Strojni{ki vestnik 48(2002)10, Journal of Mechanical Engineering 48(2002)10, ISSN ISSN UDK /.09: UDC

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1 Strjni{ki vestnik 48(2002)10, urnal f Mechanical Engineering 48(2002)10, ISSN ISSN UDK /.09: UDC /.09: Pristvnik Izvirni znanstveni A., Krpe ~lanek., ^repin{ek-lipu{ (1.01) L.: Simulacija tvrbe - A Simulatin Original f scientific the Frmatin paper (1.01) Simuliranje nastajanja gru~ dispergiranih delcev pd vplivm zunanjega magnetnega plja A Simulatin f the Cluster-Frmatin Prcess in a Dispersin f Fine Particles Under the Influence f an External Magnetic Field Andrej Pristvnik - urij Krpe - Lucija ^repin{ek-lipu{ Naprave za magnetn bdelav vde (MOV) s uèinkvita, gspdarna in dbra eklška rešitev za prepreèevanje izlèanja vdnega kamna. Na pdlagi labratrijskih preskusv s ugtvili, da je uèinkvitst MOV dvisna d sestave bdelvanega disperznega sistema in bratvalnih razmer naprave. Na temelju terije Derjagin-Landau in Verway-Overbeek (DLVO) in statistiène metde Mnte Carl Metrplis sm razvili teretièni mdel nastajanja gruè dispergiranih delcev pd vplivm zunanjega magnetnega plja. Nadalje sm p naèelu dprtega vira na pdlagi menjenega mdela razvili raèunalniški prgram za simulacij in grafièn predstavitev nastajanja gruè pd vplivm zunanjega magnetnega plja. Rezultate izraèunv sm analizirali z metd delitve in metd stpenjske prazdelitve Strjniški vestnik. Vse pravice pridržane. (Kljuène besede: bdelava vde magnetna, vdni kamen, metde Mnte Carl Metrplis, analize gruè) Magnetic water-treatment (MWT) devices fr scale cntrl can be used with gd ecnmic and eclgical benefits. Frm experimental results under well-cntrlled labratry cnditins we have established that the effects f MWT devices are very dependent n the cmpsitin f the treated dispersin system and their wrking cnditins. T investigate the effects f magnetic field n the prcess f cluster frmatin in a fine-particle dispersin under the influence f the external magnetic field f a MWT device a theretical mdel based n the Derjagin-Landau and Verway-Overbeek (DVLO) thery and the statistical Mnte Carl Metrplis methd was used. The pen-surce cmputer prgrams fr the simulatin and the graphical presentatin f clustering under the influence f an external magnetic field were develped. The results were analysed by cluster analysis, based n the partitining and hierarchical methds urnal f Mechanical Engineering. All rights reserved. (Keywrds: magnetic water treatment, scale preventin, Metrplis Mnte Carl, cluster analysis) 0 UVOD Magnetna bdelava vde je pgst uprabljana metda nekemiène bdelave vde za nadzr izlèanja vdnega kamna. Nadzr vdnega kamna se dsega z vdenjem napajane vde skzi magnetn plje. e gspdarna metda, vendar še vedn daje nezanesljive rezultate. Kljub veè desetletnim pruèevanjem na tem pdrèju še ni izdelana dknèna znanstvena terija, kak te naprave natanèn delujej in kakšni s pgji za njihv ptimaln delvanje. Naravne vde s bgati disperzni sistemi, ki vsebujej razliène klide, ine in druge sestavine. Zaradi naravne prenasièensti ali pa vzpstavitve prenasièenja b spremembah med predelav vde (npr. sprememba temperature, tlaka ali ph) lahk na stenah cevvdv in naprav pride d izlèanja nekaterih izmed teh snvi v bliki težk dstranljivih 0 INTRODUCTION Magnetic water-treatment is a frequently used nn-chemical methd fr scale cntrl. The scale preventin is achieved by passing the water thrugh the magnetic field. It is an ecnmically favrable technique, but it still gives unreliable results. Despite several decades f intensive research dne in this area, n scientifically cnfirmed theretical explanatin exists yet that adequately describes hw MWT devices wrk and the cnditins fr their effective and ptimum peratin. Natural waters are rich dispersin systems that cntain many cllids, ins, etc. Due t the natural supersaturating f water supplied t varius systems r the supersaturating as a result f changed cnditins during water prcessing (such as pressure drp, temperature and ph) a hard scale precipitates n the pipeline and the walls f the equipment. The stran 520

