Three-dimensional diffusion coefficient

The three-dimensional diffusion coefficient is predicted by the simple reptation model to be reciprocally proportional to the square of polymer molar mass ... 

The diffusion along the rod axis is one-dimensional, and the diffusion in the direction perpendicular to the axis is two-dimensional, because the degree of freedom is 1 and 2 in the two directions, respectively. The three-dimensional diffusion coefficient Dq of the center of mass is the isotropic mean of D and that is, Dq = (D -I- IDIt is expressed as... 

The theory of isotropic three-dimensional diffusion allows yss to be expressed by a product of the exciton diffusion coefficient Ds and the effective annihilation capture radius Rs [189],... 

Comparing the linear correlation coefficient and standard deviation for every kinetics mechanism function in Table 3, the NO. 19 function is the most probable one. In consequence, the anti Jander equation controlled by the three-dimensional diffusion model is the most probable mechanism function for reaction of KBL anthracite coal between 40°C and 70°C. 

We now provide an example of such an inversion from the work of Wright et al (1992) in which spatial computer simulations were used to generate data on the aggregation of fractal clusters formed by Brownian motion of colloidal particles. We consider three-dimensional diffusion under two circumstances (i) that in which the diffusion coefficient of the cluster is independent of its mass and (ii) that in which the diffusion coefficient, decreases with increasing mass. The simulated process automatically produces noisy data and the number density in cluster mass is presented in Figure 6.2.10 at three different times for both cases (i) and (ii). 

A.2.2 Fickian and Non-Fickian Diffusion The three-dimensional selfdiffusion coefficient, D, of a polymer chain in a melt is given by... 

The current situation on fast reactor calculations is that, while a combination of basic differential cross section data, adjusted by correlation methods, with two-dimensional transport or three-dimensional diffusion codes, can give reasonable agreement for quantities such as critical mass or relative reaction rates in central regions of an FBR core, more precise data are still required for accurate prediction of differential effects such as the Doppler and sodium void coefficients. These two effects, together with the influence of the delayed neutron parameters for a fast reactor, are most conveniently dealt with in the following section, which considers the dynamic behavior of the system. 

C. M. Hansen, The Three-Dimensional Solubility Parameter and Solvent Diffusion Coefficient, Danish Technical Press, Copenhagen, Denmark, 1967. 

MC simulations and semianalytical theories for diffusion of flexible polymers in random porous media, which have been summarized [35], indicate that the diffusion coefficient in random three-dimensional media follows the Rouse behavior (D N dependence) at short times, and approaches the reptation limit (D dependence) for long times. By contrast, the diffusion coefficient follows the reptation limit for a highly ordered media made from infinitely long rectangular rods connected at right angles in three-dimensional space (Uke a 3D grid). 

Pore shape is a characteristic of pore geometry, which is important for fluid flow and especially multi-phase flow. It can be studied by analyzing three-dimensional images of the pore space [2, 3]. Also, long time diffusion coefficient measurements on rocks have been used to argue that the shapes of pores in many rocks are sheetlike and tube-like [16]. It has been shown in a recent study [57] that a combination of DDIF, mercury intrusion porosimetry and a simple analysis of two-dimensional thin-section images provides a characterization of pore shape (described below) from just the geometric properties. 

More recently, three-dimensional (3D) pulse sequences with DOSY have been presented where a diffusion coordinate is added to the conventional 2D map. As in the conventional 2D spectra, these experiments reduce the probability of signal overlap by spreading the NMR frequency of the same species over a 2D plane, and distribute the diffusion coefficient. 

Contaminant precipitation involves accumulation of a substance to form a new bulk solid phase. Sposito (1984) noted that both adsorption and precipitation imply a loss of material from the aqueous phase, but adsorption is inherently two-dimensional (occurring on the solid phase surface) while precipitation is inherently three-dimensional (occurring within pores and along solid phase boundaries). The chemical bonds that develop due to formation of the solid phase in both cases can be very similar. Moreover, mixtures of precipitates can result in heterogeneous solids with one component restricted to a thin outer layer, because of poor diffusion. Precipitate formation takes place when solubility limits are reached and occurs on a microscale between and within aggregates that constitute the subsurface solid phase. In the presence of lamellar charged particles with impurities, precipitation of cationic pollutants, for example, might occur even at concentrations below saturation (with respect to the theoretical solubility coefficient of the solvent). 

Complete Mix Reactor - The complete mix reactor is also labeled a completely stirred tank reactor. It is a container that has an inhnite diffusion coefficient, such that any chemical that enters the reactor is immediately mixed in with the solvent. In Example 2.8, we used the complete mix reactor assumption to estimate the concentration of three atmospheric pollutants that resulted from an oil spill. We will use a complete mix reactor (in this chapter) to simulate the development of high salt content in dead-end lakes. A series of complete mix reactors may be placed in series to simulate the overall mixing of a one-dimensional system, such as a river. In fact, most computational transport models are a series of complete mix reactors. 

In the three-dimensional problem, it will be noticed from (71) that in X oJ2 the density of states and the diffusion coefficient occur in the denominator, as they do also in the expression given by Kawabata (1981). If the disorder broadens the band, as will occur in the Anderson model if V0 > B9 then (75) should be modified to... 

The evidence that the 1 couple can diffuse freely in the liquid domains entrapped by the three-dimensional network of the gelators has also been found in the case of a PVDF-HFP gel via steady-state voltammetry at ultramicroelectrodes. Quite surprisingly the voltammogramms of the liquid and of the gel are almost perfectly superimposable (Fig. 17.14) and the diffusion coefficient of the redox ions could be calculated to be 3.6 x 10 cm2/s and 4.49 x 10-6 cm2/s for I- and I3, respectively, using Equation 17.15,... 

Here, the first term on the right-hand side gives the net diffusive inflow of species A into the volume element. We have assumed that the diffusive process follows Fick s law and that the diffusion coefficient does not vary with position. The spatial derivative term V2a is the Laplacian operator, defined for a general three-dimensional body in x, y, z coordinates by... 

Polymer molecules in a solution undergo random thermal motions, which give rise to space and time fluctuations of the polymer concentration. If the concentration of the polymer solution is dilute enough, the interaction between individual polymer molecules is negligible. Then the random motions of the polymer can be described as a three dimensional random walk, which is characterized by the diffusion coefficient D. Light is scattered by the density fluctuations of the polymer solution. The propagation of phonons is overdamped in water and becomes a simple diffusion process. In the case of polymer networks, however, such a situation can never be attained because the interaction between chains (in... 

Diffusion into a sphere represents a three-dimensional situation thus we have to use the three-dimensional version of Fick s second law (Box 18.3, Eq. 1). However, as mentioned before, by replacing the Cartesian coordinates x,y,z by spherical coordinates the situation becomes one-dimensional again. Eq. 3 of Box 18.3 represents one special solution to a spherically symmetric diffusion provided that the diffusion coefficient is constant and does not depend on the direction along which diffusion takes place (isotropic diffusion). Note that diffusion into solids is not always isotropic, chiefly due to layering within the solid medium. The boundary conditions of the problem posed in Fig. 18.6 requires that C is held constant on the surface of the sphere defined by the radius ra. 

According to the model of random walk in three dimensions, the diffusion coefficient of a molecule i, can be expressed as one-third of the product of its mean free path A, and its mean three-dimensional velocity u, (Eq. 18-7a). In the framework of the molecular theory of gases, u, is (e.g., Cussler, 1984) ... 

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