One of the problems of utilizing sieve analyses is in the interpretation of the mean size of the particles which pass through the smallest mesh size and are held in the pan. It has however been suggested that an equation of the following form can be used to calculate such data from comminuted products:

$\frac{dx}{dD}=D^k$dots

Where $x$ represents the mass fraction of particles of size $D,$ and $$ and $k$ are constants.

In a test on galena, crushed material from a ball mill has the following analysis:

$\backslash$table$[[$Mesh sizes $(mm),$Weight fraction$],[0\mathrm{.}595-0\mathrm{.}420,0\mathrm{.}150],[0\mathrm{.}420-0\mathrm{.}297,0\mathrm{.}200],[0\mathrm{.}297-0\mathrm{.}210,0\mathrm{.}171],[0\mathrm{.}210-0\mathrm{.}149,0\mathrm{.}134],[0\mathrm{.}149-0\mathrm{.}105,0\mathrm{.}104],[0\mathrm{.}105-0\mathrm{.}074,0\mathrm{.}080],[0\mathrm{.}074-,0\mathrm{.}161]]$

Reference to the analysis shows that there is a significant proportion passing through the finest screen. Use this data to:

a$)$ Show that equation $(1)$ is valid for the smallest sizes of the material

b$)$ Calculate the quantity of the material between the following mesh sizes:

$0\mathrm{.}074mm-0\mathrm{.}050mm$

$0\mathrm{.}050mm-0\mathrm{.}030mm$

c$)$ Calculate the Sauter mean Diameter of the material.