Particle densities, definition

There are several particle density definitions available. Depending on the appKcation, one definition may be more suitable than the others. For nonporous particles, the definition of particle density is straightforward, i.e., the mass of the particle, M, divided by the volume of the particle, Vp, as shown in Eq. (15). 

The Laplace-Young equation refers to a spherical phase boundary known as the surface of tension which is located a distance from the center of the drop. Here the surface tension is a minimum and additional, curvature dependent, terms vanish (j ). The molecular origin of the difficulties, discussed in the introduction, associated with R can be seen in the definition of the local pressure. The pressure tensor of a spherically symmetric inhomogeneous fluid may be computed through an integration of the one and two particle density distributions. 

For degenerate states a problem arises with the definition of cumulants. We consider here only spin degeneracy. Spatial degeneracy can be discussed on similar lines. For S 0 there are (2S + 1) different Afs-values for one S. The n-particle density matrix p Ms) = of a single one of these states does not... 

In this section, definitions of terms used in relation to particle density are presented. As will be shown, these issues exhibit many alternative approaches. For instance, density is... 

Using the definition (1.10) of the single-particle density matrix and Eq. (1.11), it is easy to obtain an equation for the single-particle density matrix ... 

For porous solids such as coal, there are five different density measurements true density, apparent density, particle density, bulk density, and in-place density. The true density of coal is the mass divided by the volume occupied by the actual, pore-free solid in coal. However, determining mass of coal may be deemed as being rather straightforward, but determining volume presents some difficulties. Volume, as the word pertains to a solid, cannot be expressed universally in a simple definition. Indeed, the method used to determine volume experimentally, and subsequently, the density, must be one that applies measurement rules consistent with the adopted definition. 

In order to achieve simultaneous suspension of solid particles and dispersion of gas, it is necessary to define the state when the gas phase is well dispersed. Nienow (1975) defined this to be coincident with the minimum in Power number, Ne, against the aeration number, 1VA, relationship (see Fig. 12 [Sicardi et al., 1981]). While Chapman et al. (1981) accept this definition, their study also showed that there is some critical particle density (relative to the liquid density) above which particle suspension governs the power necessary to achieve a well-mixed system and below which gas dispersion governs the power requirements. Thus, aeration at the critical stirrer speed for complete suspension of solid particles in nonaerated systems causes partial sedimentation of relatively heavy particles and aids suspension of relatively light particles. Furthermore, there may be a similar (but weaker) effect with particle size. Wiedmann et al. (1980), on the other hand, define the complete state of suspension to be the one where the maximum in the Ne-Ren diagram occurs for a constant gas Reynolds number. 

Particle density may be defined as the total mass of the particle divided by its total volume however, depending upon the different definitions of the total volume (or the different ways to measure the particle volume), there are various definitions of particle density in existence (see Table 4). 

In order to get clear understanding of the subtle differences between the definitions of various particle density types, an illustration can be formed as shown in Figure 8. 

Similar to particle density, there are various definitions relating to particle surface area [16] ... 

It is worthwhile mentioning at this point that all properties of a subsystem defined in real space, including its energy, necessarily require the definition of corresponding three-dimensional density distribution functions. Thus, all the properties of an atom in a molecule are determined by averages over effective single-particle densities or dressed operators and the one-electron picture is an appropriate on ] [y)... 

The apparent particle density, pp, is the ratio of the particle weight to the particle volume, or sum of the imaccessible volumes and the internal pore volume. This is the most practical definition. However, it carmot be as accurate as densities usually are because of the difficulty associated with defining accurately the external column porosity (see previous subsection),... 

This definition simplifes considerably in the case of an isotropic liquid for which the single particle density is a constant (i.e. p(r) = N/V), resulting in the radial distribution function g(r). As said above, these definitions will form a part of our later thinking. 

If the particles are composed of multiple chemical species, then usually the fluid phase will be also. In such cases, it is necessary to introduce a vector of internal coordinates whose components are the mass of each chemical species seen by a particle. Obviously, the sum of these internal coordinates is equal to the fluid mass seen by a particle. By definition, if is the mass of component a, then integration over phase space leads to a component fluid-phase mass density ... 

Anisotropic Particle Scattering Varying Intensity Decay in Different Directions. In case of anisotropy the decay of the scattering intensity I (s) is a function of the direction chosen. The intensity extending from 5 = 0 outward in a deliberately chosen direction i is mathematically the definition of a slice (cf. Sect. 2.7.1, p. 22). Thus, the Fourier-Slice theorem, Eq. (2.38), turns the particle density function Ap (r) into a projection Ap (r) j (r,) and the scattering intensity is related to structure by... 

The above equation gives the porosity (= voidage) of the powder and whether or not this includes the pores within the particles depends on the definition of particle density used in this evaluation - see section 2.4. 

In conclusion to this section on porosity of bulk powders, it seems that the indirect method of calculating its value from other powder properties, namely the bulk and particle densities, is adequate for most practical applications. The fact whether or not the porosity should include the internal pores of the particles is taken care of by selection of a suitable method for particle density measurement (section 2.4). Care must be taken to avoid a possible confusion between the definitions of porosity and void ratio. 

Notice that because the one-particle densities are positive definite, the function f(r,60,4 ) is monotonously increasing. The uniqueness of the solutionis guaranteed by chosing the condition f(r = 0,0 , 0) = 0 for all paths. In integrated form, this transformation function is... 

Just as for Green s functions, expressions like two-particle operator and two-particle density matrix refer to the number of particles involved in the definition of the operator (two in the case of an interaction, one for a potential energy, etc.), not to the total number of particles present in the system. 

The local charge density, p(x) = j x)/c, is equivalent to a local particle density i//(x) vr(x), clearly positive definite, and j(x) = [Y x),p x),j x)) is the associated current density. Integrating (41) over the 3-space coordinates, and assuming that there is no net flow of charge at infinity, gives... 

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