Selection of Relevant Characteristic Particle Size

The selection of a relevant characteristic particle size to start any sort of analysis or measurement often poses a problem. In practice, the particles forming a powder will rarely have a spherical shape. Many industrial powders are of mineral (metallic or non-metallic) origin and have been derived from hard materials by any sort of size reduction process. In such a case, the comminuted (or size- reduced) particles resemble polyhedrons with nearly plane faces, in a number of 4 to 7, and sharp edges and corners. The particles may be compact, with length, breadth, and thickness nearly equal, but sometimes they may be plate-like or needle-like. As particles get smaller, and by influence of attrition due to handling, their edges may become smoother; thus, they can be considered to be spherical. The term “diameter” is therefore often used to refer to the characteristic linear dimension. All these geometrical features of an important number of industrial powders, such as cement, clay, and chalk, are related to the intimate structure of their forming elements, whose arrangements are normally symmetrical, with definite shapes like cubes, octahedrons, etc. On the other hand, particulate food materials are mostly organic in origin, and their individual grain shapes can have a great diversity of structures, since their chemical compositions are more complex than those of inorganic industrial powders. Shape variations in food powders are enormous, ranging from extreme degrees of irregularity (ground materials like spices and sugar), to an approximate sphericity (starch and dry yeast) or well-defined crystalline shapes (granulated sugar and salt).

Considering the aspects mentioned above, expressing a single particle size is not simple when its shape is irregular. This case is frequent in many applications, mostly when dealing with food powders of truly organic origin. Irregular particles can be described by a number of sizes. There are three groups of definitions, as listed in Tables 2.3–2.5: equivalent sphere diameters, equivalent circle diameters, and statistical diameters. In the first group, the diameters of a sphere that have the same property of the particle itself are found (e.g., same volume, same settling velocity, etc.). In the second group, the diameters of a circle that have the same property of the projected outline of the particle are considered (e.g., projected area or perimeter).

Table 2.3. A list of definitions of “equivalent sphere diameters” (adapted from Allen, 1981). Symbol Name Equivalent property of a sphere

xv Volume diameter Volume xs Surface diameter Surface

xsv Surface volume diameter Surface to volume ratio

xd Drag diameter Resistance to motion in the same fluid at the same velocity xf Free-falling diameter Free-falling speed in the same liquid, same particle density xst Stokes’ diameter Free-falling speed if Stokes’ law is used (Rep< 0.2) xA Sieve diameter Passing through the same square aperture

Table 2.4. A list of definitions of “equivalent circle diameters”. Symbol Name Equivalent property of a circle

xa Projected area diameter Projected area if particle is resting in a stable position xp Projected area diameter Projected area if particle is randomly orientated xc Perimeter diameter Perimeter of the outline

Table 2.5. A list of definitions of “statistical diameters”. Symbol Name Dimension measured

xF Feret’s diameter Distance between two tangents on opposite sides of particle xM Martin’s diameter Length of the line which bisects the image of particle xSH Shear diameter Particle width obtained with an image shearing eyepiece xCH Maximum cord diameter Maximum length of a line limited by the contour of the particle

The third group of sizes are obtained when a linear dimension is measured (usually by mi- croscopy) parallel to a fixed direction. The most relative measurements of the diameters mentioned above would probably be the statistical diameters because they are practically determined by direct microscopy observation. Thus, for any given particle, Martin’s and Feret’s diameters could be radi- cally different and, also, both different from a circle of equal perimeter or equal area (see Fig. 2.1). In practice, most of the equivalent diameters will be measured indirectly to a given number of particles taken from a representative sample and, therefore, it would be more practical to use a quick, less accurate measure on a large number of particles than a very accurate measure on few particles. Also, it would be difficult to perceive the above-mentioned equivalence of the actual particles with an ideal sphericity. Furthermore, such equivalence would depend on the method employed to determine the size. For example, Fig. 2.2 shows an approximate equivalence of an irregular particle depending on different equivalent properties of spheres.

Taking into account the concepts presented above, it is obvious that the measurement of particle size is dependent upon the conventions involved in particle size definition and also upon the physical principles employed in the determination process (Herdan, 1960). When different physical principles are used in particle size determination, it cannot be assumed that they should give identical results. For this reason, it is recommended that a characteristic particle size be selected to be measured according to the property or the process under study. For example, in pneumatic conveying or gas cleaning, it is more relevant to choose to determine the Stokes’ diameter, as it represents the diameter of a sphere of the same density as the particle itself, which would fall in the gas at the same velocity as the real particle. In particles flowing through packed or fluidized beds, on the other hand, the surface–volume

22 Food Powders Feret’s diameter Martin’s diameter Maximum linear diameter Minimum linear Circle of equal perimeter Circle of equal area Fixed direction

Figure 2.1. Methods used to measure diameter of non-spherical particles.

diameter is more useful, i.e., the diameter of a sphere having the same surface-to-volume ratio as the particle is more relevant to the aerodynamic process.