Effective terms diffusion

Ordinary diffusion involves molecular mixing caused by the random motion of molecules. It is much more pronounced in gases and Hquids than in soHds. The effects of diffusion in fluids are also greatly affected by convection or turbulence. These phenomena are involved in mass-transfer processes, and therefore in separation processes (see Mass transfer Separation systems synthesis). In chemical engineering, the term diffusional unit operations normally refers to the separation processes in which mass is transferred from one phase to another, often across a fluid interface, and in which diffusion is considered to be the rate-controlling mechanism. Thus, the standard unit operations such as distillation (qv), drying (qv), and the sorption processes, as well as the less conventional separation processes, are usually classified under this heading (see Absorption Adsorption Adsorption, gas separation Adsorption, liquid separation). 

We have seen how solutes behave in terms of dissolution, hydrophobic effects, and diffusion. With cellular automata it is possible to examine the... 

The reaction will then appear to follow first-order kinetics, regardless of the functional form of the intrinsic rate expression and of the effectiveness factor. This first-order dependence is characteristic of reactions that are mass transfer limited. The term diffusion controlled is often applied to reactions that occur under these conditions. 

Several models for diffusive transport in and among minerals have been discussed in the literature one is the fast grain boundary (FGB) model of Eiler et al. (1992, 1993). The FGB model considers the effects of diffusion between non-adjacent grains and shows that, when mass balance terms are included, closure temperatures become a strong function of both the modal abundances of constituent minerals and the differences in diffusion coefficients among all coexisting minerals. 

Behavior of trace element that can be treated as effective binary diffusion The above discussion is for the behavior of the principal equilibrium-determining component. For minor and trace elements, there are at least two complexities. One is the multicomponent effect, which often results in uphill diffusion. This is because the cross-terms may dominate the diffusion behavior of such components. The second complexity is that the interface-melt concentration is not fixed by thermodynamic equilibrium. For example, for zircon growth, Zr concentration in the interface-melt is roughly the equilibrium concentration (or zircon saturation concentration). However, for Pb, the concentration would not be fixed. 

Here, kv is an electrochemical rate constant, and F is the faraday, the charge on 1 mol ofunivalentions. It contains the exponential term for the electrode potential (assuming a cathodic reaction in a region in which the rate of the back anodic reaction can be neglected). However, it does take into account the effect of diffusion on the observed current density, i. 

In Equation (9.6), x is the direction of flux, nt [mol m-3 s 1 ] is the total molar density, X [1] is the mole fraction, Nd [mol m-2 s 1] is the mole flux due to molecular diffusion, D k [m2 s 1] is the effective Knudsen diffusion coefficient, D [m2 s 1] is the effective bimolecular diffusion coefficient (D = Aye/r), e is the porosity of the electrode, r is the tortuosity of the electrode, and J is the total number of gas species. Here, a subscript denotes the index value to a specific specie. The first term on the right of Equation (9.6) accounts for Knudsen diffusion, and the following term accounts for multicomponent bulk molecular diffusion. Further, to account for the porous media, along with induced convection, the Dusty Gas Model is required (Mason and Malinauskas, 1983 Warren, 1969). This model modifies Equation (9.6) as ... 

The dissolution controlled release matrix systems provide sustained release profiles i.e., the active drugs in these systems are released continuously at a slow rate to provide a long-term therapeutic effect. Unlike diffusion controlled release coated systems, release profiles from dissolution controlled release coated systems do not follow zero-order kinetics but fall within the classification of delayed release systems,4 pulsatile or repeat-action systems,5 and sustained release systems.3... 

This section focuses on three basic types of catalyst selectivity, termed Type I, Type II, and Type III after Wheeler [113]. The main purpose of this is to demonstrate the fundamental effects of diffusion on the apparent selectivity of the catalyst, and to explain briefly the different situation, which is found when there is no longer a single functional, the effectiveness factor, to characterize the system, but the important questions concern the relative rates of reaction and how they are affected by diffusion. 

The described treatment of mass transport presumes a simple, relatively uniform (monomodal) pore size distribution. As previously mentioned, many catalyst particles are formed by tableting or extruding finely powdered microporous materials and have a bidisperse porous structure. Mass transport in such catalysts is usually described in terms of two coefficients, a effective macropore diffusivity and an effective micropore diffusivity. 

The A term is related to multipath effects (eddy diffusion) that are independent of mobile phase flow rate. Analyte molecules can follow multiple pathways of differing lengths that spread the analyte molecules apart and cause peak broadening. Since the smaller the particles, the lower the difference among molecule walks along the column, the A term is linearly dependent on particle diameter, d, according to X, a structure factor ... 

In this form of equation (24), the effect of diffusion on the enthalpy flux is included implicitly in the first term, and the only term from the heat-flux vector that remains explicitly is that of heat conduction. The above equations serve to eliminate completely the diffusion velocities from the governing equations, replacing them by the flux fractions. The fact that equations (31) and (33) appear to be less complicated than equations (22) (with dYJdt = 0) and (24) often justifies this transformation. 

In many industrial reactions, the overall rate of reaction is limited by the rate of mass transfer of reactants and products between the bulk fluid and the catalytic surface. In the rate laws and cztalytic reaction steps (i.e., dilfusion, adsorption, surface reaction, desorption, and diffusion) presented in Chapter 10, we neglected the effects of mass transfer on the overall rate of reaction. In this chapter and the next we discuss the effects of diffusion (mass transfer) resistance on the overall reaction rate in processes that include both chemical reaction and mass transfer. The two types of diffusion resistance on which we focus attention are (1) external resistance diffusion of the reactants or products between the bulk fluid and the external smface of the catalyst, and (2) internal resistance diffusion of the reactants or products from the external pellet sm-face (pore mouth) to the interior of the pellet. In this chapter we focus on external resistance and in Chapter 12 we describe models for internal diffusional resistance with chemical reaction. After a brief presentation of the fundamentals of diffusion, including Pick s first law, we discuss representative correlations of mass transfer rates in terms of mass transfer coefficients for catalyst beds in which the external resistance is limiting. Qualitative observations will bd made about the effects of fluid flow rate, pellet size, and pressure drop on reactor performance. 

The influence of the mesh size will be seen in terms of its effect on diffusivity. Mesh size values for neutral gels have been calculated using the following expression [68] ... 

In the actual bed, solid exchange between the ascending and descending zone is significant, and the circulation is seen as a series combination of localized recirculation. If this scheme of solid mixing is more realistic, the temperature profile may be suitably expressed in terms of an effective thermal diffusivity, which will be used later for discussion of instability in the dilute phase. 

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