ap-v2.dvi


Acta Polytechnica Vol. 50 No. 2/2010

Stochastic Models of Solid Particles Grinding

G. A. Zueva, V. A. Padokhin, P. Ditl

Abstract

Solid particle grinding is considered as a Markov process. Mathematical models of disintegration kinetics are classified on
the basis of the class of Markov process that they belong to. A mathematical description of the disintegration kinetics of
polydisperse particles bymilling in a shock-loading grinder is proposed on the basis of the theory ofMarkov processes taking
into account the operational conditions in the device. The proposed stochastic model calculates the particle size distribution
of the material at any instant in any place in the grinder. The experimental data is in accordance with the predicted values
according to the proposed model.

Keywords: Markov processes, grinding kinetics models, density of distribution.

1 Introduction
Alongside the traditional approach, based on phe-
nomenological introduction of the mechanics of a con-
tinuous medium into a description of chemical en-
gineering processes, in particular, milling processes,
stochastic approaches, particularly those based on
Markov processes, have become widely applied.

To a certain extent, these two approaches are
mutually complementary. However, in many cases
the detailed formal means of stochastic theory en-
able models of milling to be constructed more ratio-
nally [1, 2, 6, 7, 12, 13]. Stochasticity, which is an
integral part of the fracture process of polydisperse
material, is automatically taken into account.

Compliance with the Markov process forms the ba-
sis for modeling the particle size distribution during
milling. This means that the future behavior of the
system is not affected by the behavior of the system
in the past [2].

It is worth noting that the methods of Markov pro-
cesses have been used rather productively in theoreti-
cal analyses of many processes in chemical technology,
e.g. in mechanoactivation [8], separation, classifica-
tion of heterogeneous systems [9]. One of the main
constructors of the statistical theory of basic processes
in chemical technology was A. M. Kutepov [10, 11].
He formulated the following positive tendencies which
stimulate the further development of the statistical
theory of processes of chemical technology as follows:
• Interest in making active use of the methods of

non-equilibrium statistical thermodynamics, the
theory of casual processes and synergetic steadily
amplifies.

• Constant enrichment of statistical theory by new,
highly-effective mathematical means.

• Rapid development of computer technologies has
enabled the creation of the automated systems for
calculating and designing equipment for chemical
manufacturing.

• To solve problems in simulating chemical pro-
cesses it is necessary to create a bank of simple,
evident mathematical models of the processes of
chemical technology which are easily solved using
a computer. Stochastic models of these processes
obtained using fundamental methods of modern
statistical theory fully satisfy these requirements.

Feller [14] reported well-arranged and inspiring re-
view of stochastic theories in his comprehensive book.

2 Classification of grinding
kinetics models

The existing variety of types of Markov processes al-
lows the construction of stochastic models of disinte-
gration kinetics of varying complexity that adequately
reflect the specific features of the process of grinding
materials in a range of milling plants.

The jump Markov process most adequately de-
scribes the impulse character of loading of solids in
a shock-loading grinder. Generally, bead vibrations
result in a time-continuous spectrum of actions on the
material. This can be approximated by a continuous
Markov process.

A classification of mathematical models of grinding
kinetics, based on their relationship to a fixed class of
Markov processes, is given in Table 1. This classifica-
tion of models is incomplete and provisional. Never-
theless, it convincingly shows the physical conciseness
of Markov models of grinding processes, and also the
potential of the Markov-based methodology.

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Acta Polytechnica Vol. 50 No. 2/2010

Table 1: Classification of Grinding Kinetics Models

Group

No.

Markov process

type
Model type

Primary

disintegration

mechanism

Grinder design

1 Markov chain
(state-discrete and
time-discrete)

Matrix free impact Shock mills (shock –
reflective, disintegrator,
etc.), roller mills

2 Markov process,
state-discrete and
time-continuous
(transitions in casual
instants)

differential –
differential

constrained impact,
abrasion, crushing

vibrational,
magnetic-vortical,
drum-type, spherical,
epicyclic mills, etc.

3 jump process
(time-continuous and
state-continuous
transition in casual
instants)

integral –
differential

free impact Shock-reflective,
disintegrators, hammer
mills, rotor, etc.

4 diffusive process
(continuous)

diffusive abrasion bead, sand mills, etc.

5 mixed process integral –
differential +
diffusive,
integral –
differential +
differential –
differential)

abrasion + free impact,
constrained impact

jet-mills:
counter-current, ring,
pulsating, centrifugal –
counter-current, etc.

3 Stochastic models
An analysis of the milling of dispersed material in a
device with a periodical action of theoretical merging,
as described by the matrix model, shows that this pro-
cess can be classified as a stationary Markov chain.

