CHARACTERISTICS OF COMPLEX SEPARATION SCHEMES AND AN ERROR OF SEPARATION PRODUCTS OUTPUT DETERMINATION

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1 Górnctwo Geonżynera Rok 0 Zeszyt / 006 Igor Konstantnovch Mladetskj * Petr Ivanovch Plov * Ekaterna Nkolaevna Kobets * Tasya Igorevna Markova * CHARACTERISTICS OF COMPLEX SEPARATION SCHEMES AND AN ERROR OF SEPARATION PRODUCTS OUTPUT DETERMINATION The separaton characterstc of the scheme s determned by separaton characterstcs of each operaton ther number and sequence of connecton. Let s assume that before mxture the quantty of partcles wth separatng sgn X was P and after mxture t was P (Fg. ) and was equal to the sum of quanttes of partcles before mxture and entered nto the gven operaton wth crculatng loadng.e. m + j j Р Р P where j s a seral number of crculatng loadng. The crculaton s formed by consecutve connecton of devces.e. n P P E j + Then m n P P + P E j + Whence n m P P E j + * Natonal Mnng Unversty of Ukrane Dnepropetrowsk Ukrana 9

2 Thus the mxture operaton separaton characterstc s P CM P m n /( P j + E Let s apply the receved formulas on examples. Fg.. Consecutve connecton of separaton and mxture operatons 0

3 We assume there s a technologcal block for separaton (Fg. ). Fg.. The scheme of a technologcal separaton block Let s make the separaton characterstc of ths connecton. Accordng to laws of consecutve connecton P β PP OCP For the feedback technologcal connecton the characterstc s Р ОС РР Makng dentcal transformatons we shall receve the requred formula: РР РР РР Р β РР Р( Р) Р+ РР Let s complcate the separaton devces connecton (Fg. ). The sequence of separaton characterstcs for such connecton s as follows Р Р РР Р Р β FB FB and the feedback characterstc s Р FB РР Р РР Р FB FB

4 Makng dentcal transformatons we shall receve: PPP P β PP 5 PP 5 We shall not carry out the further transformatons as they are trval. The connecton separaton characterstc turns out rather complex but there are dentcal transformatons resultng n an obvous knd expresson. Fg.. The scheme of a complcated technologcal separaton block However for the crcut shown n Fgure the dentcal transformatons become so bulky that the probablty of a mstake comes near to one. Therefore for such scheme t s necessary to make the equatons of fractons balance for each separaton recepton: ( ) ( ) р Р р + р + р ; р Р р + р + р ; 0 0 ( ) ( ) р Р р + р ; р Р р + р ; р Р р ; р Р р ; () р Р р ; р Р р ; p p + p OC To determne outputs of all products t s necessary to solve a system of equatons (). As t has a hgh order the soluton s carred out numercally for example by a smple teratons method. Numercal solutons demand specal formal transformaton to obtan an algorthmc form.

5 Fg.. The scheme of a technologcal separaton block wth deep feedback For a system of lnear equatons t looks lke: uur() uur( ) ur Х АХ + В () where B ur s a vector of free terms; A s a matrx of factors; Х uur s a vector of requred roots approxmatons. The matrx of factors A wll be as follows Р Р Р Р Р Р Р Р Р Р Р Р uur and the vector of free terms s Х ( РХ0 Р Х ). For the analyss of the separaton scheme separaton characterstcs of all devces are set. As the separaton s bnary each devce has two separaton characterstcs. As the result the scheme shown n Fgure s characterzed by eght separaton characterstcs. They are

6 represented as dscrete set values for example by sx values. Thus the ntal data on the separaton scheme wll be as a matrx of 6 8 sze. The calculaton of the separaton crcut can be done n the followng way. The frst column of the separaton characterstcs matrx s chosen and the system of the equatons () s solved as a result we shall receve the vector of the frst fracton of partcles separaton. Than the second column of the separaton characterstcs matrx s chosen and agan the system of the equatons s solved. The vector of the second fracton of partcles separaton s receved. The soluton s made as many tmes as many columns are n the separaton characterstcs matrx. As a result we shall receve a new matrx of 6 8 sze. To determne the -th product output t s necessary to sum the elements of the correspondng -th soluton matrx row. To determne qualty ndces t s necessary to multply row elements and the fracton qualty parameters and to sum the receved products. So for the crcut n Fgure 5 the soluton s obtaned as the matrx The ffth product s the enrched product and the eghth one s depleted. The sum of outputs of these products s equal to one as t s receved as a result of calculatons. The technologcal balance of concentraton processes s connected wth the determnaton of output parameters of each technologcal pont. When the technology has deep recycles such calculatons become nconvenent and are connected wth teratve procedures to select the requred parameters resultng n materal balance. The purpose of the gven argumentatons s to fnd formal algorthm whch would allow makng calculatons by the computer. Let s assume that we have some separaton crcut (Fg. 5). Fg. 5. The separaton crcut wth the recycle

