Dimensions defined

Therefore, iiweknowthecompactedbulkdensity,thenitispossibletocomputethemass in thebed using the mathematical statement for the conservation of mass. In this case the reactoranditsphysical dimensions define thecontrol volume. The rateofcatalyst delivery is a constant that we will call min The rate of mass flow out of the reactor is zero, that is,... 

Grassberger and Procaccia [grass83] show that 0 < Dc < Dj, where Di is the information dimension defined in equation 4.89. 

It will be noted that the relevant characteristic dimension in the Biot number is defined as the ratio of the volume to the external surface area of the particle (V/Ae), and the higher the value of V/Ae, then the slower will be the response time. With the characteristic dimension defined in this way, this analysis is valid for particles of any shape at values of the Biot number less than 0.1... 

It has been shown in Chapter 29 that the set of vectors of the same dimension defines a multidimensional space S in which the vectors can be represented as points (or as directed line segments). If this space is equipped with a weighted metric defined by W, it will be denoted by the symbol S. The squared weighted distance between two points representing the vectors x and y in is defined by the weighted scalar product ... 

Statistical dimensions number of variables (manifest or latent) taken into account in evaluation. Statistical dimensions define the type of data handling and evaluation, e.g. univariate, bivariate, multivariate... 

The development leading to Equation (66) in this section related the form factor P(6) to the radius of gyration Rg, which is one measure of the structure of a particle. We can actually get much more information from the form factor. In the following section, we discuss this and illustrate the use of P(d) for measuring the fractal dimension (defined in Chapter 1, Section 1.5b. 1) of an aggregate. 

A> Area, dimension, defined by b Equivalent radius of flow cell... 

H Cyclone dimension, defined by ro Radial location of particles at t = 0... 

H Height of the settling chamber re Cyclone dimension, defined by... 

The channel length defined by the SAP technique is not defined by a top-down patterning step, i.e. the gap dimensions are not known a priori. We have used a range of metrology tools to resolve and determine the channel dimensions defined... 

On the other hand, if we consider the so-called Fourier-Plancherel functions in one dimension defined by Eq. (3.2), all of them are situated in the complement C(u 1) as soon as the rotation angle is different from zero (a 0). It is evident that, in polar coordinates, the exponential factor exp(ikr) is going to have similar properties, since one has... 

The fractal dynamics of holes are diffusive, and the diffusivity depends strongly on the tenuous structure in fractal lattices. The fractal dimension defines the self-similar connectivity of hole motions, the relaxation spectrum, and stretched exponential... 

The strong-, weak- and intermediate regimes are all a product of the elastic constant of the basic mechanical unit (the floe, the links between the floes, or a combination of both) and the number of these units present in the direction of the externally applied force (Shih et al. 1990). Therefore, the fractal dimension defines to the size of the clusters. A large fractal dimension represents a large cluster that translates to less cluster-cluster interactions per unit volume and a decreased elastic modulus. At high volume fractions, cluster size decreases and the number of cluster-cluster interactions increases, and thus the elastic constant also increases. 

The chemical space is a multidimensional area with each dimension defined by a descriptor which can be molecular weight, polarity, solubility, membrane permeability, binding constant. H-bond-ing properties etc. and encompasses all small carbon-based molecules that could in principle be created [5]. 

The complete fractal A as displayed in Fig. 8.2, was never completely analysed. Hillermeier et al. (1992) studied this problem for one embedding dimension defined by considering a horizontal cut through the fractal at n = 1.1. The reduced fractal, then, has a Hausdorff-Besicovitch... 

Originally, the gas distributors in fluidized beds were made of perforated steel plates (see Fig. 7.70, top). The size of the holes, in most cases the diameter of circular bores, the percent open area, defined by the sum of all hole areas, sometimes the distribution pattern of the holes in the plate, and the gas pressure in the plenum below the distribution plate, which, together with the other dimensions, defined the gas flow rate, were major design parameters. To obtain a good, stable fluidized bed, the gas velocity has to be uniform across the entire area of the bed and must be adjusted such that, as a result, the solid particles are in a suspended state. 

Special extrusion tools may be necessary to feed the mass to the orifice(s.) Orifice dimensions define resistive force and cross section of extrudate. Orifice design must enable flow and assist in pressure release at discharge. Cutting devices may be necessary or desirable to divide strand into pellets. Drive must be capable of sustaining mixing, conditioning and extrusion. Individual process steps may be carried out in-line in separate equipment. 

A fractal object such as a C curve may have some unusual properties. The properties are that it has a fractal dimension but this fractal dimension is not a fraction. In the case of the C curve it is equal to two. The reason this object is still a fractal relates to a definition of fractal dimension. First, one defines two concepts of dimensions the topological dimension, which corresponds to our usual concept of a dimension, and a so-called Hausdorff-Besicovic dimension. If for a given object the two dimensions defined are different, the object is said to have a fractal dimension. In the case of the... 

The complexity of the polymer structure is reflected in the large number of dimensions needed to describe it. Alexander and Orbach [28] proposed the use of spectral or fracton dimension for the description of the density of states on a fractal. The necessity of introducing is due to the fact that the fractal dimension defined by Equation (11.1) does not reflect this parameter. The investigators made use of the fact that anomalous diffusion of particles is expected on a fractal and, hence ... 

The lattice is defined not only by the unit cell, principally by its external dimensions (defining the crystal system), but also by aspects of its internal constitution (defining the lattice type) and symmetry within the unit cell. There are seven crystal systems triclinic, monoclinic. orthorhombic, tetragonal, cubic, trigonal, and hexagonal. These are defined on the basis of the external geometry of the unit cell. The constraints defining the crystal system can easily be summarized ... 

FIGURE 1.1. Radial part of the wavefunction for a U (a) and 2p (b) orbitals showing an arbitrary cutoff beyond which R r) is less than some small value. The surface in three dimensions defined by this radial cutoff is shown in (c) for the li orbital and in (d) for the 2p orbital. 

Dimensions, defined as distances between points or surfaces. Example width, height, or length... 

Figure 1. The cyclotron resonance principle as applied to mass spectrometers. An alternating electric field whose frequency equals the cyclotron frequency (Equation 1) for a particular ion mass, excites the cyclotron motion of that ion. An oscillator is connected to the plates of a capacitor, whose dimensions define the sample volume, and gives rise to an alternating electric field within the capacitor. If the frequency of the oscillator equals the cyclotron frequency (Equation 1) of an ion located within the capacitor, the radius of the ion s cyclotron orbit will be increased (i.e., the ion cyclotron motion is excited). This phenomenon is called cyclotron resonance. The kinetic energy of the ion increases as the ion follows the spiral path shown, and the presence of cyclotron resonance is detected by measuring the signal that is induced in the plates of the capacitor by the excited ion motion.

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