Nonspherical particles, equivalent diameters

Eor randomly packed spherical particles, the constants M and B have been deterrnined experimentally to be 150 and 1.75, respectively. Eor nonspherical particles, equivalent spherical diameters are employed and additional corrections for shape are introduced. 

The particle must be spherical, smooth, and rigid, I liis assumption is not always valid as discussed previously. For nonspherical particles, the diameter calculated is an equivalent Stokes diameter, the diaiii-eter of a sphere of the same material with the same sedimentation velocity. 

The terminal velocity in the case of fine particles is approached so quickly that in practical engineering calculations the settling is taken as a constant velocity motion and the acceleration period is neglected. Equation 7 can also be appHed to nonspherical particles if the particle size x is the equivalent Stokes diameter as deterrnined by sedimentation or elutriation methods of particle-size measurement. 

The particle size deterrnined by sedimentation techniques is an equivalent spherical diameter, also known as the equivalent settling diameter, defined as the diameter of a sphere of the same density as the irregularly shaped particle that exhibits an identical free-fall velocity. Thus it is an appropriate diameter upon which to base particle behavior in other fluid-flow situations. Variations in the particle size distribution can occur for nonspherical particles (43,44). The upper size limit for sedimentation methods is estabHshed by the value of the particle Reynolds number, given by equation 11 ... 

Note - In designing a system based on the settling velocity of nonspherical particles, the linear size in the Reynolds number definition is taken to be the equivalent diameter of a sphere, d, which is equal to a sphere diameter having the same volume as the particle. 

For nonspherical particles, values for the slip correction factor are available in slip flow (MU) and free-molecule flow (Dl). To cover the whole range of Kn and arbitrary body shapes, it is common practice to apply Eq. (10-58) for nonspherical particles. The familiar problem then arises of selecting a dimension to characterize the particle. Some workers [e.g. (H2, P14)] have used the diameter of the volume-equivalent sphere this procedure may give reasonable estimates for particles only slightly removed from spherical, or in near-con-tinuum flow, but gives the wrong limit at high Kn. An alternative approach... 

Fig. 4.2.9 Histograms of the size distributions of the particles shown in Fig. 4.2.8. Original and final size distributions are shown by broken and solid lines, respectively. The diameter of an equivalent sphere having the same volume as a nonspherical particle was obtained with a Coulter counter. (From Ref. 9.)...

Besides mass concentration, atmospheric particles are often characterized by their size distribution. Aerosols are typically sized in terms of the aerodynamic equivalent diameter (dae) of the particle, usually expressed in micrometer (pm) or nanometer (nm) (Mark, 1998). Atmospheric particles are usually nonspherical and with unknown density. Therefore, the r/ae of a particle is usually defined as the diameter of an equivalent unit density sphere (p = 1 gctrf3) having the same terminal velocity as the particle in question (Mark, 1998 Seinfeld and Pandis, 1998). 

Particles used in practice for gas-solid flows are usually nonspherical and polydispersed. For a nonspherical particle, several equivalent diameters, which are usually based on equivalences either in geometric parameters (e.g., volume) or in flow dynamic characteristics (e.g., terminal velocity), are defined. Thus, for a given nonspherical particle, more than one equivalent diameter can be defined, as exemplified by the particle shown in Fig. 1.2, in which three different equivalent diameters are defined for the given nonspherical particle. The selection of a desired definition is often based on the specific process application intended. 

Note, again, that 4 is for a spherical particle. Eor nonspherical particles, the sieve diameter dp must be converted into its equivalent spherical particle by the equation mentioned in a previous paragraph. 

For nonspherical particles, the equivalent diameter used in the Reynolds and Sherwood numbers is dp = jAp/T = 0.564 J p, where A is the external surface area of the pellet. 

For nonspherical particles, the analysis employs the diameter of a sphere of equivalent volume. A correction factor, which depends upon the shape of the body and its orientation in the fluid, must be applied. 

Particle Size As particles are extended three-dimensional objects, only a perfect spherical particle allows for a simple definition of the particle size x, as the diameter of the sphere. In practice, spherical particles are very rare. So usually equivalent diameters are used, representing the diameter of a sphere that behaves as the real (nonspherical) particle in a specific sizing experiment. Unfortunately, the measured size now depends on the method used for sizing. So one can only expect identical results for the particle size if either the particles are spherical or similar sizing methods are employed that measure the same equivalent diameter. 

For this case it was also possible to show experimentally that the size Xq, calculated as the diameter equivalent to the specific surface of the particles, describes the real conditions well. Therefore, after a small correction, " equation (6) can also be applied for nonspherical particles. Then the estimated elementary tensile strength becomes... 