7 Let s wrte down a system of balance equatons for the gven crcut: Q ( Q + Q ); Q ( Q + Q ); 0 0 Q Q ; Q Q The number of the equatons s equal to the number of unknown values therefore the soluton s sngle-valued and takng nto account the low order of the system t can be solved by a substtuton. We shall determne an output of the enrched product: Q Q 0 ; Q Q0 Thus on the bass of known outputs on separate operatons and known laws of products transformaton n separaton crcuts t s possble to determne the output of any product n relaton to an ntal feedng. Indvdual outputs are calculated when requred by the Q IJ rato of charges IJ. Q0 Nowadays the mathematcal approach s developed allowng calculaton of estmatng accuracy of a product output for a rather smple technology of mneral wealth enrchment []. Such estmaton s carred out wth the help of partal dfferentaton of the product output functon. The more complcated s the crcut the more complcated s the operaton of dfferentaton. Therefore t s necessary to have a method allowng to execute the mentoned estmaton of accuracy for the crcut of any complexty. So f to calculate outputs at known parameters of the product preparaton and then to gve ncrements to these parameters and agan make calculatons t s possble to receve an estmaton of output dervatves on correspondng parameters that serves as a measure of accuracy of nterested parameters estmaton. Separaton characterstcs Р(α) and functons of partcles dstrbuton on fractons F(α) can be receved expermentally. For ths reason each fracton defnton error one can consder as known: σ σ σ σ σ σ р α р α р αп F α F α F αn The complex crcut product output wll be calculated on the bass of the repeated soluton of system of the lnear equatons of products balance. At the begnnng we assume that n the result the certan value s obtaned. Knowng t we gve an ncrement to the functon Р(α). For example we shft t upwards or downwards and agan calculate an output. Now t wll be. 5

8 For the error defnton t s necessary for functons F(α) and Р(α) to have dervatves on all the range α. In ths case the dfferental of the functon wll be: ΔP( α ) P( α ) P ( α ) ΔΔF( α ) ΔF( α ) ΔF ( α ) Then the dfferental of the argument s: Δ ( α ) P( α) ΔF( α) P( α) ΔF( α ) Δ P F( α ) P( α ) ΔF ( α ) P( α ) ΔF ( α ) The functon dervatves can be wrtten down as follows: P Δ P( α) ; P ΔP( α ) F Δ F ( α ) F ΔΔF( α ) Then the error from each varable change wll be formed from errors of each product fracton defnton as functons F(α) and Р(α) are ntegral: n ( ) Δ p α σ P σpα ; ΔP( α ) n ( ) Δ p α σ F σfα ΔF( α ) As a result the general error of the numercal output estmaton can be wrtten by the followng expresson: σ σ +σ Р F As an example we shall gve the calculatons of the output error for the separaton crcut shown n Fgure 6. ) We determne the product output of the complex crcut on the bass of the repeated soluton of the system of lnear equatons of products balance. As a result we have receved the certan value. 6 Knowng t we shft the functon Р(α) downwards and agan calculate the output.

9 Fg. 6. The technologcal separaton block ) Usng the gven equatons we have receved ther solutons whch are regstered n Table. ΔP( α ) P( α ) P ( α ) ΔΔF( α ) ΔF( α ) ΔF ( α ) TABLE α Р(α) F(α) We calculate an argument dfferental. The receved values are regstered n Table. Δ ( α ) P( α ) ΔF( α ) P ( α ) ΔF( α ) P Δ ( α ) P( α ) ΔF( α ) P( α ) ΔF ( α ) F TABLE α p (α) F (α)

