MODELLING OF COMBINED CASCADE CLASSIFIERS
Eugene Barsky
Department of Industrial Engineering and Management
Jerusalem College of Engineering, Jerusalem, Israel
Michael Barsky
The Institutes of Applied Research
Ben-Gurion University of the Negev, Beer-Sheva, Israel
Summary. Since the creation of cascade classifiers, the most prominent achievement
in the technology of loose material separation is the development and application of
combined cascade apparatuses. These apparatuses use simultaneously several
identical or different separating cascades, which makes it possible to improve
considerably the produced material quality. On the basis of generalized experience in
the application of such apparatuses, a strict mathematical model of separation
processes in these apparatuses has been developed. The adequacy of the model to
experimental results is demonstrated.
1. INTRODUCTION
Two stages of the separation process improvement can be singled out. At the
first stage, the attention is mainly concentrated on the increase in the productivity of
an apparatus with single-stage separation [1, 2]. This resulted in the development of
various designs of separators with different operation principles. However, their
efficiency is restricted. The attempts of increasing the efficiency led to the creation of
multi-stage (cascade) apparatuses realizing multiple separation of powders within the
same case.
2. CASCADE CLASSIFIERS
The designs of elements assembling a cascade are presented in Fig. 1. The
development of cascade classifiers represents the second stage in the development of
powder separation technology.
A separating cascade comprises two stages with a certain number of
connections between them.
The simplest version of such a cascade comprises identical separating
elements (stages) operating in the same mode. Such a cascade can be called regular.
Complicated cascades of the 1st order are characterized by various connections
between the stages and different separating elements operating in different modes.
The value characterizing the separation of a narrow size class in a separate element of
the cascade in a steady-state mode can be presented as
ri
k
(1)
ri
where ri is the initial content of narrow size class particles at the i-th stage; ri* is the
number of particles of the same size class passing from the i-th stage to (i – 1)th one
(the stages being counted top-down); k is the distribution coefficient.
It is established that in case of a regular cascade, the extraction of a narrow
size class into the fine product for the entire apparatus can be described by the
following relationship [1, 2]:
1-120

1   z 1i
1   z 1
z  1  i
F f ( x) 
z 1
0
k  0,5
k  0,5
(2)
k 0
where z is the number of stages in a cascade,
i* is the stage of the initial feeding (counting top-down);
χ = (1 – k)/k.
The magnitude k is determined by the separating flow structure using the
following formula:
(3)
k  1 0,4B
2
where B = (gd/w )(ρ – ρ0)/ρ0,
d is the size of narrow class particles, m;
w is the air flow velocity, m/s;
g is the gravity acceleration, m/s2;
ρ is the density of the separated material, kg/m3,
ρ0 is the density of the separated medium, kg/m3.
For a complicated cascade, at arbitrary distribution coefficients, such dependence
acquires the form [3]:
1
F f ( x) 
z
 
