SOME MATHEMATICAL MODELS OF MILLING
N. D. Orlova
Translated from Novye Ogneupory, No. 8, pp. 46 – 51, August, 2006.
Original article submitted October 14, 2005.
Various system approaches to studying milling and mixing processes formalized as components of a
physicomechanical system are discussed. Qualitative analysis of the milling processes is performed in the
context of the total spectrum of phenomena starting with the atomic-molecular level and ending with produc
A system approach to studying milling and mixing pro
cesses was described in [1, 2] where the total milling process
was formalized as a complex physicomechanical system.
The qualitative analysis of milling was performed in the con-
text of the whole spectrum of phenomena starting with the
atomic-molecular level and ending with the technological
processes. Later studies developed mathematical models of
milling for particular milling and mixing devices. Milling is
carried out in various plants based on different methods for
destroying materials; therefore, the construction of physical
and mathematical models usually takes into account the type
of the mill and the milling method.
Let us consider the construction of a mathematical model
describing the milling process using differential equations of
continuum mechanics taking into account the milling kinet
ics. At the same time, the type of the mill is not explicitly
represented in the equations.
There are several known approaches to the mathematical
description of the milling process. One of them involves
mathematical description of kinetic curves as a function of
the mean size of the milled particles and the milling time. In
our opinion, this method does not describe the destruction of
an individual particle. Despite its drawbacks, the method
makes it possible to determine with sufficient accuracy the
efficiency of milling plants for particular materials. The pro
posed model contains two phases of destruction: destruction
of a single averaged non-defective (with respect to its size)
particle and then describing the kinetic curves of milling.
The model chooses a defect-free particle, as it is always more
difficult to destroy than a particle with defects.
The second approach is energy-based. In this description
the estimated energy spent on changing the mean size of a
milled particle is related to the energy supplied to the mill.
Since the length of the energy supply chain is not identical in
different mills and the correlation between the estimated en-
ergy and the mean particle size is not really established, the
practical effect of this model is low.
The third approach involves a probabilistic estimate of
the possibility of particles penetrating the working zone of
the mill and is related to the function of the probability den-
sity function of the destruction of particles, not taking into
account the physicomechanical properties of the milled ma-
terial. This method is mainly used for estimating experimen-
The fourth approach to the milling process is related to
the concept of fractal geometry. Note that in this case we are
dealing with an alteration of some generally accepted physi
cal notions characterizing the milling process. The dispersion
of powder materials is characterized by notions characteriz
ing both a single particle and the whole set of particles com
prising the particular powder. The elements of an aggregate
are also called particles. The linear size l
for porous particles
can be introduced using fractal geometry (the analog of the
Koch curve) :
=´ = »
where n is the number of pores; D » 1.26. If a particle is po
rous, its surface area S
as well can be represented using
Refractories and Industrial Ceramics Vol. 47, No. 4, 2006
1083-4877/06/4704-0251 © 2006 Springer Science+Business Media, Inc.
Odessa National Marine Academy, Odessa, Ukraine.