For this purpose, in accordance with the terminol-
ogy used in the theory of Markov processes, we will
introduce the concept of a vector line of probabilities
of the states of the modeling system that is identical to
a vector column of the size distribution of the ground
material [2]

π(k + 1) = π(k)P, k = 0, 1, 2, . . . ; (1)

π(k)|k=0 = π0,

or
π(k) = π0P

k, (2)

where π0 – a vector of initial probability distribution;
π(k) – a vector of probabilities of states in time step
k; P – matrix of transition.

This is a model of periodical milling that can be
developed with the help of a stationary Markov chain.
However, loading particles in real conditions happens
in random time instants. It is therefore necessary to
describe grinding processes with the help of a discrete

Markov process, bringing in what happens at casual
time intervals.

It is known [4] a Markov process with continuous
time and discrete states is determined by matrix A
of intensities of transitions with time-constant compo-
nents aij and vector π0 of probabilities of states of the
system at the initial time instant.

A mathematical description of the grinding process
in this case is

dπ(t)
dt

= π(t)A; (3)

π(0) = π0.

The solution of this equation

π(t) = π0 exp(At). (4)

The matrix A of the transition intensities is differen-
tial, and it has a close connection with the stochastic
matrix P of the transitions [4].

Let us find matrix A for a time-continuous process,
for which the probabilities of conditions at moments
of time t = 0, 1, 2, . . . are the same as for the time-
discrete process, described by matrix P . The time of
one transition of a discrete process is taken as the time

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Acta Polytechnica Vol. 50 No. 2/2010

unit. Comparing the solutions of equations (2), (4) at
t = k we can see that

exp(A) = P

or
A = ln P. (5)

The procedures for finding the logarithmic and ex-
ponential function of a matrix are well known [5]. Ex-
pression (4) enables us to determine some of the par-
ticles of each fraction at any moment t while grinding
a portion of an ideal mixture in a device.

The hydrodynamic conditions in the device obvi-
ously have an essential influence on the milling process.
It should therefore be reflected in the mathematical
description. In accordance with the theory of Markov
processes we have introduced a theoretical analysis of
the grinding process in the device of theoretical ex-
truding of continuous action.

The equation describing the continuous process of
grinding in a shock-centrifugal milling device of theo-
retical displacement [5] in terms of Markov processes
is:

∂π(t, x)
∂t

= π(t, x)A − v
∂π(t, x)

∂x
. (6)

Here π(t, x) – particle size distribution at the mo-
ment t at passage of length distance x in a theoretical
extrusion device (the vector of state probabilities); v –
linear speed of a stream.

Applying Laplace transformations to equation (6)
twice [7], we receive an expression for the image of the
vector of probabilities of states

Π(s, p) =
(

Π(0, p) + v
π0
s

)
(sE − A + vpE)−1. (7)

Here L[t] = s; L[x] = p;

L [π(t, x)] = Π(t, p);

L [Π(t, p)] = Π(s, p);

L [π(0, x)] = Π(0, p),

where π(0, x) – vector of probabilities of states in the
initial moment of time in the section specified by dis-
tance x:

π(0, x) =

{
π0, x = 0,

0, x �= 0.
(8)

Here π0 – density of distribution of probabilities of
states at the initial moment, or otherwise, density of
the distribution of the number of particles in the sizes
at the moment t = 0 on an input into the device.

According to (8)

Π(0, p) =

⎧⎨
⎩

π0
p

, x = 0,

0, x �= 0.

Therefore the image of the solution of equation (6) is:
if x �= 0, then

Π(s, p) = v
π0
s

(sE − A + vpE)−1; (9)

if x = 0, then

Π(s, p) =

(
π0
p

+ v
π0
s

)
(sE − A + vpE)−1.

We are interested in the case when x �= 0, since the
density of a probability distribution at the entrance of
the grinding device (x = 0) is known at any instant; it
equals π0.

So, having applied the inverse Laplace transform
to expression (9) to a variable s, and then to a vari-
able p, we receive the density of distribution of the
probabilities of states at any point of distance x at
any moment t. In other words we receive the particle
size distribution in any section x at any moment t.

Note that under steady conditions t → ∞ and the
limiting vector of density of the probability distribu-

tion π∞ has components dependent on the value
x

v
and also on the initial density of the probability dis-
tribution π0.

Setting the stationary particle size distribution π∞
on an output of the device and having the average
speed of the stream, we can define, for example, the
necessary length of the device.

Using the constructed model, we can solve the in-
verse problem, i.e. we can find the elements of ma-
trix A of the transition intensities. Equation (6) for
vector π∞ of the limiting probabilities of states be-
comes

dπ∞(x)
dx

= π∞(x)
A

v
, (0 ≤ x ≤ l), (10)

where l is the total length of the grinder.
The boundary condition is

π∞(0) = π0. (11)

Solving equation (10) in view of boundary condition
(11), we receive the density of the limiting probabili-
ties

π∞(x) = π0 exp
(
A

x

v

)
, (0 ≤ x ≤ l).

