Equivalent diameter definitions

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). 

Re using the equivalent diameters defined in the following. This situation is, by arbitrary definition, opposite to that for the hydraulic diameter used for turbulent flow. 

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. 

The connection that has been shown in Section VIII to exist between burn-out in a rod bundle and in an annulus leads to the question of whether or not a link may also exist between, for example, a round tube and an annulus. Now, a round tube has its cross section defined uniquely by one dimension—its diameter therefore if a link exists between a round tube and an annulus section, it must be by way of some suitably defined equivalent diameter. Two possibilities that immediately appear are the hydraulic diameter, dw = d0 — dt, and the heated equivalent diameter, dh = (da2 — rf,2)/ however, there are other possible definitions. To resolve the issue, Barnett (B4) devised a simple test, which is illustrated by Figs. 38 and 39. These show a plot of reliable burn-out data for annulus test sections using water at 1000 psia. Superimposed are the corresponding burn-out lines for round tubes of different diameters based on the correlation given in Section VIII. It is clearly evident that the hydraulic and the heated equivalent diameters are unsuitable, as the discrepancies are far larger than can be explained by any inaccuracies in the data or in the correlation used. 

Modified from [2], within which particles with equivalent diameters usually between 0.01 pm and 100 pm are specified. This extends beyond the size range specified for a colloidal system. To avoid confusion the definition proposed here is reeommended. 

Particle of gel of any shape with an equivalent diameter of approximately 0.1 to 100 pm. Modified from [2]. The definition proposed here is recommended for its precision and because it distinguishes between a microgel and a nanogel. 

Other definitions are often used. For example, Stringham et cil. (S8) based Re on the area-equivalent diameter while Christiansen and Barker (C3) used cylinder length. 

As shown in Table III, several authors (Fidaleo and Moresi, 2005a Kraaijeveld et al., 1995 Kuroda et al., 1983 Sonin and Isaacson, 1974) established power function relationships between the Sherwood number (Sh) and the Reynolds (Re) and Schmidt (Sc) numbers in ED cells equipped with different eddy promoters, even if different definitions of the equivalent diameter were used to calculate the Reynolds number. 

SQ iL/KAP)yi. Equivalent diameters are not the same as hydraulic diameters. Equivalent diameters yield the correct relation between flow rate and pressure drop when substituted into Eq. (6-36), but not Eq. (6-35) because V Q/(kDe/4). Equivalent diameter De is not to be used in the friction factor and Reynolds number f 16/Re using the equivalent diameters defined in the following. This situation is, by arbitrary definition, opposite to that for the hydraulic diameter DH used for turbulent flow. 

In this chapter, the basic definitions of the equivalent diameter for an individual particle of irregular shape and its corresponding particle sizing techniques are presented. Typical density functions characterizing the particle size distribution for polydispersed particle systems are introduced. Several formulae expressing the particle size averaging methods are given. Basic characteristics of various material properties are illustrated. 

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. 

The book contains two parts each part comprises six chapters. Part I deals with basic relationships and phenomena of gas-solid flows while Part II is concerned with the characteristics of selected gas-solid flow systems. Specifically, the geometric features (size and size distributions) and material properties of particles are presented in Chapter 1. Basic particle sizing techniques associated with various definitions of equivalent diameters of particles are also included in the chapter. In Chapter 2, the collisional mechanics of solids, based primarily on elastic deformation theories, is introduced. The contact time, area, and... 

With this definition, for spheres, the use of Equation 8.39 gives just the diameter of sphere. Expressions of equivalent diameters for different particle shapes as used in packed bed reactors are presented in Table 8.1. ... 

The hydrauhc diameter method does not work well for laminar flow because the shape affects the flow resistance in a way that cannot be expressed as a function only of the ratio of cross-sectional area to wetted perimeter. For some shapes, the Navier-Stokes equations have been integrated to yield relations between flow rate and pressure drop. These relations may be expressed in terms of equivalent diameters De defined to make the relations reduce to the second form of the Hagen-Poiseulle equation, Eq. (6-36) that is, De = 2 Q. L/ nAPY . Equivalent mameters are not the same as hydraulic diameters. Equivalent diameters yield the correct relation between flow rate and pressure drop when substituted into Eq. (6-36), but not Eq. (6-35) because V Q/IkDeH). Equivalent diameter De is not to be used in the friction factor and Reynolds number / 16/Re using the equivalent diameters defined in the following. This situation is, by arbitrary definition, opposite to that for the hydrauhc diameter Dh used for turbulent flow. 

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. 