l i  1
z
l
1    l
l 1
Experimental study of various types of separating cascades of the 1st order has shown
that the effect of an increasing number of stages takes place and is rather significant,
but it has a markedly exponential character. With 8-9 stages, the effect growth reaches
a certain limit, and further addition of stages does not practically increase the
separation effect.
Therefore, further improvement of separation processes becomes possible at
the creation of cascades of the 2nd order of the type z × n.
In this case, a combined cascade consists of n separating elements (cascades of
st
the 1 order), each of them consisting of zi separating stages operating in different
modes.
3. COMBINED CASCADES OF Z × N TYPE
The simplest version of a combined cascade of z × n type comprises n
identical separating cascades of the 1st order consisting of the same number of stages
z, with a fixed place of the initial material inlet into one of the apparatuses and all the
apparatuses operating in the same mode. Various combined apparatuses are shown by
way of example in Fig. 2.
Even in this extremely simplified case, such combined separator can comprise
a large number of structural connection schemes between separate elements. As
experimental studies have demonstrated, some of these schemes realize a higher order
of the process organization in comparison with the separating cascade of the 1st order.
1-121
They are not equivalent to a mere increase in the number of elements in a cascade
apparatus [3].
To avoid cumbersome repeated explanations, it seems expedient to define the
following notions:
- free outlet – a local free flow outflowing into a combined fine or coarse
product from a separate column;
- connection – a local restricted outlet from one column to another;
- structural scheme – a scheme of outlets, inlets and connections between
individual separating elements in a combined cascade;
- isomorphic schemes – schemes with identical connecting functions;
- transposed schemes – schemes obtained from the given ones by transposing
some columns with a complete conservation of the previous outlets and
connections; apparently, transposed schemes are isomorphic;
- F0 – fractional extraction of fine product in a single column;
- F – fractional extraction into fine product for the entire combined cascade;
- F(F0) – connective function corresponding to each specific structural scheme;
- inverted scheme – a scheme with all outlets and connections for the fine
product become identical to the outlets and connections for the coarse one, and
vice versa. The connective function for an inverted scheme is F-1(F0).
Apparently, for an inverted scheme, the connective function for the coarse
product is identical to F(F0) function with the argument F0 substituted with (1
- F0):
Fc1 ( F0 )  F ( F0  1  F0 );
- working schemes – efficient schemes for the combined cascade (Fig. 2f);
- defective schemes – schemes with either some elements excluded out of the
process (Fig. 2a, b, c) or some elements devoid of active connections with
other elements (Fig. 2d), or else some elements playing the part of carriers
(Fig. 2e). Schemes with several of the mentioned defects also exist.
4. WORKING SCHEMES FOR COMBINED CASCADES OF Z × N TYPE
It is necessary to examine all possible variants of working schemes, since only
their complete analysis makes it possible to find the most advanced schemes.
If a combined cascade comprises n elements, the total number of free outlets
and connections equals 2n.
The minimal number of free outlets is
Pmin  2
Hence, the maximal possible number of connections between n elements amounts to
S max  2n  2
The minimal number of connections is:
S min  n  1
Taking this into account, the maximal number of outlets is
Pmax  n  1
At a fixed number of free outlets in a combined scheme P, the number of connections
between n elements amounts to
(4)
S  2n  P
In the general case, the number of schemes is equal to the number of ways allowing
the organization of S connections. For any column, any connection with any of the
1-122
remaining (n – 1) columns can be organized. Since there are S connections, and each
of them has (n – 1) directions, the total number of various schemes amounts to
(5)
N   (n  1) S
Taking (4) into account, we can write:
(6)
N   (n  1) 2n P
In the general case, we can obtain the total number of non-isomorphic schemes of all
kinds including direct inverted, transposed and all kinds of defective schemes:
n 1
P 1


(7)
N
  (n  1) 2 n  P  (cnm cnP m )

P 2 
m 1

where c nm  = n!/m!(n – m)! is the binomial coefficient (m implies the number of free
outlets into the fine product). It is noteworthy that the number of schemes determined
using the expression (7) greatly exceeds the number of working schemes.
Thus, at n = 2, according to (7)
N
8,

and the number of working schemes is only
N d  4.
At n = 3 we, respectively, obtain:
N
 348;
N d  47

At n = 4,
N
 30348; N d  904

Structural schemes of all working variants for n = 2 are presented in Fig. 2f.
5. CONNECTIVE FUNCTIONS FOR COMBINED CASCADES
A connective function for a combined separator represents an equation of the
type
F  F ( F0 )
where F is fractional extraction for the entire apparatus;
F0 is fractional extraction in a single element of a combined cascade.
First we consider the simplest case of z × 2 apparatus under the condition that
both elements are identical regular cascades of the 1st order.
Fig. 2f shows all of the four possible schemes of this apparatus. It is
noteworthy that the schemes I and II are inverted with respect to each other, as well as
the schemes III and IV. For the scheme I,
F2  (1  F0 ) F0
F1  F0 ,
In this case, the connective function is:
F ( F0 )  F1  F2  F0  (1  F0 ) F0  1  (1  F0 ) 2
For the inverted scheme II, the connective function can be written as
F ( F0 )  F02
For the scheme III, the total outlet of fine product can be written in the form of an
infinite series:
F ( F0 )  F0  F02 (1  F0 ) 2  F03 (1  F0 ) 2    F0n (1  F0 ) n 1
We obtain a geometric progression, and it is clear that
1-123
F0
1  F0  F02
For the inverted scheme IV, one can obtain from similar considerations:
F02
F ( F0 ) 
1  F0  F02
Here we present expressions for the simplest case. It is difficult to derive a general
expression for more complicated schemes using this method.
Therefore, we used another method of formalization of combined cascades. It
has turned out that a combined cascade of any degree of complexity consisting of n
independent elements can be represented by a square matrix with the dimensions n ×
n.
Figure 3 shows a combined cascade separator comprising three elements. Its
connective function determined by the previously used method was computed and
amounted to
F 3  F 2  F0
F ( F0 )  0 3 0
F0  F0  1
We represent this scheme in the form of a connective matrix:
_______________
j
1
2
3
i
_______________
1 F0 1  F0 0
_______________
2
0 1  F0 F0
_______________
F0 1  F0 0
3
_______________
Matrix element aij at i  j denotes feeding from the i-th apparatus to the inlet
of the j-th one. Diagonal elements aii denote the outlet from the apparatus; j-column
represents feeding from all the apparatuses; i-th line represents the outlet from the i-th
apparatus to all other apparatuses.
Such presentation of a combined cascade makes it possible to describe not
only any combined facility, but also to rule out a great number of inoperable schemes,
because such matrix should possess the following features:
1) matrix element aij should acquire one of the following four values: 0; F0; 1 –
F0; 1;
F ( F0 ) 
n
2)
 (a
i 1
ij
 aii )  0
j2
It means that something should be necessarily fed into each apparatus except the first
one (the apparatus fed with initial material taken as a unit).
n
3)
a
j 1
ij
 1,
Each apparatus has an outlet of both fine (F0) and coarse (1 - F0) products;
1-124
n
4)
a
i 1
ii
 0,
A combined facility should necessarily have an outlet. F0 and (1 - F0) or 1 should be
necessarily simultaneously present among the elements aii (i = 1, 2, 3, … n), since the
material is not accumulated in the apparatus.
In should be noted that a matrix of such kind is functional, since F0 is a
function of the size and velocity of particles, of the place of the material introduction
into the apparatus, etc. For instance, if all the three elements of a combined cascade
are different or have different modes (Fig. 3b), then such non-uniform cascade is also
described by a similar matrix. In this case all i-th elements acquire a respective index:
1  F1
0
 F1


