How is the Volume Mean Diameter (VMD) calculated?

The calculation of the volume mean diameter is based on the definition of moments of a distribution. The kth moment of a distribution q of order r (r=0 Number, ... r=3 Volume) over the particle size x is defined as a weighted integral of that distribution as follows:

Moment Formula 01

For finite size classes, the integration in the above formula is replaced by a sum over n elements:

Moment Formula 02

x_i_mean

arithmetic mean value of particle size interval i

x_i_upper

upper limit of particle size interval i

x_i_lower

lower limit of particle size interval i

n

number of particle size intervals

In terms of the definitions above, the VMD is the first moment of a q3(x) (particle volume over particle size) distribution.

VMD_Formula

A comprehensive explanation of the calculation including an example is given in ISO/FDIS 9276-2: Representation of results of particle size analysis — Part 2: Calculation of average particle sizes/diameters and moments from particle size distributions