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)
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.
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. )