Diffusion modeling with

Weiner and Rosenbluth had used a discrete diffusion model with excitable kinetics to study phenomenologically observed spatial phenomena in cardiac muscle tissue. 

The terms Jga and Jsa are the diffusive fluxes of species a in the gas and solid phases, respectively. Note that in addition to molecular-scale diffusion, these terms include dispersion due to particle-scale turbulence. The latter is usually modeled by introducing a gradient-diffusion model with an effective diffusivity along the lines of Eqs. (149) and (151). Thus, for large particle Reynolds numbers the molecular-scale contribution will be negligible. The term Ma is the... 

One of the central problems in air pollution research and control is to determine the quantitative relationship between ambient air quality and emission of pollutants from sources. Effective strategies to control pollutants can not be devised without this information. This question has been mainly addressed in the past with source-oriented techniques such as emission inventories and predictive diffusion models with which one traces pollutants from source to receptor. More recently, much effort has been directed toward developing receptor-oriented models that start with the receptor and reconstruct the source contributions. As is the case with much of air pollutant research, improvements in pollutant chemical analysis techniques have greatly enhanced the results of receptor modeling. 

Fig. 9.11. Two stationary-state profiles for the diffusive model with precursor, D = 0.05, and ku = 0.01 (a) n0 = 0.6885, from the lowest branch on the stationary-state locus (b) pt0 = 0.9226,...

We will then compare the dynamics of the radial diffusion model with a first-order exchange model which gives the same half-life as the radial model (Eq. 18-37). A preview of this comparison is given in Fig. 18.7b, which shows that the linear model underpredicts the exchange at short times and overpredicts it at long times. 

A critical comparison by van Konyenberg and Steele [230] and Jones et al. [231] of extended diffusion models with Brownian motion and other continuum models strongly favours the former treatment. More detailed analysis is given by Berne and Pecora [232]. 

So far, it appears that the gas transport properties of glassy polymer membranes, manifested in a decreasing P(a), or increasing D(C), function can be adequately represented by the above dual diffusion model with constant diffusion coefficients Dl5 D2 (or Dtj, DX2). We now consider the implications of this model from the physical point of view ... 

The question about the difference between the macroscopic and microscopic values of the quantities characterizing the translational mobility (viscosity tj, diffusion coefficient D, etc.) has often been discussed in the literature. Numerous data on the kinetics of spin exchange testify to the fact that, with the comparable sizes of various molecules of which the liquid is composed, the microscopic translational mobility of these molecules is satisfactorily described by the simple Einstein-Stokes diffusion model with the diffusion coefficient determined by the formula... 

Depending on the distribution chosen, as few as three fitting parameters may be required to define a distribution of diffusion rates. In some cases, a single distribution was used to describe both fast and slow rates of sorption and desorption, and in other cases fast and slow mass transfer were captured with separate distributions of diffusion rates. For example, Werth et al. [42] used the pore diffusion model with nonlinear sorption to predict fast desorption, and a gamma distribution of diffusion rate constants to describe slow desorption. 

Water flux through the membrane is represented by Equation 4.3. This flux is based on the solutions - diffusion model with the added term to reflect transport due to the imperfections. 

In this section we analyze experimental data and make comparisons with theory. Data were obtained for 100 CdSe-ZnS nanocrystals at room temperature.1 We first performed data analysis (similar to standard approach) based on the distribution of on and off times and found that a+= 0.735 0.167 and v = 0.770 0.106,2 for the total duration time T = T = 3600 s (bin size 10 ms, threshold was taken as 0.16 max I(t) for each trajectory). Within error of measurement, a+ a k 0.75. The value of a 0.75 implies that the simple diffusion model with a = 0.5 is not valid in this case. An important issue is whether the exponents vary from one NC to another. In Fig. 13 (top) we show the distribution of a obtained from data analysis of power spectra. The power spectmm method [26] yields a single exponent apSd for each stochastic trajectory (which is in our case a+ a apSd). Figure 13 illustrates that the spread of a in the interval 0 < a < 1 is not large. Numerical simulation of 100 trajectories switching between 1 and 0, with /+ (x) = / (x) and a = 0.8, and with the same number of bins as the experimental trajectories, was performed and the... 

Polnaszek and Bryant (1984a,b) measured the frequency dependence of water proton relaxation for solutions of bovine serum albumin reacted with a nitroxide spin label (4.6 mol of nitroxide per mol of protein). The relaxation is dominated by interaction between water and the paramagnetic spin label. The data were best fit with a translational diffusion model, with the diffusion constant for the surface water in the immediate vicinity of the nitroxide being five times smaller than that for... 

