1. Characterization of single particles

Particle characterization is important in all aspects of particle production, manufacturing, handling, processing, and applications. Characterization of particles is the first necessary task required in a process involving solid particles. The required characterization includes not only the intrinsic static parameters (such as size, density, shape, and morphology) but also their dynamic behavior in relation to fluid flow (such as drag coefficient and terminal velocity). In this chapter, the characterization of single particle with different available techniques is introduced. The dynamic behavior of a single particle in the flow field at Stokes flow regime is then discussed.

The complete characterization of a single particle requires the measurement and definition of the particle characteristics such as size, density, shape, and surface morphology. Because the particles of interest are usually irregular in shape and different in surface morphology, there are many different ways and techniques to characterize the particles. 1.1 Definitions of particle size

The most basic characteristic of a single particle is the geometric characteristic, such as the particle size, shape, surface structure and surface morphology, etc. On the particle shape, it may be symmetric, such as the sphere 、 cylinder or cube; or it may be asymmetric. In order to describing the particle size and shape, the equivalent diameter, special surface area and shape factor should be defined first. Particles of shapes other than spherical Spherical particlesuniquely characterized by its diameter

颗粒的有效直径也称当量直径。常用的当量直径有三种定义方式： Equivalent diameter （颗粒的有效直径） Volume-equivalent diameter （等体积当量直径） d v The volume -equivalent diameter, d v, is defined as the diameter of a sphere having the same volume as the particle and can be expressed mathematically as 假定一球形颗粒具有与被考察颗粒相同的体积，则该球形颗粒的 直径即为被考察颗粒的等体积当量直径 d v :

where V p = the volume of the particle

Surface area-equivalent diameter （等表面积当量直径） d s 假定一球形颗粒具有与被考察颗粒相同的表面积，则该球形颗粒的直 径即为被考察颗粒的等表面积当量直径 ds:ds: Where S p = surface area of the particle The surface area-equivalent diameter, d s, is defined as the diameter of a sphere having the same surface area of the particle. Mathematically it can be shown to be

假定一球形颗粒具有与被考察颗粒相同的比表面积（定义见下）， 则该球形颗粒的直径即为被考察颗粒的等比表面积当量直径 d sv : Specific area-equivalent diameter ( 等比表面积当量直径 ) d sv The surface–volume diameter, d sv, also known as the Sauter diameter, is defined as the diameter of a sphere having the same external-surface-area-to-volume ratio as the particle. This can be expressed as

1.1.1 equivalent diameter The choice of any particular diameter for characterization of an irregular particle depends on the intended application. For example, the surface area-equivalent diameter d sv can be used in the study of adsorption; the volume equivalent diameter d v or specific area equivalent diameter d sv will be used in the study of the two-phase flow model The most relevant diameter for application in a fluidized bed is the surface– volume diameter, d sv.

Where, the “ = ” applies to spherical particles. As for non- spherical particles, the more different from the spherical, the bigger the difference between the three equivalent diameters. Relationship : 颗粒的有效直径 (equivalent diameter)( 续 ) Relationship among the three equivalent diameter

1.1.1 颗粒的有效直径 (equivalent diameter)( 续 ) There are other definitions of equivalent diameter in addition to these three, such as arithmetic mean diameter 、 geometric mean diameter 、 projected area diameter 、 perimeter diameter 、 drag diameter 、 sieve diameter, Stockes diameter, etc. They will be applied in different occasions

1.1.2 Definitions of Particle Shape Natural and man-made solid particles occur in almost any imaginable shape, and most particles of practical interest are irregular in shape. A variety of empirical factors have been proposed to describe nonspherical shapes of particles. These empirical descriptions of particle shape are usually provided by identifying two characteristic parameters from the following four: (1) volume of the particle, (2) surface area of the particle, (3) projected area of the particle, and (4) projected perimeter of the particle. The projected area and perimeter must also be determined normal to some specified axis.

