Size distribution arithmetic-normal

If the particle size distribution is normal or log normal, then the data can be linearized by plotting the particle frequency as a function of particle rize on arithmetic or logarithmic probability graph p r respectively. The 50% value of sudi plots yields the geometric median diameter and the geometric standard deviation is the ratio of the 84.1% m the 50% values. 

The reader will recognize that the mean particle size determined from Eq. (3.3) is the arithmetic mean. This is not the only mean size that can be defined, but it is the most significant when the particle size distribution is normal. The geometric mean Xg is the nth root of the product of the diameter of the n particles and is... 

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-... 

A size distribution that fits the normal distribution equation can be represented by two parameters, the arithmetic mean size, x, and the standard deviation, cr. The mean size, x, is the size at 50% of the distribution, also written as jcgo. The standard deviation is easily obtained from the cumulative distribution as... 

For quality control purposes, ceramists are often required to determine if the particle size distribution of one batch of powder is the same or different from another. This determination is difficult when the two batches of powder have similar mean sizes. A statistical method [19] must be used to make this distinction. To determine if two particle size distributions are the same or different. Student s t-test is used by applying the null hypothesis to the two sample means. For normal distributions the f-statistic is defined as tl rati of the difference between the two sample arithmetic means (A and A2) to the standard deviation of the difference in the means [20] ... 

The arithmetic mean size of a number distribution (x ) is the sum of the sizes of the separate particles divided by the number of particle it is most significant when the distribution is normal. 

Drop size distributions are often plotted as frequency v. logarithm of diameter to test the symmetry around the first arithmetic mean value or first moment m of an a.s.sumed log normal di-strihiition, which is a quite common ca.se. 

The remaining task is to determine Xa from the distribution by surface our size distribution is by mass, however, and a conversion has to be made. Rather than converting the whole distribution (from mass to surface) in this case, because the distribution is log-normal (see Figure 2.8), we can use the fact that the arithmetic mean by surface is equal to the harmonic mean by mass... 

COMMON METHODS OF DISPLAYING SIZE DISTRIBUTIONS 1.7.1 Arithmetic-normal Distribution... 

To check for a arithmetic-normal distribution, size analysis data is plotted on normal probability graph paper. On such graph paper a straight line will result if the data fits an arithmetic-normal distribution. 

To illustrate some basic principles of particle size representation and distribution, an experimentally derived size distribution of a sample of sodium starch glycolate suspended in water is shown in Figure 10 along with its cumulative distribution. " Where the size axis transformed to a logarithmic scale, one would observe that the sample is characterized by an effective log-normal distribution. For this particular sample, the median particle size (i.e., the dso value) was calculated to be 78.7 pm, the mode was found to be 90.1 pm, and the arithmetic mean was calculated to be 89.4 pm. 

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.

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. 

Here, Joi and parameters defining the log-normal distribution. Joi is the median diameter, and cumulative-distribution curve has the value of 0.841 to the median diameter. In Joi and arithmetic mean and the standard deviation of In d, respectively, for the log-normal distribution (Problem 1.3). Note that, for the log-normal distribution, the particle number fraction in a size range of b to b + db is expressed by /N(b) db alternatively, the particle number fraction in a parametric range of Info to Info + d(lnb) is expressed by /N(lnb)d(lnb). 

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

The original log-normal distribution has a median size of 7.29 pm with a geometric standard deviation of 1.78 the unbiased distribution has a median size of 7.29 pm with a geometric standard deviation of 1.80 and the biased distribution has a median size of 6.99 pm (an error of 4%) with a standard deviation of 1.79. For measurements in arithmetic progression of sizes the effect is small, provided sizing is carried out at 10 or more size intervals, and for a log-normal distribution the position of the mode is only slightly affected. 

Sigma g (geometric standard deviation, ug is the arithmetic antilog of the standard deviation of a normal population of logs of aerosol diameters, surface areas, volumes, or masses. Each of these different measures of aerosol size is log-normaUy distributed, o-g is dimensionless. Using antilogs to transform them to arithmetic numbers makes the distribution easier to understand. 

There are a great number of different mean sizes and a question arises which of those is to be chosen to represent the population. The selection is of course based on the application, namely what property is of importance and should be represented. In liquid filtration for example, it is the surface volume mean Xsv (surface arithmetic mean Xa surface) because the resistance to flow through packed beds depends on the specific surface of the particles that make up the bed (see equation 9.36). It can be shown that Xsv is equal to the mass harmonic mean Xh (see Appendix 2.2). For distributions that follow closely the log-normal equation (see section 2.5) the geometric mean Xg is equal to the median. 

Although the distribution in the log-normal representation is completely specified by the geometric median particle size and the geometric mean standard deviation, a number of other average values have been derived to define useful properties. These are especially useful when the physical significance of the geometric median particle size is not clear. The arithmetic mean (d v) particle size is defined as the sum of all particle diameters divided by the total number of particles, and is calculated using ... 

---