(8)
1  F2
F2   A
0
F
1  F3
0 
 3
Such a form of presentation makes it possible to obtain the connective
function F(Fi) and find simple algorithms for the enumeration and analysis of n-link
cascades in a general case.
Thus, if we denote the matrix representing a specific combined cascade by A,
then its connective function can be written as
 r1    1
   
 r2   0 
T
(9)
A   r3    0 
   
   
r   
 n  0 
T
where A * is a matrix obtained from A by transposition and replacement of all aii
elements with (-1);
r1; r2; r3… rn is the material flow through a respective element (initial content for each
element).
Thus, for the matrix (8) we can write:
0
F0 
 1



(10)
AT  1  F0
1
1  F0 
 0

F0
1 

It follows from (9) that the connective function is based on the balance of material
flows. In compliance with (1) we can write
rf
F fi  i
(11)
ri
where Fi is the extraction of a narrow class into the fine product in the i-th element of
the combined cascade;
rfi is the quantity of a narrow class extracted into the fine product;
ri is the initial quantity of this class in the i-th element.
Taking this into account, we can write:
r f1  F1 r1
r f 2  F2 r2

r f n  Fn rn
1-125
It follows from the balance condition that
r1  1  a r1 r2  a31r3    a n1 rn
r2  a12 r1  a32 r3    a n 2 rn

(12)
rn  a1n r1  a 2 n r2    a n 1 rn 1
In the general case, the connective equation is written as
a 21
a31  a n1  r1    1
 1

   
1
a32  a n 2  r2   0 
 a12
(13)

   
      

   
a
 r   0 
a
a


1
2n
3n
 1n
 n   
In a particular case, for an arbitrary three-element cascade we can write:
 r1  a 21r2  a31r3  1
a12 r1  r2  a32 r3  0
a13 r1  a 23 r2  1  0
Taking this into account, the matrix form of the connective equation will acquire the
form:
a 21
a31  r1    1
 1

   
(14)
1
a32  r2    0 
 a12
a





a 23
 1  r3   0 
 13
The left-hand side of the equation (14) comprises a product of the matrix reflecting a
specific scheme of a combined cascade by the vector of the material flow through
separate elements of the apparatus. The right-hand side of the equation (14) comprises
the vector of the material inlet into the apparatus. If the material enters each element
of the combined cascade by portions a, b, c, then, assuming their sum to be a unity,
the right-hand side of the equation (14) should be rewritten as:
 a
 
 b
 c
 
Connective equation makes it possible to calculate material flows r1; r2; r3… rn
through all the elements of the combined cascade, as well as to determine the
connective function F(F0).
Thus, for the scheme presented in Fig. 3, we can write:
F ( F0 )  F0 r1
From the solution of the matrix equation:
1  F0  F02
r1 
1  F0  F03
Hence,
F  F02  F03
F ( F0 ) 
,
(15)
1  F0  F03
which corresponds to the connective function for this scheme of a combined cascade
obtained by another, more complicated method.
1-126
In the general case of n-element combined cascade, the matrix equation is
written as
a 21
a31  a n1  r1    a 
 1

