Space-Fractional Transport Equation

In this section we use the idea of subordination to obtain the space-fractional transport equation. Since r(t) is a nonnegative Levy process, the Laplace exponent l(s) defined in (3.161) can be written as 

3 Random Walks and Mesoscopic Reaction-Transport Equations 

Y-stable subordimtor. As an example, consider a strictly y-stable random time Ty t), the stable subordinator, for which a = Q and the Levy measure is 

The strictly stable process Ty(t) has a nice scaling property Ty(t) = t 7 j,(l) for all t. Scaling arguments lead to 

Since the asymptotic decay of the tail of gy(x) is x that Pt(x, t ) has a power-law tail 

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In turbulent reactive flows, the chemical species and temperature fluctuate in time and space. As a result, any variable can be decomposed in its mean and fluctuation. In Reynolds-averaged Navier-Stokes (RANS) simulations, only the means of the variables are computed. Therefore, a method to obtain a turbulent database (containing the means of species, temperature, etc.) from the laminar data is needed. In this work, the mean variables are calculated by PDF-averaging their laminar values with an assumed shape PDF function. For details the reader is referred to Refs. [16, 17]. In the combustion model, transport equations for the mean and variances of the mixture fraction and the progress variable and the mean mass fraction of NO are solved. More details about this turbulent implementation of the flamelet combustion model can also be found in Ref. [20],... 

Rieckmann and Keil (1997) introduced a model of a 3D network of interconnected cylindrical pores with predefined distribution of pore radii and connectivity and with a volume fraction of pores equal to the porosity. The pore size distribution can be estimated from experimental characteristics obtained, e.g., from nitrogen sorption or mercury porosimetry measurements. Local heterogeneities, e.g., spatial variation in the mean pore size, or the non-uniform distribution of catalytic active centers may be taken into account in pore-network models. In each individual pore of a cylindrical or general shape, the spatially ID reaction-transport model is formulated, and the continuity equations are formulated at the nodes (i.e., connections of cylindrical capillaries) of the pore space. The transport in each individual pore is governed by the Max-well-Stefan multicomponent diffusion and convection model. Any common type of reaction kinetics taking place at the pore wall can be implemented. 

Jakobsen et al. [95] and Lindborg et al. [135] did apply similar fractional step algorithms combined with FVM discretizations in space for variable density flows to simulate the performance of chemical processes in fixed bed reactors. The fractional steps defining the elements of these algorithms are sketched in the following. In a set of introductory steps, the mixture composition and the temperature were calculated. A set of scalar transport equations on the form (12.236) is generally solved for the species mass densities and the mixture enthalpy at the next time level n + 1. However, in reactor simulations, the enthalpy balance is frequently expressed in terms of temperature. The discrete form of the governing equations is thus written as ... 

Equation (9.41) constitutes a fundamental solution for purely convective mass burning flux in a stagnant layer. Sorting through the S-Z transformation will allow us to obtain specific stagnant layer solutions for T and Yr However, the introduction of a new variable - the mixture fraction - will allow us to express these profiles in mixture fraction space where they are universal. They only require a spatial and temporal determination of the mixture fraction/. The mixture fraction is defined as the mass fraction of original fuel atoms. It is as if the fuel atoms are all painted red in their evolved state, and as they are transported and chemically recombined, we track their mass relative to the gas phase mixture mass. Since these fuel atoms cannot be destroyed, the governing equation for their mass conservation must be... 

Here, we present an approach for the description of such anomalous transport processes that is based on the continuous-time random walk theory for a power-law waiting time distribution w(t) but which can be used to find the probability density function of the random walker in the presence of an external force field, or in phase space. This framework is fractional dynamics, and we show how the traditional kinetic equations can be generalized and solved within this approach. 

Thus concentration c must be replaced by ccm, where porosity c, as noted, is the fraction of total volume occupied by the mobile phase and thus the fractional volume serving as conduit space for the solute transport. The last equation thus becomes... 

Thus, the fractional equilibrium state (99) can be considered as a consequence of anomalous transport of phase points in the phase space resulting in the anomalous continuity equation (104). Note that the usual form of the evolution (93) is a direct consequence of the canonical Hamiltonian form of the microscopic equations of motion. Thus, the evolution of (105) implies that the microscopic equations of motion are not canonical. The actual form of these equations has not yet been investigated. However, there are strong indications that dissipative effects on the microscopic level become important. 

The generalized reaction-diffusion equation (2.82) can be written in a form using fractional derivatives for subdiffusive transport, where the waiting PDF of species i is given in Laplace space by (2.52), (i) 1 —. In that case... 

This equation has boundary conditions Z = 1 in the fuel stream and Z = 0 in the oxidant stream. The diffusion coefficient D is arbitrary, but because the maximum temperature determines the location of the reaction zone, the enthalpy diffusion is the most important transport process in the mixture fraction space. Therefore, the thermal diffusivity is a good choice to be the diffusion coefficient in (4.22). 

Before casting Equation 7.1 in dimensionless form, the inclusion of terms to describe adsorption and velocity enhancement of the transported species will be considered. These phenomena are, of course, more appropriate to polymer transport than tracer transport but the form of the equation is very similar. The velocity enhancement referred to concerns the effect of the excluded volume or inaccessible pore volume effect which the polymer shows (Chauveteau, 1982, Dawson and Lantz, 1972) and which is discussed in more detail below. However, the physical observation on polymer transport is that there appears to be a fraction of the pore space—either the very small pores (Dawson and Lantz, 1972) or regions close to the wall of the porous medium (Chauveteau, 1982)—which is inaccessible to polymer transport. This leads to an enhancement of the average velocity of the polymer through the porous medium. When there is both adsorption of transported polymer onto the rock matrix and a fraction of the pore volume is apparently inaccessible to the polymer, Equation 7.1 must be extended to ... 

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