Volume diameter

In many important cases of reactions involving gas, hquid, and solid phases, the solid phase is a porous catalyst. It may be in a fixed bed or it may be suspended in the fluid mixture. In general, the reaction occurs either in the liquid phase or at the liquid/solid interface. In fixed-bed reactors the particles have diameters of about 3 mm (0.12 in) and occupy about 50 percent of the vessel volume. Diameters of suspended particles are hmited to O.I to 0.2 mm (0.004 to 0.008 in) minimum by requirements of filterability and occupy I to 10 percent of the volume in stirred vessels. 

The size of a spherical particle is readily expressed in terms of its diameter. With asymmetrical particles, an equivalent spherical diameter is used to relate the size of the particle to the diameter of a perfect sphere having the same surface area (surface diameter, ds), the same volume (volume diameter, dv), or the same observed area in its most stable plane (projected diameter, dp) [46], The size may also be expressed using the Stokes diameter, dst, which describes an equivalent sphere undergoing sedimentation at the same rate as the sample particle. Obviously, the type of diameter reflects the method and equipment employed in determining the particle size. Since any collection of particles is usually polydisperse (as opposed to a monodisperse sample in which particles are fairly uniform in size), it is necessary to know not only the mean size of the particles, but also the particle size distribution. 

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. 

At short distances, approximately equal to the excluded volume diameter, effective pair forces obtained from force matching exhibit unphysically large fluctuations. This is largely due to inadequate sampling of configurations at short distances in... 

As we just suggested, particle size and shape are important physical properties influencing powder flow and compaction. Particle size is a simple concept and yet a difficult one to quantitate. Feret s diameter, Martin s diameter, projected area diameter, specific surface diameter, Stokes diameter, and volume diameter are but several of the measurements that have been used to quantify particle size using a variety of methods. 

Another mean size based on volume is the mean volume diameter d v. If all the particles are of diameter d v, then the total volume of particles is the same as in the mixture. 

The apphcation of the laser diffraction technique is sometimes questioned becanse it measures geometric instead of aerod5mamic particle diameters. However, the aerod5mamic diameter can be calcnlated when the dynamic shape factor and density are known. Moreover, the dynamic shape factor (x) of micronized particles will often be only shghtly higher than 1 and so is the tme particle density (1.0 < pp < 1.4 g cm ). As a conseqnence, the aerodynamic diameter differs only slightly from the eqnivalent volume diameter (see Eq. 3.1). 

The definition of the surface-volume mean diamefer given by equation 1.11 must be modified for use wifh dafa from a sieve analysis. By assuming that the shape and density of fhe particles are constant for all size fractions, a number distribution can be transformed fo a mass distribution (Smith, 2003) and therefore the surface-volume diameter becomes... 

TABLE 9.3 Values of the Constant b in the Hatch - Choate Equations for Converting the Count Geometric Mean Diameter to Mass, Surface, or Volume Diameters... 

Not only do CCN affect the number of cloud droplets formed, but they also affect the size distribution of these droplets. This also affects cloud albedo and its sensitivity to changes in the number concentration (see Eqs. (JJ) and (KK)). Figure 14.47, for example, shows the size distribution for cloud droplets measured in urban and nonurban air around Denver, Colorado (Al-kezweeny et al., 1993). The median volume diameter was 14 jj,m for the urban air cloud, and this was only 50% of that of the much larger droplets in the... 

Fig. 27. Progressive dispersion of calcium carbonate in polypropylene within a co-rotating intermeshing twin-screw extruder. Filler dispersion is expressed in terms of a mean volume diameter determined by image analysis...

Since the desired shape of a pellet is a sphere, shape factors have been used to describe the pellets. These are characterized variously as sphericity, roundness, shape coefficient, elongation index, and aspect ratio (63-67). Using the volume diameter, d, and projected diameter, d, a good measure... 

Abbreviations, d y, surface volume diameter dy, volume diameter dp, projected diameter = 0.99y/L.W. S, surface area of pellet V, volume of pellet R, ratio of max. diameter dmax to min. diameter dmin- M, ratio of width (W) to depth (X) of particle K-S,y, specific surface area , obtained by microscopy. 

Seam-to-seam vessel length should be determined from the geometry once liquid volume diameter and height are known. 

Figure 1.2. Schematic illustration of multidimensions of a particle and its equivalent volume diameter, surface diameter, and sieve diameter.

Surface Diameter, Volume Diameter, and Sauter s Diameter The surface diameter, ds, volume diameter, d, and Sauter s diameter, d, are defined such that each of them reflects a three-dimensional geometric characteristic of an individual particle. A surface diameter is given as the diameter of a sphere having the same surface area as the particle, which is expressed by... 

The volume diameter of a particle may be useful in applications where equivalent volume is of primary interest, such as in the estimation of solids holdup in a fluidized bed or in the calculation of buoyancy forces of the particles. The volume of a particle can be determined by using the weighing method. Sauter s diameter is widely used in the field of reacting gas-solid flows such as in studies of pulverized coal combustion, where the specific surface area is of most interest. 

A fly ash sample from a fluidized bed coal combustor is analyzed to obtain particle size data. Table PI. 1 shows the distributions of the projected area equivalent diameter of the particle dA obtained by the image analysis and the volume diameter of the particle d obtained by the electrozone technique [Ghadiri et al., 1991]. 

The volume-equivalent diameter, or volume diameter (see Eq. (1.3)), of an initial bubble formed at a single upward orifice, dbi, may be related to the volumetric gas flow rate through the orifice Q0[ by [Davidson and Harrison, 1963 Miwa et al., 1972]... 

---