurs under various conditions of relative humidity (RH) and temperature. The RH determines the equilibrium moisture concentration, whereas higher temperatures will accelerate the moisture sorption process. In order to predict the moisture profile in a particular structure, it is assumed that Fickian diffusion kinetics operate. It will be seen later that many matrix resins exhibit non-Fickian effects, and other diffusion models have been examined. However, most resin systems in current use in the aerospace industry appear to exhibit Fickian behaviour over much of their service temperatures and times. Since the rate of moisture diffusion is low, it is usually necessary to use elevated temperatures to accelerate test programmes and studies intended to characterize the phenomenon. Elevated temperatures must be used with care though, because many resins only exhibit Fickian diffusion within certain temperature limits. If these temperatures are exceeded, the steady state equilibrium position may not be achieved and the Fickian predictions can then be inaccurate. This can lead to an overestimate of the moisture absorbed under real service conditions. 

One of the problems with the model of SGZ and other diffusion models, is that under eonditions close to full hydration (Figure 4.2 c), there is essentially no water coneentration gradient, and diffusion models are unable to produce... 

The methods presented in the last two sections can be applied to any other diffusion models in the pellet. Readers are encouraged to apply the method to their specific systems. Despite of the simplicity suggested by the method, the extraction of the micropore diffiisivity (second order process) requires a very careful collection of experimental data. If micropore diffusion is dominating the dynamic uptake, the batch adsorber provides a better means to extract the micropore diffiisivity, and this will be discussed in Chapter 15. 

Diffusion within the largest cavities of a porous medium is assumed to be similar to ordinary or bulk diffusion except that it is hindered by the pore walls (see Eq. 5-236). The tortuosity T that expresses this hindrance has been estimated from geometric arguments. Unfortunately, measured values are often an order of magnitude greater than those estimates. Thus, the effective diffusivity D f (and hence t) is normally determined by comparing a diffusion model to experimental measurements. The normal range of tortuosities for sihca gel, alumina, and other porous solids is 2 < T < 6, but for activated carbon, 5 < T < 65. 

When a polymer film is exposed to a gas or vapour at one side and to vacuum or low pressure at the other, the mechanism generally accepted for the penetrant transport is an activated solution-diffusion model. The gas dissolved in the film surface diffuses through the film by a series of activated steps and evaporates at the lower pressure side. It is clear that both solubility and diffusivity are involved and that the polymer molecular and morphological features will affect the penetrant transport behaviour. Some of the chemical and morphological modification that have been observed for some epoxy-water systems to induce changes of the solubility and diffusivity will be briefly reviewed. 

If the. /-diffusion model is valid but only the energy relaxation time is known then Eq. (1.57) may be used to find the other ... 

The important fact is that the number of collisions Zr increases with temperature. It may be attributed to the effect of attraction forces. They accelerate the molecule motion along the classical trajectories favouring more effective R-T relaxation. This effect becomes relatively weaker with increase of temperature. As a result the effective cross-section decreases monotonically [199], as was predicted for the quantum J-diffusion model in [186] (solid line) but by classical trajectory calculations (dotted and broken lines) as well. At temperatures above 300 K both theoretical approaches are in satisfactory mutual agreement whereas some other approaches used in [224, 225] as well as SCS with attraction forces neglected [191] were shown to have the opposite temperature dependence for Zr [191]. Thus SCS results with a... 

In gridpoint models, transport processes such as speed and direction of wind and ocean currents, and turbulent diffusivities (see Section 4.8.1) normally have to be prescribed. Information on these physical quantities may come from observations or from other (dynamic) models, which calculate the flow patterns from basic hydrodynamic equations. Tracer transport models, in which the transport processes are prescribed in this way, are often referred to as off-line models. An on-line model, on the other hand, is one where the tracers have been incorporated directly into a d3mamic model such that the tracer concentrations and the motions are calculated simultaneously. A major advantage of an on-line model is that feedbacks of the tracer on the energy balance can be described... 

The second assumption has been effectively invalidated by the discovery of the hydrated electron. However, the effects of LET and solute concentration on molecular yields indicate that some kind of radical diffusion model is indeed required. Kuppermann (1967) and Schwarz (1969) have demonstrated that the hydrated electron can be included in such a model. Schwarz (1964) remarked that Magee s estimate of the distance traveled by the electron at thermalization (on the order of a few nanometers) was correct, but his conjecture about its fate was wrong. On the other hand, Platzman was correct about its fate—namely, solvation—but wrong about the distance traveled (tens of nanometers). 

