Diffusion coefficients influencing factors

Multicomponent Diffusion. In multicomponent systems, the binary diffusion coefficient has to be replaced by an effective or mean diffusivity Although its rigorous computation from the binary coefficients is difficult, it may be estimated by one of several methods (27—29). Any degree of counterdiffusion, including the two special cases "equimolar counterdiffusion" and "no counterdiffusion" treated above, may arise in multicomponent gas absorption. The influence of bulk flow of material through the films is corrected for by the film factor concept (28). It is based on a slightly different form of equation 13 ... 

Figure 10 shows that Tj is a unique function of the Thiele modulus. When the modulus ( ) is small (- SdSl), the effectiveness factor is unity, which means that there is no effect of mass transport on the rate of the catalytic reaction. When ( ) is greater than about 1, the effectiveness factor is less than unity and the reaction rate is influenced by mass transport in the pores. When the modulus is large (- 10), the effectiveness factor is inversely proportional to the modulus, and the reaction rate (eq. 19) is proportional to k ( ), which, from the definition of ( ), implies that the rate and the observed reaction rate constant are proportional to (1 /R)(f9This result shows that both the rate constant, ie, a measure of the intrinsic activity of the catalyst, and the effective diffusion coefficient, ie, a measure of the resistance to transport of the reactant offered by the pore stmcture, influence the rate. It is not appropriate to say that the reaction is diffusion controlled it depends on both the diffusion and the chemical kinetics. In contrast, as shown by equation 3, a reaction in solution can be diffusion controlled, depending on D but not on k. 

The mass transport influence is easy to diagnose experimentally. One measures the rate at various values of the Thiele modulus the modulus is easily changed by variation of R, the particle size. Cmshing and sieving the particles provide catalyst samples for the experiments. If the rate is independent of the particle size, the effectiveness factor is unity for all of them. If the rate is inversely proportional to particle size, the effectiveness factor is less than unity and

experimental points allow triangulation on the curve of Figure 10 and estimation of Tj and ( ). It is also possible to estimate the effective diffusion coefficient and thereby to estimate Tj and ( ) from a single measurement of the rate (48). 

In addition, it was concluded that the liquid-phase diffusion coefficient is the major factor influencing the value of the mass-transfer coefficient per unit area. Inasmuch as agitators operate poorly in gas-liquid dispersions, it is impractical to induce turbulence by mechanical means that exceeds gravitational forces. They conclude, therefore, that heat- and mass-transfer coefficients per unit area in gas dispersions are almost completely unaffected by the mechanical power dissipated in the system. Consequently, the total mass-transfer rate in agitated gas-liquid contacting is changed almost entirely in accordance with the interfacial area—a function of the power input. 

To summarize, the kinetics of the silanization reaction are strongly influenced by the efficiency of the devolatilization process. The degree of devolatilization mainly depends on processing conditions (e.g., rotor speed and fill factor), mixer design (e.g., number of rotor flights, size of the mixer), and material characteristics. The diffusion coefficient of the volatile component in the polymeric matrix is of minor influence. 

The ionic mobilities Uj depend on the retarding factor 0 valid for a particular medium [Eq. (1.8)]. It is evident that this factor also influences the diffusion coefficients. To find the connection, we shall assume that the driving force of diffusion is the chemical potential gradient that is, in an ideal solution,... 

The dynamical properties of polymer molecules in solution have been investigated using MPC dynamics [75-77]. Polymer transport properties are strongly influenced by hydrodynamic interactions. These effects manifest themselves in both the center-of-mass diffusion coefficients and the dynamic structure factors of polymer molecules in solution. For example, if hydrodynamic interactions are neglected, the diffusion coefficient scales with the number of monomers as D Dq /Nb, where Do is the diffusion coefficient of a polymer bead and N), is the number of beads in the polymer. If hydrodynamic interactions are included, the diffusion coefficient adopts a Stokes-Einstein formD kltT/cnr NlJ2, where c is a factor that depends on the polymer chain model. This scaling has been confirmed in MPC simulations of the polymer dynamics [75]. 

Diffusion of small molecular penetrants in polymers often assumes Fickian characteristics at temperatures above Tg of the system. As such, classical diffusion theory is sufficient for describing the mass transport, and a mutual diffusion coefficient can be determined unambiguously by sorption and permeation methods. For a penetrant molecule of a size comparable to that of the monomeric unit of a polymer, diffusion requires cooperative movement of several monomeric units. The mobility of the polymer chains thus controls the rate of diffusion, and factors affecting the chain mobility will also influence the diffusion coefficient. The key factors here are temperature and concentration. Increasing temperature enhances the Brownian motion of the polymer segments the effect is to weaken the interaction between chains and thus increase the interchain distance. A similar effect can be expected upon the addition of a small molecular penetrant. 

In addition to temperature and concentration, diffusion in polymers can be influenced by the penetrant size, polymer molecular weight, and polymer morphology factors such as crystallinity and cross-linking density. These factors render the prediction of the penetrant diffusion coefficient a rather complex task. However, in simpler systems such as non-cross-linked amorphous polymers, theories have been developed to predict the mutual diffusion coefficient with various degrees of success [12-19], Among these, the most notable are the free volume theories [12,17], In the following subsection, these free volume based theories are introduced to illustrate the principles involved. 

