Macroscale defined

Hybrid mixture theory is a hybridization of classical volume averaging of field equations (conservation of mass, momenta, energy) and classical theory of mixtures [8] whose theory of constitution results from the exploitation of the entropy inequality in the sense of Coleman and Noll [9], In [4] the microscale field equations for each species of each phase, modified appropriately to include charges, polarization, and an electric field, are averaged to the macroscale, defined to be the scale where the phases are indistinguishable. Thus at the macroscale the porous media is viewed a mixture, with each thermodynamic property for each constituent of each phase defined at each point in space. 

In order to design a zeoHte membrane-based process a good model description of the multicomponent mass transport properties is required. Moreover, this will reduce the amount of practical work required in the development of zeolite membranes and MRs. Concerning intracrystaUine mass transport, a decent continuum approach is available within a Maxwell-Stefan framework for mass transport [98-100]. The well-defined geometry of zeoHtes, however, gives rise to microscopic effects, like specific adsorption sites and nonisotropic diffusion, which become manifested at the macroscale. It remains challenging to incorporate these microscopic effects into a generalized model and to obtain an accurate multicomponent prediction of a real membrane. 

The shear rate between liquid layers moving at the average velocity is an important factor in macroscale processes involving large blobs. The velocity fluctuations are responsible for microscale processes of homogenization at the molecular level. The data for turbine agitators show that the mean velocity fluctuation, defined as... 

The description of a gas flow is well established from the micro- to macroscales. The length scale of a gas flow can be characterized by the local Knudsen number Kn, which is defined as... 

Measurements of macromixing by, for example, a motionless mixer are based on the coefficient of variation (CoV), which is a statistical measure of radial homogeneity at the macroscale. It is defined as the standard deviation of concentration measurements made at the exit of a mixer divided by the mean concentration ... 

There is a need to distinguish at this point how the shear rate in the impeller zone differs from the shear rate in the tank zone. To do this, however, one must carefully define shear rate and the corresponding concepts of macroscale shear rate and microscale shear rate. When one studies the localized fluid velocity through utilization of a small dimension probe, or as is currently used, a laser Doppler velocity meter device, one sees that at any point in the... 

This section furnishes a brief overview of the general formulation of the hydrodynamics of suspensions. Basic kinematical and dynamical microscale equations are presented, and their main attributes are described. Solutions of the many-body problem in low Reynolds-number flows are then briefly exposed. Finally, the microscale equations are embedded in a statistical framework, and relevant volume and surface averages are defined, which is a prerequisite to describing the macroscale properties of the suspension. 

Recently reported meso- and macroscale self-assembly approaches conducted, respectively, in the presence of surfactant mesophases [134-136] and colloidal sphere arrays [137] are highly promising for the molecular engineering of novel catalytic mixed metal oxides. These novel methods offer the possibility to control surface and bulk chemistry (e.g. the V oxidation state and P/V ratios), wall nature (i.e. amorphous or nanocrystalline), morphology, pore structures and surface areas of mixed metal oxides. Furthermore, these novel catalysts represent well-defined model systems that are expected to lead to new insights into the nature of the active and selective surface sites and the mechanism of n-butane oxidation. In this section, we describe several promising synthesis approaches to VPO catalysts, such as the self-assembly of mesostructured VPO phases, the synthesis of macroporous VPO phases, intercalation and pillaring of layered VPO phases and other methods. 

To explain why the boiling number seems to govern the transition between heat flux increasing a and vapour quality increasing a, the following interpretation is proposed, based on macroscale boiling mechanisms. From the Rohsenow (1952) and Kew and Cornwell (1997) analysis, an inertial characteristic time "Ccv for the liquid layer and a characteristic time Xb for bubbles leaving the wall can be defined. Then, from the Kutaleladze (1981) and Rohsenow (1952) analysis it can be shown that the ratio of these two characteristic times can be written ... 

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. 

Traditional chemical reaction engineering deals with smaller sizes, typically with tubing and pipe sizes ranging from about 1 mm to 10 m. This will be termed the macroscale. Table 16.1 provides definitions of three smaller scales meso, micro, and nano. These terms have been used in a variety of ways in the literature. The term mi-croreactor is sometimes used for systems defined here as mesoscale. The definitions in Table 16.1 are rational if somewhat arbitrary. They are based on the characteristic size of flow channels. 

Local thermodynamic equilibrium in space and time is inherently assumed in the kinetic theory formulation. The length scale that is characteristic of this volume is i whereas the timescale is xr. When either L i, ir or t x, xr or both, the kinetic theory breaks down because local thermodynamic equilibrium cannot be defined within the system. A more fundamental theory is required. The Boltzmann transport equation is a result of such a theory. Its generality is impressive since macroscopic transport behavior such as the Fourier law, Ohm s law, Fick s law, and the hyperbolic heat equation can be derived from this in the macroscale limit. In addition, transport equations such as equation of radiative transfer as well as the set of conservation equations of mass, momentum, and energy can all be derived from the Boltzmann transport equation (BTE). Some of the derivations are shown here. 

Abstract We here treat a diffusion problem coupled with water flow in bentonite. The remarkable behavior originates from molecular characteristics of its constituent clay mineral, namely montmorillonite, and we show the behavior based on a unified simulation procedure starting with the molecular dynamic (MD) method and extending the obtained local characteristics to a macroscale behavior by the multiscale homogenization analysis (HA Sanchez-Palencia. 1980). Not only the macroscale effective diffusion property but also the adsorption behavior is well defined based on this method. 

We assume mesoscale and microscale domains are periodic (Figure 5). Let ar , and jc be coordinate systems defined in macro-domain 2o, meso-domain Qi and micro-domain If we perform a laboratory seepage/consolidation test on a typical Japanese bentonite, Kunigel VI, the macroscale specimen is of scale lO m, the mesoscale quartz grains are of lO m, and the microscale clay minerals are of 10 m. So, we estimate the scale factor =10 , and introduce the relations x = x°/z, x = x /e. 

The mechanical properties of a material serve to define the macroscale performance of a sports material. Analyzing mechanical properties is the first step to fully understand how a polymer system will hold up to the stresses and strains during end use. The... 

By using microgels with well-defined properties as building blocks, macroscale hydrogels with unique spatial properties, such as gradients in mechanical and/or biomolecular characteristics, may be built from the bottom up. Microsphere-based scaffolds additionally offer 3D pore interconnectivity and desirable pore size. For instance, chitosan microsphere scaffolds have been produced for cartilage and osteochondral tissue engineering [81]. 

The eddy turbulence model, or simply eddy model, assumes that the small-scale eddies control surface renewal and, subsequently, mass transfer. This model acknowledges a scale dependence. Macroscale movements, those represented by the Reynolds number. Re, are assumed to have a small impact on surface renewal, where the Reynolds number is defined as... 

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