The ideas of Overton are reflected in the classical solubility-diffusion model for transmembrane transport. In this model [125,126], the cell membrane and other membranes within the cell are considered as homogeneous phases with sharp boundaries. Transport phenomena are described by Fick s first law of diffusion, or, in the case of ion transport and a finite membrane potential, by the Nernst-Planck equation (see Chapter 3 of this volume). The driving force of the flux is the gradient of the (electro)chemical potential across the membrane. In the absence of electric fields, the chemical potential gradient is reduced to a concentration gradient. Since the membrane is assumed to be homogeneous, the... 

As noted in Chapter 1, the composition PDF description utilizes the concept of turbulent diffusivity (Tt) to model the scalar flux. Thus, it corresponds to closure at the level of the k-e and gradient-diffusion models, and should be used with caution for flows that require closure at the level of the RSM and scalar-flux equation. In general, the velocity, composition PDF codes described in Section 7.4 should be used for flows that require second-order closures. On the other hand, Lagrangian composition codes are well suited for use with an LES description of turbulence. 

As we have seen, the electric state of a surface depends on the spatial distribution of free (electronic or ionic) charges in its neighborhood. The distribution is usually idealized as an electric double layer one layer is envisaged as a fixed charge or surface charge attached to the particle or solid surface while the other is distributed more or less diffusively in the liquid in contact (Gouy-Chapman diffuse model, Fig. 3.2). A balance between electrostatic and thermal forces is attained. 

While first-order models have been used widely to describe the kinetics of solid phase sorption/desorption processes, a number of other models have been employed. These include various ordered equations such as zero-order, second-order, fractional-order, Elovich, power function or fractional power, and parabolic diffusion models. A brief discussion of these models will be provided the final forms of the equations are given in Table 2. 

The gradient of this graph therefore permits the determination of n, and the intercept allows k to be calculated. The advantage of using a Sharp-Hancock plot rather than a least squares fitting process with the Avrami equation is that if Avrami kinetics are not applicable, this can be seen in the former plot, and hence other kinetic models may be investigated. Purely diffusion controlled processes can be identified using a Sharp-Hancock plot n is foimd to be 0.5 in such cases. 

Trends in air pollutant concentrations can be predicted with simple empirical models based on atmospheric and laboratoiy data. Concentrations of nonreactive pollutants from point sources can be predicted vfith accuracy well within a factor of 2 predictions are more likely to be too high than too low, especially predictions of concentration peaks. Concentrations of reactive pollutants, such as ozone and other photochemical oxidants, can be predicted reasonably well with photochemical-diffusion models when detailed emission, air quality, and meteorolc c measurements are available most such predictions of air pollution in Los Angeles, California, have been accurate to within approximately 50% for ozone. Detailed performance analyses are found elsewhere in this chapter. 

The literature contains reviews of air quality modeling that stress special purposes. Some concentrate on meteorologic aspects, and others combine this with air chemistry. Proceedings of several conferences are another information resource. Recent surveys have been addressed specifically to photochemical modeling problems. It may be concluded that, although they are relatively complex, the photochemical-diffusion models perform as well as, if not better than, available inert-species models. 

Other Springer model derivatives include those of Ge and Yi, ° van Bussel et al., Wohr and co-workers, and Hertwig et al. Here, the models described above are slightly modified. The model of Hertwig et al. includes both diffusive and convective transport in the membrane. It also uses a simplified two-phase flow model and shows 3-D distributions... 

On the other hand, when the membrane is saturated, transport still occurs. This transport must be due to a hydraulic-pressure gradient because oversaturated activities are nonphysical. In addition, Buechi and Scherer found that only a hydraulic model can explain the experimentally observed sharp drying front in the membrane. Overall, both types of macroscopic models describe part of the transport that is occurring, but the correct model is some kind of superposition between them. - The two types of models are seen as operating fully at the limits of water concentration and must somehow be averaged between those limits. As mentioned, the hydraulic-diffusive models try to do this, but from a nonphysical and inconsistent standpoint that ignores Schroeder s paradox and its effects on the transport properties. 

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