Solids Bounded by Curved Surfaces

Vibrational Frequency Spectrum of a Continuous Solid.—To find the specific heat, on the quantum theory, we must superpose Einstein specific heat curves for each natural frequency v1y as in Eq. (1.3). Before we can do this, we must find just what frequencies of vibration are allowed. Let us assume that our solid is of rectangular shape, bounded by the surfaces x = 0, x = X, y — 0, y = F, z = 0, z = Z. The frequencies will depend on the shape and size of the solid, but this does not really affect the specific heat, for it is only the low frequencies that art very sensitive to the geometry of the solid. As a first step in investigating the vibrations, let us consider those particular waves that arc propagated along the x axis. 

Figure 3 shows a 3D view of the topological atoms in hydrogen cyanide. The solid curves appearing in Fig. 1, which run broadly vertically, are so-called interatomic surfaces. They reoccur in Fig. 2 as surfaces in 3D. These surfaces are the sharp boundaries between atoms inside a molecule. Note that the boundary between molecules within a molecular assembly is also sharp, because a molecule is simply the union of its topological atoms. A molecule in condensed matter is thus fully bounded by interatomic surfaces. It is clear that there are no gaps between the atoms and also that they do not overlap. 

In summary, we refer to Figure 5.5, which may be considered as the projection of the entire equilibrium surface on the entropy-volume plane. All of the equilibrium states of the system when it exists in the single-phase fluid state lie in the area above the curves alevd. All of the equilibrium states of the system when it exists in the single-phase solid state lie in the area bounded by the lines bs and sc. These areas are the projections of the primary surfaces. The two-phase systems are represented by the shaded areas alsb, lev, and csvd. These areas are the projections of the derived surfaces for these states. Finally, the triangular area slv represents the projection of the tangent plane at the triple point, and represents all possible states of the system at the triple point. This area also is a projection of a derived surface. 

Fig. E6.3a The synthesis of a hollow cylinder pump, (a) The building block (b) a rotating hollow cylinder forms the moving plane (c) the stationary plane is formed by the outer surface of a solid stationary shaft. A channel block separates inlet and outlet. Feeding and discharge are carried out through slits in the shaft leading to axial holes drilled in the shaft (d) the two surfaces that form a shallow curved channel are bounded by a sidewall or flight running along the circumference of the shaft.

The solid curves in Figures 5 and 6 are the theoretical curves obtained for indicated values of thermal diffusivity of the solid. The following treatment has been applied. The temperature gradient within the zone bounded by the deflagrating surface on the one hand and ambient temperature on the other can be written ... 

In the drying of materials it is necessary to remove free moisture fi om the surface as well as bound moisture fi-om the interior. The drying characteristics of wet solids can be described by plotting the rate of drying against the corresponding moisture content. A typical drying curve is shown in Fig. 1 and it can easily be seen that this is subdivided into four distinct sections ... 

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