MATHEMATICAL MODELS FOR A CLOSED-CIRCUIT GRINDING

Transcription

1 HUNGARIAN JOURNAL OF INDUSTRIAL CHEMISTRY VESZPRÉM Vol. 37( pp (009 MATHEMATICAL MODELS FOR A CLOSED-CIRCUIT GRINDING P. B. KIS 1, CS. MIHÁLYKÓ, B. G. LAKATOS 1 College of Dunaújváros, Department of Appled Mathematcs, Dunaújváros, Táncscs M. u. 1/a, HUNGARY E-mal: buzane.ks.proska@mal.duf.hu Unversty of Pannona, Veszprém, Egyetem u. 10, HUNGARY New conuous and dscrete mathematcal models are elaborated for descrbng some types of closed-crcut grndng mll-classfer systems. Based on the dscrete model computer smulaton s also developed for nvestgaton of grndng processes. The starg pont n developng the model s the conuous grndng equaton for the open-crcut grndng whch s a partal ntegro-dfferental equaton descrbng aal mng and breakage of partcles n the grndng mll. The convectve flow n the mll s modeled as partcle sze-dependent process. The boundary condtons at the nlet and outlet of the mll, the ntal condtons, and the equatons descrbng the operaton of the classfer and ts mass balance are the addtonal equatons of the model. The propertes and capabltes of the new computer model are demonstrated and analyzed by smulaton developed n Matlab envronment. The effects of all parameters of the grndng process, among others of parameters of the classfer, are studed va smulaton. The man statstcal characterstcs of the steady states, the hold-up of the grndng devce and the operaton of the classfer are also studed. The models presented appear to be very fleble and useful tools for analyss and desgn of closed-crcut grndng processes, and are sutable for the deeper understandng of the amed processes. Keywords: Closed-crcut grndng, computer smulaton, grndng-classfyng system, mathematcal modelng, mllclassfer system. Introducton Grndng s a wdely used operaton n the ndustral processes. Due to the fact that there are numerous types of grndngs many knds of grndng devces have been developed based on varous operaton prncples. At the same tme, grndng s an energy consumng operaton therefore the cost-effcency of the operaton can be mproved by the decreasng the grndng energy. One of the man urgngs of the mathematcal descrpton and modelng of grndng s eactly the mprovement of the cost-effcency. Numerous ecellent papers have been devoted to the descrpton of the batch grndngs [1,, 3]. Mhálykó, Blckle and Lakatos [4] developed determnstc conuous and dscrete models for the open crcut grndng descrbng processes n long ball mlls. Further development and generalzaton of these models provded mathematcal descrpton also for closed-crcut grndng systems wth classfyng devces [5, 6]. In the lterature, grndng wth nternal classfcaton of the product has also been modeled [7, 8]. The am of ths paper s to present conuous and dscrete mathematcal models for descrbng some further types of closed-crcut grndng mll-classfer systems. The model s formed by a conuous grndng equaton for the open-crcut grndng process, beng a partal ntegro-dfferental equaton descrbng aal mng and breakage of partcles n the mll. The convectve flow of ground materal n the mll s modeled as szedependent process. The boundary condtons at the nlet and outlet of the mll, the ntal condtons, and the equatons descrbng the operaton of the classfer and ts mass balance are the addtonal equatons of the model. Usng the dscrete model computer smulaton s carred out nvestgag the effects of parameters and operatonal condtons of the grndng system. Conuous and dscrete mathematcal models The closed-crcut grndng models of the authors whch have been prevously publshed descrbe such a grndngclassfyng system whch returns the large partcles from the classfer to the nlet of the mll [5, 6, 9]. Ths paper presents mathematcal models referrng to another group of the closed-crcut grndngs. The consdered process s as follows: the fresh partcles to be ground are entered nto the mll through the nlet of the mll; the partcles are grndng whle they are flowng along the mll; the partcles are classfed; only the fne

