Three-dimensional distribution

Figure 4 shows a three-dimensional distribution of the Sauter mean diameter, measured 38.1-mm downstream from the nozzle using a Delavan 1 GPH-80°A pressure atomizer. The operating pressure was 690 kPa (100 psi). TypicaUy, the mean diameters graduaUy increase with an increase in radial... 

Fig. 4. Three-dimensional distribution of Sauter mean diameter (SMD) in a typical hoUow-cone spray.

Mijnarends, P.E. (1977) Reconstruction of three-dimensional distribution, In Compton Scattering, Williams, B. (Ed.), McGraw-Hill, New York, pp. 323-345. 

Similar treatments of the terms in Eq. (19) containing vy and vz will yield exact analogues of Eq. (29) for f(vy) and f(vz). Since the product of the three one-dimensional distribution functions gives the three-dimensional distribution function, i.e., F(v) = f(vx)f(vy)f(vz), we may combine Eq. (29) with similar expressions for the y and z velocity components to give... 

To obtain pictures of the orbital ip = R0< >, we would need to combine a plot of R with that of 0, which requires a fourth dimension. There are two common ways to overcome this problem. One is to plot contour values of ip for a plane through the three-dimensional distribution as shown in Figures 3.8a,c another is to plot the surface of one particular contour in three dimensions, as shown in Figures 3.8b,d. The shapes of these surfaces are referred to as the shape of the orbital. However, plots of the angular function 0 (Figure 3.7) are often used to describe the shape of the orbital ip = RQ because they are simple to draw. This is satisfactory for s orbitals, which have a spherical shape, but it is only a rough approximation to the true shape of p orbitals, which do not consist of two spheres but rather two squashed spheres or doughnut shapes. 

The great advantage of these methods is the possibility of creating three-dimensional distributions, also referred to as tomographs, which is not possible with FTIR spectroscopy and not always possible with Raman spectroscopy, if, for instance the sample is opaque. 

Ideally, the site characterization study has defined the vertical and horizontal extent of the contamination. Contoured site maps showing the (three-dimensional) distribution of the contaminants allow identification of areas that require extensive restoration, or may be allowed to be monitored to closure under natural attenuation. Knowledge of how much contamination exists and its location is the important first step in the remediation process. Evaluation of these data will permit consideration of the various remediation remedies available. Where the contaminant is contained within the shallow (<6 m) unsaturated zone and is recalcitrant (not readily biodegradable), excavation for off-site treatment or disposal may be the most expeditious procedure. Alternatively, depending on the contaminant, a variety of in situ procedures, including bioremediation, air sparging, soil vapor extraction, and fixation, may be applicable. 

The structure of a molecule is given by the three-dimensional distribution of atomic cores and valence electrons. This structure has been elucidated for many molecules with the use of X-ray or electron diffraction data. Chemical properties of molecules are observed under conditions which permit internal motions. Such observations yield views which may differ markedly as a function of time. Thus, observable properties are determined from equilibrated ensembles of species differing in geometry and energy. 

This transmission method provides an image of a section through the crystal and as such enables the experimenter to study the three-dimensional distribution of defects. The beam from the spot of a fine-focus or microfocus source is collimated into a ribbon beam of width approximately 10 //m before the singlecrystal specimen. This provides an incident beam of width small compared with that of the base of the Borrmaim fan formed by extremes of the diffracted and transmitted beams with the crystal surface. [In other words, the beam width must... 

Three-dimensional distributions of the micro-residual stresses are very complicated, and are affected by the elastic properties, local geometry and distribution of the composite constituents within a ply. Many analytical (Daniel and Durelli, 1962 Schapery, 1968 Harris, 1978 Chapman et ah, 1990 Bowles and Griffin, 1991a, b Sideridis, 1994) and experimental (Marloff and Daniel, 1969 Koufopoulos and Theocaris, 1969 Barnes et ah, 1991 Barnes and Byerly, 1994) studies have been performed on residual thermal stresses, A two-dimensional photoelastic study identified that the sign and level of the residual stresses are not uniform within the composite, but are largely dependent on the location (Koufopoulos and Theocaris,... 

A series of so-called compact dimers has also been described [4]. These compounds were characterized by an even three-dimensionally distributed hydrophilicity due to the perpendicular ring conformation, and stabiHzation of hydrophilicity by hydroxylated amido groups characterized by high energy rotational barriers for (E/Z) isomers. Furthermore, their viscosity was low, allowing injection at 350 mgl mL (at this concentration and at 20°C, their viscosities were found to be between 19 and 26 mPa.s vs 44.5 mPa.s for iodixanol) [4]. 

A cumulant of rank s is a symmetric tensor with (s2 + 3s + 2)/2 unique elements for a three-dimensional distribution. Like the moments p and the quasimoments c, the cumulants are descriptors of the distribution. For a onedimensional distribution, the relations between the cumulants and the moments are defined by equating the two expansions ... 

Figure 4.3-3 Three-dimensional distribution function of chloride ions relative to dimethylimidazolium ions.

Geothermal reservoir rocks are typically fractured and therefore exhibit variable and anisotropic permeability. For that reason it is neither possible to predict with confidence how an injection well may perform with respect to its injectivity nor with respect to which way the injected fluid will flow once it is in the reservoir. Because of this complication, the success of injection varies between fields and it is anticipated that a special injection scheme must be developed for each field depending on its characteristics, mainly the three-dimensional distribution of permeability and the waste fluid composition. Injection may require drilling of special wells. Alternatively, wells drilled for the purpose of production may not have adequate yield but can be used successfully as injection wells. When this is the case, no special wells need to be drilled for injection purposes, which reduces road building and therefore scenery spoliation. 

Z-series images allow samples to be studied at different depths, and thus provide information about the three-dimensional distribution of solids and liquid. The number of fields (z series) that can be viewed depends on pinhole position, objective, and sample brightness. Fewer planes can be observed in darker samples. Images taken at steps of 3pm are needed to obtain a three dimensional representation of the microstructure. 

Current observational data seem to confirm this picture when the red shifts of a large number of galaxies were measured, it turned out that their three-dimensional distributions have large voids, and this confirms a cellular structure of matter distribution. However, the question of the objectivity of a cellular or mesh structure is a subtle one since the eye tends to organize patterns even in a random distribution of points. 

The primary sequence dictates higher order structures. Therefore, a collection of sequences actually corresponds to a collection of shapes, each one displaying a unique three-dimensional distribution of elementary groups able to engage diverse types of interaction with any kind of target molecule. In vitro selection of an oligonucleotide sequence should, therefore, be viewed as the selection of an RNA or DNA structure complementary to a portion of the target. 

Progress in all interfacial science is limited by both the availability and the effective application of analytical techniques. In the vicinity of an interface, information is sought ideally with the best possible spatial resolution concerning the absolute value and three-dimensional distribution of the chemical identity, crystal structure, the nature of bonding, and electri-... 

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