Geometric conditions

Knudsen diffusion, but the dependence on physical and geometric conditions... 

Abstract—The geometrical conditions pertaining to closure, helicity, and interlayer distance between successive layers with circular cross-sections in carbon tubules (nanotubes) have been examined. Both the intralayer length of the C—C bonds and the interlayer distance between successive layers must vary with the radius of the layers. The division into groups of the sheets in nanotubes is found to be due to the reciprocal interaction of the interlayer distance variations and of the conditions required to maintain constancy of the pitch angle. 

The mechanical properties of natural fibers depend on cellulose type because each type of cellulose has a specific cell geometry and the geometrical conditions determine the mechanical properties. 

There is only very few process design literature for the calculation of power input for shake flasks [14-17]. Only the recent publication of Biichs et al. [17] provides a suitable correlation for the operating and geometric conditions... 

The crystal structures of two compounds are isotypic if their atoms are distributed in a like manner and if they have the same symmetry. One of them can be generated from the other if atoms of an element are substituted by atoms of another element without changing their positions in the crystal structure. The absolute values of the lattice dimensions and the interatomic distances may differ, and small variations are permitted for the atomic coordinates. The angles between the crystallographic axes and the relative lattice dimensions (axes ratios) must be similar. Two isotypic structures exhibit a one-to-one relation for all atomic positions and have coincident geometric conditions. If, in addition, the chemical bonding conditions are also similar, then the structures also are crystal-chemical isotypic. 

Two structures are homeotypic if they are similar, but fail to fulfill the aforementioned conditions for isotypism because of different symmetry, because corresponding atomic positions are occupied by several different kinds of atoms (substitution derivatives) or because the geometric conditions differ (different axes ratios, angles, or atomic coordinates). An example of substitution derivatives is C (diamond)-ZnS (zinc blende)-Cu3SbS4 (famatinite). The most appropriate method to work out the relations between homeotypic structures takes advantage of their symmetry relations (cf. Chapter 18). 

Pattern transition in vertical adiabatic flow. Upward vertical flow has been studied intensively, both because of the simplicity of the geometric condition and the relevance in applications. The map shown in Figure 3.4 is the result of rather recent and relevant studies into the interpretation of regime transition mechanisms. In this figure, the transition between bubbly flow and slug flow occurs be-... 

Channel techniques employ rectangular ducts through which the electrolyte flows. The electrode is embedded into the wall [33]. Under suitable geometrical conditions [2] a parabolic velocity profile develops. Potential-controlled steady state (diffusion limiting conditions) and transient experiments are possible [34]. Similar to the Levich equation at the RDE, the diffusion limiting current is... 

C5 cyclization requires stricter geometric conditions than aromatization. This is in favor of the dual-site mechanism of C5 cyclic reactions (25). All metals catalyzing it have an fee lattice, and their atomic diameter lies between 0.269 and 0.277 nm. These two criteria must be fulfilled simultaneously. With such a distance between the two sites, the screening of the C—C bond adjacent to the preferably adsorbed tertiary C atom becomes evident. Figure... 

When X-rays illuminate a crystalline material, the atoms in the crystal act as scattering centers. Because of the periodic nature of crystals, the scatterers can be considered to be associated with periodically spaced parallel planes a distance d apart. For certain angles of incidence to these planes the X-rays are scattered coherently and in phase. The coherent scattering is known as X-ray diffraction and the geometric condition required for diffraction, the Bragg equation, is given by... 

Figure 11 Power characteristics of a Rushton turbine stirrer under given geometric conditions, measured in two differently scaled vessels (scale 1 2) and fitting the flow behavior of the viscoelastic fluid [polyacrylamide (PAA) solution] by changing its viscosity. Source From Ref 13.

Figure 11.10 Schematic representation of co-Clapeyron vector orientations of (a) base T), P) vectors and (b) resultant Clapeyron vectors X) (dotted), illustrating the geometrical condition (11.165a) for coexistence of a (solid line) and /3 (dashed line) phases. The coordinate origin is marked with a small square in each panel. [Vector lengths, angles, ya, and ASap, AVap, values are chosen purely for illustrative purposes the Clapeyron-matching condition cannot be sensibly illustrated in Si-based units for any real system (Sidebar 11.6).]...

Another set of classical cases when the varieties rt p 1(Y)l are considered arise if we wish to impose geometric conditions to certain families. The prototype of these examples are the linear systems of hypersurfaces of a given degree d in Pr to which we have imposed the condition of passing through a given set of distinct points p1>...,P t the linear system in question is... 

Nevertheless, it is possible to obtain a constant relationship between the current at both microelectrodes for certain geometrical conditions. Thus, for microbands and microhemicylinders fulfilling rc = w/4, a constant ratio is obtained, but in this case it is necessary to use the same experimental timescale [10] ... 

Observation of H-F coupling has been used in postulating evidence for intramolecular C-F- H-0 hydrogen bonding in specific cases where geometrical conditions allow it (Scheme 2.20) 22-24... 

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