# Optimization of the two-stage cyclone-type dust concentrator – design and operating parameters

This article presents experimental research results data and optimization of a two-stage cyclone-type dust concentrator and the design and operating parameters data. To resolve the assigned optimization task, SUMT (Sequential Unconstrained Minimization Technique) was applied. The Davidon-Fletcher-Powell method was used to find the minimum of a function, afterwards the iterative procedure was realized. The results were processed using joint analysis of the characteristic curve with regression equation of dust collecting efficiency and liquid resistance.

## 1 Introduction

A large part of the technological processes of building materials enterprises, as well as a number of other branches of industry are the mechanical manipulation and material handling processes, related to the gravity flow of large amounts of granular materials on closed gutters and accompanied by intensive release of dust.

Traditional aspiration systems, characterized by volumetric emission of dusty air, are widely used for localization of sources of dust emissions. Reduction of the power consumption and the dust emission of the system can be achieved as a result of purification of the aspirated air in centrifugal dust concentrators – cyclones, the conclusion of the trapped dust of which is carried out with part of the carrier gas stream [1].

The working principle of the two-stage cyclone-type dust concentrator is ejection of particles from the gas-dust flow by centrifugal force (Figure 1). Through the inlet pipe into the cyclone part of the dust concentrator between the body and the cylindrical insert a polluted gas and dust stream is fed. Particles of dust by centrifugal force hit the walls of the body and are carried out from the internal cavity of the two-stage dust concentrator by the air flow through the annular cavity and then enter the recirculation line. Part of the air flow, which remained after the first stage of purification, is directed to the countercurrent cyclone for the second stage of purification through the annular gap between the upper end of the cylindrical insert and the lid.

## 2 Methodology of determination of the

## two-stage cyclone-type dust concentrator

## optimal design parameters

As a result of the experimental research, it was found that the flow rate of the air supplied for cleaning exerts a global influence on the main part of the efficiency of the two-stage dust concentrator capture and the value of h_{TSDC} increases with the growth in the flow rate of the filtered air due to the first and second stages of purification. Increase in dust collection efficiency of the first h_{I} and second h_{II} stages of purification with a growth in the flow rate of the gas supplied for purification have a positive impact on the dust separation process [2, 3].

Growth of the factor of the relative consumption of recirculation air in the first stage of cleaning h_{I} increases the efficiency of air purification, the cleaning efficiency is reduced due to the decrease in the velocity of the gas flow in the second stage h_{II}. Increase the geometric characteristics of the height of the entrance pit in the second stage and the depth of immersion of an exhaust pipe can induce a random decrease or increase in the overall efficiency of the two-stage dust concentrator. Growth of the flow rate of the filtered gas increases the hydraulic resistance of the two-stage dust concentrator in the first stage and lowers in the second stage of purification. Therefore, the use of factors X_{1}, X_{2} and X_{3} limits the overall efficiency of the two-stage dust concentrator [4].

Optimization of the two-stage dust concentrator design and operating parameters was reduced to the determination of the values of the factors entering into the regression equation (1), when the maximum value of the response function is achieved – the efficiency of dust collection of the two-stage dust concentrator [5].

[eta]_{TSDC} = 91.15 – 0.18 · X_{1} + 0.26 · X_{2}

+ 1.08 · X_{3} + 0.29 · X_{4} + 0.22 · X_{1} · X_{2}

– 0.88 · X_{1} · X_{4} – 0.79 · X_{12} –0.5 · X_{22}

– 0.44 · X_{12} – 0.13 · X_{42 ⇥}(1)

However, the maximum possible efficiency of dust collection of the two-stage dust concentrator can take place at the values of the factors X_{1}, X_{2}, X_{3}, X_{4}, which are unacceptable for the operation conditions for the following reasons:

1. Optimal relative recirculation airflow rate (X_{4}) in most cases, in practice, will not correspond to the required flow rate in the recirculation line of the aspiration system.

2. Optimal values of factors may be unacceptable, because they can create the unacceptably high hydraulic resistance of the machine. This, above all, refers to the flow of filtered air (X_{3}), which exerts a dominant influence both on the cleaning efficiency (1), and on the hydraulic resistance (2) of the two-stage dust concentrator:

ΔP_{TSDC} = 1605 – 64 · X_{1} + 66 · X_{2} + 532 · X_{3}

– 71 · X_{4} + 69 · X^{2}_{1} + 62 · X^{2}_{2} + 78 · X^{2}_{3} ⇥(2)

Therefore, in the future factor X_{4} was excluded from consideration and fixed at certain levels of variation. And the optimal values of the factors X_{1}, X_{2}, X_{3} were limited to the maximum possible level of the hydraulic resistance of the machine [4].

## 3 Optimization task solution

To resolve the assigned optimization task, SUMT (Sequential Unconstrained Minimization Technique) was applied firstly proposed by Carroll [6] and which has been developed further by Fiacco and McCormick [7, 8], which is based on the idea of converting the task of minimizing a function:

⇥

⇥(3)

with the appropriate restrictions imposed on x, in the problem of finding the minimum without constraints of the function:

⇥

⇥(4)

The function P(x) is a penalty function. It is necessary that, in violation of restrictions, it “fine” the function Z, increasing its value. In this case, the minimum of the function Z will be within the constraint area.

