For example, if you wish to maximise the recovery of gold particles from a river sediment then this is the distribution to serve that end. Distributions of solid surface area are relevant in processes involving surface interactions, such as catalysis or separations using packed columns (i. e flow through porous media). Distributions of particle number are relevant in processes where particles interact with each other in aggregation processes (such as coagulation and flocculation). Conversion between frequency and cumulative forms Frequency to Cumulative Form.
Frequency data presented in a table for various characteristic sizes are simply summed stepwise in a consistent direction across the size range e. g. Size f(x) Fu(x) Fo(x) 1 0. 2 0. 2 0. 2+0. 7+0. 1 2 0. 7 0. 2+0. 7 0. 7+0. 1 3 0. 1 0. 2+0. 7+0. 1 0. 1 If the summation proceeds from x = 0 to x = ? then the summation to any given size provides an estimate of the total frequency below that size. If it proceeds in the opposite direction then it gives the frequency above that size. These two forms of the cumulative distribution are termed undersize Fu(x) and oversize Fo(x) respectively.
Cumulative to Frequency Form If the distribution is provided as a curve, then differentiate it numerically to produce a range of size class divisions, x, x + 1dx, x + 2dx, etc. and determine the portion of frequency in each interval (i. e. the area under the curve between x and x + 1dx etc. If data are provided as a Table then subtract frequencies of adjacent classes e. g. Size Fu(x) f(x) (x(0)) 0 (nothing below zero) x(1) 0. 13 Fu(x(1)) ? Fu(x(0) x(2) 0. 26 Fu(x(2)) ? Fu(x(1) x(3) 0. 34 Fu(x(3)) ? Fu(x(2) x(n? 1) Fu(x(n? 1)) ? Fu(x(3) x(n) 1. 00 Fu(x(n)) ?.
Fu(x(n? 1)) Interconversions of cumulative distributions to and from frequency distributions employing different bases can generate numerical error and are best avoided. It is recommended that appropriate experimental techniques be employed to provide size distribution data of the form suitable for application to the process of interest. Charcteristics of Size Distributions (Statistical models and parameters) Mode The most commonly occurring size (the maximum value of f(x). Median The size which divides F(x) in half. (i. e. the value of x when F(x) =0. 5).
Mean There are various kinds of the general form (related to the distributed property) :- _ g(x) = g(x)f(x)dx  where f(x) can be on any basis (number, surface etc. ) and g(x) is some function of particle size (x) which may take the following forms: Form of g(x) Mean Diameter (x) x1 Arithmetic mean x (number mean diameter) x2 Quadratic mean x x3 Cubic mean x (volume mean diameter) log x Geometric mean x x-1 Harmonic mean x The arithmetic mean of f(x) relates the total length of all particles (places end to end in a line) to the total number of particles.
Polydispersity may be quantified by the ratio of the volume mean diameter to the number mean diameter. Model descriptions of particle size distributions High speed computers allow analytical models to be fitted to experimental data so that subsequent (hand) calculations can often be streamlined. Normal distribution: (2 parameter model)  NB. Negative sizes are (theoretically) possible with this distribution (which is, therefore, often of little use) ? is the standard deviation and xa (bar) the arithmetic mean size. Log normal distribution: (2 parameter model)
This is a widely used model which is skewed to the right (large sizes) and gives equal probability to ratios of sizes rather than size differences. The model is obtained by substitution of ln x for x, ln xg for xa and ln ? g for ? in equ 6. (i. e. use a log scale for the size axis. [7a] A plot of dF/d(ln x) against ln x represents a symmetrical normal distribution with a geometric mean xg that is in this case equal to the median size x0. 5. This is often expressed in a more convenient form in terms of dF/dx and the modal size (xm) via the substitution:
ln xm = ln xg – ln2 ? g [7b] to give [7c] Probability and log-probability graph paper are available to assist parameter estimation for best fit models. Software packages (often those associated with particle size analysers) are available which will numerically estimate model parameters using simplex or other minimisation routines. Rosin-Rammler distribution: (2 parameter model) This gives the cumulative percentage oversize as a function of a size range parameter (xg) and the steepness of the curve (n). 
The frequency distribution is given by differentiation of this curve. Harris’s distribution. (3 parameter model) (See Harris C. C. Trans. SME, 244 No. 6 187-190 (1969). ) Harris showed that most two-parameter models are special cases of a more general model:-  where F(x) is the cumulative percentage oversize, xo is the maximum size in the sample, s is a parameter which reflects the slope of the log-ln plot in the fine region and r is concerned with the shape of the log-ln plot in the coarse region. General Comment.
Whilst there is some convenience to be gained from describing a particle size distribution in terms of an analytical function, the purpose of processing particles is often to effect the separation of materials possessing different properties (and distributed properties). Many particle size distributions are effectively the result of the superposition of the distributions of the individual and often various components of the material. There is some virtue to describing homogeneous suspensions using analytic functions, especially for uni-modal size distributions.
However, modern computational facilities allow rapid processing of sets of discrete size data with the advantage of preserving accuracy. Manipulation of size data can lead to accumulation of significant numerical error. Numerical differentiation of cumulative distributions can be prone to error – so any data processing protocol which reduces the total number of such operations will provide greater accuracy. (See Grade Efficiency Analysis. ) Particulate Systems: Characterisation Page 1 of 10 ce321_1. doc.