Particle diameter definition

In most applications, the systems and processes contain large amounts of particles with size distribution each size may also possess a distinguished shape. To describe properly these systems and processes for design and analysis, they need to be adequately characterized to reflect their physical and chemical potentials. In the following sections, different average particle diameter definitions are introduced along with statistical descriptions of particle size distribution. Depending on applications, one definition may be more suitable than others. Thus care must be exercised to select the proper characterization for each process. 

Several diameter definitions are used in particle image measurements (Fig. 7). Martin diameter, the chord length which divides the projected particle into two equal areas with respect to a fixed dkection (29) Feret diameter, the projected length with respect to a reference dkection (30) and the diameter of equivalent surface area, the diameter of a ckcle the area of which is equivalent to the projected area of the particle in question (3). 

Because of the close similarity in shape of the profiles shown in Fig. 16-27 (as well as likely variations in parameters e.g., concentration-dependent surface diffusion coefficient), a contrdling mechanism cannot be rehably determined from transition shape. If rehable correlations are not available and rate parameters cannot be measured in independent experiments, then particle diameters, velocities, and other factors should be varied ana the obsei ved impacl considered in relation to the definitions of the numbers of transfer units. 

Equation (1) points to a number of important particle properties. Clearly the particle diameter, by any definition, plays a role in the behavior of the particle. Two other particle properties, density and shape, are of significance. The shape becomes important if particles deviate significantly from sphericity. The majority of pharmaceutical aerosol particles exhibit a high level of rotational symmetry and consequently do not deviate substantially from spherical behavior. The notable exception is that of elongated particles, fibers, or needles, which exhibit shape factors, kp, substantially greater than 1. Density will frequently deviate from unity and must be considered in comparing aerodynamic and equivalent volume diameters. 

H is the plate height (cm) u is linear velocity (cm/s) dp is particle diameter, and >ni is the diffusion coefficient of analyte (cm /s). By combining the relationships between retention time, U, and retention factor, k tt = to(l + k), the definition of dead time, to, to = L u where L is the length of the column, and H = LIN where N is chromatographic efficiency with Equations 9.2 and 9.3, a relationship (Equation 9.4) for retention time, tt, in terms of diffusion coefficient, efficiency, particle size, and reduced variables (h and v) and retention factor results. Equation 9.4 illustrates that mobile phases with large diffusion coefficients are preferred if short retention times are desired. 

The growth rate G is the temporal change of the particle diameter and the particle flux N is the temporal change of the number of particles. By this definition, the differential rate of change of particles leads to the following equations... 

A particle is generally imagined to be spherical or nearly spherical. Either particle radius or particle diameter can be used to describe particle size. In theoretical discussions of particle properties, the radius is most commonly used, whereas in more practical applications the diameter is the descriptor of choice. Thus one should carefully ascertain which definition is being used when the term particle size is used. In this text particle diameter is used throughout. 

Figure 1.1 Illustration of three common definitions of particle diameter. In general, Martin s diameter is less than the equivalent area diameter, which in turn is less than Feret s diameter.

The review published by Ergun (E2) provides a definitive description of pressure drop in packed tubes when the ratio of particle diameter to tube diameter is sufficiently low. In addition, although the complicated relationship between the diameter ratio, the fraction void and the friction factor can not be accurately represented without some explicit dependence of the friction factor on the diameter ratio, Ergun showed that his correlation does work for a wide variety of experimental conditions. The friction factor is calculated from the expression... 

FIGURE 2.2 Multitude of particle diameters (a) definition of unrolled radius R (b) unrolled curve. Taken fiiton Figure 4.1 in Allen [1]. 

A very important form of such disturbances is caused by the presence of the wall of the tube containing the packed bed. Vortmeyer and Schuster (1983) have used a variational approach to evaluate the steady two-dimensional velocity profiles for isothermal incompressible flow in rectangular and circular packed beds. They used the continuity equation, Brinkman s equation (1947), and a semiempirical expression for the radial porosity profile in the packed bed to compute these profiles. They were able to show that significant preferential wall flow occurs when the ratio of the channel diameter to the particle diameter becomes sufficiently small. Although their study was done for an idealized situation it has laid the foundation for more detailed studies. Here CFD has definitely contributed to the improvements of theoretical prediction of reactor performance. 

Definitions Nu = h dp/kf, Pr = Cpfi/k)f, h =, wall coefficient, dp = particle diameter = 6VplAp, kf = fluid molecular conductivity, E = porosity. Re = dpG/fi, G = superficial mass velocity per unit cross section. 

Definitions of particle diameters derived by different methods have been described in detail [4]. The aerodynamic diameter is defined as the diameter of a unit-density sphere having the same settling velocity, generally in air, as the particle. This encompasses particle shape, density, and physical size, all of which influence the aerodynamic behavior of the particle. As a dynamic parameter, it can generally be linked with aerosol deposition and specifically with that in the lung [5]. 

Chung and Wen (1968) and Wen and Fan (1975) have proposed a dimensionless equation using the dependency of the dispersion coefficient on the (particle) Reynolds number Re (Eq. 6.169) for fixed and expanded beds. It is an empirical correlation based on published experimental data and correlations from other authors that covers a wide range of Re. Owing to two different definitions of the Reynolds number, the actual appearance varies in the literature. Since the particle diameter dp, is the characteristic value of the packing, Eq. 6.168 based on the (particle) Peclet number Pe (Eq. 6.170) is used here ... 

---