2 blg vdnega kamna. MOV je še psebej betavna metda za uravnavanje interakcij med klidnimi delci v vdnih disperzijah [1]. Èeprav mngi pstpki vplivaj na klidni sistem, pmeni ravntežje med interakcijami privlakv in dbjev (pri dlèeni termièni dejavnsti) kriterij za stabilnst klidnih disperzij v naravnih vdah. Natanènejše razumevanje tega vzajemnega delvanja pdaja terija DLVO [2]. Težnji delcev, da bi se združili zaradi delvanja van der Waals Lndnvih sil kratkega dsega, nasprtuje elektrièni nabj na trdni pvršini, ki deluje tudi na veèjih razdaljah med delci. Tak je celtna interakcijska energija med dvema klidnima delcema (E t ) vsta energije elektrstatskega dbja (E r ; k se elektrièni dvjni plasti delcev prekrivata) in energije medmlekularnega privlaka (E a ; delvanje van der Waals Lndnvih sil) [2]: MOV is a particularly prmising technique fr cntrlling the interactins amng cllidal particles in water dispersins [1]. Althugh many prcesses affect cllidal behavir, the balance between attractin and repulsin interactins (fr a particular level f thermal activity) is a criterin fr the stability in cllid dispersins. A detailed understanding f this interplay is the basis f the DLVO thery [2]. The tendency f particles t aggregate as a result f the shrt-range van der Waals Lndn frces is cunter acted by the electrically charged layer n the particles surfaces. Thus, the ttal interactin energy (E t ) between tw cllidal particles is the sum f the repulsin energy (E r ; electric repulsin, when duble layers f tw particles verlap) and the attractin energy (E a ; the particles interactin as a result f the van der Waals Lndn frces) [2]: Et = Er + Ea (1). Energija elektrstatskega dbja (E r ) med delcema je dvisna d plmera teh delcev (a), razdalje med njunima centrma (R), elektrkinetskega ptenciala (j d ) in dielektriène knstante disperznega medija (e r ). Med enakima, krglima, razmerma majhnima delcema s širk elektrièn dvjn plastj (majhen ka) je energija dbja: kjer sta: s razmerje med razdalj med središèem in premerm delca (s=r/a) in k Debye-Hückelv parameter (njegva bratna vrednst je v prvem približku enaka debelini elektriène dvjne plasti): en 0 k= Za kside, dispergirane v vdi, je elektrièni ptencial na trdni pvršini (j d ) dlèen s ph disperzije ([2] in [3]). Energija medmlekularnega privlaka (E a ) med dvema enak velikima, krglima delcema je dlèena z enaèb: k æ H Ea 6 è -4 kjer je k H Hamakerjeva knstanta. Magnetstatiène interakcije med delci spreminjaj bnašanje tekèine in lahk destabiliziraj klidni sistem. K je disperzija klidnih delcev izpstavljena zunanjemu magnetnemu plju, se pjavij magnetne sile, ki zmanjšujej stabilnst klidnega sistema. Energija magnetnega privlaka (E m ) med dvema krglima delcema na razdalji R ij je dvisna d gstte (B) in smeri delvanja zunanjega magnetnega plja, veliksti delcev (a) in magnetnih lastnsti minerala. Pri vzpredni usmeritvi magnetnih 2 ee r 0aj Er = d exp( -ka( s-2)) s The repulsin energy (E r ) between tw particles depends n the radius (a), the distance between the centres f the particles (R), the electrkinetic ptential (j d ) and the dielectric cnstant (e r ) f the dispersin medium. Fr