OPTIMIZATION CRITERION FOR COMBINED CASCADE
CLASSIFIERS
Eugene Barsky
The Department of Industrial Engineering and Management
Jerusalem College of Engineering
Jerusalem, Israel
Abstract. Combined classifiers consisting of several interconnected independent
apparatuses have been recently used in industrial practice. We make at attempt to
develop an objective quality criterion for such apparatuses on the basis of certain
boundary conditions. Using this criterion, we have analyzed operating schemes of
two- and three-element combined classifiers and determined the most efficient ones.
INTRODUCTION
Since the development of cascade separators, combined cascade classifiers
have been the most prominent achievement in the present-day technique of pourable
materials separation [1]. Their application makes it possible, on the one hand, to
improve significantly the quality of powdered materials, and on the other, to produce
powders of preset grading in the course of separation proceeding from any initial
composition [2].
QUALITY CRITERION FOR COMBINED CLASSIFIERS
Speaking about combined classifiers, we imply a number of separators (most
frequently of the same type) interconnected in a certain manner in the process of
separation [1]. There are many interconnection schemes in combined classifiers.
Thus, in case of two separators (elements of a combined classifier), the number of
operating schemes is N2 = 4, in case of three elements the number of such schemes
amounts to N3 = 47, and in case of four elements N4 = 904. Therefore, it seems rather
urgent to evaluate the quality of operating schemes in an unambiguous and unbiased
manner from the standpoint of perfection of the separation process in them, i.e.
establishing optimal interconnection of elements within the apparatus.
It has been shown that for a combined classifier of any complicacy it is
possible to determine an analytical interconnection function for various elements in
the form of the following dependence [3]:
F  F ( F0i )
(1)
where F is the fractional extraction of a narrow size class into the fine product for the
entire combined apparatus,
F0 is the fractional extraction of the same class into the fine product in one element
comprised in this apparatus.
When evaluating a combined cascade, it is necessary to single out the
perfection degree of the interconnection of its elements. Naturally, a connection
function should be assumed as the basis of such evaluation.
To evaluate objectively a combined cascade using connection functions, it is
necessary to formulate the quality criterion which should satisfy a number of
requirements:
1. It should be independent of the characteristics of individual elements of a
combined classifier, i.e. of F0 i , because it evaluates connections between the
elements, but not the elements.
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2. The criterion should be the same for the direct and inverted schemes because
in both cases the connection scheme is the same.
It should be noted that the terminology used in the present paper is taken from
paper [3], where we have introduced the principal notions and definitions in this field.
Remind that an inverted scheme can be obtained from a direct one by substituting all
the connections all the connections for fine product with those of coarse product
F0  1  F0 , and vice versa.
The connection function for an inverted scheme is obtained by a formal
substitution F0  1  F0 . Although, as a result, we obtain a different connection
function for the inverted scheme, their quality criteria should give the same numerical
evaluation.
3. The index should equal a unity for schemes consisting of one or several
elements when
(2)
F ( F0 )  F0
4. The index should equal infinity when an ideal cascade is realized in it.
F  0,5
1

(3)
F ( F0 )  0,5
F  0,5
0
F  0,5

5. This parameter should be monotonous.
6. It should be integral.
7. The quality criterion should take into account the displacement of optimal
separation boundary for a combined cascade with respect to a similar
boundary in an individual element.
To explain the generalized quality criterion, Fig. 1 shows some structural
schemes of a three-element combined cascade, heir connection functions and the
respective graphs.
Quality criterion for the evaluation of the perfection of a combined scheme
satisfying all above conditions has the following form:
4
J ( F0 )  F0 1  F0  F ( F0 )
(4)
3
where J ( F0 ) is the quality criterion;
4/3 is a normalized factor;
F ( F0 ) is the connection function (fractional extraction into a fine product for a
combined cascade);
F0 is fractional extraction for which
1
F F0 
(5)
2
The parameter E(F) is determined as


1
(6)
E ( F )  
 1

 1  E ( F0 ) 
where


 

