Inhomogeneous diffusion model

A spatially varying diffusivity can model not only inhomogeneities in the medium but also hydro-dynamic interactions between the Brownian particles and channel walls. The diffusivity is then replaced by a diffusion tensor D(r), and instead of Eq. 1, one obtains... 

Using the previous equations we can derive the microscopic structure (/. e., the kinds of defects and their concentrations) from the experimentally determined lifetimes and intensities if the problem is homogeneous. This premise, however, does not hold for RPV steels completely. In inhomogeneous problems, the diffusion of positrons from the various implantation sites to the trapping centres must also be considered [125,130]. However, the mathematical difficulties associated with the corresponding diffusion-trapping model (DTM) [73] have so far prevented exact solutions from being obtained for all but the simplest problems [116,117], Thus, it is impossible to qualitatively analyse the very detailed experimental results obtained with a pulsed positron beam. 

Figure 2.8 depicts the Nemst diffusion layer model which shows that beyond the critical distance, <5, the solution is well mixed such that the concentration of the electroactive species is maintained at a constant bulk value. In this vicinity, the mixing of the solution to even out inhomogeneities is due to natural convection induced by density differences. Additionally, if the electrochemical arrangement is not sufficiently thermostated, slight variation throughout the bulk of the solution can provide a driving force for natural convection. 

Meanwhile, computational methods have reached a level of sophistication that makes them an important complement to experimental work. These methods take into account the inhomogeneities of the bilayer, and present molecular details contrary to the continuum models like the classical solubility-diffusion model. The first solutes for which permeation through (polymeric) membranes was described using MD simulations were small molecules like methane and helium [128]. Soon after this, the passage of biologically more interesting molecules like water and protons [129,130] and sodium and chloride ions [131] over lipid membranes was considered. We will come back to this later in this section. 

A spectral model similar to (3.82) can be derived from (3.75) for the joint scalar dissipation rate eap defined by (3.139), p. 90. We will use these models in Section 3.4 to understand the importance of spectral transport in determining differential-diffusion effects. As we shall see in the next section, the spectral interpretation of scalar energy transport has important ramifications on the transport equations for one-point scalar statistics for inhomogeneous turbulent mixing. 

In general, liquid-phase reactions (Sc > 1) and fast chemistry are beyond the range of DNS. The treatment of inhomogeneous flows (e.g., a chemical reactor) adds further restrictions. Thus, although DNS is a valuable tool for studying fundamentals,4 it is not a useful tool for chemical-reactor modeling. Nonetheless, much can be learned about scalar transport in turbulent flows from DNS. For example, valuable information about the effect of molecular diffusion on the joint scalar PDF can be easily extracted from a DNS simulation and used to validate the micromixing closures needed in other scalar transport models. 

The conditional velocity also appears in the inhomogeneous transport equation for x. / ), and is usually closed by a simple gradient-diffusion model. Given the mixture-fraction PDF, (5.316) can be closed in this manner by first decomposing the velocity into its mean and fluctuating components ... 

Of all of the methods reviewed thus far in this book, only DNS and the linear-eddy model require no closure for the molecular-diffusion term or the chemical source term in the scalar transport equation. However, we have seen that both methods are computationally expensive for three-dimensional inhomogeneous flows of practical interest. For all of the other methods, closures are needed for either scalar mixing or the chemical source term. For example, classical micromixing models treat chemical reactions exactly, but the fluid dynamics are overly simplified. The extension to multi-scalar presumed PDFs comes the closest to providing a flexible model for inhomogeneous turbulent reacting flows. Nevertheless, the presumed form of the joint scalar PDF in terms of a finite collection of delta functions may be inadequate for complex chemistry. The next step - computing the shape of the joint scalar PDF from its transport equation - comprises transported PDF methods and is discussed in detail in the next chapter. Some of the properties of transported PDF methods are listed here. 

Because the conditional scalar Laplacian is approximated in the FP model by a non-linear diffusion process (6.91), (6.145) will not agree exactly with CMC. Nevertheless, since transported PDF methods can be easily extended to inhomogeneous flows,113 which are problematic for the CMC, the FP model offers distinct advantages. 

Note that the turbulent diffusivity Tt(x, t) must be provided by a turbulence model, and for inhomogeneous flows its spatial gradient appears in the drift term in (6.177). If this term is neglected, the notional-particle location PDF, fx>, will not remain uniform when VTt / 0, in which case the Eulerian PDFs will not agree, i.e., i=- f0. 

As discussed in Chapter 7, when modeling inhomogeneous flows the value of At must be chosen also to be smaller than the minimum convective and diffusive time scales of the flow. 

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