identical, spherical and relatively small particles with a wide electric duble layer (ka is small) the repulsin energy is: (2), where parameter s is the rati between the distance and the radius (s=r/a) and k is the Debye-Hückel parameter (its reciprcal value is cnsidered as a first apprximatin fr the length f the electric duble layer): åzc i (3). kt A i i ee r =- ç s s s 0 B Fr xide minerals in water, the surface ptential (j d ) f the particles is determined by the ph f the dispersin ([2] and [3]). The energy f attractin (E a ) between tw spherical particles with an identical radius is defined by the equatin: 2 s - ö ln (4), ø where the parameter k H is the Hamaker cnstant. Magnetstatic particle interactins mdify the behavir f the fluid and can affect the cllidal stability. When a dispersin f cllid particles is placed in an external magnetic field an additinal magnetic frce arises, which decreases the stability f the cllid system. The energy f the magnetic attractin (E m ) between tw spherical particles separated by a distance R ij depends n the magnetic density (B) and the angle f the external magnetic field, the radius f the particles (a) and the magnetic prperties f the particles. Fr the stran 521

3 mmentv dveh enakih, krglih delcev je energija magnetnega privlaka: 32ð =- Celtna energija (E t ) interakcij med klidnima delcema v sistemu, izpstavljenemu zunanjemu magnetnemu plju, je tak: E m parallel-rientated magnetic mments f tw identical particles, the energy f the magnetic interactin is: a c B 3 (5). 9m0Rij Thus, the ttal energy f the interactin (E t ) fr cllid particles in an external magnetic field is: Et = Er + Ea + Em (6). Statistiène numeriène metde, znane kt metde Mnte Carl, za simulacij uprabljaj zapredja nakljuènih števil. Mdel za pruèevanje lastnsti klidnih vdnih disperzij pd vplivm magnetnega plja ([2] d [8]) temelji na metdi Mnte Carl Metrplis ([4] in [13]). Osnva mdela je dvrazsežna kvadratna celica z dlžin stranice A, z N-timi nakljuèn razprejenimi delci (sl.1). Vsi delci s krgli in enaki. Lega slehernega delca je znana in znaèena s krdinatami (x, y,q), kjer je q kt med magnetnim mmentm delca in zunanjim magnetnim pljem [6] (sl.2). Celtna energija sistema (E S ) je enaka vsti vseh energij interakcij med klidnimi delci v prstru: Metda vsebuje izraèunavanje energijskih sprememb (DE), medtem k se krdinate enega delca v analizirani celici nakljuèn spremenij za majhne N S=åEti () i1 = E Statistical numerical methds, knwn as Mnte Carl methds, are methds that utilize sequences f randm numbers t perfrm the simulatin. The presented mdel is based n the Mnte Carl Metrplis ([4] and [13]) methd and has been used t investigate the prperties f cllid particle dispersins in water under the influence f a magnetic field ([2] t [8]). The mdel is based n a tw-dimensinal square cell with a side f length A, cntaining N randmly distributed particles (Fig. 1). All the particles are spherical and identical. The psitin f any particle is knwn and can be specified with crdinates (x, y,q), where q is the angle between the magnetic mment f the particle and the applied magnetic field [6] (Fig. 2). The ttal energy f the system (E S ) is the sum f the ttal energies f the interactins amngst all the cllid particles: (7). The methd cnsists f calculating the energy change (DE) when the crdinates f ne particle in the analyzed cell are changed, at randm, by small Sl. 1. Nakljuèna prazdelitev krglih delcev v dvrazsežnem prstru kvadratne blike (A=70 mm, a=0,5 mm, N=300) Fig. 1. Randmly distributed spherical particles in a tw-dimensinal square cell (A=70 mm, a=0.5 mm, N=300) Sl. 2. Par delcev (i,j) - krgle blike v dvrazsežni celici b delvanju zunanjega magnetnega plja pd ktm y [9] Fig. 2. A pair f spherical particles (i,j) in a twdimensinal square cell under a magnetic field applied at an angle y[9] stran 522

