Diffusion based model

In order to understand the physical basis for turbulent-diffusivity-based models for the scalar flux, we first consider a homogeneous turbulent flow with zero mean velocity gradient18 and a uniform mean scalar gradient (Taylor 1921). In this flow, velocity fluctuations of characteristic size... 

The turbulent diffusivity defined by (4.74) is proportional to the turbulent viscosity defined by (4.46). Turbulent-diffusivity-based models for the scalar flux extend this idea to arbitrary mean scalar gradients. The standard gradient-diffusion model has the form... 

The fact that diffusion models describe a number of chemical processes in solid particles is not surprising since in most cases, mass transfer and chemical kinetics phenomena occur simultaneously and it is difficult to separate them [133-135]. Therefore, the overall kinetics of many chemical reactions in soils may often be better described by mass transfer and diffusion-based models than with simple models such as first-order kinetics. This is particularly true for slower chemical reactions in soils where a fast reaction is followed by a much slower reaction (biphasic kinetics), and is often observed in soils for many reactions involving organic and inorganic compounds. 

A second and vitally important conclusion that came out of the literature in the 1970s was that linear driving force models also produced solutions that were indistinguishable from diffusion based models. 

That this was known in industry at least 15 years earlier is one of the unfortunate discrepancies between academic research and commercial industrial research and development. Not all that is known is necessarily published. This realization subsequently lead to the development of both solid phase and gas phase linear driving force models that each provide very good representations of measured data without the excess labor involved with the diffusion-based models. For trace systems there are quite a few analytical solutions that are available and quite tractable for both design work and the analysis of adsorption column performance. 

Because the export machinery selectively binds transport-competent mRNPs, i.e., mRNAs associated with the appropriate nuclear proteins, the diffusion-based model for intranuclear RNA movement provides an additional checkpoint for gene expression at the level of mRNA export. This type of control has been demonstrated for tRNA export only mature tRNAs are transported to the cytoplasm because incompletely processed tRNAs do not efficiently bind to exportin-t, the tRNA-specific export receptor. 

Aucour et al. (1999) applied a diffusion-based model to account for their observations of the 5 C value of DIC in the Rhone River, France. Their model predicts that 15-60 min are required to estabhsh equilibrium between dissolved and atmospheric CO2 in the Rhone. Using 6 C = —12.5%o for the starting riverine DIC and 6 C = —8%o for atmospheric CO2, they predict that it would take between 0.2 d and 2.0 d to attain the observed average 5 C value of —5%o in the downstream river DIC, depending upon the mean depth of the river. In addition, Aucour et al. (1999) observed an inverse trend between 5 C of DIC and the concentration of DIC in the Rhone River. They attribute this to... 

There is a correlation between the mechanical and chemical data of figures 7-10. It is clear from the multilayer or lamellar structure of these in vitro lesions that they are formed by a complex demineralization process that cannot be explained by simple, diffusion-based models. The surface layer, which is extremely weak, has lost almost all of its Ca and P, except for a very small amount close to the surface. This region close to the surface is stronger than the body of the lesion, but still very weak when compared to the underlying enamel. The body of the lesion is extremely compliant and mechanically very weak. The weak interior and surface layers of the lesion make it particularly prone to damage when the surface is mechanically loaded. Collectively, the mechanical, chemical and structural data indicate that even the less demineralized surface zone (A on fig. 7, 8) does not have the same microstructure or mechanical strength as sound enamel. 

In Chapter 7 we define mass transfer coefficients for binary and multicomponent systems. In subsequent chapters we develop mass transfer models to determine these coefficients. Many different models have been proposed over the years. The oldest and simplest model is the film model this is the most useful model for describing multicomponent mass transfer (Chapter 8). Empirical methods are also considered. Following our discussions of film theory, we describe the so-called surface renewal or penetration models of mass transfer (Chapter 9) and go on to develop turbulent eddy diffusivity based models (Chapter 10). Simultaneous mass and energy transport is considered in Chapter 11. 

A major disadvantage associated with the diffusion-based models is that detailed information on the structure of the porous medium is required such detailed information is not required for the mass-transfer model. In addition, use of diffusion-based models assumes a priori knowledge of the nonequilibrium mechanism a commitment to a particular mechanism is required in designing/selecting the model to be used. Such a requirement is desirable for situations where the mechanism is fully understood. However, for situations where the mechanism involved is not fully elucidated, the use of a model that is not mechanism-unique, such as the first-order ma.ss-transfer model. 

The first-order mass transfer model can be readily interpreted in terms of the various diffusion-based models and several researchers have done so Isee Brusseau and Rao (1989a) and references cited therein]. A straightforward means of equating the two models is to define the mass transfer constant in terms of the aqueous diffusion coefficient, shape factor, and diffusion path length characterizing the porous medium. Ball (1989) reported the following equation, equating k2 from the first-order bicontinuum model to the RIPD model... 

Theoretical studies on the interaction of hydrogen with zeolites fall into three main categories statistical/diffusion based models binding site models and modelling to aid interpretation of specific experiments/experimental techniques. A large number of such studies have been reported a brief survey is given below but a comprehensive review is not attempted here. 

Yoshizuka et al. [105] studied the extraction kinetics and mechanism of metal extraction in a hollow-fiber contactor, by using a diffusion-based model with interfacial reaction and by considering the laminar flow of the aqueous and organic solutions through the hollow fiber. The rate constants for various steps were calculated by the experimental kinetic data. 

Neipp C., Gallego S, Ortuno M., Marquez A., Alvarez M., Belendez A. Pascual I. (2003). First-harmonic diffusion-based model applied to a polyvinyl-alcohol-acrylamide-based photopolymer, J. Opt. Soc. Am. B, Vol. 20, No. 10, 2052-2060. 

Fainerman and Miller [35] found that displacement of an initially adsorbed surfactant by a second, more surface-active species allowed measurement of the desorption rate of the former. For example, competitive adsorption of sodium decyl sulfate and the nonionic Triton X-165 gave a desorption rate constant for the former of 40 s". Mul-queen and coworkers [36] recently developed a diffusion-based model to describe the kinetics of surface adsorption in multicomponent systems, based upon the Ward-Tor-dai equation. Experimental work with a binary mixture of two nonionic alkyl ethoxy-late surfectants [37] showed good agreement with the model, demonstrating a similar temporal adsorption profile to that found by Diamant and Andehnan [34],... 

LAW 01b] Lawrence J.R., O Neil F.T., Sheridan J.T., Photopolymer holographic recording material parameter estimation using a nonlocal diffusion based model , Journal of Applied Physics, vol. 90, pp. 3142-3148, 2001. 

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