Substituting for x = l, the elements of matrix A can
be determined by solving the system of the equations
constructed according to the condition

π∞(l) = π0 exp

(
A

l

v

)
(12)

and the condition of equality to zero of the sum of
elements in the matrix lines.

Matrix A has following appearance

A =

⎛
⎜⎜⎜⎜⎝

0 0 . . . 0

a21 a22 . . . 0

. . .

aN1 aN2 . . . aN N

⎞
⎟⎟⎟⎟⎠ ,

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Acta Polytechnica Vol. 50 No. 2/2010

where aij = 0, if i < j; a11 = 0, and

a21 + a22 = 0;

. . . (13)

aN1 + aN2 + . . . + aN N = 0.

Thus, having received experimentally steady state
distribution of particles in the sizes π∞ on an output
of the device, we can unequivocally find elements of
matrix A and also elements of a matrix of transitions
P. The common view of matrix P :

P =

⎛
⎜⎜⎜⎜⎝

1 0 . . . 0

p21 p22 . . . 0

. . .

pN1 pN2 . . . pN N

⎞
⎟⎟⎟⎟⎠ .

Considering that the elements of matrix P are formed
as follows

Pij = Piϕij ,

We can find probabilities Pi of destruction of the parti-
cles of each fraction i and probabilities ϕij of formation
of particles of the j-th fraction at destruction of larger
particles of the i-th fraction (i = 2, N ).

Similarly, a model can be constructed for contin-
uous milling of a dispersed material in the device of
theoretical merging, modeling it by a Markov process
with discrete states and in continuous time. In this
case, the equation for π(t) is [6]:

dπ(t)
dt

= π(t)A +
Q

V
(π(0) − π(t)) . (14)

Here Q – volumetric flow of dispersed material; V –
operating volume of the device.

In the case of an intermediate hydrodynamic mode,
the process of grinding can be simulated by means of a
cell model. Thus, the process will be described by sys-
tem of differential equations of type (14). The number
of the equations should be equal to number of cells
of the ideal mixture into which the device is broken
down.

Using a set of blocks that simulate milling in a
continuous action device with various hydrodynamic
modes, it is possible to solve problems of modeling, op-
timization and constructive framing of processes com-
bined with grinding.

The mathematical model (6, 8) of particle milling
in the device of theoretical displacement of continuous
action has been used to describe the process of milling
in a rotor-pulse grinder. Note that the two-level shock-
reflective grinder working on a single passage is close
to the hydrodynamic structure of a dispersed parti-
cle stream to the device of theoretical displacement of
continuous action.

If we know a vector π, describing distribution of
the state probabilities, it is possible to find the density
of probability distribution f , m−1 (or %/m−1). This

procedure is well-known in probability theory. Obvi-
ously f is identical to density of size distribution.

4 A check on the adequacy of
the mathematical model

In order to obtain the experimental density of the size-
particle distribution, we used the results of research on
the process of grinding in a patch-centrifugal mill. The
check on the adequacy of the mathematical model of
grinding involves comparing the calculated density of
size-particle distribution Equations (10 and 11) with
the experimental density of size-particle distribution.
The experiments proved that the time necessary to
reach steady state conditions was a few seconds and
less. As an example the results of a check on the ade-
quacy of the mathematical model of grinding in miller
are shown. Such equipment function can be described
by Eqs. (6 to 13). For reasons of convenience, the
calculations and the experiments used quartz of sand
and benzoic acid. The linear speed of material was
taken v = 24 m · s−1 in both calculations and experi-
ments.

To obtain calculated values for the density of the
size-particle distribution, it is necessary to make use of
expression (12). Having determined the probabilities
of particle destruction Pi and the value of the dis-
tributive functions ϕij , and also the residence time of

the particles τ =
l

v
corresponding to the regime and

design parameters, we find matrix P and then ma-
trix A (A = ln P ). Having substituted residence time
τ , elements aij of matrices A and the coordinates of
initial vector π0 in the system of equations (12), we
calculate the values for particles distribution density
against particle size on an output of the device.