With QENS, both the rotational and translational motions of CH4 can be observed. It was found that the rotational motion of CH4 in ZSM-5 can be described by an isotropic rotational diffusion model with a rotational diffusion time constant, Dr (72). The values of Dr for CH4 adsorbed at 250 K in ZSM-5 are of the order of 5 x 10 s . In MD simulations at 400 K, Dr was found to be of the order of 10 s (73). This difference is due to the fact that a radius of gyration of 0.15 nm was used in the computer fits of the QENS profiles. This radius is intermediate between a simple rotation model, with 0.11 nm for the distance between the protons and the center of mass of the methane molecule, and the radius of the channel in which the molecule performs oscillations. 

An extension of previous diffusion models with the incorporation of reaction equilibrium. This model includes the reversibility in the reaction of solute with the reagent present in the internal droplets. 

Axial dispersion in the bubble column is usually well expressed by the following one-dimensional diffusion model with respect to liquid concentration c ... 

Figure 2. Comparison of the extent of delignificatlon predicted by the reaction-diffusion model with the measured Klason lignin contents In the residues obtained from supercritical methylamlne extraction of red spruce at 276 bar, 1 g/mln solvent flow rate, and four different temperatures (170, 175, 180, and 185 C).

This equation contains three new terms, namely flux of scalar variance, production of variance and dissipation of scalar variance, which require further modeling to close the equation. The flux terms are usually closed by invoking the gradient diffusion model (with turbulent Schmidt number, aj, of about 0.7). This modeled form is already incorporated in Eq. (5.21). The variance production term is modeled by invoking an analogy with turbulence energy production (Spalding, 1971) ... 

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

Recent analysis shows that another parameterization of Dd(w) giving a lower Arf at w < 10, would generate a better agreement of the diffusion model with experimental data [95]. However, any feasible approach has to be based on the proper experimental determination of the... 

For this steady problem, use eddy diffusion model with constant diffusivities Kc, Kh in the canyon and the upper flow, respectively. The equations are ... 

However, a solution in the Laplace domain has been derived by Kucera [30] and Kubin [31]. The solution cannot be transformed back into the time domain, but from that solution, these authors have derived the expressions for the first five statistical moments (see Section 6.4.1). For a linear isotherm, this model has been studied extensively in the literature. The solution of an extension of this model, using a macro-micropore diffusion model with external film mass transfer resistance, has also been discussed [32]. All these studies use the Laplace domain solution and moment analysis. 

The solid-diffusion model with a concentration-dependent Pick matrix of diffusivities ... 

Figure 15. Half width at half maximum of the broadened component of the neutron quasielastic spectra obtained with an acid Nafion membrane containing 15% H20 by weight. The points are the widths of the best fit Lorentzian lines from spectra obtained with an incident wavelength of 10 A (Q), 11 A and 13 A (A). The full line is the theoretical width predicted by the model with diffusion in a sphere (with D = 1.8 X 10 5 cm2/s and a = 4.25 A). The two theoretical asymptotes for Q O and Q 00 are also shown (compare with Figure 2 of Reference 2 for more details). Half width at half maximum of the best fit Lorentzian lines to the spectra obtained with bulk water at 28° C (incident wavelength 10 A) is denoted by +. The straight line passing through the points (+) is the theoretical width predicted by the simple self diffusion model with Dt = 2.5 X 10-5 err /s. Note the different vertical scales for the Nafion sample and the bulk water sample.

The Fickian diffusion models with constant effective diffusivities presented earlier and the rigorous dusty gas model presented in this section are not the only alternatives for modelling diffusion and reaction in porous catalyst pellets. Fickian models with effective diffusion coefficients which are varying with the change of concentration of the gas mixture can also be used. This is certainty more accurate compared with the Fickian model with constant diffusivities although of course less accurate than the dusty gas model. The main problem with these models is the development of relations for the change of diffusivities with the concentration of the gas mixture without solving the dusty gas model equations. Two such techniques are presented in this section and their results are compared with dusty gas model results. 

To represent the partially premixed turbulent combustion of a refinery gas in the heater, a combination of the flamelet formulations for premixed and nonpremixed combustion was used [16]. The standard k-e model was used for turbulent flow calculations. The effect of turbulence on the mixture fraction was accounted for by integrating a beta-PDF derived from the local mixture fraction and mixture fraction variance, which were in turn obtained by solving their respective transport equations. A relatively simple approach was used to compute radiant heat transfer—a diffusion model with a constant absorption coefficient (0.1 m i). 

Comparisons of the extended diffusion model with experimental spectra on small molecules have been performed by Steele, who concludes that the model, although not very successful, is still superior to any other model studied (van Koynenberg and Steele, 1974 Jones, 1974). 

In the case of the axial diffusivity model with step perturbation, the following equation was used (Levenspiel, 1999 Smith, 1986) ... 

Fig. 3 Transient temperature response for CH30H-NaX, pressure step 48-80 Pa (step A, run 13) showing conformity between experimentally observed temperature and theoretical curve, calculated from diffusion model with Do = 2.6 x 10 m s h = 2.3 Wm" From Grenier et al. [27]...

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