Steady-state diffusion modeling

Fig. rV-26. Steady-state diffusion model for film dissolution. (From Ref. 293.)... 

For the case of a two-phase system with two parallel noninteracting paths that both contribute to the diffusion of the solute and where the diffusion coefficients of the solute of interest are different in the two phases, the solution of the two isolated one-dimensional steady-state diffusion models gives... 

Using the steady-state diffusion model described in section 4.3.2 one may define the parameter q as ... 

A simple, quantitative, steady-state diffusion model (36) demonstrates the importance of physical processes in shaping the vertical distribution of phytoplankton. This model uses values of the eddy diffusion coefficient K from the theoretical model of James (35), which reproduces accurately the annual cycle of vertical temperature structure for this area of the Celtic Sea. The submodels for photosynthetic production, light, and grazing can be varied to any of the established models nutrient luxury or nutrient limitation of growth can be included. The model reproduces the main features of the UOR observations in the Celtic Sea and English Channel. 

The conversion of the solid reactant B is obtained from integration of the pseudo-steady-state diffusion model with reaction at the boundary ... 

Other Models for Mass Transfer. In contrast to the film theory, other approaches assume that transfer of material does not occur by steady-state diffusion. Rather there are large fluid motions which constantiy bring fresh masses of bulk material into direct contact with the interface. According to the penetration theory (33), diffusion proceeds from the interface into the particular element of fluid in contact with the interface. This is an unsteady state, transient process where the rate decreases with time. After a while, the element is replaced by a fresh one brought to the interface by the relative movements of gas and Uquid, and the process is repeated. In order to evaluate a constant average contact time T for the individual fluid elements is assumed (33). This leads to relations such as... 

Gal-Or and Hoelscher (G5) have recently proposed a mathematical model that takes into account interaction between bubbles (or drops) in a swarm as well as the effect of bubble-size distribution. The analysis is presented for unsteady-state mass transfer with and without chemical reaction, and for steady-state diffusion to a family of moving bubbles. 

Surface Renewal Theory. The film model for interphase mass transfer envisions a stagnant film of liquid adjacent to the interface. A similar film may also exist on the gas side. These h5q>othetical films act like membranes and cause diffu-sional resistances to mass transfer. The concentration on the gas side of the liquid film is a that on the bulk liquid side is af, and concentrations within the film are governed by one-dimensional, steady-state diffusion ... 

Two rather similar models have been devised to remedy the problems of simple film theory. Both the penetration theory of Higbie and the surface renewal theory of Danckwerts replace the idea of steady-state diffusion across a film with transient diffusion into a semi-inhnite medium. We give here a brief account of surface renewal theory. 

Figure 2.1 Dependence of the effectiveness factor on the Thiele modulus for a first-order irreversible reaction. Steady-state diffusion and reaction, slab model, and isothermal conditions are assumed.

Neal and Nader [260] considered diffusion in homogeneous isotropic medium composed of randomly placed impermeable spherical particles. They solved steady-state diffusion problems in a unit cell consisting of a spherical particle placed in a concentric shell and the exterior of the unit cell modeled as a homogeneous media characterized by one parameter, the porosity. By equating the fluxes in the unit cell and at the exterior and applying the definition of porosity, they obtained... 

This chapter starts with a short introduction on the skin barrier s properties and the methods employed for analyzing experimental data. This is followed by an overview of several selected approaches to predict steady-state diffusion through the skin. Then a few approaches that approximate the structural complexity of the skin by predicting drug diffusion in biphasic or even multiphasic two-dimensional models will be presented. Finally, the chapter concludes with a short summary of the many variables possibly influencing drug permeation and penetration. 

Although there is plenty of experimental evidence that ultrasound improves leaching the exact mechanism is not fully understood. Swamy and Narayana [60] have suggested models for leaching in the presence and absence of ultrasound (Fig. 4.4). Normal leaching takes place as the solvent front moves inward and a steady state diffusion occurs through the depleted outer region and is equal to the rate of reaction within the reaction zone itself (Fig. 4.4a). 