E ( F0 ) 
F0
1
 1  F ( F )dF   F ( F )dF
0
0
0
0
0
F0
If we use the plots given in Fig. 1 for these calculations, it becomes clear that
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(7)
E  ( F0 )  1  S t ,
where S t is Trompf’s area modified for these conditions [4].
Let us dwell on the conditions stated above.
The first condition is due to the purpose of the index to evaluate the perfection
degree of the separation scheme from the standpoint of approaching the ideal process
independently of the characteristics of individual elements.
The second condition is similar to an unambiguity requirement. From the
standpoint of the problem under study, the direct and inverted schemes are
symmetrical, and therefore their perfection degree is equivalent. The formulated
criterion satisfies this requirement, which can be checked using formula (4) with
substitution F0  1  F0 .
Normalization conditions for the conditions 3, 4 are chosen for the sake of
clearness and convenience, as well as for reasons of the index sensitivity.
If J(F0) ≤ 1, such combined cascade has no advantages over a single column.
S t  1  0,5  0,5  0,5  0,5  0,5  0,5  0,5  1  0,5  0,5  0,5  0,5  0,5  1  3  0,25  0,25
For the connection function F(F0) = F0, the plot is symmetrical (Fig. 2).
Taking this into account,
3
E  ( F0 )  1  S t  1  0,25 
4
Hence, a normalizing factor equal to 4/3 arises. In this case, the total index amounts to
4
 1
 1
J ( F0 )   0,5  0,5
 1   3  1
3
 1  0,75  3
For an ideal combined cascade, the modified Trompf’s area tends to zero, i.e.
S t  0, and this means that E  ( F0 )  1, E ( F )  . . Thus,
J (E )  
under the condition that
1
F0  F0

F ( F0 )  0,5
F0  F0
0
F0  F0

For a two-element cascade, there are only four operating schemes (Fig.3). Their
principal parameters are given in Table 1.
Table 1.
No. in Fig. 3
I
II
Connection function
F ( F0 )  1  (1  F0 ) 2
F ( F0 )  F02
E ( F0 )
F0
0,2929
0,7971
4,1213
4,1213
J ( F0 )
1,221
1,221
III
F ( F0 ) 
F0
1  F0  F02
0,38197
4,1990
1,3384
IV
F ( F0 ) 
F02
1  F0  F02
0,6180
4,1990
1,3384
It follows from Table 1 that:
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1) direct and inverted schemes are characterized by the same quality index value,
although they ensure separation over different boundaries;
2) combined classifiers with recycling of fine or coarse product are more efficient
than simple schemes.
For a three-element cascades there are 47 operating schemes. They were
analyzed from the standpoint of the described method. Fourteen of these schemes
have the parameter J ( F0 )  1. These schemes are of no interest. The most efficient
schemes are shown in Fig. 4, and their principal parameters are summarized in Table
2.
Table 2. Principal parameters for combined three-element cascades
No. in Fig. Connection function
1
2 F0  2 F02  F03
F
1  F0  F 2
2
F
F03
1  F0  F02
E ( F0 )
F0
0,261
0,739
5,1249
5,1249
J ( E0 )
1,5912
1,5912
3
F
2 F0  F02
1  F02 (1  F0 )
0,7549
5,4204
1,4474
4
F
F03
1  F02 (1  F0 )
0,2541
5,4204
1,4474
5
F
F03
F03  F0  1
0,6823
F0
(1  F0 ) 3  F0
0,3177
6
F
5,3799
5,3799
1,5905
1,5905
7
F
F0 1  F0 (1  F0 )
1  2F0 (1  F0 )
8
F
F03
2 F02  2 F0  1
0,6478
5,1061
1,5634
9
F
F02
1  2 F0 (1  F0 )
0,5
5,5178
1,8393
0,3522
5,1061
1,5634
10
F
F02
1  F0  F03
0,5549
5,1803
1,7127
11
F
F0 (2  F0 )
1  F0  F02
0,5
4,7262
1,5754
12
F
F02 (2  F0 )
(1  F0 ) 3  F0
0,4450
5,1803
1,7127
It follows from Table 2 that here the direct and inverted schemes have also different
connection functions and values of the parameters F0 , , but receive the same value of
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the quality criterion according to the proposed method. On the basis of this Table, we
can arrange all combined three-element cascades by their perfection degree. In this
case the apparatus schematically presented in Fig. 4, 9 will occupy the first place.
Using this method, combined apparatuses comprising any number of elements
can be analyzed, and the most efficient ones can be determined
.
CONCLUSION
A method of unbiased evaluation of any complicated combined classifiers is
developed, which makes it possible to determine principal parameters characterizing a
combined scheme. On the basis of these parameters, the most efficient scheme can be
chosen among those under study.
REFERENCES
1. Barsky, M.D. Fractionating of Powders. Moscow: “Nedra”, 1980.
2. Knory, F.M. Predicting the Performance of Multistage Separation Process. CRC Press,
Boca Raton, Florida, 2000.
3. Barsky, E. and Barsky, M. Combined Separators. (manuscript)
4. Trompf, K.F. Nene Wege fur die Beurteiling der Aufbereiting Von Steinkohle, Gluckauf,
No. 73, 1937, 125-131.
5. Barsky, E. and Buikis, M. Mathematical model for cascade separation at identical stages of
the separator. Latvian Journal of Physics and Technical Science, No. 5, 2001, 22-32.
FIGURES
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