2 154 partcles are allowed to leave the mll through the outlet; due to the classfcaton the large partcles are retaned n the mll. Let symbol stands for the partcle sze, let symbol ma denote the largest partcle sze. The convectve flow of the partcles along the devce s characterzed va the convectve flow velocty and denoted by u(, whle symbol D stands for the aal dsperson of the partcles. The operaton of the classfer s characterzed by the classfcaton functon ψ(. Let the mass densty functon m( characterze the sze dstrbuton of the partcles where m( d epresses the mass of the partcles at the aal coordnate y of the mll at tme t wthn the partcle sze nterval ( +d n a unt mass of the partcles. At the model-constructon let us suppose that 1 the transport of the partcles n the mll s descrbed by the aal dsperson model; the grndng knetcs s descrbed by the frst order law of breakage. The conuous mathematcal model s gven by the followng equatons (1 (6. The operaton of the grndng mll s descrbed va equaton (1 whch s a partal ntegral-dfferental equaton: m( m( m( t = D u( t ma S ( m( + S( L b( L m( L, dl. (1 Equatons ( and (3 (4 epress the ntal and the boundary condtons, respectvely: m ( 0 = m0 ( y, ( m( f f ( = u( m( D, f y = 0, (3 m( = 0, f y = Y. (4 In equaton (1 functon S(L represents the rate of breakage of partcles of sze L, usually termed selecton functon, whle functon b( L s the breakage densty functon. In equaton (3 functon f f ( denotes the mass flow rate of the feed to the mll. Equaton (3 epresses the assumpton that there s no back-mng from the mll nto the transfer ppe. Symbol Y stands for the length of the mll, therefore y = 0 epresses the nlet of the mll n equaton (3 and y = Y means the outlet of the mll n equaton (4. Equatons (5 and (6 descrbe the operaton of the classfer. The mass flow of the large partcles s gven by equaton (5: f r ( = ψ( f( = ψ( u( m( Y,. (5 In equaton (5 f( s the mass flow densty functon at the outlet of the mll. We suppose that the large partcles whch are not allowed to leave the mll due to the operaton of the classfer reman n the grndng devce. Ths assumpton s epressed by equaton (6: f( = f l ( + ψ( f(, (6 where functon f l ( s the mass flow densty functon of that partcles whch flowed n the grndng devce to the outlet of the mll. Usng the conuous model equatons (1 (6 a dscrete mathematcal model has been derved smlar way nvolved n the paper [6]. Let Δt denote the length of the tme step, let t n stand for t n = nδt. Let us subdvde the mll nto J equal sectons and denote h y the length of a secton of the mll, thus h y = Y/J. Let us sort the partcles nto sze fractons. Let mn denote the smallest partcle sze. If I denotes the total number of sze fractons of partcles and we ntroduce the notaton h = ( ma mn /I, then = mn + h stands for the th partcle sze fracton. Let symbol denote the mean sze of the th sze fracton,.e. = ( -1 + / and let t ( stand for the mean resdence tme of partcles belongng to the th sze nterval,.e. t ( = Y / u(. Let µ(, y j, t n denote the mass of the partcles belongng to the th sze fracton and the j th secton at the moment of tme t n. Let us ntroduce symbols V F ( and V B ( (=1,,, I va the followng formulae: V F Pe( τ ( 1 + J ( =, Pe( V ( = B J τ (, Pe( J where Pe ( = u( Y / D( s the Peclet number for the th sze fracton of partcles. In the above formulae u ( and D ( denote the convectve flow velocty and the aal mng coeffcent of that sze fracton, respectvel whle τ ( = Δt / t (. The breakage dstrbuton functon let t denote symbol B(X, descrbes of the rubbles when the partcle of sze X breaks. The connecton between the breakage dstrbuton functon and the densty functon s as follows: B( X, = b( l, X dl. mn After ntroducng the notaton pk, = ΔtS( k [ B( k, B( k, 1 ] then the equatons of the dscrete model can be descrbed by equatons (7 (9. The development of the dscrete model can be manpulated smlarly to those publshed n the paper [6]. The left-hand sdes of equatons (7, (8, (9 contan the mass of the partcles belongng to the th sze fracton at the moment of tme t n+1. The frst group of equatons descrbes the sze composton of partcles n the frst secton; the second group reflects to the nner sectons

