Models gradient transport

Models for transport distinguish between the unsaturated zone and the saturated zone, that below the water table. There the underground water moves slowly through the sod or rock according to porosity and gradient, or the extent of fractures. A retardation effect slows the motion of contaminant by large factors in the case of heavy metals. For low level waste, a variety of dose calculations are made for direct and indirect human body uptake of water. Performance assessment methodology is described in Reference 22. 

The gradient transport model is most appropriate when the turbulence is confined to scales that are small relative to the pollutant volume. It is therefore most applicable to continuous line and area sources at ground... 

Modelling oxygen transport in the aeration system is important as it can be used as a reference for overall process performance improvement as well as process design and simulation. The oxygen transfer process mentioned above is based on the concentration gradient between... 

The presentation in this paper concentrates on the use of large-scale numerical simulation in unraveling these questions for models of two-dimensional directional solidification in an imposed temperature gradient. The simplest models for transport and interfacial physics in these processes are presented in Section 2 along with a summary of the analytical results for the onset of the cellular instability. The finite-element analyses used in the numerical calculations are described in Section 3. Steady-state and time-dependent results for shallow cell near the onset of the instability are presented in Section 4. The issue of the presence of a fundamental mechanism for wavelength selection for deep cells is discussed in Section 5 in the context of calculations with varying spatial wavelength. 

The diffusion layer theory, illustrated in Fig. 15B, is the most useful and best-known model for transport-controlled dissolution. The dissolution rate here is controlled by the rate of diffusion of solute molecules across a diffusion layer of thickness h, so that kT kR in Eq. (40), which simplifies to kx = kT. With increasing distance, x, from the surface of the solid, the concentration, c, decreases from cs at x = 0 to cb at x = h. In general, c is a nonlinear function of x, and the concentration gradient dddx becomes less steep as x increases. The hyrodynamics of the dissolution process has been fully discussed by Levich [104]. In a stirred solution, the flow velocity of the liquid dissolution medium increases from zero at x = 0 to the bulk value at x = h. 

Corrsin, S. (1974). Limitations of gradient transport models in random walks and in turbulence. Adv. Geophys. 18A, 25-59. 

Notwithstanding any particular structural model, water transport in PEMs, in general, should be considered a superposition of diffusion in gradients of activity or concentration and hydraulic permeation in gradients of liquid or capillary pressure. Hydraulic permeation is the predominant mechanism xmder conditions for which water uptake is controlled by capillary condensation, whereas diffusion contributes significantly if water strongly interacts with the polymeric host. The molar flux of liquid water in the membrane, N, is thus given by... 

Over the last four decades or so, transport phenomena research has benefited from the substantial efforts made to replace empiricism by fundamental knowledge based on computer simulations and theoretical modeling of transport phenomena. These efforts were spurred on by the publication in 1960 by Bird et al. (6) of the first edition of their quintessential monograph on the interrelationships among the three fundamental types of transport phenomena mass transport, energy transport, and momentum transport. All transport phenomena follow the same pattern in accordance with the generalized diffusion equation (GDE). The unidimensional flux, or overall transport rate per unit area in one direction, is expressed as a system property multiplied by a gradient (5)... 

This model assumes the diffusive flows combine by the additivity of momentum transfer, whereas the diffusive and viscous flows combine by the additivity of the fluxes. To the knowledge of the authors there has never been given a sound argument for the latter assumption. It has been shown that the assumption may result in errors for certain situations [22]. Nonetheless, the model is widely used with reasonably satisfactory results for most situations. Temperature gradients (thermal diffusion) and external forces (forced diffusion) are also considered in the general version of the model. The incorporation of surface diffusion into a model of transport in a porous medium is quite straightforward, since the surface diffusion fluxes can be added to the diffusion fluxes in the gaseous phase. 

The correlation between fluctuating velocity and hold-up is modeled using a gradient transport approximation as ... 

For LES performed in physical space, the basic sub-grid stress model is the eddy-viscosity model proposed by Smagorinsky . The Smagorinsky model is based on the gradient transport hypothesis and the sub-grid viscosity concept, just as the Reynolds stress models based on the Boussinesq eddy viscosity hypothesis, and expressed as ... 

Corrsin S (1974) Limitations of Gradient Transport Models in Random Walks and in Turbulence. In Landsberg HE, van Mieghem J (eds) Advances in geophysics, Academic Press, New York, 18A... 

However, as discussed in chap 1.2.7, the gradient-diffusion models can fail because counter-gradient (or up>-gradient) transport may occur in certain occasions [15, 85], hence a full second-order closure for the scalar flux (1.468) can be a more accurate but costly alternative (e.g., [2, 78]). 

Cell-free system. The first experiment with cell-free systems containing labeled nuclei and nonlabeled cytoplasm (Schneider, 1959 Scholtissek and Potter, 1960) indicated the removal of newly formed RNA from the nuclei during the incubation. The newly synthesized RNA was found in the incubation medium in particles with different sedimentation coefficients. These results have been confirmed by others (Samarina and Zbarsky, 1964 Ishikawa et al., 1969 Lukanidin, 1969). Sometimes ATP and other donors of energy enhance the loss of RNA from nuclei (Ishikawa et al., 1969). However, this process seems not to differ from the simple extraction of D-RNP from nuclei and thus may not be a model of transport. Lukanidin (1969) analyzed the particles in a CsCl density gradient and found that almost all labeled material leaving the nucleus has a buoyant density of 1.40 g/cm and does not interact with the cytoplasmic ribosomes which band at p = 1.55 to 1.58 g/cm. Only a small amount of labeled material banded in the intermediate zone, but it had a base composition similar to rRNA and thus may represent the ribosomal RNA precursors. The addition to the system of a variety of factors necessary for protein synthesis did not influence the results. 

Besides the diffusive transport, the viscous transport due to pressure gradient also contributes to the total flux, and can be conveniently represented by Darcy s formula. Therefore, choosing the right model for transport and reaction in porous medium is highly important in... 

Solution-diffusion model In the solution-diffusion model, permeates dissolve in the membrane material and then diffuse through the membrane down a concentration gradient. Separation is achieved between different permeates because of differences in the amount of material that dissolves in the membrane and the rate at which the material diffuses through the membrane. The solution-diffusion model is the most widely accepted transport mechanism for many membrane processes [209,210]. Selectivity and permeability of a pervaporation membrane mainly depend on the first two steps, that is, the solubility and diffusivity of the components in the membrane. According to this model, mass transport can be divided into the three steps the mechanism is shown in Fig. 3.11 ... 

Heterogeneous intra and external gradients are included and catalyst effectiveness factors are calculated continuously during the integration. The kinetic rate expressions are intrinsic rates. This is definitely the preferred model, but rigorous modelling of transport processes is tedious for all catalyst sizes and shapes in syngas processes. 

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