   
1
a32an 2  r2    b 
 a12
   
(16)
  r3     c 

   
      
   
a
a2n
a3n   1 rn   0 
 n1
Thus, this method based only on the equations of balance of material quantity in each
element of a combined cascade, makes it possible to calculate an apparatus of any
level of complexity with arbitrary connections, modes and material feeding into its
different elements.
Figure 4 shows schemes of three-element combined cascades of the highest
practical interest and connective functions for each of them determined according to
the described method.
6.
EXPERIMENTAL
CHECK-UP
OF
THE
ADEQUACY
MATHEMATICAL MODELS OF COMBINED CASCADES
OF
To check the estimated dependencies, several sets of experiments were carried
out under laboratory and industrial conditions. The appara- tuses of z × 2 type with
consecutive refining of coarse and fine products were studied under laboratory
conditions (Fig. 2f, I, II).
Industrial tests were carried out on a combined cascade of z × 8 type with
consecutive refining of coarse product.
The following elements constituting the cascade were used in these
apparatuses:
1) a polycascade classifier (Fig. 1c) with the number of stages z = 9 and feeding
of the initial material into the stage i* = 3;
2) a classifier with radial grates;
3) a shelf classifier (Fig. 1a) with z = 6; i* = 6; n = 8.
Under laboratory conditions, the experiments were carried out on grinded
quartzite and foundry sand. Under industrial conditions, the expe- riments were
carried out on potassium salt. The elements of the appa- ratuses in each set of
experiments were identical, and equal air flow rates were specified in each of them.
The material concentration in the flow was maintained constant or varies within the
limits of self-similarity region from 1.1 to 2.5 kg/m3. In each set of experiments five
different air flow rates were specified, namely 2.8, 3.07, 4.13, 4.98 and 6.14 m/s.
From these experiments, fractional extraction of a narrow class by one element
of the apparatus F0(l) and fractional extraction by the entire apparatus F(l) were
determined.
The same estimated parameters were determined using the connection
functions:
F = 1 – (1 – F0) – for the coarse product refinement;
F = F02 – for the fine product refinement;
F = 1 – (1 – F0)8 – for eight-fold fine product refinement.
The magnitude of the parameter F0 was determined by the relationship (2). On
the basis of experimental data, it was established that the magnitude F0(l) is constant
for each element of the apparatus.
1-127
By way of example, Fig. 5 shows F0(l) magnitudes for each element of an
eight-row apparatus with the productivity of 40 t/h fractionating fine-grain potassium
chloride at the air flow rate w = 6.14 m/s.
The relationship of estimated (F) and experimental [F(l)] magni- tudes for the
given apparatus at different flow rates is presented in Fig. 6.
As follows from the graph, the discrepancy between the estimated and
experimental data does not exceed 5.0%, which is within the accu- racy of
measurements under industrial conditions. A similar coincidence was observed in all
laboratory studies.
7. CONCLUSION
The present paper substantiates the progressive nature of combined cascade
classifiers. A detailed strategy of their computation is developed, and the conducted
studies confirmed the adequacy of the proposed model for computing parameters of
combined cascade schemes of loose material separation.
REFERENCES
1. Barsky, M.D. Powder Fractionating, “Nedra”,Moscow, 1980.
2. Barsky, E. and Barsky, M. Processes of Gravitational Classification. Urals
State Technical University Press, Ekaterinburg, 2003.
3. Barsky, M. and Barsky, E. Criterion for Efficacy of Separation of a Pourable
Material into N Components. In: “Computers Application in the Minerals
Industries”, “A.A. Balkema Publishers” Zisse (Abington), Extom (Tokyo),
2001, p. 507-509.
4. Barsky, M.D., Govorov, A.V. and Shishkin, S.F. Degree of organization of
combined separation schemes, Yekaterinburg, Russia,Collection of articles
No. 1001x n-D80, Cherkassy, 1983, p. 199-207.
5. Barsky, M.D., Lar’kov, N.S. and Govorov, A.V. Analysis of the results of
industrial dust removal from potassium chloride using a multi-row classifier.
Proceedings of VNIIG, Leningrad, 1978, p. 21-35.
FIGURES
Figure 1. Stages of a cascade: a – shelves for pouring over; b – zigzag channel; c –
polycascade stages; d – radial grates; e – polycascade stages with radial grates.
1-128
Figure 2. Examples of structural schemes of combined cascades: a, b, c, d – defective
schemes; e – conveying scheme; f – four working versions of a two-element
combined cascade; s – initial material; f – fine material; c – coarse material
Figure 3. Connection scheme of a three-element combined cascade.
1-129
Figure 4. Connective function for 12 schemes of three-element combined cascades.
Figure 5. F0(l) dependence for various size classes in the elements of a seven-row classifier.
Ordinate axis: experimental F(l) magnitude; abscissa axis: No. of columns, n. Designation of
the average particle size in microns: 1 – 82; 2 – 130; 3 – 180; 4 – 258; 5 – 358; 6 – 450; 7 –
565; 8 – 715.
Figure 6. Relationship of estimated F and experimental F(l) magnitudes of fractional
extraction for an eight-row classifier. Ordinate axis: estimated F magnitude; abscissa axis:
experimental F(l) magnitude.
1-130