Path distance

The vapor-layer model developed in Section IV.A.2 is based on the continuum assumption of the vapor flow. This assumption, however, needs to be modified by considering the kinetic slip at the boundary when the Knudsen number of the vapor is larger than 0.01 (Bird, 1976). With the assumption that the thickness of the vapor layer is much smaller than the radius of the droplet, the reduced continuity and momentum equations for incompressible vapor flows in the symmetrical coordinates ( ,2) are given as Eqs. (43) and (47). When the Knudsen number of the vapor flow is between 0.01 and 0.1, the flow is in the slip regime. In this regime, the flow can still be considered as a continuum at several mean free paths distance from the boundary, but an effective slip velocity needs to be used to describe the molecular interaction between the gas molecules and the boundary. Based on the simple kinetic analysis of vapor molecules near the interface (Harvie and Fletcher, 2001c), the boundary conditions of the vapor flow at the solid surface can be given by... 

The gas molecules fly about and among each other, at every possible velocity, and bombard both the vessel walls and collide (elastically) with each other. This motion of the gas molecules is described numerically with the assistance of the kinetic theory of gases. A molecule s average number of collisions over a given period of time, the so-called collision index z, and the mean path distance which each gas molecuie covers between two collisions with other molecules, the so-called mean free path length X, are described as shown below as a function of the mean molecule velocity c the molecule diameter 2r and the particle number density molecules n - as a very good approximation ... 

Finally, the plot in Fig 11 shows their calculations for the ratio of instantaneous velocity to terminal as a function of flight path distance in terms of expl thickness ie, 0.5 on the plot is a flight distance equal to 0.5 of the expl thickness, etc... 

Equation (3.299) is identical to Fourier s law of heat conduction, k = LqJT2. The validity of Eq. (3.299) is the same as the validity of Fourier s law, and the equation is valid when the relative variation of temperature is small within the mean free path distance A in the case of gases... 

Searching the minimum paths over the entire set of node pairs yields a huge amount of path-distance couples, stored into a path-distance matrix. Its elements count the frequency of the corresponding path-distance pairs therefore, the matrix expresses the probability factor for the existence of t vo points on the molecular surface at a specific distance and path. 

Consequently, elements of the matrix corresponding to nonexisting path-distance pairs contain zero frequency value, ivhereas all nonzero values indicate the molecular elongation, size, and variegation of the surface. 

The frequency of path-distance pairs for a single molecule can be viewed by means of three-dimensional plots in which the frequency distribution (z axis) is reported versus distances [x axis) and the path-distance (y axis), see Fig. 5.2. Molecular shape peculiarities are condensed in the frequency distribution graph in which low path-distance values (path = distance), represented by points close to the distance axis of the plot in Fig. 5.2, characterize more planar surfaces, whereas high path-distance values (path > distance) characterize more wrinkled surfaces. 

Figure 5.2. GRID positive MIF (a) and path—distance frequency distribution (b) for the methotrexate. Two nodes are used as an example their euclidean distance and minimum path are highlighted with green lines (a) and the point representing that node pair is indicated by P (b).

These qualitative statements are due to a simple graphical analysis of the cavity frequency distribution plot compared in Fig. 5.8. However, each cavity can be inspected and compared in detail. For example, the peak indicated by the arrow in Fig. 5.8 for 3A4 corresponds to the path-distance pairs reported in red color in Fig. 5.8(d). They end up far away from the heme in a subpocket region generated by the residues Leu 211 and Tyr 307. This subpocket is not present in the other CYPs and can be involved in a selective recognition of the substrate molecule. 

Since PathFinder works in path/distance space, the frame of reference for every molecule is internal and, therefore, no pairwise alignment is necessary when molecules are compared. PathFinder at the same time incorporates information on both overall shape (long distances) and local topology (shorter distances). 

Flow path distance from core inlet [cm]... 

Fig. 21 Predicted corrosion potential (ECP) versus flow path distance from the bottom of the core for 0 and 1.2 ppm of hydrogen added to the feedwater of the Leibstadt BWR [19]. The locations correspond to those given in Fig. 19.

FIGURE 4.1 Matrices can be used for describing chemical structures in different fashions. The adjacency matrix (a) of thionyl chloride shows whether are bonded to each other. The distance matrix (b) describes the number of bonds between two elements of a structure. The Cartesian distance matrix (c) contains the real three-dimensional (Euclidean) distances between atoms calculated from Cartesian coordinates of the atom positions. The bond path distance matrix (d) contains the sum of bond length between two atoms and is, in contrast to the Cartesian matrix, independent of the conformation of the molecule. 

We have seen before that different types of matrices can be used for characterizing a molecule. Depending on which matrix is used, the distance r j in a radial function can represent either the Cartesian distance, a bond-path distance, or simply the number of bonds between two atoms. Consequently, we yield three groups of RDF descriptors. 

FIGU RE 5.6 Scheme of the calculation of bond-path distances introduced in Equation 5.27. k represents the bond-path levels for spheres between atoms i and j. 

In addition, three distance modes — Cartesian, bond-path, and topological-path distances — are compared. Cartesian RDF descriptors are usually quite sensitive to small constitntional changes in the molecule. The bond-path descriptors exhibit less sensitivity, whereas topological bond-path descriptors only indicate extreme changes in the entire molecnle or in the size of the molecule. 

RDF. The distance mode dehnes the mode for distance calculation available modes are Cartesian distances, bond-path distances, and topological distances. Descriptors may be calculated on particular atoms. Exclusive mode restricts the calculation to the atom type, and with ignore mode the selected atom type is ignored when calculating the descriptor. In partial-atom mode an atom number has to be given instead of the atom type. The second atom property is available if 2D RDF is selected as code method. 

RDF descriptors may be used in any combination to fit the required task. For instance, it is possible to calculate a multidimensional descriptor based on bond-path distances and restricted to nonhydrogen atoms in the shape of a frequency pattern. Consequently, more than 1,400 different descriptors are available. A final summary of RDF descriptor types, their properties, and applications is given in Table 5.1. This section summarizes typical applications, some of which are described in detail in the next chapter. 

The path-distance map matrix, denoted as PD, resembling the bond length-weighted distance matrix of a molecular graph, is defined as [Bajzer, Randic et al, 2003]... 

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