Numerical Analysis Based on the Model

In this Section explicit solutions to the equations of motion, (5.53, 54), are presented for some important combinations of parameters. Firstly, in Sect. 5.5.1 numerical experiments are conducted in order to explore the model s dynamic behaviour for a hypothetical economy. Secondly, in Sect. 5.5.2 the model is employed in an econometric analysis of changes in volume and composition of industrial investment in a real economy, that of the Federal Republic of Germany. 

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In conclusion, it can be said that the model depicted in Fig. 17.7 gives a good description of the observed results. A numerical analysis based on the same model is seen from Fig. 17.10 to also allow the radial growth rate dependence to be predicted and the agreement with experimental observations is reasonable. 

This model has been successfully used by Chin and Sabde [25] for crevice cathodic protection using numerical analysis based on the dilute solution theory and reduction reaction of dissolved oxygen and Aa+, Cl, and OH ions at the crevice surface. Hence, the Nemst-Plank equation, eq. (4.2), can be generalized as a differentiable and continues scalar diffusion molar flux function... 

In the following, we first discuss the situations where EC occurs as a primary forward bifurcation and where the standard model is directly apphcable (cases A and B). Then we discuss configurations where EC sets in as a secondary instability upon an already distorted Freedericksz ground state and compare it with experiments (cases C and D). Note that in this case the linear analysis based on the standard model already becomes numerically demanding. Finally, we address those combinations of parameters where a direct transition to EC is not very robust, since it is confined to a narrow Ca range around zero. For cases E and H this range may be accessible experimentally while for cases F and G it is rather a theoretical curiosity only. 

Initial stress and pore pressure in the foundation were firstly generated in the numerical model with the backfill elements being deactivated. Consolidation analysis then was performed on staged-construction of backfill. Safety analysis based on the SRM was carried out after the calculation of the final construction stage was finished. 

The two-particle nature of Coulomb interaction in equation (10.27) is the reason that among the third-order contributions to the transition amplitude, in addition to one particle effective operators (as in the standard J-O approach), two particle objects are also present. However, the numerical analysis based on ab initio calculations performed for all lanthanide ions, applying the radial integrals evaluated for complete radial basis sets (due to perturbed function approach), demonstrated that the contributions due to two-particle effective operators are relatively negligible [11,44-58]. This is why here they are not presented in an explicit tensorial form (see for example Chapter 17 in [13]). At the same time it should be pointed out that two-particle effective operators, as the only non-vanishing terms, play an important role in determining the amplitude of transitions that are forbidden by the selection rules of second- and the third-order approaches. This is the only possibility, at least within the non-relativistic model, to describe the so-called special transitions like, 0 <—> 0 in Eu +, for example, as discussed above. 

In Spite of the existence of numerous experimental and theoretical investigations, a number of principal problems related to micro-fluid hydrodynamics are not well-studied. There are contradictory data on the drag in micro-channels, transition from laminar to turbulent flow, etc. That leads to difficulties in understanding the essence of this phenomenon and is a basis for questionable discoveries of special microeffects (Duncan and Peterson 1994 Ho and Tai 1998 Plam 2000 Herwig 2000 Herwig and Hausner 2003 Gad-el-Hak 2003). The latter were revealed by comparison of experimental data with predictions of a conventional theory based on the Navier-Stokes equations. The discrepancy between these data was interpreted as a display of new effects of flow in micro-channels. It should be noted that actual conditions of several experiments were often not identical to conditions that were used in the theoretical models. For this reason, the analysis of sources of disparity between the theory and experiment is of significance. 

This chapter describes a DML model proposed by the authors, based on the expectation that the Reynolds equation at the ultra-thin film limit would yield the same solutions as those from the elastic contact analysis. A unified equation system is therefore applied to the entire domain, which gives rise to a stable and robust numerical procedure, capable of predicting the tribological performance of the system through the entire process of transition from full-film to boundary lubrication. 

Numerous commercial oxygen analysers are available, based on the principle of the Hersch cell, but all being protected from the analyte medium, either gaseous or liquid, by means of a membrane (usually Teflon). This membrane detector is known as the Clark cell (see later under membranes as measurement aids, p. 352) for gas analysis we may mention the Beckman Models 715 (also for liquids), 741, 743 (for flue gas), 755 and 778119. 

Here 0O is the characteristic temperature at volume V0. An average value for the volume dependence of the standard entropy at 298 K for around 60 oxides based on the Einstein model is 1.1 0.1 J K-1 cm-3 [15]. A corresponding analysis using the Debye model gives approximately the same numeric value. 

The Aspen Properties implementation of the NRLT-SAC method is available as a template. aprbkp file to license holders of Aspen Properties or Aspen Plus release 12.1 or above, by contacting Aspen s support centre or regional sales offices. The template is distributed with an Excel interface to simplify the data regression process and is suitable for non-expert users of Aspen Properties. Numerous Excel templates are available for data analysis and design calculations, based on the NRTL-SAC model. 

There are numerous equations in the literature describing the concentration dependence of the viscosity of dispersions. Some are from curve fitting whilst others are based on a model of the flow. A common theme is to start with a dilute dispersion, for which we may define the viscosity from the hydrodynamic analysis, and then to consider what occurs when more particles are added to replace some of the continuous phase. The best analysis of this situation is due to Dougherty and Krieger18 and the analysis presented here, due to Ball and Richmond,19 is particularly transparent and emphasises the problem of excluded volume. The starting point is the differentiation of Equation (3.42) to give the initial rate of change of viscosity with concentration ... 

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