It is apparent from early observations [93] that there are at least two different effects exerted by temperature on chromatographic separations. One effect is the influence on the viscosity and on the diffusion coefficient of the solute raising the temperature reduces the viscosity of the mobile phase and also increases the diffusion coefficient of the solute in both the mobile and the stationary phase. This is largely a kinetic effect, which improves the mobile phase mass transfer, and thus the chromatographic efficiency (N). The other completely different temperature effect is the influence on the selectivity factor (a), which usually decreases, as the temperature is increased (thermodynamic effect). This occurs because the partition coefficients and therefore, the Gibbs free energy difference (AG°) of the transfer of the analyte between the stationary and the mobile phase vary with temperature. 

In another review, Hoffert discussed the social motivations for modeling air quality for predictive purposes and elucidated the components of a model. Meteorologic factors were summarized in terms of windfields and atmospheric stability as they are traditionally represented mathematically. The species-balance equation was discussed, and several solutions of the equation for constant-diffusion coefficient and concentrated sources were suggested. Gaussian plume and puff results were related to the problems of developing multiple-source urban-dispersion models. Numerical solutions and box models were then considered. The review concluded with a brief outline of the atmospheric chemical effects that influence the concentration of pollutants by transformation. 

The permeability, P (P = Pc x D), of a nonpolar substance through a cell membrane is dependent on two physicochemical factors (1) solubility in the membrane (Pc), which can be expressed as a partition coefficient of the drug between the aqueous phase and membrane phase, and (2) diffusivity or diffusion coefficient (D), which is a measure of mobility of the drug molecules within the lipid. The latter may vary only slightly among toxicants, but the former is more important. Lipid solubility is therefore one of the most important determinants of the pharmacokinetic characteristics of a chemical, and it is important to determine whether a toxicants is readily ionized or not influenced by pH of the environment. If the toxicant is readily ionized, then one needs to understand its chemicals behavior in various environmental matrices in order to adequately assess its transport mechanism across membranes. 

Diffusion NMR experiments performed by Newkome et al. show that the diffusion coefficients - and hence the hydrodynamic radii - can yield information about the influence of external factors (e.g. pH values) on the size and shape of dissolved dendrimers [26]. Since the spatial structure of a dendrimer in solution... 

Several factors can be identified as being crucial for the foaming of immiscible polymer blends the blend morphology, the phase size of the blend constituents, the interfacial properties between the blend partners, and, last but not least, the properties of the respective blend phases such as the melt-rheological behavior, the glass transition temperature, the gas solubility, as well as the gas diffusion coefficient. Most of these factors also individually influence the melt-rheological behavior of two-phase blends. 

Interest in the use of SC solvents as a reaction media is founded upon recent advances in our understanding of their unique thermo-physical and chemical properties. Worthy of special note are those thermophysical properties (6) which can be manipulated as parameters to selectively direct the progress of desirable chemical reactions. These properties include the solvent s dielectric constant (7), ion product (8,9), electrolyte solvent power (10,11), transport properties"[viscosity (12), diffusion coefficients (13) and ion mobilities (14)], hydrogen bonding characteristics (15), and solute-solvent "enhancement factors" (6). All these properties are strongly influenced by the solvent s density P in the supercritical state. 

It is worth mentioning that physicochemical considerations predict the opposite influence of the degree of deficiency of a chemical compound on the values of the reaction- and self-diffusion coefficients. The former must decrease with increasing compound deficiency, while the latter is known to increase with its increasing. This seems to be the case, for example, with oxides like FeO, Fe3C>4, MnO, CoO, etc., though complicating factors often mask these effects. 

Unsteady state diffusion in monodisperse porous solids using a Wicke-Kallenbach cell have shown that non-equimolal diffusion fluxes can induce total pressure gradients which require a non-isobaric model to interpret the data. The values obtained from this analysis are then suitable for use in predicting effectiveness factors. There is evidence that adsorption of the non-tracer component can have a considerable influence on the diffusional flux of the tracer and hence on the estimation of the effective diffusion coefficient. For the simple porous structures used in these tests, it is shown that a consistent definition of the effective diffusion coefficient can be obtained which applies to both the steady and unsteady state and so can be used as a basis of examining the more complex bimodal pore size distributions found in many catalysts. 

Diffusion is a physical process that involves the random motion of molecules as they collide with other molecules (Brownian motion) and, on a macroscopic scale, move from one part of a system to another. The average distance that molecules move per unit time is described by a physical constant called the diffusion coefficient, D (in units of mm2/s). In pure water, molecules diffuse at a rate of approximately 3xl0"3 mm2 s 1 at 37°C. The factors influencing diffusion in a solution (or self-diffusion in a pure liquid) are molecular weight, intermolecular... 

Particle radius affects both the potential energy of interaction and the diffusion coefficient. Hence particle size influences the rate under all conditions. Under reaction-control conditions, Fig. 4 shows that a l-pm increase in particle radius reduces the deposition rate by a factor of 5, while under mass-transfer... 

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