Arithmetic mean diameter

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 ... 

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. 

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. 

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.

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

The arithmetic mean diameter d is the averaged diameter based on the number density function of the sample d is defined by... 

The sizes of a powder sample are found to follow a log-normal distribution with surface mean diameter, (c) the volume mean diameter, (d) the Sautermean diameter, and (e) the DeBroucker mean diameter. 

Arithmetic Mean Diameter The average diameter of all the droplets in the spray sample. 

The arithmetic mean diameter is the sum of the weights retained on each sieve multiplied by the corresponding diameter and divided by the total weight, thus... 

The method of sizing may also be incorporated into the symbol. Hence, for particle sizing by microscopy, the arithmetic mean diameter becomes da NLa- l volume-moment mean diameter calculated from the... 

Smaller dimension in case of wheat. Arithmetic-mean diameter is used in Part A reciprocal-mean diameter in Part B. 

If direct measurements are used and followed by statistical processing of the measured results, the cell size is conventionally represented via the so-called mean cell diameter which is defined differently by various authors If there are N cells of the same shape but of different diameters Dj, the mean probable cell diameter will be a number Dp which has the following property one half of all cells have diameters below D and the other half above it. Among other mean cell diameter definitions the more generally used are arithmetical mean diameter = XDp./N, geometrical... 

The value obtained is the arithmetic mean diameter. An approximation could be reached by allocating the spheres into categories defined by size, and counting the number in each category counted. Using standard statistical terminology, the arithmetic mean diameter would be given by ... 

Figure 12.2 Effects of low- and high-shear swirl distribution on arithmetic mean diameter, number density, and volume flux for (a) nonburning and (6) burning cases for equal combustion airflow distribution between inner and outer annulus of the burner 1 — 50°/30° and 2 — 65 /30°. The upper panels correspond to X = 30 mm, and the lower panels correspond to X = 5 mm.

Other averages are sometimes useful. The arithmetic mean diameter Df, is... 

Average particle size, mm, pm, or ft arithmetic mean diameter [Eq. 

Calculate the arithmetic mean diameter Df, for the —4- to H-200-mcsh fractions of the material analyzed in Table 28.2, How does Df, differ qualitatively from the volume mean diameter 5 ... 

The arithmetic mean diameter during the process is evaluated by... 

Equation 17 was also found to apply in the continuous unit where one deals with clouds of falling particles instead of single falling particles. In this case, dolomite acceptor particles varying in mean diameter from 0.0165 to 0.0215 in. were allowed to shower through fluidized char beds. The diameter used here is the arithmetic mean diameter calculated from terminal velocity of the particles in air. Eight measurements were made, and the standard deviation of the measured terminal velocity from that calculated by Equation 17 was 7.9%, and the maximum deviation was 15.2%. 

The arithmetic mean diameter (or number mean) is calculate... 

Taking into account the size of all the particles observed by CTEM, a calculation was made to estimate the arithmetic mean diameter of these particles, the equivalent diameter dgq comparable to the mean diameter obtained by chemisorption methods and calculated by comparing surface and volume of the particles, the specific surface area, and the dispersion, which represents the munber of accessible metal atoms [14-16],... 

By taking sequential cross-sections of images such as Ae one shown in Figure 5, it is possible to obtain a particle size distribution for the nanocrystals. Figure 7 shows one such distribution, in this case revealing an arithmetic mean diameter of 14 nm with a sample standard deviation of 5 nm for a sample comprised of 124 particles. The best-fit gaussian function for the histogram shown was ... 

Statistically, the particle size distribution can be characterized by three properties mode, median, and mean. The mode is the value that occurs most frequently. It is a value seldom used for describing particle size distribution. The average or arithmetic mean diameter, d, is affected by all values actually observed and thus is influenced greatly by extreme values. The median particle size, is the size that divides the frequency distribution into two equal areas. In practical application, the size distribution of a typical dust is typically skewed to the right, i.e., skewed to the larger particle size. The central tendency of a skewed frequency distribution is more adequately represented by the median rather than by the mean (see Fig. 9). Mathematically, the relationships among the mean, median, and mode diameter can be expressed as... 

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