Log-normal

From the probability distributions for each of the variables on the right hand side, the values of K, p, o can be calculated. Assuming that the variables are independent, they can now be combined using the above rules to calculate K, p, o for ultimate recovery. Assuming the distribution for UR is Log-Normal, the value of UR for any confidence level can be calculated. This whole process can be performed on paper, or quickly written on a spreadsheet. The results are often within 10% of those generated by Monte Carlo simulation. 

The only two distributions we shall consider are the Gaussian distribution ( normal law ) and the log-normal distribution. 

To obtain the expression for the log-normal distribution it is only necessary to substitute for I and a in Equation (1.52) the logarithms of these quantities. One thus obtains... 

The log-normal curve is obtained by plotting the frequency against In / instead of against I itself (cf. Fig. 1.15). 

The distribution curves may be regarded as histograms in which the class intervals (see p. 26) are indefinitely narrow and in which the size distribution follows the normal or log-normal law exactly. The distribution curves constructed from experimental data will deviate more or less widely from the ideal form, partly because the number of particles in the sample is necessarily severely limited, and partly because the postulated distribution... 

Fig. 1.15 A log-normal plot. Note the irregular shape, arising from the smallness of the sample. (Courtesy, DallaValle )...

The particle sizes of fillers are usually collected and ordered to yield size distributions which are frequendy plotted as cumulative weight percent finer than vs diameter, often given as esd, on a log probabiUty graph. In this manner, most unmodified fillers yield a straight-line relationship or log normal distribution. Inspection of the data presented in this manner can yield valuable information about the filler. The coarseness of a filler is often quantified as the esd at the 99.9% finer-than value. Deviations from linearity at the high and low ends of the plot suggest that either fractionation has occurred to remove coarse or fine particles or the data are suspect in these ranges. 

Another frequentiy used average is the geometric mean, which is particularly usehil for log-normal or wider (spanning over a decade) distributions. The geometric mean diameter, d is calculated usiag the logarithm values of the measured diameters ... 

There are a variety of ways to describe the droplet population. Figures 14-88 and 14-90 illustrate one of the most common methods, the plot of cumulative volume against droplet size on log-normal graph paper. This satisfies the restraint of not extrapolating to a negative drop size. Its other advantages are that it is easy to plot, the results are easy to visualize, and it yields a nearly straight line at lower drop sizes. 

When a distribufion of particle sizes which must be collected is present, the aclual size distribution must be converted to a mass distribution by aerodynamic size. Frequently the distribution can be represented or approximated by a log-normal distribution (a straight line on a log-log plot of cumulative mass percent of particles versus diameter) wmich can be characterized by the mass median particle diameter dp5o and the standard statistical deviation of particles from the median [Pg.1428]

With the log-normal probability law, it is ratios of equal amounts which are equally likely. In order to obtain a symmetrical beU-shaped frequency cui ve it is therefore necessary to plot the population density per log (micron) against log (size) [Hatch and Choate, J. Franklin Inst., 207, 369(1929)] ... 

To measure a residence-time distribution, a pulse of tagged feed is inserted into a continuous mill and the effluent is sampled on a schedule. If it is a dry miU, a soluble tracer such as salt or dye may be used and the samples analyzed conductimetricaUy or colorimetricaUy. If it is a wet mill, the tracer must be a solid of similar density to the ore. Materials hke copper concentrate, chrome brick, or barites have been used as tracers and analyzed by X-ray fluorescence. To plot results in log-normal coordinates, the concentration data must first be normalized from the form of Fig. 20-15 to the form of cumulative percent discharged, as in Fig. 20-16. For this, one must either know the total amount of pulse fed or determine it by a simple numerical integration... 

FIG. 20-16 Log -normal plot of residence-time distrihiition in Phelps Dodge mill. 

There are many complications with interpreting MWCO data. First, UF membranes have a distribution of pore sizes. In spite of decades of effort to narrow the distribution, most commercial membranes are not notably sharp. What little is known about pore-size distribution in commercial UF membranes fits the Poisson distribution or log-normal distribution. Some pore-size distributions may be polydisperse. 

The graphite microstmcture is assumed to contain a log-normal distribution of pores. Under these circumstances, for a specific defect, the probability that its length falls between c and c+dc is f(c)dc, with f(c) defined as ... 

The average nonuniform permeability is spatially dependent. For a homogeneous but nonuniform medium, the average permeability is the correct mean (first moment) of the permeability distribution function. Permeability for a nonuniform medium is usually skewed. Most data for nonuniform permeability show permeability to be distributed log-normally. The correct average for a homogeneous, nonuniform permeability, assuming it is distributed log-normally, is the geometric mean, defined as ... 

This means that the average permeability for this heterogeneous medium is the area-weighted average of the average permeability of each of the elements. If the permeability of each element is log-normally distributed, these are the geometric means. 

It calculates one-dimensional heat conduction through walls and structure no solid or liquid ciMiibustion models are available. The energy and mass for burning solids or liquids must be input. It has no agglomeration model nor ability to represent log-normal particle-size distribution. 

Uncertainly estimates are made for the total CDF by assigning probability distributions to basic events and propagating the distributions through a simplified model. Uncertainties are assumed to be either log-normal or "maximum entropy" distributions. Chi-squared confidence interval tests are used at 50% and 95% of these distributions. The simplified CDF model includes the dominant cutsets from all five contributing classes of accidents, and is within 97% of the CDF calculated with the full Level 1 model. 

The size of inhaled particles varies markedly. The size distribution approximates a log-normal distribution that can be described by the median or the geometric mean, and by the geometric standard deviation. For fibers, both... 

FIGURE 14.5 Log -normal probability size distributions, illustrating geometrical transposition... 

FIGURE 14.6 Log-normal probability size distributions, illustrating geometrical transposition between number, mass, and linear momentum curves and the mean size particle d,, which can be used in estimating the free-falling velocity of the particle group. 

FIG. 28 Log-normal plot of relaxation time t2 vs bias for three different chain lengths (given as a parameter) and a series of medium densities [21]. 

TABLE 6.15. Estimated Means and Standard Deviations for Log-Normal Range Distributions (base e) for Six Event Groups... 

A nonnegative random variable X has a log-nonnal distribution whenever In X, the natural logariilun of X, has a normal distribution. The pdf of a random variable x liaving a log-normal distribution is specified by... 

The mean and variance of a random variable X having a log-normal distribution are given by... 

Figure 20.5.3 plots tlie pdf of the log-noniial distribution for a = 0 and (3=1. Probabilities concerning random variables liaving a log-normal distribution can be calculated using tables of the normal distribution. If X lias a log-normal distribution witli parameters a and p, then In X lias a normal distribution with p = a and o = p. Probabilities concerning X can tlierefore be converted into equivalent probabilities concerning In X. Suppose, for example, tliat X lias a log-nonnal distribution with a = 2 and p = 0.1. Then... 

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