3 155 of the mll; fnall equaton (9 epresses the sze composton n the last secton of the mll. μ (, y1, + 1 = = ( 1 VF ( τ ( t ( S( μ(, y1, + + V μ (, y, t + B ( n I + p k, μ ( k, y1, + a ( =1,,, I. (7 k= The frst term on the rght-hand sde of equaton (7 epresses the mass of partcles belongng to the th sze nterval at the moment of tme t n whch nether moved forward nor broke; the second term represents that partcles of the same sze nterval that moved backwards from the second secton; the thrd term gves the mass of partcles that remaned n the frst secton and broke from some greater or equal sze than to ths very sze nterval; the last term denotes the mass of the feed partcles belongng to the th sze nterval. μ (, y j, + 1 = = 1 V ( V ( τ ( t ( S( ( F B μ (, y j, + VF ( μ(, y j 1, + VB ( μ (, y j+ 1, + + I + pk, μ ( k, y j, k= j=,, J 1, =1,,, I. (8 The frst term on the rght-hand sde of equaton (8 gves the mass of partcles of the th sze fracton that dd not changed at all durng a unt of tme; the second term represents the fracton of partcles belongng to the th sze nterval that moved forward from the prevous secton; the thrd term epresses that part of partcles whch moved backward from the net secton; whle the last term means the mass of those partcles that remaned n the secton and broke from some sze greater or equal than to the nterval n queston. Fnall equaton (9 descrbes the sze composton of the partcles as μ (, yj, + 1 = = ( 1 VB ( τ ( t ( S( μ(, yj, + + V μ (, y, t + F ( J 1 n I k, k J n k= ( ( VF ( VB( μ(, yj, + p μ (, y, t + + ψ =1,,, I. (9 where the frst term on the rght-hand sde descrbes the mass of those partcles whch dd not changed at all; the second term represents the partcles whch moved forward from the last but one secton; the thrd term represents the mass of partcles that remaned n the last secton and broke to the nterval n queston; the last epresson,.e. ψ ( ( VF ( VB ( μ(, y J, gves the mass of that partcles of that sze were retaned n the mll due to the classfcaton. The set of recursve equatons (7, (8, and (9 provde, n prncple, the dscrete mathematcal model of the closed-crcut grndng system. In the model the consttutve relatons D(, u(, and ψ( are of arbtrary form, and can be formulated arrangng the smulaton n accordance wth the equpment and operatonal condtons. Numercal eperments and results Based on the dscrete mathematcal model computer smulaton has been elaborated n Matlab envronment for eamnaton the capabltes and propertes of the model. The partcle sze dstrbuton of the materal to be ground was chosen from the lterature [10] as well as the parameters n the model whch are the knetc and process parameters. The parameters of the selecton functon and breakage dstrbuton functon form the knetc parameters, whle the convectve flow veloct the aal dsperson, and the parameters of the classfcaton functon consttute the process parameters. The forms of the selecton functon and breakage dstrbuton functon are as follows: S( = K s α, where K s and α are the parameters of the functon; γ B ( X, = Φ + (1 Φ, X X where β, γ, Φ are the parameters [11]. The sze dependency of the convectve flow velocty of partcles was characterzed va the formulae u( = k u( 1, (=1,,, I where 1 k = 0 k λ 1, 1+ ( / 50 where u( 1 stands for the velocty of flow of the fnest partcles. In the formula for k symbol 50 denotes that partcle sze for whch the convectve flow velocty s 0.5u( 1, and λ s an approprate constant [1]. The classfer was modeled usng the Molerus curve approach so the operaton of the classfer was characterzed by the followng effcency curve [13]: ψ ( = 1 1+ cut 1 e c(1 In the formula for ψ( symbol denotes the partcle sze. Functon ψ( has two parameters; c and cut. Parameter c characterzes the classfer, whle cut means the cut sze. It s very lkely that the classfer retans a partcle n the mll f the sze of a partcle s greater than the cut sze. Parameter c epresses the sharpness. Wth the ad of the computer smulaton the operaton of the grndng-classfyng system was nvestgated va cut. β

