Micro-scale modelling fraction

Hydraulic permeability f as a function of fibre volume fraction. Results of micro-scale model at two values of disorder measure (a) are verified against the literature and present numerical data. 

A key application of the multi-scale modeling formulation discussed previously is to determine munerically the appropriate macro-scale material parameters for use in a macroscale model, see for example Zohdi and Wriggers [15]. The solution of the macro-scale model can then be performed using mature finite element software in a fraction of the time that it would take to do a full multi-scale simulation. The motivation for adopting this strategy would be to capture as closely as possible the micro-scale material parameters, for use at the macro-scale. 

Mineral standards were hand crushed to -1/4 inch, then ground to a fine powder in a ball mill (alumina elements) or Bleuler Model 526/LFS678 puck mill. The resultant powder was aerodynamically classified in a Bahco Model 6000 micro particle classifier and the finest fraction ( 18 throttle) was collected. A size criterion of 90% or more by weight of particles 5 micron and smaller in diameter was used for the mineral standards. Sizes were verified by Coulter Counter. Duplicate 13 mm KBr pellets were prepared and the spectra were weight-scaled by techniques similar to those reported by Painter (3) and Elliot (4). With one exception, all the mineral standard spectra were averages of spectra from duplicate pellets. The one exception was the iron sulfate spectrum, which was obtained as the difference spectrum by subtracting the spectrum of HCl-washed weathered pyrite from that of the weathered pyrite. A weight correction was applied to the difference spectrum. 

P 28] A 3-D solid model of the cross-shaped micro mixer is meshed to a sufficiently fine scale with brick elements of 2 pm for the simulations [71]. Simulation results were intended at very short time scales, e.g. in intervals of 50 ps, to verify the mixing patterns at the initial state after application of pressure. The numerical values of the mass fraction are taken to give quantitative measures of the mixing efficiency. The pre-processor fluidics solver and post-processor of ConventorWare were used for the simulations. The software FLUENT 5 was used for verification of these results, since the former software is so far not a widely established tool for fluid dynamic simulation. 

For gases, 5c 1, for hquids. Sc 1. This implies that in turbulent flow of liquids, the species concentration field contains smaller scale structures than the velocity field. Similar to the decay time of the turbulent eddies in the velocity field, Tu, (12.2-1), the decay time of the eddies in the species concentration field, the previously introduced micro-mixing time, xy, can be modeled in terms of the correlation of the species mass fraction fluctuations, the so-called scalar (co-) variance, (K F), and its dissipation rate, the so-called scalar dissipation rate, sy. 

Finally, although the critical concentrations of B in the alloy A-B fox both transition models are inversely proportional to where Dis the diffusivity of B in the alloy, their physical meanings are quite different. In Wagner s theory, the diffusion boundary for B is at the internal oxidation front an increase of D can increase the internal oxide volume fraction in the alloy substrate. On the other hand, in the present analysis, the diffusion boundary for B is at the oxide scale/alloy interface an increase of D increases the formation of external oxide scale on the alloy surface. For example, surface micro- or nano-crystallization of M-Cr alloys can increase the outward diffusion of Cr along grain boundaries with a higher density, thus the diffusivity in Eq. (3.28) increases, which decreases Ng and promotes the formation of permanent Cr203 scale on M-Cr alloys [31,32]. 

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