10 Functon dervatves are wrtten down as follows: P Δ P( α) ; P ΔP( α ) F Δ F ( α ) F ΔΔF( α ) The values receved at ther calculaton are gven n Table. TABLE α p (α)/ Р F (α)/ F The error from each varable change s calculated as follows: n ( ) Δ p α σ P σpα ; ΔP( α ) n ( ) Δ p α σ F σfα ΔF( α ) We have receved the values of each varable and have wrtten them n Table. TABLE σ P σ F Now t s possble to calculate the general error of numercal output estmaton: σ σ +σ Р F σ σ 5% 8

11 Ths crcut s not too complex and so we stll can calculate an output error by a classcal method wth the purpose to check the offered numercal approach. Devces separaton characterstcs are dentcal and are gven n Table 5. There also the dstrbuton functon of jonts of raw materal gong to separaton s gven. TABLE 5 α Р ΔF Frst t s necessary to determne outputs of all products. For ths purpose the separaton characterstcs for each technology pont are set. The feedback separaton characterstc s Р FB РРР The feedback separaton characterstc s Р FB РРРР For each crcut pont the separaton characterstc s formed by the scheme of multplcaton of separaton characterstcs from the nput up to a pont. Pont : Р Ι РРР РРРР + РРРРРРР The denomnator at all separaton characterstcs wll be the same therefore further we shall wrte down: pont : Р РР Ι pont : Р РР Р Ι pont : Р V РР Р Р Ι pont 5: РV РРРРР Ι pont 6: Р V Р Р Ι pont 7: Р pont 8: Р pont 9: Р V РР Р Ι V РР Р Р Ι РРРРР Ι Х 9

12 Let s calculate a denomnator of separaton characterstcs (Table 6): TABLE 6 α Р Denomnator Let s wrte down all separaton characterstcs calculated accordng to the above-stated expressons (Table 7). TABLE 7 α Р Р Р Р Р 5 Р 6 Р 7 Р 8 Р We calculate the output and qualty n each pont of technology. Pont (Table 8). TABLE 8 α Р ΔF

13 α n β 0. Smlarly we calculate for other ponts of technology. Therefore we have: 0.5 β β β β β β β β Let s calculate partal outputs for each separaton recepton. The frst recepton: The second recepton: The thrd recepton:

14 The fourth recepton: The functon of the concentrate output s made smlarly to the separaton characterstcs and looks lke: ( ) ( ) + ( )( ) Let s generate expressons for partal dervatves. Accordng to the calculated partal outputs we shall wrte down the expressons: 0.9 β β ( ) 0.6 β β Let s calculate dervatves on correspondng outputs. 0.9( ) 0.9 ( ) ( ) 0 6( ) 0.6 ( ) ( ) 097( ) ( ) 079( ( )) ( ( ))

15 α ν Let s further calculate partal dervatves from partal outputs on correspon- β ν dng varables α α α α α α α α β β β ( β 0.6) Let s assume σ α σ β σν. To determne ther numercal values whch wll be the same as for numercal determnaton of errors; let s do the next actons (Fg. 7). Fg. 7. The crcut of recepton of output functon arguments ncrements

16 Accordng to functons ncrements we can determne an argument ncrement Δα for each accepted dscrete value of the functon usng the plot. Then we can fnd the average value of such ncrement on all a range of α change. It wll be the requred Δα. In ths case t we have β 0.5 Δα β β ( β 0.5) β β β ( β 0.6) β 07 0 ν 06 ν 06 ν 076 ν (0.76 ν)( ) + (0.6 ν) ν ν (0.76 ν) 076 ν 08 ν (0.8 ν)( ) + (0.76 ν) ν ν (0.8 ν) 08 ν 085 ν (0.85 ν)( ) + (0.8 ν) ν ν (0.85 ν)

17 α β ν σ σ + σ + σ α β ν σ (. 0.0) + ( ) + ( ) 0 00 σ 0.0 α β ν σ σ + σ + σ α β ν σ 0.0 α β ν σ σ + σ + σ α β ν σ 0.0 α β ν σ σ + σ + σ α β ν σ 0.0 The error functon s: σ σ + σ σ + σ Though the dvergence s about 5% we beleve that the approach of the error s numercal analyss s comprehensble as opportuntes of such analyss are practcally unlmted. Concluson The offered method of the output determnaton of the accuracy estmaton for enrchment products can be appled for enterprse technologcal balance formng. REFERENCES [] Kozn V.Z.: Approbaton and control of technologcal processes of enrchment. М. Nedra