All proposed shape factors to date are open to criticism, because a range of bodies with different shapes may have the same shape factor. This is really inevitable if complex shapes are to be described only by a single parameter. Thus in selecting a particular shape factor for application, care must be taken to assure its relevance.

Question: what is the range of the Sphericity? Between 0 ～ 1 (Wadell, 1933) proposed the “degree of true sphericity” be defined as It is a dimensionless parameter. Sphericity ( 颗粒的球形度 )

For a true sphere, the sphericity is thus equal to 1. For nonspherical particles, the sphericity is always less than 1. The drawback of the sphericity is that it is difficult to obtain the surface area of an irregular particle and thus it is difficult to determine Φ directly. Usually the more the aspect ratio departs from unity, the lower the sphericity.

Circularity ( 圆形度 ) Wadell (1933) also introduced the ‘‘degree of circularity’’, defined as Ψ

Unlike the sphericity, the circularity can be determined more easily experimentally from microscopic or photographic observation. For an axisymmetric particle projected parallel to its axis, ψ is equal to unity. Use of ψ is only justified on empirical grounds, but it has the potential advantage of allowing correlation of the dependence of flow behavior on particle orientation

Since the sphericity and circularity are so difficult to determine for irregular particles, Wadell (1933) proposed that Φ and Ψ be approximated by ‘‘operational sphericity and circularity:’’

For a true sphere, the sphericity is thus equal to 1. For nonspherical particles, the sphericity is always less than 1 Where d cs is the diameter of the smallest circumscribing sphere of the particle.

Ψ Operational circularity Where d cs = the diameter of the smallest circumscribing circle of the particle.

Definition of particle density There are several particle density definitions available. Depending on the application, 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, V p, as shown Apparent particle density ( 颗粒密度 )

For porous particles with small pores, the particle volume in above equation should be replaced with the envelope volume of the particle as if the particles were nonporous. This would be more hydrodynamically correct if the particle behavior in the flow field is of interest or if the bulk volume of the particles is to be estimated.

Skeletal density,ρ sk, ( 骨架密度 ) The skeleton density is defined as the mass of the particle divided by the skeletal volume of the particle.

The specific surface area is defined as the surface area per unit volume of particles, that is the ratio of the surface area to the volume of the particle Specific surface area Question : The smaller the particle, the bigger the specific surface area. When the volume of the particles is a constant, the bigger the specific surface area of the particle, the farther the shape coefficient deviates from sphere

1.1.4 porosity the volume which has been filled with other materials divided by the whole particle volume : Where,V i is the total volume of all the pores within the particle (including all the closed and opened pores).

equivalent diameter ( 颗粒的有效直径 ) ： Volume-equivalent diameter （等体积当量直径） Surface area-equivalent diameter （等表面积当量直径） Specific area-equivalent diameter ( 等比表面积当量直径 ) Summary: Characterization of a single particle shape factor( 颗粒的形状系数 ) ： Circularity ( 圆形度 ) Sphericity ( 球形度 ) specific surface area( 颗粒的比表面积 ) porosity of particle( 颗粒的孔隙率 )

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Fluid dynamics of a single particle The required characterization includes not only the intrinsic static parameters (such as size, density, shape, and morphology) but also their dynamic behavior in relation to fluid flow (such as drag coefficient and terminal velocity).

重力 F g 阻力 F D 浮力 F b Gravitational force Buoyancy force Drag force Analysis of forces acting on a smooth sphere

Particle terminal velocity of smooth sphere

u t : mean velocity of fluid relative to that of particle

Flow regimes Stoke’s law Intermediate Newton’s law

1.2.4 The flow behavior of non-spherical particles For non-spherical particles, different researchers have different terminal velocity formula. Here only a brief introduction Since the surface area of non-spherical particles is greater than that of the sphere with the same volume. In the laminar flow region, fluid resistance of non-spherical particles is greater than that of the sphere with the same volume.Therefore, terminal settling velocity of non-spherical particles is less than that of the sphere with the same volume

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