4 vrednsti. V primeru, da je nva vrednst celtne energije sistema nižja d predhdne, delec stane v nvi legi; v nasprtnem primeru se izraèuna faktr P in primerja z nakljuèn vrednstj x, x Î[0,1]: P= exp -D amunts. If the new ttal energy f the system is less than the previus ne the particle stays in its new psitin; therwise a factr P is calculated and cmpared with the randm number x, x Î[0,1]. ( E k T) / B (8). Èe je faktr P veèji d nakljuènega števila, delec hrani nv leg, sicer se vrne na prvtn leg. Opisani pstpek se izvede za vseh N delcev v kvadratni celici. 1 RAÈUNALNIŠKI PROGRAM ZA SIMULACIO NASTAANA GRUÈ Pri pruèevanju magnetne bdelave vde sm uprabili terij klidv, matematike in statistike. Raèunalniški prgram Open Surce (MCM) [15] sm priredili za dvrazsežn simulacij pstpka nastajanja gruè v finih disperzijah pd vplivm zunanjega magnetnega plja. D takšnega nastajanja naj bi namreè prišl v napravah MOV, kt predvidevaj nekateri avtrji ([10] in [11]). Za grafièn predstavitev sm uprabili sklp raèunalniških prgramv MCM View [9] in za analiz gruè Fanny in Twins [12]. Pdatki s vneseni z dvema vhdnima dattekama. V prvi datteki s pdani: velikst celice, števil in fizikalne lastnsti delcev; dlèena je tudi struktura vmesnih izhdnih dattek. V drugi vhdni datteki je dlèena izhdišèna lega delcev. Rezultati simulacije s zapisani v razliènih blikah in s lahk predstavljeni s prgrami, kakršna sta Micrsft Excel [15] ali Micrcal Origin [15]. Za grafièn predstavitev je razvit CM View prgram Cmpaq Array Visualizer [15]. Rezultati s bili nadalje vrednteni z metd delitve in metd stpenjske prazdelitve za analiz gruè p Kaufmanu in Russeauwu [12]. V ta namen sta bila uprabljena prgrama Fanny in Twins. 2 REZULTATI SIMULACIE Kt primerjalni pdatki za primarne numeriène izraèune s bile uprabljene fizikalne lastnsti hematitnih delcev (Fe 2 O 3 ) v vdni disperziji. Na pdlagi tega s bili izvedeni numerièni izraèuni za nekatere minerale, ki ustvarjaj vdni kamen v vdi: diamagnetni kalcijev karbnat (CaCO 3 ), kalcijev sulfat (CaSO 4 ), silicijev diksid (SiO 2 ), antifermagnetna hematit (Fe 2 O 3 ), getit (FeOOH) in fermagnetni magnetit (Fe 3 O 4 ). Za vse naštete minerale je bila simulacija nastajanja gruè izvedena v naslednjih razmerah: - razlièn števil delcev v kvadratni celici in razlièni plmer a, If the P factr is greater than the randm number the particle retains its new psitin, therwise it is returned t its riginal psitin. This prcedure is applied fr all N particles in a square cell. 1 A COMPUTER PROGRAM FOR THE SIMULATION OF CLUSTER FORMATION Well-knwn theries frm the field f cllid science, statistics and mathematics have been taken int cnsideratin and applied in magnetic watertreatment research. An pen-surce cmputer prgram (MCM) [15] fr the tw-dimensinal simulatin f the cluster-frmatin prcess in a fine-particles dispersin under the influence f an external magnetic field was develped. Accrding t sme authrs ([10] and [11]) such a cluster frmatin might ccur in MWT devices. A set f cmputer prgrams, MCM View [9] and Fanny and Twins [12], were used fr the graphical presentatin and the cluster analysis, respectively. The data are entered with tw input files. In the first input file the size f the cell, the number and the physical prperties f the particles are determined and the structure fr the intermediate utput files is defined. In the secnd input file the riginal psitin f the particles is determined. The results f the simulatin are written in varius frmats and can be presented with the prgrams such as Micrsft Excel [15] r Micrcal Origin [15]. Fr the graphical presentatin, the MCM View prgram was develped n the basis f the Cmpaq Array Visualizer [15]. The results were further analysed with the partitining and hierarchical methds fr cluster analysis f Kaufman and Russeauw [12]. Fr this purpse we used the Fanny and Twins prgrams. 