Effective coefficient of diffusion

Since this depth is significantly smaller than the dimensions of the piece of catalyst, and at the same time is much greater than the diameter of individual pores, the phenomenon can be represented schematically by introducing the effective coefficient of diffusion through the porous substance (which depends on the number of the pores and their diameters), and by examining a layer of the catalyst of indefinite depth with a flat surface. 

In the moving system of coordinates, the flux J is similar to the relative flux introduced in Section II. Therefore the expression (6.130) can be considered as Pick s law with the effective coefficient of diffusion... 

For the turbulent flow regime, the flux of bubbles of volume co toward the test bubble of volume V can be considered as a diffusion flux with the effective coefficient of diffusion Dr. Consider a bubble of volume V, placed in a turbulent flow of liquid containing bubbles of volume co with the number concentration n. Assuming that the process is stationary and spherically symmetric, we have the following equation describing the distribution n(r) ... 

Furthermore, for calculating the effective coefficient of quasi-diffusion in a composite (D) with the corresponding limitation of the entire process of heterogeneous mass-exchange, equations reported in Section 5.1 may be used. The high kinetic permeability of cellosorbents for large organic ions are listed in Table 16. 

Since the left side of Eq. (7) represents the release rat of the system, a true controlled-release system with a zero-order release rate can be possible only if all of the variables on the right side of Eq. (7) remain constant. A constant effective area of diffusion, diffusional path length, concentration difference, and diffusion coefficient are required to obtain a release rate that is constant. These systems often fail to deliver at a constant rate, since it is especially difficult to maintain all these... 

The studies of Hasinoff [53] on the recombination rate of carbon monoxide and the heme units after photodissociation of carboxy ferrous microperioxidase come close to satisfying the requirements for observing the effects of anisotropic reactivity and rotational diffusion on the rate of a translational diffusion-limited reaction. In Chap. 2, Sect. 5.6, the details of this study were briefly mentioned. Hasinoff found that the rate of recombination was substantially diffusion-limited in all three aqueous solvents used at 260 K, but at higher temperatures, the rate of reaction of the encounter pair, feact, was a significant factor in determining the overall rate of recombination (see Fig. 9). The observed rate coefficient of recombination, feobs, was separated into the rate coefficient of diffusive formation of encounter pairs, feD, and the rate coefficient of reaction of encounter pairs, fcact, with the Collins and Kimball expression, eqn. (26)... 

We imagine a distribution of a which is characterized by an amplitude o0 and a length scale L which exceeds the maximum scale of the turbulent pulsation, l. We denote the pulsating velocity by u the turbulent coefficient of diffusion, the coefficient of thermal conductivity and the effective turbulent kinematic viscosity are all expressed by the formula k = ul. For an initial uniform distribution of a, obviously,... 

Thus, it is not the absolute value of D, but its ratio with k that determines the character of the phenomena. Convection blurs the effect in convective motion the particles of gas which carry quantities of material and heat are in the ratio of the concentration to the product of the specific heat and temperature, which corresponds to equality of the effective (related to the gas motion) coefficients of diffusion and thermal diffusivity. In all cases radiation from the surface of the catalyst lowers its temperature Tr. 

The dependence of the friction coefficient Cm of the macromolecule on its length M is affected by exclude-volume effects and effects of draining or nondraining (permeability of macromolecular coils). Taking into account equation (2.14), the coefficient of diffusion can be written as... 

Identify the minimum value of the dispersion, a(Ds), in order to obtain the best value of the effective coefficient of the surface diffusion. 

However, since the solution in which the transport takes place is confined, the kinetics are affected in various ways (i) the coefficient of diffusion can differ from the coefficient of diffusion in the bulk solution (ii) a diffusion path is imposed by the geometry of the pore network and, (iii) liquid flow is slowed down and its effect generally becomes negligible as compared with the effect of diffusion. 

At a set temperature and concentration, the coefficient of diffusion D is constant and its temperature dependence, calculated by the Arrhenius law [36], can be ignored. However, in the presence of external forces F and F stipulating the appearance of an additional flow of the gaseous components by electrodiffusion and by thermodiffusion (Soret effect [37]), respectively, the coefficient d in Equation (2.7) describes the force of the convective diffusion as follows ... 

Figure 16a and b shows the effect of L on the radial dependence of the steady-state concentration and flux at the substrate/solution interface for a first-order dissolution process characterized by Ki = 10 and L = 0.1, 0.32, and 1.0. As the tip-substrate separation decreases, the effective rate of diffusion between the probe and the surface increases, forcing the crystal/so-lution interface to become more undersaturated. Conversely, as the UME is retracted from the substrate, the interfacial undersaturation approaches the saturated value, since the solution mass transfer coefficient decreases compared to the first-order dissolution rate constant. Movement of the tip electrode away from the substrate also has the effect of promoting radial diffusion, and consequently the area of the substrate probed by the UME increases. 

Transport processes are concerned with the flow of mass, momentum, and energy in fluids in nonuniform states. For normal liquids near equilibrium, the transport rates are proportional to the gradients of concentration, mass velocity, and temperature and the coefficients of diffusion, viscosity, and thermal conductivity are the respective proportionality constants. Various cross coefficients such as those of binary and thermal diffusion arise in Reciprocal processes expressing the effects of combined gradients of concentration and temperature. 

Thus, the true charge-transfer current can be calculated from the ordinate at the origin in the plot between the reciprocal of the measured current density, j"1, as a function of w. The slope (B 1) is the reciprocal value of the Levich constant, 0.620nFCJoj, because it is the only portion that strictly depends on the co value [107], where D, is the coefficient of diffusion of they-particle. With the currents corrected from the mass transport effects, we can depict the Tafel lines, from which the values of j0 and a can be calculated. 

Wellinghoff et al. (1995) derived a correlation directly from a pore diffusion model based on Pick s law and obtained an equation which predicts the effective distribution coefficient of diffusion washing as a function of four dimensionless numbers. The apparent differences between the two approaches vanish upon closer examination. Namely, when neglecting the factors that cannot be arbitrarily influenced due to process inherent restrictions the two approaches are essentially alike. They only differ in that Wellinghoff et al. (1995) additionally consider a dimensionless area which corresponds to the ratio of the pore surface to the surface of the crystal layer. This factor can only be determined experimentally yet which is fairly complicated and thus limits the applicability of the approach. 

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