Farrell TJ, Patterson MS, Wilson B. A diffusion-theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo. Medical Physics 1992, 19, 879-888. 

In a previous paper (7), we have illustrated that diffusion in FCC takes place in the non-steady regime and that this explains the failure of several attempts to relate laboratory measurements on FCC catalysts to theories on steady state diffusion. Apart from the diffusion aspects, Nace (13) has also indicated the limited accessibility of the zeolite portal surface area by comparing the cracking rates of various model compounds with an increasing number of naphthenic rings on zeolite and amorphous FCC catalysts, figure 2. 

Theoretical. The theory of steady state diffusion in a hollow sphere has been described by Crank (16). Because each frustum-shaped cell in the system closely approximates a spherical sector of a hollow sphere, a theoretical model can be developed on this basis to predict the release characteristics for this sytem. This assumption should be valid until the point is reached such that the curved interface (r in Figure 2) touches the flat impermeable backing, which should represent ca. 90% of the release. 

Despite the fact that the skin is a heterogeneous membrane, Fick s laws of diffusion have been successfully used to analyze skin permeation data. Solutions to the second law have been used in mechanistic interpretations (see later) and in considering concentration profiles within the skin. Fick s first law has been used to analyze steady-state diffusion rates and in the development of predictive models for skin permeability. 

Under the simplifying assumptions of the Krogh-Erlang model, the steady state oxygen distribution in the tissue at any position z is governed by the steady state diffusion equation in radial coordinates... 

The conceptual model for diffusive soil transport down a hillslope is shown in Eigure 24 (Heimsath et al., 1997, 1999). In any given section of the landscape, the mass of soil present is the balance of transport in, transport out, and soil production (the conversion of rock or sediment to soil). If it is assumed that the processes have been operating for a sufficiently long period of time, then the soil thickness is at steady state. The model describing this condition is... 

The carbon isotope ratios of pedogenic carbonates have been used to infer atmospheric CO2 concentrations from calculations based on a model for steady-state diffusive mixing with soil-respired CO2 in the soil profile (Cerling, 1991 Cerling, 1992 Ekart et al., 1999 Ghosh et al., 2001). Significant sources of uncertainty in this approach include the dependence of soil CO2 diffusion on temperature and moisture, the contribution of C3 versus C4 plants to respired CO2, and the somewhat arbitrary choice of values for the mole fraction of respired CO2 at depth in the soil. 

As our first application of the linearized theory we consider steady-state, one-dimensional diffusion. This is the simplest possible diffusion problem and has applications in the measurement of diffusion coefficients as discussed in Section 5.4. Steady-state diffusion also is the basis of the film model of mass transfer, which we shall discuss at considerable length in Chapter 8. We will assume here that there is no net flux = 0. In the absence of any total flux, the diffusion fluxes and the molar fluxes are equal = J. ... 

To illustrate the application of the film model for nonideal fluid mixtures we consider steady-state diffusion in the system glycerol(l)-water(2)-acetone(3). This system is partially miscible (see Krishna et al., 1985). Determine the fluxes Ap A2, and A3 in the glycerol-rich phase if the bulk liquid composition is... 

In the film model we assume that all the resistance to mass and heat transfer is concentrated in a thin film and that transfer occurs within this film by steady-state diffusion and... 

In order to develop a continuous separation process, Kataoka et al. [54] simulated permeation of metal ion in continuous countercurrent column. They developed the material balance equation considering back mixing only in the continuous phase and steady-state diffusion in the dispersed emulsion drops which is similar to the Hquid extraction situation. Bart et al. [55] also modeled the extraction of copper in a continuous countercurrent column. They considered only the continuous phase back mixing in the model and assumed that the reaction between copper ions and carrier is slow, so that the differential mass balance equation for external phase in their model is... 

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