2 RESULTS OF THE SIMULATION As the reference data fr the primary numerical calculatins, the physical prperties f hematite particles (Fe 2 O 3 ) in an aqueus dispersin were used. On this basis, the numerical calculatins fr sme f the scale-frming minerals in the water, such as diamagnetic calcium carbnate (CaCO 3 ), calcium sulfate (CaSO 4 ), silicn dixide (SiO 2 ), antiferrmagnetic hematite (Fe 2 O 3 ), gethite (FeOOH) and ferrmagnetic magnetite (Fe 3 O 4 ), were carried ut. Fr all these minerals a simulatin f cluster frmatin was carried ut fr the fllwing cnditins: - a different number f particles in the square cell and a different radius a stran 523

5 - razlièni kt y uprabljenega magnetnega plja (0, 30, 60 in 90 0 ), - razlièni elektrkinetski ptencial j d (10, 20 in 30 mv), - razlièna gstta B uprabljenega magnetnega plja (d 0 d 1T) in - razliène ph vrednsti vdnih disperzij in drug [9]. Celtna energija interakcije je bila izraèunana za delce hematita (N=40 d 300) s plmerm a=0,5 d 5 mm, vlumsk magnetn susceptibilnstj c=0,02 in Hamakerjev knstant k H = v kvadratni celici z dlžin stranice A=25 d 70 mm. Abslutna temperatura disperzije je bila nastavljena na 300 K in ph na 7,0. Knvergenca je glede na prejšnje raziskave [6] naravnana na 600 premikv na delec. Izhdišèna lega delcev je bila naravnana z matrik 5 8. Slika 3 prikazuje leg delcev pri 10 in 600 premikih pd vplivm magnetnega plja z gstt 0,5 T in ktm a different angle y f the applied magnetic field (0, 30, 60 and 90 0 ) - a different electrkinetic ptential j d (10, 20 and 30 mv) - a different density B f the applied magnetic field (frm 0 t 1T) - different ph values f the aqueus dispersins and thers [9]. The ttal interactin energy fr the hematite particles (N = 40 t 300) with the radius a=0.5 t 5 mm the vlume magnetic susceptibility c=0.02 and the Hamaker cnstant k H = in a square unit cell with side length A=25 t 70 mm was cmputed. The abslute temperature was set t 300 K and the ph was set t 7.0. Accrding t earlier research [6], the recmmended rate f cnvergence is up t 600 shifts per particle. The riginal psitin f the particles was set up as a 5 8 matrix. Figure 3 shws the psitins f the particles after 10 and 600 shifts under a magnetic field f 0.5 T, applied at an angle f 30. Sl. 3.a. Prazdelitev delcev p 10 premikih [9] Fig. 3.a. Psitin f the particles after 10 shifts [9] Na sliki 3.b je razvidn nastajanje ene veèje in dveh manjših gruè, ki s razprejene v smeri magnetnega plja. Iz slike ni mè nepsredn vredntiti intenzivnsti in veliksti gruè, zat je treba rezultate analizirati z ustrezn matematièn metd. V raèunalniškem prgramu Fanny je bila uprabljena metda lgièn mehke tvrbe gruè, ki je psplšitev metde delitve. Intenzivnst pstpka nastajanja gruè je bila vredntena z nrmaliziran vrednstj razdelitvenega keficienta ( k 5 s = ). Slika 5 prikazuje intenzivnst pstpka nastajanja gruè delcev hematita pri razliènih elektrkinetskih ptencialih in razliènih ktih uprabljenega magnetnega plja. V skladu z eksperimentalnimi rezultati [13] s uèinki naprav za magnetn bdelav vde mèn dvisni d sestave bdelvanega disperznega sistema in bratvalnih razmer. Vrednst ph v