Diffusion characteristic functions

As illustrated, here a single variable (the maximum temperature) is chosen as a characteristic function of the solution. For the premixed twin flame, this is a good choice. However, in other circumstances, like an opposed-flow diffusion flame, the choice of a characteristic scalar is less clear. Vlachos avoids the need for a choice by using a norm of the full-solution vector to characterize the solution in the arc length [415,416], The Nish-... 

Although the previous equation signifies the importance of the diffusion characteristics of the reactant species, it cannot be used to describe adequately the rate of the reaction. The reason is that the concept of global concentrations for the riA and ng molecules is meaningless, since a unit volume cannot be conceived due to the local fluctuations of concentrations. Hence, the local concentrations of the reactants determine the rate of the reaction for diffusion-limited reactions. Accordingly, local density functions with different diffusion coefficients for the reactant species are used to describe the diffusion component of reaction-diffusion equations describing the kinetics of diffusion-limited reactions. 

Therefore, the power-law behavior itself is a self-similar phenomenon, i.e., doubling of the time is matched by a specific fractional reduction of the function, which is independent of the chosen starting time self-similarity, independent of scale is equivalent to a statement that the process is fractal. Although not all power-law relationships are due to fractals, the existence of such a relationship should alert the observer to seriously consider whether the system is self-similar. The dimensionless character of a is unique. It might be a reflection of the fractal nature of the body (both in terms of structure and function) and it can also be linked with species invariance. This means that a can be found to be similar in various species. Moreover, a could also be thought of as the reflection of a combination of structure of the body (capillaries plus eliminating organs) and function (diffusion characteristics plus clearance concepts). 

The characteristic function of this diffusion process is by definition the Fourier transform of p(x, t), namely... 

We would like to attract the attention of the reader to the case when the environment is a source of anomalous diffusion. Paz et al. [116] studied the decoherence process generated by a supra-ohmic bath, but they did not find any problem with the adoption of the decoherence theory. It is convenient to devote some attention to the case when the fluctuation E, is a source of Levy diffusion [59]. If the fluctuation E, is an uncorrelated Levy process, the characteristic function again decays exponentially, and the only significant change is that the... 

Fig. 7. The characteristic functions of rj vs.

Effect of reaction order on diffusion factor y. Calculation of the characteristic function of y applicable to the case of an n order reaction yields similar functional relationships, in which the modulus

concentration term. For example, the case of second-order reaction involves the modulus... 

The measured responses were fitted using the characteristic function of micropore diffusion in isotropic spherical particles of uniform size (see the symbols for the measured responses and the best fit curves in Figure 5), The larger the deviation of the data from the best fit the wider is the particle-size distribution (cf Figures 2 and 5). 

The existing models can be grouped in two principal categories, black box models and structural models. Within the empirical black box models the membrane is considered a continuous, nonporous phase in which water of hydration is dissolved. An effective diffusion coefficient which is a characteristic function of the water content controls the water flux. 

Respiration, Rg (ml of gas G/hr kg of fruit), of a respiring product depends on number of factors such as gaseous composition surrounding the products, characteristics of the product (gas diffusion characteristics discussed earlier), storage period and temperature, etc. Thus, Rg may be represented as a function f of those factors as given by ... 

The overall in-phase and out-of-phase characteristic functions on the RHS of Eqs. 1 and 2 depend on the theoretical models describing the overall kinetic processes involved in the system. By fitting the experimental data on the LHS with the theoretical models on the RHS of the equations, diffusion coefficients and some other kinetic parameters which are included in the overall FR characteristic functions can be extracted. 

When only a single intracrystalline diffusion process occurs in a microporous system, the characteristic functions are [2,3]... 

Fig. 3 The ideal shape of the phase lag, z b, and amplitude ratio, Pb/Pz> curves vs. frequency (a) and the relevant theoretical characteristic functions, KS n and JC3out> for a single diffusion process occurring in microporous spherical crystals when I = 10 p.m, K= 1, and D = 10 m s ...

Figure 8 displays some typical FR data of Ci - Ce n-alkanes diffusing in coffin shaped crystals of silicalite-1 (40 x 40 x 260 p,m ). All the spectra in Fig. 8a-f,l can be fitted by the theoretical in-phase and out-of-phase characteristic function curves of the single diffusion model described by Eqs. 3-6, implying that only a simple, single diffusion process is involved in these systems. The diffusivities calculated from the best fit are presented in Fig. 9 and Tables 1 and 2. Equations 5 and 6 were applied since the channel framework structure of sihcahte-1 is comprised of near circular (0.54 x 0.56 nm)... 

Fig. 8 FR spectra of methane (a), ethane (b), propane (c), n-butane (d), n-pentane (e) and n-hexane (f) in sUicalite-l (cf. [65]). (n.o) indicate the experimental in-phase and out-of-phase KSqm characteristic functions, respectively. A single diffusion process model was used to fit the data in (1), while the non-isothermal diffusion model was used to fit the data in (2) except (f, 2) which was fitted using the two independent diffusion processes model. Solid lines denote the theoretical overall characteristic functions, and dash and dash-dot lines denote the theoretical diffusion processes occurring in the straight channels (dash) and the sinusoidal channels (dash-dot). Note 1 Torr= 133.33 Pa...

Fig. 10 FR spectra of benzene in sUicalite-l (cf. [65]). Continuous lines are the fits of the theoretical single diffusion process model (a-c) and the two diffusion processes model (d-f). The symbols ( , o) are the experimental in-phase and out-of-phase characteristic function data, respectively. Note 1 Torr = 133.33 Pa...

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