Dimensionality, inherent difficulties

For accuracy, in power number Np calculations, the power of the load on the impeller rather than the mixer motor should be used. Before attempting to use dimensional analysis, one has to measure/estimate power losses for empty bowl or dry stage mixing. Unlike power consumption of the impeller (based on torque measurements), the baseline for motor power consumption does not stay constant and changes significantly with load on the impeller, mixer condition, or motor efficiency. This may present inherent difficulties in using power meters instead of torque. Torque, of course, is directly proportional to power drawn by the impeller (the power number can be determined... 

At the end of chapter 1, an inherent difficulty became apparent. How is it possible to conveniently specify a crystal structure in which the unit cell may contain hundreds or even thousands of atoms In fact, crystallographers make use of the symmetry of crystals to reduce the list of atom positions to reasonable proportions. However, the application of symmetry to crystals has far more utility than this accountancy task. The purpose of this chapter is to introduce the notions of symmetry, starting with two-dimensional patterns. 

The variety of trends in heat transfer data and the inherent difficulties in performing experimental work on these small systems have made it very challenging to develop a well-established understanding of boiling in micro-channels. Several authors have correlated their experimental results through different sets of generally non-dimensional groups. Others, on the other hand, have attempted to either extend methods previously developed for conventional macro-scale systems to the micro-scale, or define new approaches specifically for micro-channel two-phase flows. 

This example shows the degree of complication inherent in the nonlocal extension of the continuum theory even for the simplest Born-like case. In accord with Equation (1.141), the dimensionless parameter A/a measures the importance of nonlocality effects the local Born limit is recovered when A/a - 0. The opposite strongly nonlocal limit a/A - 0 corresponds to the unscreened solvation Usoly = -Q2( 1 - 1 /e /la. For the general form of the dielectric function a(k) a numerical solution for one-dimensional Equation (1.144) is straightforward [19]. However, there exists a principal difficulty hindering such solution when a(k) has poles on the real A-axis (see Sections 1.6.7 and 1.6.8). This creates oscillating kernels a f — r ) in the real space. 

The primary difficulty inherent in this issue is the small niunber of materials with suitable crystal structures and lattice constants. Some transition metals and ceramics, such as Ni, Cu, Fe, and cBN (Table 5, Ch. 3), are the few isostructural materials with sufficiently similar lattice constants (mismatch <5%). In addition, the extremely high surface energies of diamond (ranging from 5.3 to 9.2 J m for the principle low index planes) and the existence of interfacial misfit and strain energies between diamond films and non-diamond substrates constitute the primary obstacles in forming oriented two-dimensional diamond nuclei. Earlier attempts to grow heteroepitaxial diamond on the transition metals were not successful. The reasons may be related to the high solubility/ mobility of C in/on the metals (for example, Fe, Co, or the... 

The findings clearly show that chemically modified SPG maintains its inherent ability as a one-dimensional host. The wrapping of the chemically modified SPG provides a novel strategy to create functional polymer composites in a supramolecular manner. Considering a general difficulty in introducing fimctional groups into the functional polymer backbones, the present system can be a new potential path to develop fimctional polymeric materials. 

Given that elegant advances are being made in automated protein structure classification and even with the soon-to-be-initiated production stage of the PSI (called PSI-2), the difficulties inherent in protein crystallization imply that not all possible protein structures will be known in the near future. Accordingly, there is a need to predict at atomic resolution the three-dimensional (3-D) shape of novel designer proteins and proteins whose... 

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