Macroscale dispersion

Rubin and Gdme -Hemdndez (1990) and Indelman and Dagan, (1993) have developed analytical expressions fotyupscaling both the mean and the variance of spatially autocorrelated lnOfc fdatcL Since macroscale dispersion can be related to the variance of ln( sat), it may be possible to apply their expressions for predicting variance as a function of block size to the problem of upscaling estimates of K determined in the laboratory to the field scale. Unfortunately, numerical simulations based upon this approach are likely to be computationally expensive. 

These descriptive terms are used to classify the degree of nauoscale dispersion as well as global micro- and macroscale dispersion of the nanoparticles in the polymer matrix. Since no numerical standards exist for rating the degree of nanoparticle dispersion in the polymer matrix, use of these terms is strictly qualitative and continues to be area of some controversy, as the classification of dispersion is mostly the opinion of the user. Unfortunately, not all researchers in the nanocomposite area use these terms in the same ways. The definition of intercalated comes from an... 

The main reason for the discrepancy between the one-dimensional tracer test and the radial tracer test lies in the dimension of the source zone. One-dimensional tracer tests are characterized by a large source zone (i.e., the full cross-section of the medium), whereas convergent tracer tests have a point source. In the latter case, solute plumes do not sample the full variability of aquifer properties, and therefore undergo smaller dispersion processes. This mostly highlights that, even if theories are available to predict macroscale dispersion coefficients, they are bounded to certain limitations which could make them unsuited to given situations. 

The flow of the continuous phase is considered to be initiated by a balance between the interfacial particle-fluid coupling - and wall friction forces, whereas the fluid phase turbulence damps the macroscale dynamics of the bubble swarms smoothing the velocity - and volume fraction fields. Temporal instabilities induced by the fluid inertia terms create non-homogeneities in the force balances. Unfortunately, proper modeling of turbulence is still one of the main open questions in gas-liquid bubbly flows, and this flow property may significantly affect both the stresses and the bubble dispersion [141]. 

During the last decade considerable attention has been put on the macroscale modeling of bubble breakage in gas-liquid dispersions (e.g., [92, 43, 99,... 

The term macroscale will be used to denote multiphase models that employ a hydro-dynamic description of the disperse phase. Such models are also called multi-fluid models (because the disperse phase is treated as an effective fluid), or Euler-Euler models. The name of the latter comes from the numerical treatment of the disperse phase (i.e. discretization on a fixed grid), as opposed to Euler-Lagrange models wherein the disperse phase is tracked in a Lagrangian framework as discrete entities. We should note that, in the... 

The process of formulating mesoscale models from the microscale equations is widely used in transport phenomena (Ferziger Kaper, 1972). For example, heat transfer between the disperse phase and the fluid depends on the Nusselt number, and mass transfer depends on the Sherwood number. Correlations for how the Nusselt and Sherwood numbers depend on the mesoscale variables and the moments of the NDF (e.g. mean particle temperature and mean particle concentration) are available in the literature. As microscale simulations become more and more sophisticated, modified correlations that are based on the microscale results will become more and more common (Beetstra et al, 2007 Holloway et al, 2010 Tenneti et al, 2010). Note that, because the kinetic equation requires mesoscale models that are valid locally in phase space (i.e. for a particular set of mesoscale variables) as opposed to averaged correlations found from macroscale variables, direct numerical simulation of the microscale model is perhaps the only way to obtain the data necessary in order for such models to be thoroughly validated. For example, a macroscale model will depend on the average drag, which is denoted by... 

As mentioned above, macroscale models are written in terms of transport equations for the lower-order moments of the NDF. The different types of moments will be discussed in Chapters 2 and 4. However, the lower-order moments that usually appear in macroscale models for monodisperse particles are the disperse-phase volume fraction, the disperse-phase mean velocity, and the disperse-phase granular temperature. When the particles are polydisperse, a description of the PSD requires (at a minimum) the mean and standard deviation of the particle size, or in other words the first three moments of the PSD. However, a more complete description of the PSD will require a larger set of particle-size moments. 

The transport equations appearing in macroscale models can be derived from the kinetic equation using the definition of the moment of interest. For example, if the moment of interest is the disperse-phase volume fraction, then it suffices to integrate over the mesoscale variables. (See Section 4.3 for a detailed discussion of this process.) Using the velocity-distribution function from Section 1.2.2 as an example, this process yields... 

It is very interesting to note that the virtual-mass force mainly acts through the effective volume coefficient y, and tends to reduce the momenffim-exchange terms. The final step is to use the definitions of the mass-average moments introduced in Chapter 4 to derive the force terms in the macroscale momenffim balance. For the disperse-phase momentum balance (see Eq. (4.85) on page 123), this procedure leads to... 

The CAEM technique was also used to characterize the carbonaceous deposits produced in the macroscale studies. In this case fragments of the various deposits were dispersed onto transmission specimens of graphite and subsequently heated in 5 Torr 02 During this reaction the various carbonaceous components of the deposit oxidized at different rates and numerous details became evident. At 600°C the amorphous carbon was removed leaving behind a predominantly filamentous carbon residue. These structures, which formed an interconnected network, varied in width from 5 to 35 nm, and were up to lxlO nm in length. Upon further oxidation up to 750°C, virtually all the carbonaceous material had disappeared leaving a residue of metal or metal oxide particles. 

In general in complex media, the convection and diffusion of a solute is a difficult problem to analyze at the microscale and the moment method enables the analysis to be carried out at the macroscale leading to the replacement of the convective-diffusion problem by an effective global velocity and an effective dispersion tensor as in Eq. (4.6.30). Brenner s procedure analyzes the time evolution of the spatial moments (cf. Eq. 4.6.36) of the conditional probability density that a Brownian particle is located at a given position at a specific time knowing the position from which it was initially released into the fluid. 

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