Arithmetical diameter

In his experiments, Martin used the average arithmetic diameter dav as a measure of surface. Using this diameter he obtained values of ag for crushed quartz ranging from 2.0-2.5. These values of a3 are considerably less than the value of t = 3.1416 taken for spheres. 

This corresponds to the ratio of arithmetic mean volume and arithmetic mean droplet surface and comprises the droplet size distribution of the investigated emulsions. For a theoretical monomodal size distribution [36] the Sauter diameter is approx, two times higher than the arithmetic diameter, because the influence of larger but less numerous droplets is more pronounced. 

The great disadvantage of the second definition is that the hypothetical spheres with the arithmetical diameter are quite snudler than the r packing... 

The arithmetical diameter of the packmg Da is the diameter of a spherical packing with the same value of e and a as the real packing in the column. It is easy to see that it is calculated by the equation ... 

The simplest calculation of the mean, referred to as arithmetic mean (count mean diameter) for data grouped in intervals, consists of the summation of all diameters forming a population, divided by the total number of particles. It can be expressed mathematically by equation 1 ... 

The mean volume (mass diameter) is the arithmetic mean diameter of all the particle volumes or masses forming the entire population and, for spherical particles, can be expressed as in equation 2 ... 

The diametei of average mass and surface area are quantities that involve the size raised to a power, sometimes referred to as the moment, which is descriptive of the fact that the surface area is proportional to the square of the diameter, and the mass or volume of a particle is proportional to the cube of its diameter. These averages represent means as calculated from the different powers of the diameter and mathematically converted back to units of diameter by taking the root of the moment. It is not unusual for a polydispersed particle population to exhibit a diameter of average mass as being one or two orders of magnitude larger than the arithmetic mean of the diameters. In any size distribution, the relation ia equation 4 always holds. 

Thep and q denote the integral exponents of D in the respective summations, and thereby expHcitiy define the diameter that is being used. and are the number and representative diameter of sampled drops in each size class i For example, the arithmetic mean diameter, is a simple average based on the diameters of all the individual droplets in the spray sample. The volume mean diameter, D q, is the diameter of a droplet whose volume, if multiphed by the total number of droplets, equals the total volume of the sample. The Sauter mean diameter, is the diameter of a droplet whose ratio of volume-to-surface area is equal to that of the entire sample. This diameter is frequendy used because it permits quick estimation of the total Hquid surface area available for a particular industrial process or combustion system. Typical values of pressure swid atomizers range from 50 to 100 p.m. 

Coefficient of Variation One of the problems confronting any user or designer of crystallization equipment is the expected particle-size distribution of the solids leaving the system and how this distribution may be adequately described. Most crystalline-product distributions plotted on arithmetic-probability paper will exhibit a straight line for a considerable portion of the plotted distribution. In this type of plot the particle diameter should be plotted as the ordinate and the cumulative percent on the log-probability scale as the abscissa. 

The volume flow rate is calculated as the arithmetical mean of the measured velocities multiplied by the duct cross-sectional area. The number of diameters along which the traversing occurs is not defined. If a near-symmetrical velocity profile is expected, an even travetse along one diameter may be sufficient. In case of a more disturbed profile, traversing along two or more diameters is recommended. 

The solution of the previous equations require careful attention to the sequence of the arithmetic. Perhaps one difficult requirement is the need to establish the L or G in Ib/hr/ft of tower cross-section, requiring an assumption of tower diameter. The equations are quite sensitive to the values of A and B. 

Figure 8. Relative size distribution for the MPIF Na /Pd catalyst particles before (left, 35 particles) and after catalytic tests (right, 75 particles, magnetic stirring). Arithmetic average diameters 319 and 276mm were determined for particles before and after catalytic tests, respectively. Image Pro Plus program. (Reprinted from Ref [29], 2003, with permission from Elsevier.)...

It may be mentioned here that the mode which represents the most commonly occurring size in a given distribution is not of much use in mineral processing since it does not describe fully the characteristics of a group of particles. The arithmetic mean diameter suffers from the same limitation except when the distribution is a normal one. The harmonic mean diameter is related to the specific surface area. It is, therefore, useful in such mineral processing operations where surface area is an important parameter. 

The second method is more elegant, because it only involves the numerical computation of moments (cf. Sect. 1.3) of the smeared CLDg2 (rn) followed by moment arithmetics [200], The first step is the computation of the Mellin transform102 of the analytical function gc (rn) which we have selected to describe the needle diameter shape. This is readily accomplished by Mathematica [205], Because the Mellin transform is just a generalized moment expansion, we retrieve for the moments of the normalized chord distribution of the unit-disc103... 

Fig. 2 Normal, or Gaussian, size-frequency distribution curve. Percentage of particles lying within 1 and 2 standard deviations about the arithmetic mean diameter are indicated.

Consider a volume containing c A molecules of A (mass mA) and c B molecules of B (mass mB) per unit volume. A simple estimate of the frequency of A-B collisions can be obtained by assuming that the molecules are hard spheres with a finite size, and that, like billiard balls, a collision occurs if the center of the B molecule is within the collision diameter dAB of the center of A. This distance is the arithmetic mean of the two molecular diameters dA and dB ... 

Plotting droplet size data on an arithmetic-probability graph paper will generate a straight line if the data follow normal distribution. Thus, the mean droplet diameter and standard deviation can be determined from such a plot. 

In many applications, a mean droplet size is a factor of foremost concern. Mean droplet size can be taken as a measure of the quality of an atomization process. It is also convenient to use only mean droplet size in calculations involving discrete droplets such as multiphase flow and mass transfer processes. Various definitions of mean droplet size have been employed in different applications, as summarized in Table 4.1. The concept and notation of mean droplet diameter have been generalized and standardized by Mugele and Evans.[423] The arithmetic, surface, and volume mean droplet diameter (D10, D2o, and D30) are some most common mean droplet diameters ... 

Some other correlations that have been specifically developed for liquid metals include those proposed by Nichiporenko)488 Schmitt)489 Thompson)491 and Date et al)494] Nichiporenkol488 correlated the diameter of particles (powder) of predominant fraction, D h with the arithmetic mean particle diameter (linear average diameter), Z)10, and the Reynolds number as well as the Weber... 

A mean length diameter or arithmetic mean diameter may also be defined by ... 

Arithmetic mean values of the diameter, R, Kihara diametisr, and and 9 are used in the evaluation of the interaction between two o lecules A and B. These values, and an associated parameter t, are defined as... 

Most readers will be familiar with the bell-shaped normal distribution plotted in Fig. 9.12. When applied to the size distribution of particles, for example, such a distribution is fully characterized by the arithmetic mean D and the standard deviation a, where a is defined such that 68% of the particles have sizes in the range D a In the log-normal distribution, the logarithm of the diameter D is assumed to have a normal distribution. (Either logarithms to the base 10 or loga-... 

The geometric number mean diameter, D%N, is related to the arithmetic mean of In (diameter) ... 

Proposed additional new standard for particulate matter, diameter <2.5 pm (PM-2.5) 3 y annual arithmetic mean, 15 pg/ m3 arithmetic mean of 24 h 98th percentile averaged over 3 y, 65 pg/m3, Data from EPA (2003c). 

---