Molecular mixing

As seen above, the mean chemical source term is intimately related to the PDF of the concentration fluctuations. In non-premixed flows, the rate of decay of the concentration fluctuations is controlled by the scalar dissipation rate. Thus, a critical part of any model for chemical reacting flows is a description of how molecular diffusion works to damp out concentration fluctuations at the SGS level. 

The first term on the right-hand side is the expected value of the scalar Laplacian conditioned on the scalars having values 0 33 An example of the time evolution of the conditional scalar Laplacian, corresponding to the scalar PDF in Fig. 1.11, is plotted in Fig. 1.12 for an initially non-premixed inert-scalar field. The closure of the conditional scalar Laplacian is discussed in Chapter 6. For the time being, it suffices to note the similarity between (1.36) and the IEM model, (1.16). Indeed, the IEM model is a closure for the conditional scalar Laplacian, i.e., 

In other closures for the chemical source term, a model for the conditional scalar dissipation rate (e 0 = ip) is required. (An example is plotted in Fig. 1.13 for the scalar PDF shown in Fig. 1.11.) Like the conditional scalar Laplacian, the conditional scalar 

Note that all terms in (1.37) can be directly extracted from DNS data for turbulent-scalar mixing. Thus, unlike the CRE approach, the FM approach allows for hie direct validation of micromixing models. 

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Ordinary diffusion involves molecular mixing caused by the random motion of molecules. It is much more pronounced in gases and Hquids than in soHds. The effects of diffusion in fluids are also greatly affected by convection or turbulence. These phenomena are involved in mass-transfer processes, and therefore in separation processes (see Mass transfer Separation systems synthesis). In chemical engineering, the term diffusional unit operations normally refers to the separation processes in which mass is transferred from one phase to another, often across a fluid interface, and in which diffusion is considered to be the rate-controlling mechanism. Thus, the standard unit operations such as distillation (qv), drying (qv), and the sorption processes, as well as the less conventional separation processes, are usually classified under this heading (see Absorption Adsorption Adsorption, gas separation Adsorption, liquid separation). 

As explained in Chapter 5, these materials are essentially non-volatile solvents for PVC. Because of their molecular size they have a very low rate of diffusion into PVC at room temperature but at temperatures of about 150°C molecular mixing can occur in a short period to give products of flexibility varying according to the type and amount of plasticiser added. 

The rubber and the polystyrene should not be compatible. If they are there will be molecular mixing and no improvement in toughness. 

Molecular mixing method for improvement of the molecular orientation in monomolecular layer... 

In this study, we demonstrated that high orientational order of a polar azobenzene-linked amphiphile was introduced to monolayers by means of the molecular mixing with a homologous amphiphile. Further, enhancement of SHG was observed in the mixed monolayer owing to the improvement of orientation of polar molecules. The mixing technique is expected to develop the construction of noncentrosymmetric LB films with highly efficient optical nonlinearity. 

Figure 1.11. A non-premixed scalar PDF as a function of time for inert-scalar mixing. Note that at very short times the PDF is bi-modal since all molecular mixing occurs in thin diffusion layers between regions of pure fluid where = 0 or 1. On the other hand, for large times, the scalar PDF is nearly Gaussian.

As discussed in Section 3.4, differential-diffusion effects will decrease with increasing Reynolds number. A single molecular mixing time scale T[Pg.168]

In the presence of molecular mixing, when the mixture fraction is less than fmax, F2co is no longer bounded above by ) 2 (c ). Instead, as noted earlier, the upper bound moves to the mixing line shown in Fig. 5.12. Thus, in a turbulent reacting flow with reaction and mixing, the theoretical upper bound on F2co is... 

Unlike presumed PDF methods, transported PDF methods do not require a priori knowledge of the joint PDF. The effect of chemical reactions on the joint PDF is treated exactly. The key modeled term in transported PDF methods is the molecular mixing term (i.e., the micromixing term), which describes how molecular diffusion modifies the shape of the joint PDF. 

We have used Fick s law of diffusion with separate molecular diffusivities for each species. However, most PDF models for molecular mixing do not include differential-diffusion effects. 

The composition PDF thus evolves by convective transport in real space due to the mean velocity (macromixing), by convective transport in real space due to the scalar-conditioned velocity fluctuations (mesomixing), and by transport in composition space due to molecular mixing (micromixing) and chemical reactions. Note that any of the molecular mixing models to be discussed in Section 6.6 can be used to close the micromixing term. The chemical source term is closed thus, only the mesomixing term requires a new model. 

Before discussing in detail specific molecular mixing models, it is useful to first state a few important constraints that can be derived by computing expected values. The first constraint follows from45... 

I) The molecular mixing model must leave the scalar mean unchanged. 

A second constraint can be found by multiplying the molecular mixing term by 4> p and averaging ... 

Note, however, that in the presence of a mean scalar gradient the local isotropy condition is known to be incorrect (see Warhaft (2000) for a review of this topic). Although most molecular mixing models do not account for it, the third constraint can be modified to... 

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