The penalty function P(x) is written as follows:

⇥ (5)

where C_{j}(x)>0 – the constraints imposed on x,

m – number of constraints,

r – the positive value.

If the imposed constraints are of the form h(x)≤ 0, then they are written as – h(x) ≥ 0.

With allowance for (5), the function Z=j( , r) takes the form:

⇥ (6)

If takes valid values for which C_{j}( )≥0, then Z takes values that are greater than the corresponding values of f ( ), and the difference can be reduced by the fact that r can be a very small value. But if

x takes a value that, while acceptable, is close to the boundary of the constraint area, or at least one of the functions C_{j}( ) is close to zero, then the value of the function P( ) and hence the value of the function Z will become very large. Thus, the effect of the function P( ) is to create a “crest with steep edges” along each boundary of the constraint area. Therefore, if the search starts from an admissible point and the minimum of the function j( ,r) is searched without any restrictions, then the minimum of the original function (3) will, of course, be reached within the admissible domain for the constrained task. Assuming that r is a sufficiently small value for the effect of the function P( ) to be small at the minimum point, we can make the minimum point of the function j( ,r) without constraints coinciding with the minimum point of the function f(x) with constraints.

Optimization of the two-stage dust concentrator design and operating parameters involves finding the maximum of the function (3) .

SUMT, the block diagram of which is shown in Figure 2, assumes minimization of the function f( ). Therefore, as a function f(x) a function

⇥(7)

was adopted, with the factor X_{4} being fixed at certain levels (-1…+1).

The penalty function with eight imposed constraints

⇥(8)

where DP_{TSDC} – the hydraulic resistance of the two-stage dust concentrator, calculated by the expression (2);

A and B – respectively, the lower and upper levels of the set hydraulic resistance of the machine.

The sum of the expressions (7) and (8) is the initial equation j( ,r), the solution of the minimization task of which makes it possible to obtain the optimal values of the factors X_{1}, X_{2}, X_{3} for a given value X_{4} and is performed as follows.

Given the initial point _{0} defined by the values X_{1}, X_{2}, X_{3}, and by checking for its admissibility, we form the function j( ,r), assuming that r=r_{0}=1. After finding the minimum of the function j( _{0},r_{0}) at the point _{k}^{*} using Davidon-Fletcher-Powell meth_{0}/10, and = _{k}^{*}. After that a repeated operation of minimization follows.

_{k}) is minimized, the minimum of which is at the point ^{*}_{k} and is used as a subsequent point of the iterative procedure for minimizing the function_{k-1}), where r_{k+1}=r_{k}/10.

_{k} decreases and tends to zero, hence the sequence of points of minima converges to the solution of the task of minimizing the function (7).

_{1}, X_{2}, X_{3} of the regression equation (1) the efficiency of the dust collecting of the two-stage dust concentrator.

_{1}, X_{2}, X_{3}, for a number of fixed values of the relative recirculation airflow rate (X_{4}) and various limitations of the maximum possible hydraulic resistance of the two-stage dust concentrator are presented in the Table 2.

_{1}, X_{2}, X_{3}. Moreover, the values of these factors vary in a rather wide range.

_{1}) and the depth of immersion of an exhaust pipe (X_{2}) of the two-stage dust concentrator will depend on the specified operational parameters of the machine: hydraulic resistance _{TSDC}, flow of filtered air (X_{3}) and relative recirculation airflow rate (X_{4}). This greatly complicates the technique of engineering calculation of the proposed machine. Therefore in the future an attempt will be made to fix these factors at certain levels with the exception of factor X_{4}, because its value will be equal to the required flow rate in the recirculation line of the aspiration system [9].

## 4 Implications

Analysis of the dependence of the maximum possible dust collection efficiency of the TSDC on the set level of hydraulic resistance (Figure 3) [10] showed that the optimal value of the hydraulic resistance of the TSDC is 1800...1900 Pa.

Given a level of DP_{TSDC }= 1800 Pa, combinations of the optimal values of factors X_{1}, X_{2}, X_{3} were determined with X_{4} varying from -1 to +1 in steps of 0.1 (Figure 4).

After a joint analysis of the obtained graphical dependencies with the regression equations of the dust collection efficiency (1) and hydraulic resistance (2) of the TSDC, the following combination of factor values was adopted: X_{1 }= 0.15; X_{2 }= 0.25; X_{3 }= 0.4.

_{TSDC} = 95.46 + 0.07 · X_{4} – 0.13 · X^{2}^{}_{4}⇥(9)

_{TSDC} = 1830 – 71 · X_{4⇥}(10)

5 Conclusions

_{4}, which is confirmed by the obtained dependence (9). The increase in hydraulic resistance is insignificant (5.5 %), and in the area of high values of the relative recirculation airflow rate there is even a drop in the hydraulic resistance of the machine, which explains a small decrease in the overall efficiency of dust collection.

_{1}) – h_{s }= 133 mm, the depth of immersion of an exhaust pipe (_{2}) – h_{out }= 108 mm, the airflow rate for cleaning (_{3}) - Q_{a} = 940 m^{3}/hours.

_{k}is shown in Table 4. The optimal air velocity in the section of the TSDC was determined by the formula:

## Acknowledgments

The article was prepared within the development program of the Belgorod State Technological University (BSTU), using the equipment of the High Technology Center at BSTU.