4 156 numercal eperments. The effects of both the knetc and process parameters were eamned. The statstcal characterzaton of the ground materal at the steady state was also performed. Let us see an eample,.e. an eperment whch demonstrates the effect of the classfcaton to the steady state characterstcs of the materal at the outlet from the mll. The am of the numercal eperment s to show the nfluence sze of the classfer to the mean partcle sze and dsperson of the partcle sze dstrbuton of the ground materal whch leaves the mll at the outlet. The calculaton results are nvolved n the Table 1 and Table. In the numercal eperments the value of the parameter c was 5 and 10, respectvely. The grade effcency curves of the classfer s llustrated by Fg. 1, whle Fg. shows the dstrbuton of the feed materal and the dstrbuton of the ground materal at the outlet from the mll at the steady state. The values of the remanng parameters were: Y = 5.4 m, α = 1., β =.6, γ = 0.8, Φ = 0.4, K s = 0.08 s -1, mn = 0 µm, ma = 1540 µm, u( 1 = m/s, D = 0.00 m /s, λ = 3.5, 50 = 650 µm. also demonstrated by the data nvolved n the Table 1 and Table. Fgure 1: Grade effcency curves of the classfer, c = 5 and c = 10 Table 1: The steady state characterstcs of the materal at the outlet from the mll dependng on the cut sze of the grade effcency curve, where c = 5 Cut sze (c = 5 The steady characterstcs of the materal at the outlet from the mll Mean partcle sze (µm Dsperson of the partcle sze dstrbuton Table : The steady state characterstcs of the materal at the outlet from the mll dependng on the cut sze of the grade effcency curve, where c = 10 Cut sze (c = 10 The steady characterstcs of the materal at the outlet from the mll Mean partcle sze Dsperson of the partcle sze dstrbuton Table 1 and Table llustrate well how the average partcle sze and dsperson at the outlet from the mll keep growng wth ncreasng the cut sze of the classfer. At the same tme, the tables show the effect of the sharpness of the classfcaton. The greater s the value of parameter c the sharper s the classfcaton functon,.e. the better s the classfcaton. In the case of a good classfcaton the large partcles have a very lttle chance to leave the mll. Therefore the greater value of the parameter c yelds smaller mean partcle sze and dsperson of the partcle sze dstrbuton. Ths fact s Fgure : The sze dstrbuton of the feed materal and the sze dstrbuton of the ground materal at the outlet from the mll where cut = 00, and c = 10 The hold-up of the grndng devce can also be nvestgated va numercal eperment. Let us see how the operaton of the classfer and one of the knetc parameters K s nfluence the hold-up of the mll at the steady state. The effects sze, sharpness of the cut, and parameter K s were studed. It was supposed that the amount of the partcles n the mll was a unt at the start of the grndng process. The results of the calculatons are seen n the Table 3 and Table 4 where K s = s -1 and K s = s -1, respectvely. Fg. 3 and Fg. 4 also llustrate the hold-up of the mll at the steady state. The values of the remanng parameters were: Y = 5.4 m, α = 1., β =.6, γ = 0.8, Φ = 0.4, mn = 0 µm, ma = 000 µm, u( 1 = m/s, D = 0.00 m /s, λ = 3.5, 50 = 650 µm. Comparng Table 3 and Table 4, the results ndcate that the ncrease of the value of parameter K s causes the decrease of the hold-up of mll.

