Geometric quantities

An important purpose of tensor analysis is to describe any physical or geometrical quantity in a form that remains invariant under a change of coordinate system. The simplest type of invariant is a scalar. The square of the line element ds of a space is an example of a scalar, or a tensor of rank zero. 

Geometric quantities length, L, as in the use of a dilatometer described above area, A, as in characterizing the extent of surface in a solid catalyst (Chapter 8) and volume, V, as in describing the size of a vessel ... 

Various examples of two-angle fibers were already displayed by Woodcock et al. [12] in their Fig. 2, namely fibers with 0=150° and many different values of (p, corresponding to a vertical trajectory on the right-hand side of Fig. 3. Three different configurations with a fixed value of (p and different values of 6 are displayed in Fig. 3c in another paper by these authors [16]. Schiessel et al. [17] were able to obtain analytical expressions for all the geometrical quantities that characterize... 

A useful feature of the algebraic representations of geometric quantities is the ease with which one can work in dimensions higher than three. Although it is difficult to visualize the angle between two five-dimensional vectors, there is no particular problem involved in taking the dot product between two vectors of the form (xi,X2,xs,X4,X5). 

Since the measurements most commonly used as an index of long branching are those of intrinsic viscosities, whereas the most readily calculable theoretical measures of branching are the purely geometrical quantities or g0, and since the theoretical problem of relating these is not completely solved (Section 4), one of the applications of model branched polymers is to test proposed relationships. 

Equation (12.53) gives the desired evaluation of the general thermodynamic derivative V in a system of/degrees of freedom, expressed in terms of known geometrical quantities. As in the two-dimensional case, other expressions for V would be possible in other special choices of basis. Equation (12.53) is suitable for machine computation in multicomponent thermodynamic systems of arbitrary complexity. 

At first glance there is a remarkable resemblance between Eqs. (63 ) and (65) in spite of their quite different origin. Closer scrutiny, however, reveals two important differences. First, the coefficient of M in Eq. (65), i. e., (ba2lm%), is essentially a geometric quantity and consequently under some restriction. For instance, a and b may not both be as small as 0.1 A. At the same time, the ratio ajb must be quite large, say 50 or... 

The ratio dA/dN is a geometric quantity, determined by the curvature of the adsorbate/vapour interface. When dealing with a concave hemispherical meniscus of the capillary condensed liquid in a cylindrical tube, it can easily be calculated that... 

It was mentioned earlier that a number of special purpose routines, which do not appear in the VPLIB index, have been developed for use in structural chemistry. The most frequent requirements encountered in this area are those concerned with molecular geometry and, more specifically, with the calculation of interatomic distances, angles and torsion angles. These geometric quantities are best evaluated by vector algebra and this will always involve the calculation of vector components, lengths, direction cosines, vector cross products and vector dot products. Attention should therefore be directed at the best possible way of implementing the calculations described in the latter list on the MVP-9500. 

The other fundamental requirement for the calculation of the geometric quantities mentioned above is a routine to facilitate the derivation of interatomic distances. This may be achieved in the main by a small extension of the VSUBI routine. Consider the state of the MVP-9500 when all of the APU s present have executed the subtract operation but have not been unloaded. 

Figure 3.19. Geometrical quantities used to describe the shape of a liquid surface, (a) principal curvature radii (b) vertical position z of the point Q of the surface and joining angle

The geometrical quantities used to describe the profile of a meniscus formed on a cylinder of radius r0 are plotted on Figure E. 1. Using the cartesian coordinates, the Laplace equation becomes ... 

We may immediately evaluate the geometrical quantities that enter Eq. (5-10), to obtain... 

Figure 1.10 Geometrical quantities in defining the proximal radial distribution function gp j(r) of Eq. (1.14). The surface proximal to the outermost carbon (carbon /), with area fi, (r) r, permits definition of the mean oxygen density in the surface volume element, conditional on the chain configuration p gprox ( )-...

To account for the effects of orientation on radiation heat transfer between two surfaces, we define a new parameter called the vieu factor, which is a purely geometric quantity and is independent of the surface properties and temperature. It is also called the shape factor, configuration factor, and angle factor. The view factor based on the assumption that the surfaces are diffuse emitters and diffuse reflectors is called the diffitse view factor, and the view factor based on the assumption that the surfaces are diffuse emitters but specular reflectors is called the specular view factor. In lliis book, we consider radiation exchange between diffuse surfaces only, and ihu.s the term view factor simply means diffuse view factor. 

One way to assess the precision of a structure determination is to use the estimated standard deviation (e.s.d.) of the geometric quantity of interest. This is obtained from the least-squares refinement (see Chapter 10). The more precisely a measurement is made, the smaller is the e.s.d. of that measurement. The equations for calculating the e.s.d. values of a bond length and of a bond angle are given in Figure 11.8. 

This study is part of a continuing effort to investigate the characterization of porous materials by adsorption. Using realistic computer models and Monte Carlo simulations to obtain adsorption isotherms, we can critically evaluate standard characterization techniques by comparing isotherm-derived results with more precisely defined geometrical quantities. 

Velocity and temperature fields are therefore only similar when also the dimensionless groups or numbers concur. These numbers contain geometric quantities, the decisive temperature differences and velocities and also the properties of the heat transfer fluid. The number of dimensionless quantities is notably smaller than the total number of all the relevant physical quantities. The number of experiments is significantly reduced because only the functional relationship between the dimensionless numbers needs to be investigated. Primarily, the values of the dimensionless numbers are varied rather than the individual quantities which make up the dimensionless numbers. 

Whilst heat transfer in convection can be described by physical quantities such as viscosity, density, thermal conductivity, thermal expansion coefficients and by geometric quantities, in boiling processes additional important variables are those linked with the phase change. These include the enthalpy of vaporization, the boiling point, the density of the vapour and the interfacial tension. In addition to these, the microstructure and the material of the heating surface also play a role. Due to the multiplicity of variables, it is much more difficult to find equations for the calculation of heat transfer coefficients than in other heat transfer problems. An explicit theory is still a long way off because the physical phenomena are too complex and have not been sufficiently researched. 

The concept of view factors is quite convenient in the analysis of diffuse and gray radiation exchanges. Under these assumptions, the view factor, Fu2, is purely a geometric quantity. Physically, it means the fraction of radiative energy leaving surface 1 that reaches surface 2. In other words, it describes how much surface 1 sees surface 2, thus the name view factor. Due to the restricted nature of this chapter, the expression for the view factors will not be derived here. Instead, the expression will be given here and the reader will be referred to a more-detailed discussion in References 2, 18, and 19. Mathematically, the view factor is defined as... 

IR spectroscopy does not directly measure geometrical quantities. In Ch. 4 it is shown that measurements of the wavenumber of the centre of the most intense stretching band (X H - Y) of a H-bond X-H Y can be relatively precisely correlated to the X Y equilibrium distance of this H-bond. IR measmements are most easy, in any conditions gas, solid, liquid, solution, etc., and IR spectrometers are routine instruments in many laboratories. IR spectroscopy is consequently a basic general method to rapidly obtain geometrical parameters of H-bonds. Let us note that theoretical methods may also occasionally be used to estimate geometrical quantities of H-bonds made of small molecules. 

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