Scale-independent model

CFD reduces scale-up problems, because the models are based on fundamental physics and are scale independent. Models of the actual unit can be simulated just as easily as models of lab scale versions, so predictions, and indeed optimization of the actual unit, can be achieved. 

It is then assumed that due to this separation in scales, the so-called subgrid scale (SGS) modeling is largely geometry independent because of the universal behavior of turbulence at the small scales. The SGS eddies are therefore more close to the ideal concept of isotropy (according to which the intensity of the fluctuations and their length scale are independent of direction) and, hence, are more susceptible to the application of Boussinesq s concept of turbulent viscosity (see page 163). 

The compressed loudness-time-pitch representation Cy(t, z) of the output of the audio device is scaled independently in three different pitch ranges with bounds at 2 and 22 Bark. This operation performs a global pattern matching between input and output representations and already models some of the higher, cognitive, levels of sound processing. 

Bridging the gap between micro- and macro-scale is the central theme of the first contribution. The authors show how a so-called Energy-Minimization Multi-Scale (EMMS) model allows to do this for circulating fluid beds. This variational type of Computational Fluid Dynamics (CFD) modeling allows for the resolution of meso-scale structures, that is, those accounting for the particle interactions, and enables almost grid-independent solution of the gas-solids two-phase flow. 

The correct starting point for a new cosmology is to accept that the universe is closed in space and time. The mathematical description of such a closed system automatically introduces the notions of periodicity and scale independent symmetries. The incorporation of scientific data into the model should be consistent with appropriate mathematics. 

Dimensionless numbers are scale independent and contain a characteristic length term (L) that adjusts their magnitude to match the scale of the model. The velocity, v (m/sec), in these definitions is the superficial velocity of the fiuid. 

If a scaled molecular model of a peptide chain be constructed with freedom of rotation about the valence bonds, it is of course possible for the model chain to assume a great variety of configurations. However, if the model peptide chain be placed on a plane surface and confined to this surface, it will be found that the chain has become very rigid and the only freedom allowed will be in the side chains beyond the j3-carbon atom. Probably once a peptide chain has been spread on a water surface its structure is fixed and is, short of collapse of the film, independent of the degree of compression. 

Figure 21 illustrates the ability of the mechanistic model to match complicated suspended solids density profiles from the 152 mm x 152 mm higher temperature pilot plant. Increas suspension densities at the top of the unit are due to considerable internal inertial separation at the exit. The profiles are used to find best fit values of the scale independent "wall-to-core flux coefficient" and "wall-layer disturbance factor". The model effectively predicts the variation of suspension density with height, solids circulation rate and gas velocity using these best fit values. 

Two groups of RER models were elaborated as presented in formulas 5 and 6. The parameters of the equations were calculated using STATIS-TICA (StatSoft,2008). Table 1 shows the equation parameters for 4 models divided into two groups (group a—GDPPC as the parameter of scale). The models use from one to eleven independent variables representing country characteristics and the relevant transport systems. The variables used in the models are only those with numerical values in the equation parameters. 

The fundamental principle in scaling models is to keep all dimensionless variables constant. The dimensionless equations, e.g. Equations (4.12)-(4.15), are scale independent, and keeping Re, Pr, Eu, and Fr numbers constant results in dimensionless variables, G, p, f, and C versus t that is the same at all scales. In the simplest case, scaling with a constant Re, i.e. an increase in length scale by a factor of 10, requires that the characteristic velocity decreases by a factor of 10. As a result, assuming no density difference, i.e. Fr = 0, Eu will be constant and the pressure drop is scaled with pU. Keeping Re and Pr constant should also result in the same dimensionless heat transfer coefficient, i.e. the Nusselt number, as obtained in the dimensionless correlations in Example 4.1. 

Coarse-grained models have a longstanding history in polymer science. Long-chain molecules share many common mesoscopic characteristics which are independent of the atomistic stmcture of the chemical repeat units [4, 5 and 6]. The self-similar stmcture [7, 8, 9 and 10] on large length scales is only characterized by a single length scale, the chain extension R. 

To move up the scale of complexity one now needs to consider the energetics o rotation about each bond. The simplest approach is to assume that each bond can be treatec independently 2md that the total energy of the chain is the sum of the individual torsiona energies for each bond. However, this particular model has some serious shortcoming arising from the assumption of independence. 

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