Relation to macroscale models

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 

25) reduces to the continuity equation for the disperse-phase volume fraction. However, the disperse-phase mean velocity Up is unknown, and must be found by solving a separate transport equation. 

The disperse-phase mean momentum equation is found from the first-order moment of the velocity distribution function  

28) reduces to the transport equation for the disperse-phase mean momentum  

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These relationships are valid for isolated bubbles moving under laminar flow conditions. In the case of turbulent flow, the effect of turbulent eddies impinging on the bubble surface is to increase the drag forces. This is typically accounted for by introducing an effective fluid viscosity (rather than the molecular viscosity of the continuous phase, yUf) defined as pi.eff = Pi + C pts, where ef is the turbulence-dissipation rate in the fluid phase and Cl is a constant that is usually taken equal to 0.02. This effective viscosity, which is used for the calculation of the bubble/particle Reynolds number (Bakker van den Akker, 1994), accounts for the turbulent reduction of slip due to the increased momentum transport around the bubble, which is in turn related to the ratio of bubble size and turbulence length scale. However, the reader is reminded that the mesoscale model does not include macroscale turbulence and, hence, using an effective viscosity that is based on the macroscale turbulence is not appropriate. 

Chapter 1 introduces key concepts, such as flow regimes and relevant dimensionless numbers, by using two examples the PBE for fine particles and the KE for gas-particle flow. Subsequently the mesoscale modeling approach used throughout the book is explained in detail, with particular focus on the relation to microscale and macroscale models and the resulting closure problems. 

New combinatorial schemes are needed where various reaction variables can be altered simultaneously and where important cooperative effects can be carefully probed to provide better understanding of variation sources. New performance models must be developed that can relate intrinsic and performance properties and can provide transfer functions of scalability of performance of chemical reactors from nano- to macroscale. Advances in... 

The development of the model for the wave phenomena is based on a hierarchy of length scales [61 On the microscale this is the bubble radius R, on the mezoscale it is the cell radius A, and on the macroscale we deal with a characteristic wave length L. We make the assumption that the bubble distribution is monodisperse, that bubbles keep their spherical shape and do not fragment. The void fraction (p is then related to the bubble number density n and the radius R by... 

Transport Criteria in PBRs In laboratory catalytic reactors, basic problems are related to scaling down in order to eliminate all diffusional gradients so that the reactor performance reflects chemical phenomena only [24, 25]. Evaluation of catalyst performance, kinetic modeling, and hence reactor scale-up depend on data that show the steady-state chemical activity and selectivity correctly. The criteria to be satisfied for achieving this goal are defined both at the reactor scale (macroscale) and at the catalyst particle scale (microscale). External and internal transport effects existing around and within catalyst particles distort intrinsic chemical data, and catalyst evaluation based on such data can mislead the decision to be made on an industrial catalyst or generate irrelevant data and felse rate equations in a kinetic study. The elimination of microscale transport effects from experiments on intrinsic kinetics is discussed in detail in Sections 2.3 and 2.4 of this chapter. 

Since in the macroscale model, the reaction rate and diffusion coefficient are effective ones that are obtained on an ensemble-averaged basis, the internal diffusion will not appear in the controlling equations explicitly. The effective reaction rate already includes the influence of internal diffusion inside catalyst pellets. The external mass transfer term, which mainly accounts for the species transport outside catalyst pellets, is used in the controlling equations in macroscale models. So, the diffusion mentioned in macroscale model normally represents species diffusion outside catalyst pellets. In fluidized bed, species diffusion is closely related to the flow regime in the reactor (Abba et al., 2003). Abba et al. (2003) summarized the formulae for calculating diffusion coefficients in different flow regimes in fluidized bed. 

It is meaningful to examine the relation between microscale model, mesoscale model, and micromodel. For reaction kinetics, microscale and mesoscale models adopt the same kinetics that based on element reaction system. For diffusion, mesoscale model embodies two diffusion mechanisms (one for micropores and another for mesopores and macropores), and microscale model considers one diffusion mechanism since it only has micropores. No diffusion was considered within the macropores. It is obvious that the mesoscale model possesses the same theoretical foundation as the microscale model, but its application scope has been enlarged compared to the microscale model. Therefore, it could be reliably used as a tool to derive some parameters, such as effective chemical kinetics and effective diffusion parameters, for macroscale model. In the section following, we discuss the method on how to link the microscale kinetics to the lumped macroscale kinetics via the mesoscale modeling approach. 

As described in this chapter, the physieal theory and molecular modeling of catalyst layers provide various tools for relating the global performance metrics to local distributions of physical parameters and to structural details of the complex composite media at the hierarchy of scales from nanoscale to macroscale. The subsisting challenges and recent advances in the major areas of theoretical catalyst layer research include (i) structure and reactivity of catalyst nanoparticles, (ii) selforganization phenomena in catalyst layers at the mesoscopic scale, (iii) effectiveness of current conversion in agglomerates of carbon/Pt, and (iv) interplay of porous structure, liquid water formation, and performance at the macroscopic scale. 

Nanofibre enhances the capture of nanoparticles such as viruses, bacteria, and man-made particles, such as soot from diesel exhaust. As soon as a fluid (air or liquid) contacts a nonwoven web, molecules are subjected to various forces, such as Brownian di siou, direct interception, partial impact, electrostatic forces, and sedimentation. For nanometre-scale fibres, a second factor has to be taken into account the effect of slip flow at the fibre surface. For macroscale fibrous materials, filtration mechauisms rely on continuous flow around the fibre, with a no-slip condition at the fibre surface. The theory starts to break down when the scale of the fibre becomes small enough that the molecular movements of the air molecules are significant in relation to the size of the fibres and the flow field. Using a slip-flow model at the fibre surface can extend the usefiil range of continuous flow theory. The Knudseu number (Kn) is used to describe the importance of the molecular movements of air... 

A main distinction has been made between deterministic and stochastic modeling techniques. A further distinction has been proposed based on the scale for which the mathematical model must be derived (eg, micro-, meso-, and/or macroscale). Notably, the complexity of the model approach depends on the desired model output. Detailed microstractural information is only accessible using advanced modeling tools but these are associated with an increase high in computational cost. The advanced models allow one to directly relate macroscopic properties to the polymer synthesis procedure and, thus, to broaden the application market for polymer products, based on a fundamental understanding of the polymerization kinetics and their link with polymer processing. 

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