Fig. 1: Comparison of experimental density (solid lines)
and calculated (dotted line) density of the distribution of
quartz sand particles in sizes on an output of the device
(n = 3000 rpm, G = 100 g/min): 1 – initial material;
2 – t =1 min

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Acta Polytechnica Vol. 50 No. 2/2010

Fig. 2: Comparison of experimental (solid lines) and cal-
culated (dotted lines) of density of distribution of ben-
zoic acid particles in sizes on an output of the device
(G = 50 g/min, t = 1 min): 1 – initial material, 2 –
n =2000 rpm, 3 – n =3000 rpm, 4 – n =4000 rpm

A comparison was made of the calculated and ex-
perimental particle size density distributions of the
crushed material using the root-mean-square criterion
of conformity [6]. Figs. 1 and 2 show the experimen-
tal and calculated particle size distribution density
changes with time. Fig. 1 shows crushing of quartz
sand particles, whereas Fig. 2 shows the effect of rotor
rotational speed on the grinding of benzoic acid par-
ticles. Satisfactory agreement between experimental
and calculated data is observed. The calculated root-
mean-square criteria of the given curves do not exceed
15 %. We can conclude that the mathematical model
adequately agrees with the experimental data on the
crushing process.

5 Conclusion
The following conclusions can be drawn:
• Mathematical models of disintegration kinetics

have been classified on the basis of their belonging
to a certain class of Markov process.

• Solid particle grinding can be considered as a
Markov process.

• A mathematical description of the disintegration
kinetics of polydisperse particles when milled in
a shock-loading grinder is proposed on the basis
of the theory of Markov processes, taking into ac-
count the operating conditions in the device. The
stochastic model enables the particle size distribu-
tion of a material to be calculated at any instant
in any position in the grinder.

• The experimental data is in agreement with the
predicted values according to the model.

Acknowledgement

This research has been executed within the framework
of the analytical departmental target program Devel-

opment of the Scientific Potential of the University
(2009–2010). Action 2.1.2.6492 and within the frame-
work of the agreement on cooperation between the
Czech Technical University in Prague and the Ivanovo
State University of Chemistry and Technology.

References

[1] Nepomniatchiy, E. A.: Kinetics of milling. The-
oretical Fundamentals of Chemical Technology,
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[2] Venttsel, E. S., Ovcharov, L. A.: The Theory
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[3] Padokhin, V. A., Zueva, G. A.: Discrete Markov
Models of Process of Dispergation. Engineer-
ing and Technology of Loose Materials. Ivanovo,
1991, p. 55–59.

[4] Howard, R. A.: Dynamic Programming and
Markov Processes. M.: Sovietskoye Radio, 1964.

[5] Gantmaher, F. R.: The Theory of Matrixes. M.:
Science, 1988.

[6] Kafarov, V. V., Dorohov, I. N., Arutyunov, S. J.:
System Analysis of Processes of Chemical Tech-
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Loose Materials. M.: Science, 1985.

[7] Ditkin, V. A., Prudnikov, A. P.: Operation Cal-
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[8] Padokhin, V. A., Zueva, G. A.: StochasticMarkov
Models of Mechanostructural Transformations in
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tures, Ivanovo, 1, 2004, p. 215–224.

[9] Mizonov, V.: Application of Multi-dimensional
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[10] Kutepov, A. M.: Stochastic Theory of Processes
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Acta Polytechnica Vol. 50 No. 2/2010

Prof. Galina A. Zueva
E-mail: galina@isuct.ru
Ivanovo State University of Chemistry and Technology
pr. F. Engelsa 7, 153000 Ivanovo, Russia

Prof. Valeriy A. Padokhin
Institute of Solution Chemistry
Russian Academy of Sciences
ul. Akademicheskaya 1, 153045 Ivanovo, Russia

Prof. Ing. Pavel Ditl, DrSc.
Phone: +420 224 352 549
E-mail: pavel.ditl@fs.cvut.cz
Department of Process Engineering
Faculty of Mechanical Engineering
Czech Technical University in Prague
Technická 4, 166 07 Prague 6, Czech Republic

Nomenclature

A matrix of intensities of transitions

aij components of matrix of transition intensities, s
−1

d diameter of particle, m

f density of distribution of probabilities of states which is identical to particle size distribution density, m−1 or
% · m−1

G mass flow of a material, kg · sec−1

i, j index of state

k time step, sec

l device length, m

L Laplacian

n rotation speed of the grinder rotor, sec−1, rpm

P matrix of transition

Pi probability of particle destruction for the i-th fraction

Q volumetric flow rate of a crushed material, m3 · sec−1

V operating volume of the grinder, m3

v linear speed of a stream of material, m · sec−1

t time, sec

x variable, distance coordinate, m

Greek letters

ϕij probability of formation of particles of the j-th fraction at destruction of larger particles of the i-th fraction

π(k) vector of state probabilities in time step k, 1 or %

π0 vector of initial probability distribution, m
−1, 1 or %

π(t, x) particle size distribution density at the moment t at passage of distance x, m−1

π∞ stationary particle size distribution density on a device output, m
−1

τ residence time, sec

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