Mathematically, the one-dimensional PSD function can be expressed as m(L), where n is the population density function and L is the characteristic crystal length. For cube-Uke (or spherical) crystals, the characteristic length is approximately the diameter of the crystal. For crystals of other shape there are various definitions for the characteristic length. Most commonly, the characteristic length of the particle with an irregular shape is defined as the equivalent diameter of a sphere which has the same behaviors under the measurement conditions, for example sieving, laser scattering, and sedimentation (Mullin 2(X)1, Chapter 2). 

SURFACE-SHAPE COEFFICIENT is the coefficient of proportionality relating the surface area of the particle with the square of its measured diameter, the latter being one of the many possible definitions of particle equivalent diameter (this has to be defined when quoting values). This description of particle shape is useful in applications when particle surface is important. 

For circular pipes, Rh = R- The reader is cautioned that some definitions of Rh omit the factor of 2 shown in Equation 3.22 so that the result must be multiplied by 2 for use in equations such as 3.18 and 3.19. The use of Rh is not recommended for laminar flow, but alternatives are available in the literature. Also, the method of false transients applied to PDEs in Chapter 16 can be used to calculate laminar velocity profiles in ducts with noncircular cross sections. For turbulent, low-pressure gas flows in rectangular ducts, the American Society of Heating, Refrigerating and Air Conditioning Engineers recommends use of an equivalent diameter defined as... 

FRICTION FACTOR IN FLOW THROUGH CHANNELS OF NONQRCULAR CROSS SECTION. The friction in long straight channels of constant noncircular cross section can be estimated by using the equations for circular pipes if the diameter in the Reynolds number and in the definition of the friction factor is taken as an equivalent diameter, defined as four times the hydraulic radius. The hydraulic radius is denoted by r and in turn is defined as the ratio of the cross-sectional area of the channel to the wetted perimeter of the channel ... 

Substituting the above definition of the specific area into Eq. (4.7.10), we obtain the following expression for the equivalent diameter ... 

It is of interest to note that, by judicious definition of the characteristic diameter of nonspherical bodies, good agreement with the equations for spherical solids was obtained. A diameter defined by the total surface area of the body, divided by the perimeter normal to flow, was successfully used for spheres, hemispheres, cubes, prisms, and cylinders (PI), yielding a = 0 b - 0.692 m = 0.514 and n = [Eq. (4)]. Similar results were obtained for spheroids (S14), namely a = 0 6 = 0.74 w = 0.5 and n =. The commonly used equivalent diameter of a sphere of the same volume as the body yields transfer coefficients increasing with eccentricity (SI4). 

Before discussing the flow behavior of polymeric nanocomposites (PNCs), the nature of these materials should be outlined. As the name indicates, PNCs must contain at least two components, a polymeric matrix with dispersed nanoparticles [Utracki, 2004]. PNCs with thermoplastics, thermosets, and elastomers have been produced. The nanoparticles, by lUPAC s definition, must have at least one dimension that is not larger than 2 nm. They can be of any shape, but the most common for structural PNCs are sheets about 1 nm thick with the aspect ratio p=D/t= 20 to 6000, where D is the inscribed (or equivalent) diameter and t is the thickness of the sheet. These inorganic lamellar solids might be either natural or synthetic [Utracki et al., 2007]. 

Other techniques of deducing shape (or size) include using sieve analysis or sedimentation. The equivalent diameter for a sieve analysis is the mesh size of the upper sieve through which particles pass. (A more precise definition of the sieve diameter is the mean between the mesh size through which the particles pass and on which the particles are retained.) The characteristic diameter of a sedimentation technique would be the diameter of a sphere that has the same settling velocity. 

Despite these difficulties of definition and measurement it is most convenient for classification purposes, if a single-length parameter can be ascribed to an irregular solid particle. The most frequent expression used in connection with particle size is the equivalent diameter , i.e. the diameter of a sphere that behaves exactly like the given particle when submitted to the same experimental procedure. Several of these equivalent diameters are defined below. 

The term particle (or grain) size refers to the structural make-up of such substances as granulates, powders, dusts, granular mixes, and suspensions. Knowledge of the particle size, in conjunction with the comminution process, determines such details as grinding efficiency and ultimate product fineness. To establish particle sizes and their distribution within powdered systems, the user can have recourse to a number of different measuring processes designed to indicate, with appropriate particle definition, details of the probable equivalent diameter of a particle. 

Only four of the above particle size definitions are of general interest for applications in packed beds and fluidized beds. They are the sieve diameter, d, the volume diameter, d, the surface diameter, (f, and the surface-volume diameter, d. The most relevant diameter for application in a fluidized bed is the surface-volume diameter, 4v For applications in catalytic reactors with diflferent isometric catalyst shapes. Rase (1990) suggested that we use the equivalent diameters summarized in Table 1. 

---