5 157 Table 3: The hold-up of the grndng devce at the steady state dependng on the cut sze and sharpness, where K s = s -1 The hold-up of the mll Cut sze c = 3 at the steady state c = 5 c = Table 4: The hold-up of the grndng devce at the steady state dependng on the cut sze and sharpness, where K s = s -1 The hold-up of the mll at the steady state Cut sze c = 3 c = 5 c = Table 3 and Table 4 show well that the ncrease of the cut sze nduces the decrease of the hold-up for a fed value of the sharpness generally; as well as the ncrease of the sharpness gves the ncrease of the hold-up for a fed cut sze mostly. Fgure 4: The hold-up of the grndng devce at the steady state dependng on the cut sze and sharpness of the cut, where K s = 0.050s -1 We can fnd some equal values of the hold-ups n Table 3 and Table 4 for just the same value of parameter c, for eample, c = 3 and the hold-up s.1. Table 5 contans the statstcal characterstcs at the outlet from the mll n both cases,.e. K s = s -1 and K s = s -1. In order to produce equal hold-ups the hgher value of the parameter K s connects to smaller cut sze. As a consequence the values of the mean partcle sze and the dsperson of the partcle sze dstrbuton are also smaller as t s seen n Table 5. Table 5: The values of the mean partcle szes and the dspersons of the partcle sze dstrbuton where c = 3 and the hold-up s.1 Parameter Cut sze Mean Dsperson K s partcle sze Summary Fgure 3: The hold-up of the grndng devce at the steady state dependng on the cut sze and sharpness of the cut, where K s = s -1 Numerous papers have been devoted to the mathematcal descrpton of the batch grndngs. At the same tme the mathematcal descrpton and modelng of conuous grndng processes, n partcular the closed-crcut grndngs stll offer a lot of work for the process engneers and eperts who nvestgate ths area. The newly elaborated conuous and dscrete mathematcal models proved to be sutable to the descrpton and analyss of some types of closed-crcut mll-classfer systems. In partcular, the newly developed mathematcal models presented n the paper nvolve the descrpton of both the batch grndng and the open-crcut grndng systems. Notce that the dscrete mathematcal model of the closed-crcut grndng system,.e. the set of recur-

6 158 sve equatons (7, (8, and (9, can also be wrtten n matr form. The matr form of the model s sutable for some further, among others stablty nvestgatons. REFERENCES 1 KOSTOGLOU M., KARABELAS A. J.: An assessment of low-order methods for solvng the breakage equa-ton, Powder Technology 17 (00, REID K. J.: A soluton to the batch grndng equaton, Chem. Eng. Sc., 0 (1965, FAN L. S., SRIVASTAVA R. C.: A stochastc model for partcle dsntegraton, Chem. Eng. Sc., 36 (1980, MIHÁLYKÓ CS., BLICKLE T., LAKATOS G. B.: A smulaton model for analyss and desgn of conuous grndng mlls, Powder Technology 97 (1998, BUZÁNÉ KIS P., MIHÁLYKÓ CS., LAKATOS G. B.: Computer smulaton for closed-crcut grndngs (n Hungaran, Proc. of Conference of Chemcal Engneerng 05 (005, KIS P. B., MIHÁLYKÓ CS., LAKATOS G. B.: Dscrete model for analyss and desgn of grndng mll-classfer systems, Chemcal Engneerng and Processng 45 (006, MIZONOV V. E., BERTHIAUX H., ZHUKOV V. P., BERNOTAT S.: Applcaton of multdmensonal Markov chans to model knetcs of grndng wth nternal classfcaton, Proc. of 10th European Symposum on Commnuton, Hedelberg, (00 CD- ROM, A3. 8 HINTIKKA V., MÖRSKY P., KUUSISTO M., KNUUTINEN T.: Conuous classfyng laboratory mll new possbltes for research work, Mnerals Eng-neerng, 9 (1996, KIS P. B., MIHÁLYKÓ CS., LAKATOS G. B.: Mathematcal models for smulaton of conuous grndng process wth recrculaton, Acta Cybernetca, 15 (00, BRAUN R. M., KOLACZ J., HOYER D. I.: Fne dry commnuton of calcum carbonate n a Hcom mll wth an Inprosys classfer, Mnerals Engneerng, 15 (00, AUSTIN L. G.: A dscusson of equatons for the analyss of batch grndng data, Powder Technology 106 (1999, CHO H., AUSTIN L. G.: A study of the et classfcaton effect n wet ball mllng, Powder Technology (004, MOLERUS O., HOFFMANN H.: Darstellung von Wnd-schtertrennkurven, Cheme-Ing. Techn., 41 (1969,