The aerodynamic diameter dj, is the diameter of spheres of unit density po, which reach the same velocity as nonspherical particles of density p in the air stream Cd Re) is calculated for calibration particles of diameter dp, and Cd(i e, cp) is calculated for particles with diameter dv and sphericity 9. Sphericity is defined as the ratio of the surface area of a sphere with equivalent volume to the actual surface area of the particle determined, for example, by means of specific surface area measurements (24). The aerodynamic shape factor X is defined as the ratio of the drag force on a particle to the drag force on the particle volume-equivalent sphere at the same velocity. For the Stokesian flow regime and spherical particles (9 = 1, X drag... 

This expression defines the particle size distribution function nttidp, r, 0 where the particle diameter may be some equivalent size parameter for nonspherical particles. In theoretical applications, especially coagulation (Chapter 7), it is convenient to introduce a size distribution with particle volume as the size parameter... 

Up to this point we have considered spherical particles of a known diameter Dp and density pp. Atmospheric particles are sometimes nonspherical and we seldom have information about their density. Also a number of techniques used for atomospheric aerosol size measurement actually measure the particle s terminal velocity or its electrical mobility. In these cases we need to define an equivalent diameter for the nonspherical particles or even for the spherical particles of unknown density or charge. These equivalent diameters are defined as the diameter of a sphere, which, for a given instrument, would yield the same size measurement as the particle under consideration. A series of diameters have been defined and are used for such particles. 

The volume equivalent diameter Dve is the diameter of a sphere having the same volume as the given nonspherical particle. If the volume Vp of the nonspherical particle is know then ... 

To account for the shape effects during the flow of nonspherical particles, Fuchs (1964) defined the shape factor % as the ratio of the actual drag force on the particle FD to the drag force F g on a sphere with diameter equal to the volume equivalent diameter of the particle ... 

Stokes Diameter What is the relationship connecting the volume equivalent diameter and the Stokes diameter of a nonspherical particle with dynamic shape factor X for Re < 0.1 ... 

Calculate the Stokes diameter of the NaCl particle of the previous example. The two approaches (dynamic shape factor combined with the volume equivalent diameter and the Stokes diameter) are different ways to describe the drag force and terminal settling velocity of a nonspherical particle. The terminal velocity of a nonspherical particle with a volume equivalent diameter Dve is given by (9.104),... 

How then do we deal with nonspherical particles The usual approaches are to assume the particles arc spherical and then to proceed as we would with spheri-cal particles, or we can convert the measuretf quantity into that of an equivalent sphere. For example, if we obtain the mass m of a particle, we can convert this into the mass of a sphere because m = (4/3)irr p, where r is the particle radius and p is its density. This allows the particle size to be described by only its diameter (d = 2r). This diameter then represents the diameter of a sphere of the same mass as the particle of interest. Similar conversions can be made with other measured quantities, such as surface area or volume. 

The Andreasen pipette introduced in the 1920s is perhaps the most popular manual apparatus for sampling from a sedimenting suspension. Determination of the change in density of the sampled particle suspension with time enables the calculation of size distribution of the particles. As Stokes law applies only to spherical particles, the nonspherical particles give a mean diameter referred to as Stokes equivalent diameter. The size range measurable by this method is from 2 to 60 pm (8). The upper limit depends on the viscosity of liquid used while the lower limit is due to the failure of very small particles to settle as these particles are kept suspended by Brownian motion. 

For nonspherical particles, Yu et al. (1993) suggested to substitution of the particle diameter by the packing equivalent particle diameter calculated by... 

Most particle sizing methods are developed on the basis of imiform (monodisperse) spherical particles. Hence, nonspherical particles usually must be defined by their equivalent spherical diameter (BSD) - the diameter the particles should have, assuming they are spheres. However, the BSD is strongly dependent on the physical method underlying a given particle sizing technique. A small subset of the many different diameter values that can be used to define the size of a particle is shown in Table 8.1. 

Grashof number for mass transfer L is a characteristic dimension, i.e., the diameter of a spherical particle, or the equivalent diameter of a nonspherical particle, etc. v is the kinematic viscosity D is the binary diffusion coefficient U is the linear velocity of the gas stream flowing past the particle (measured outside the boundary layer surrounding the particle) g is the acceleration due to gravity is a characteristic concentration difference, and... 

Liquid aerosol particles are nearly always spherical. Solid aerosol particles usually have complex shapes, as shown in Figs. 1.1—1.5. In the development of the theory of aerosol properties, it is usually necessary to assume that the particles are spherical. Correction factors and the use of equivalent diameters enable these theories to be applied to nonspherical particles. An equivalent diameter is the diameter of the sphere that has the same value of a particular physical property as that of an irregular particle. For approximate analysis, shape can usually be ignored, as it seldom produces more than a twofold change in any property. Particles with extreme shapes, such as long, thin fibers, are treated as simplified nonspherical shapes in different orientations. The complex shape of some fiime and smoke particles can be characterized by their fractal dimension. (See Section 20.2.)... 

For irregular (nonspherical) solid particles, the usual method of particle characterization is to introduce an "equivalent dituneter —that is, the diameter of a spherical particle that would give the same behavior in the experimental system of interest. 

---