Parametric effects

Parametric Excitation.—The so-called parametric effects or phenomena have been known for a long time, but it is only recently that their study has been carried out systematically. 

Mudawar I, Bowers MB (1999) Ultra-high critical heat flux (CHF) for subcooled water flow boiling. I CHF data and parametric effects for small diameter tubes. Int J Heat Mass Transfer 42 1405-1428... 

Figure 5.51 Parametric effects (D, L, G) on burnout. (From Macbeth, 1963b. Reprinted with permission of Office of Official Publications of European Communities, Luxembourg.)...

The following parametric effects on density wave instability are summarized, as these effects have been often observed in the most common type of two-phase flow instability (Boure et al., 1973) ... 

Pan, C., andT. L. Lin, 1991, Prediction of Parametric Effects on Transition Boiling under Pool Boiling Conditions, Int. J. Heat Mass Transfer 34(6) 1355— 1370. (2)... 

Figure 3. Parametric effects of particle size and gas conductivity. (Data of Jacob andOsberg, 1957).

The interaction of parametric effects of solid mass flux and axial location is illustrated by the data of Dou et al. (1991), shown in Fig. 19. These authors measured the heat transfer coefficient on the surface of a vertical tube suspended within the fast fluidized bed at different elevations. The data of Fig. 19 show that for a given size particle, at a given superficial gas velocity, the heat transfer coefficient consistently decreases with elevation along the bed for any given solid mass flux Gs. At a given elevation position, the heat transfer coefficient consistently increases with increasing solid mass flux at the highest elevation of 6.5 m, where hydrodynamic conditions are most likely to be fully developed, it is seen that the heat transfer coefficient increases by approximately 50% as Gv increased from 30 to 50 kg/rrfs. 

The experiments of Dou et al. (1991) also indicate that the heat transfer coefficient varied with radial position across the bed, even for a given cross-sectional-averaged suspension density. Their data, as shown in Fig. 20, clearly indicate that the heat transfer coefficient at the bed wall is significantly higher than that for vertical surfaces at the centerline of the bed, over the entire range of suspension densities tested. Almost certainly, this parametric effect can be attributed to radial variations in local solid concentration which tends to be high at the bed wall and low at the bed centerline. 

Figure 18. Parametric effects of solid flux and particle diameter on heat transfer in fast fluidized beds. (From Furchi et al, 1988).

The data of Fig. 20 also point out an interesting phenomenon—while the heat transfer coefficients at bed wall and bed centerline both correlate with suspension density, their correlations are quantitatively different. This strongly suggests that the cross-sectional solid concentration is an important, but not primary parameter. Dou et al. speculated that the difference may be attributed to variations in the local solid concentration across the diameter of the fast fluidized bed. They show that when the cross-sectional averaged density is modified by an empirical radial distribution to obtain local suspension densities, the heat transfer coefficient indeed than correlates as a single function with local suspension density. This is shown in Fig. 21 where the two sets of data for different radial positions now correlate as a single function with local mixture density. The conclusion is That the convective heat transfer coefficient for surfaces in a fast fluidized bed is determined primarily by the local two-phase mixture density (solid concentration) at the location of that surface, for any given type of particle. The early observed parametric effects of elevation, gas velocity, solid mass flux, and radial position are all secondary to this primary functional dependence. 

The parametric effect of system pressure on the heat transfer coefficient was studied by Wirth (1995). They obtained experimental measurements of the heat transfer Nusselt number for fast fluidized beds... 

The parametric effect of bed temperature is expected to be reflected through higher thermal conductivity of gas and higher thermal radiation fluxes at higher temperatures. Basu and Nag (1996) show the combined effect (Fig. 23) which plots heat transfer coefficients as a function of bed temperature for data from four different sources. It is seen that for particles of approximately the same diameter, at a constant suspension density (solid concentration), the heat transfer coefficient increases by almost 300% as the bed temperatures increase from 600°C to 900°C. 

Another parametric effect is the apparent dependence of the heat transfer coefficient on the physical size of the heat transfer surface. Figure 24, from Burki et al. (1993), graphically illustrates this parametric effect by showing that the effective heat transfer coefficient can vary by several hundred percent with different vertical lengths of the heat transfer surface, for circulating fluidized beds of approximately the same particle diameter and suspension density. This size effect significantly contributed to confusion in the technical community since experimental measurements by inves-... 

X(n) parametrizes effective zero-cycles of degree n on X, i.e. formal linear combinations n,-[zj] of points X in X with coefficients n, IN fulfilling ni =n. X has a natural stratification into locally closed subschemes ... 

Remark 1 Fory = 0, v 0) corresponds to the optimal value of the primal problem (P). Values of the perturbation function v(y) at other points different than the origin y = 0 are useful on the grounds of providing information on sensitivity analysis or parametric effects of the perturbation vector y. 

A number of inorganic materials, e.g., KDP, KTP, LiNb03, LiI03, and borate crystals (2), are readily available on the market for parametric effects. However, these materials are very difficult to fabricate and are not easily integrated with semiconductor materials into monolithic circuits. 

Parametric Effects in Colour-Difference Evaluation, CIE Publication 101, CIE Central Bureau, Vienna, Austria, 1993. 

Surface Fluxes and Boundary Layer Parametrization Effect... 

MVF methods fail in any flow where the nature of the turbulence is altered by some parametric effect, such as rotation, which does not appear parametrically in the equations of mean motion. Such effects can be included in MVF methods only by alteration of the I or rr specification, and hence MTE or MRS methods are clearly to be preferred for such cases. 

Facilitating the insight into the complex inter-linked phenomena of chemical reactions and turbulent flow field behaviour as well as for the investigation of parametric effects, the simulation program AIOLOS is used as an effective tool for the investigation of the combustion processes in small and medium scale wood combustion systems. 

A further reduction of the computational effort in investigations of electronic structure can be achieved by the restriction of the actual quantum chemical calculations to the valence electron system and the implicit inclusion of the influence of the chemically inert atomic cores by means of suitable parametrized effective (core) potentials (ECPs) and, if necessary, effective core polarization potentials (CPPs). Initiated by the pioneering work of Hellmann and Gombas around 1935, the ECP approach developed into two successful branches, i.e. the model potential (MP) and the pseudopotential (PP) techniques. Whereas the former method attempts to maintain the correct radial nodal structure of the atomic valence orbitals, the latter is formally based on the so-called pseudo-orbital transformation and uses valence orbitals with a simplified radial nodal structure, i.e. pseudovalence orbitals. Besides the computational savings due to the elimination of the core electrons, the main interest in standard ECP techniques results from the fact that they offer an efficient and accurate, albeit approximate, way of including implicitly, i.e. via parametrization of the ECPs, the major relativistic effects in formally nonrelativistic valence-only calculations. A number of reviews on ECPs has been published and the reader is referred to them for details (Bala-subramanian 1998 Bardsley 1974 Chelikowsky and Cohen 1992 Christiansen et... 

Zhang, C., W. Zhao, T. Ye, S. H. Frankel, and J. P. Gore. 2002. Parametric effects on combustion instability in a lean premixed dump combustor. AIAA Paper No. 02-4014. 

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