Modelling of Combined Cascade Classifiers

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

(1)

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]:

(2)

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)

where B = (gd/w2)(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]:

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 the 1st 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. 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):

-  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

Hence, the maximal possible number of connections between n elements amounts to

The minimal number of connections is:

Taking this into account, the maximal number of outlets is

At a fixed number of free outlets in a combined scheme P, the number of connections between n elements amounts to

(4)

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 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)

Taking (4) into account, we can write:

(6)

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:

(7)

where = 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)

,

and the number of working schemes is only

At n = 3 we, respectively, obtain:

At n = 4,

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

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,

In this case, the connective function is:

For the inverted scheme II, the connective function can be written as

For the scheme III, the total outlet of fine product can be written in the form of an infinite series:

We obtain a geometric progression, and it is clear that

For the inverted scheme IV, one can obtain from similar considerations:

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

We represent this scheme in the form of a connective matrix:

______

1 2 3

______

1 0

______

2 0

______

3 0

______

Matrix element aij at i ¹ 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;

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).

3) 

Each apparatus has an outlet of both fine (F0) and coarse (1 - F0) products;

4) 

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:

(8)

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

(9)

where AT* 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:

(10)

It follows from (9) that the connective function is based on the balance of material flows. In compliance with (1) we can write

(11)

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:

It follows from the balance condition that

(12)

In the general case, the connective equation is written as

(13)

In a particular case, for an arbitrary three-element cascade we can write:

Taking this into account, the matrix form of the connective equation will acquire the form:

(14)

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:

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: