Improving the Computational Model for Approximation of ... - Core

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ScienceDirect Procedia Engineering 150 (2016) 2073 – 2079

International Conference on Industrial Engineering, ICIE 2016

Improving the Computational Model for Approximation of Particle Functions over Diameter of Dust in the Work Area and at the Border of the Sanitary Protection Zone A.V. Azarov*, N.S. Zhukova, V.F. Sidorenko Volgograd State University of Architecture and Civil Engineering, Akademicheskaya Str., 1, Volgograd, 400074, Russia

Abstract This paper describes the basic ways of solving the problem of experimental data approximation. Disperse composition of crushed gypsum dust was analyzed; the analysis results are presented herein. Approximating function of particulate dust composition function is a piecewise function, defined as a three-tier spline "direct-parabola-hyperbole." The approximation aims at finding seven function factors and two nodal points. The least square method was used to estimate the unknown parameters of regression models for the sample data. To apply this method to experimental data, a program complex calculation models for the approximation of the integral representation of the mass distribution function of particles in the dust diameter in CAS Maple was presented. It defines a function that describes the distribution of the particulate composition of the dust released from the open warehouse storage of crushed gypsum rock with the smallest error. © 2016 2016The TheAuthors. Authors. Published by Elsevier © Published by Elsevier Ltd.Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility ofthe organizing committee of ICIE 2016. Peer-review under responsibility of the organizing committee of ICIE 2016 Keywords: approximation; least square method; dispersed composition; dust; distribution function; error

1. Introduction During the dust study, as a rule, an interest in taken in the degree of dispersion and in character of particle size distribution. Knowing this distribution the size of the smallest and largest particles; particle size of the most by volume; percentage of dust particles in the specified range of particle diameters equivalent [1-7].

* Corresponding author. Tel.: +7-968-283-80-00 E-mail address: [email protected]

1877-7058 © 2016 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of ICIE 2016

doi:10.1016/j.proeng.2016.07.241

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In most cases, the particulate composition of the dust cannot be described in probabilistic-logarithmic coordinates of the same line, i.e. log-normal law [8-10]. Nowadays variety of means for approximation algorithms is widely used. One of these software systems is computer algebra system (CAS) Maple. This software package contains computational model implemented in Maple computer algebra system (version 10-18) and is intended for automatic processing of the results of the experimental data by the method of least squares. The software package is implemented three types of approximation of presenting experimental studies results by piecewise features that appear at certain intervals: two straight lines; straight and parabola; line, parabola and hyperbola [9-12]. The result of the software system is to determine an approximation representation of functions coefficients that describe the distribution of particulate dust composition with the lowest error and a visual representation of piecewise functions CAS Maple visualization tools. 2. Main part In the production of gypsum binder, while conducting research, dust sampling was conducted in the working area (various dust emission sources, technological equipment), as well as on the border of the sanitary protection zone. To investigate dust sampling microscopic dust particulate composition determination using a PC in software product "Dust 1" technique is applied [13-15]. The program calculates the mean diameter and defines a number of different sized particles, area occupied by the dust particle, and also determines the total D(dp). After scanning micrographs disperse composition of dust general population is determined. The micrographs of dust samples were obtained on a two-beam scanning electron microscope Versa 3D. It should be noted that the error of measurement information transmission and data comprise values close to zero. Later in the course of research there is a problem identifying the true nature of the dependence of the studied parameters, for this approximation is used - an approximate description of the correlation variables fit equation of the functional dependence, transmitting basic tendency dependence [16]. Solution of the problem of approximation [17-19] of experimental data is as follows: x identification of functions for approximation of experimental data; x finding the coefficients of functions based on the minimization of error representation of the experimental data functions; x summing up of what kinds of functions are approximated by the experimental data in a better way. Almost kind of approximating function y = D (dp) can be determined as follows. According to the table, namely the number of pixels, particle diameters, amount of said particle diameter of program "Dust 1" scatter plot function is constructed, and then held in the picture smooth curves, possibly reflecting the well location point character. According to information received this way, by the curve is set kind of approximating functions (usually among the simplest in form of analytic functions) (Figure 1). Fig. 1 considers procedure for determining the type of function for approximating the graph of values of integral distribution function of the mass of the particle diameters of the dust samples released from the open warehouse storage of crushed gypsum. Approaching the distribution function of the particulate composition of the dust of crushed gypsum stone is a piecewise function defined on intervals: before Dcr1 – straight, from Dcr1 to Dcr2 – parabola, from Dcr2 hyperbole. Approximation problem is to find seven functions coefficients a, c, f, k, b, w, r and the two nodal points Dcr1, Dcr2. One of the basic methods of regression analysis to estimate the unknown parameters of the regression model by sampling is least squares method [8]. This method is based on minimizing the sum of squared residuals. For processing of the results of experimental data by the method of least squares program complex calculation models for the approximation of the integral representation of the mass distribution function of particles in the dust diameter CAS Maple was developed. Consider the procedure for calculating a selected sample of crushed gypsum rock dust, graphic cumulative distribution function of which, is shown in Fig. 1.

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Fig.1. Approximating function defined on the intervals for the submission of approximating distribution of mass particles in accordance with diameter

The task source data, i.e. total number of points n := 62, an array diameter Xd := [2,257; 3,192;…;70,793], an array of the number of particles corresponding to each diameter of Yd := [14; 1;…;1]. Next, we find the total volume of the particles Ysum: 3 § § ·· § Xd >i @ · ¨ ¨ Yd >i @ ˜ 4 ˜ Pi˜ ¨ ¸¸ ¸ ¨ 2 ¹ ¨ ¸¸ © Ysum : evalf ¨ sum ¨ , i 1...n ¸ ¸ 3 ¨ ¨ ¸¸ ¨ ¨ ¸¸ © ¹¹ ©

(1)

The calculation result is: Ysum:=9,624 · 105 We set a cycle "for i from 1 to n do Ydd[i]" the result of the calculation of which is the values of massive Ydd[i], which determines the percentage of the volume of each particle from the total: 3 § · § Xd >i @ · ¨ Yd >i @ ˜ 4 ˜ Pi˜ ¨ ¸ ¸ 2 ¹ ¨ ¸ © for i from 1 to n do Ydd >i @ : evalf ¨ ˜100 ¸ ; end do; ˜ 3 Ysum ¨ ¸ ¨ ¸ © ¹

(2)

We get: Ydd1:=0,009 Ydd2:=0,002 … Ydd62:=19,362 Then set a cycle "for i from 1 to n do Y[i]", the result of the calculation of which is the value of the array Y[i] determines the percentage of the volume of particles İ each diameter Xd from the total volume:

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for i from 1 to n do Y >i @ : sum Ydd > j @ , j

1...i ; end do;

(3)

We get: Y1:=0,009 Y2:=0,011 … Y62:=100,0 Since all the constructions carried out on a logarithmic grid, we set the loop "for i from 1 to n do X[i]" to calculate the decimal logarithm of the diameter of each particle X [i]:



§ ln  Xd >i @ for i from 1 to n do X >i @ : evalf ¨ ¨ ln 10 ©

·¸ ; end do;

(4)

¸ ¹

We get: X1:=0,354 X2:=0,504 … X62:=1,850 The next step is to determine the logarithm of the critical diameter LDcr1, LDcr2, LDcr3. Since the value Dcr3 is the maximum particle diameter (Figure 1), then: Dcr3 := max(Xd) ;

(5)

§  ln  LDcr 3 · LDcr 3 : evalf ¨ ¸; ¨ ln 10 ¸ © ¹

(6)

We have: LDcr3 := 1,85 Approximate points by three functions: straight line, parabola, hyperbole each of these functions are unknown parameters to be determined. For direct k and b; for parabola a, c, f; for hyperbole w, r. Critical points of these functions have the same value and the same value of derivatives, so it is possible to record a series of equations to determine the coefficients 3

w:

 a,c,Dɫr 2 o 2aLDɫr 2  Dɫr 2

r:

 a, c, f , Dɫr 2 o 3aLDɫr 2  Dɫr 2

2

 cLDɫr 2  Dɫr 2 ;

2

 2cLDɫr 2  Dɫr 2  f ;

LDɫr 3 : 0, 434ln  Dɫr 2 ;

(7) (8) (9)

k:

 a, c, Dɫr1 o 2aLDɫr1 Dɫr1  c ;

r:

 a, f , Dɫr1 o aLDɫr1 Dɫr1

2

f ;

Define some piecewise function Ȟ that will approximate the point:

(10) (11)

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k (a, c, Dɫr1) ˜ ln(d ) § · 0  d  Dɫr1,  b(a, f , Dɫr1); ¨ ¸ ln(10) ¨ ¸ ¨ ¸ 2 § ln  d · § ln  d · ¨ ¸ Dɫr1  d d Dɫr 2, a ¨  c¨  f; ¸ ¸ ¸ Q :  a, c, f , Dɫr1, Dɫr 2 o piecewise ¨ ln(10) ¹ ln(10) ¹ © © ¨ ¸ w(a, c, Dɫr 2) ˜ ln(10) ¸ ¨ ;¸ ¨ Dɫr 2  d d Dɫr 3, r  a, c, f , Dɫr 2  ln  d ¸ ¨ ¸ ¨ d ! Dɫr 3, 100; © ¹

(12)

Since the parameters of the piecewise function Ȟ approximating point are unknown, then we define the parameters based on minimizing of error function [20], i.e., using the least squares method.

E:

n

 a, c, f , Dɫr1, Dɫr 2 o ¦ Y j  v  a, c, f , Xd j , Dɫr1, Dɫr 2

2

;

(13)

j 1

I.e. from the formula (13), it follows that the values of Ȟ := (a, c, f, d, Dcr1, Dcr2) is subtracted from the points obtained experimentally Y[i] and obtain the value of the error, which put in a square, summarize error values for all points and obtain the total error function Ȟ := (a, c, f, d, Dcr1, Dcr2) at all points. So, as the values of the parameters a, c, f, k, b, w, r, Dcr1, Dcr2, Dcr3 are not known, it is possible to know under what combinations of factors the function E:=( a, c, f, Dcr1, Dcr2) reaches minimum for this, command "minimize" is used. In order to simplify the calculations, it is necessary that the critical points are selected automatically. All domain diameters may "pass" with certain "steps" pre-defined starting points: Shag1:= 10 ; Shag 2 := 10

StartDɫr 2 : min  Xd  Shag 2 ;

(14)

StartDɫr2 := 12,257.

StartDɫr1: min  Xd ;

(15)

StartDɫr1 := 2,257. RealDɫr1 : StartDɫr1 ;

(16)

RealDɫr2 : StartDɫr2 ;

(17)

RealDɫr2:= 12,257; RealDɫr1:= 2,257. As a result, we can determine the value of "RealResultE" starting error for calculation: Re al Re sultE : minimize  E  a, c, f , StartDɫr1, StartDɫr 2 ;

We obtain: RealResultE := 12628,943 Defining the selection of the critical points of the cycle values:

(18)

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for iter1 from StartDɫr1 by Shag1 to  Dɫr 3  Shag 2 do StartDɫr 2 : : iter1  Shag 2 : for iter 2 from StartDɫr 2 by Shag 2 to Dɫr 3 do CurResultE : : minimize  E  a, c, f , iter1, iter 2 : if CurResultE  RealResultE then RealResultE :

(19)

: CurResultE; RealDɫr1: iter1; RealDɫr2 : iter2;end if end do end do; As a result: StartDɫr2 := 12,257 StartDɫr2 := 22,257 StartDɫr2 := 22,257 RealResultE; 331,831 RealDɫr1; 12,257 RealDɫr2; 62,257 Then error is calculated. If the obtained error is smaller than "RealResultE", it substitutes the value of the past error and each such operation critical points at which this error has been reached are stored. Values that were got after the work cycle, can be substituted into the function E:=(a, c, f, RealDɫr1, RealDɫr2) values RealDɫr1, RealDɫr2 and determined under what factors a, c, f it is minimized. We determine the remaining unknown coefficients: a := 181,529; c := - 403,010; f := 221,414; k(a,c,RealDɫr1) = -7,864; b(a,c,RealDɫr1) = 6,378; w(a,c,RealDɫr2) = -799,574; r(a,c,RealDɫr2) = 528,346; Consequently, the distribution function of dust particulate composition can be written as follows:

D d p

­ °7 ,864 ˜ lg(d p )  6 ,378 under d p d 12 , 257 ° ° 2 ®181,529 ˜ lg (d p )  403, 010 ˜ lg(d p )  221, 414 under 12 , 257 d d p d 62 , 257 ° °528,346  799 ,574 under 62 , 257 d d p  70 , 793 °¯ lg(d p )

(20)

Figure 2 shows the distribution function of the particulate composition of dust, built in the software package developed by the formula (20) 3. Conclusions

Thus, the proposed algorithm allows defining a function that describes the distribution of the particulate composition of the dust released from the open warehouse storage of crushed gypsum rock with the smallest error. It is possible to use in further calculations to determine the fractional relationships.

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Fig. 2. The approximation of the integral distribution function in D(dp) three-tier spline

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