sistemu b. Prazdelitev delcev p 600 premikih[9] b. Psitin f the particles after 600 shifts [9] In Figure 3.b the frmatin f ne majr and tw minr clusters is very clear, these are well aligned in the directin f the applied magnetic field. Frm the same figure the intensity and the size f the cluster cannt be evaluated directly, thus the results have t be analyzed with an apprpriate mathematical methd. The fuzzy clustering methd, as a generalizatin f partitining, was used in the Fanny cmputer prgram. The intensity f the cluster-frmatin prcess was measured with the value f the nrmalized versin f the partitin cefficient ( k 5 s = ). Figure 5 shws the intensity f the clusterfrmatin prcess f the hematite particles at the different electrkinetic ptentials and under a magnetic field applied at different angles. Accrding t the experimental results [13], the effects f magnetic water-treatment devices are very dependent n the cmpsitin f the treated dispersin system and wrking cnditins. The ph value f stran 524

6 0,65 0,55 0,45 0,35 0, HM Ply. (HM) ph (/) Sl. 4. Intenzivnst nastajanja gruè v dvisnsti d ph bdelvanega sistema in aprksimacija krivulje s plinmm druge stpnje [9] Fig. 4. The intensity f the cluster-frmatin prcess versus ph and a graphical apprximatin with a plynmial f the secnd rder [9] 0,25 0,75 1,00 0,25 0,75 1,00 Y= 30, j= 20mV Y= 30, j= 30mV 0,25 0,75 1,00 0,25 0,75 1,00 Y= 60, j= 20mV Y= 60, j= 30mV 0,25 0,75 1,00 0,25 0,75 1,00 Y= 90, j= 20mV Y= 90, j= 30mV Sl. 5. Intenzivnst nastajanja gruè (nrmalizirana vrednst Dunnvega keficienta s k= 5 ) pd vplivm zunanjega magnetnega plja gstte 0,5 T pri razliènih ktih in razliènih vrednstih elektrkinetskega ptenciala [9] Fig. 5. The intensity f the cluster-frmatin prcess (nrmalized versin f Dunn s partitin cefficient, s k= 5 ) under a magnetic field f 0.5 T applied at different angles and different retardatin ptentials [9] stran 525

7 je eden najblj vplivnih parametrv v pstpku nastajanja gruè. Za uspešn nastajanje gruè delcev hematita je ptimalni ph v bdelvani disperziji 5,6 d 10,3 (sl. 4). 3 SKLEP V zadnjih desetletjih je bil pravljenih velik raziskav na pdrèju magnetne bdelave vde, a je še vedn vprašljiv sam mehanizem delvanja naprav za MOV. Mehanizem je zapleten in nepsredn dvisen d kemiène sestave vde in bratvalnih razmer naprave MOV. Zaradi slabega pznavanja mehanizma delvanja staja tak njihva uèinkvitst nakljuèna. Mdel simulacije nastajanja gruè sm priredili za pdrèje magnetne bdelave. Zaèetni izraèuni s temeljili na delcih hematita. Mdel je utemeljen na teriji DLVO, statistièni metdi Mnte Carl Metrplis in na teriji analize gruè. Na temelju predstavljenega mdela sm z raèunalniškim prgramm simulirali in analizirali nastajanje gruè za veèin snvi, ki ustvarjaj vdni kamen. Predstavljeni rezultati dbr pnazarjaj pgje delvanja naprav za MOV b znani kemièni sestavi vde. the system is ne f the mst influential parameters fr the cluster-frmatin prcess. Fr the successful clustering f hematite particles, the ptimum ph range f the treated dispersin is frm 5.6 t 10.3 (Fig. 4). 3 CONCLUSION Despite the large amunt f research wrk ver past decades a theretical understanding f the MWT mechanism is still incmplete. This is the main prblem when it cmes t the design f efficient MWT devices. The MWT mechanism is cmplex and directly depends n the chemical cmpsitin f the water and the wrking cnditins. A theretical mdel fr the simulatin f cluster frmatin with hematite particles as reference data was supplemented t the regin f magnetic watertreatment. The mdel is based n the DLVO thery, the Mnte Carl Metrplis methd and the cluster-analysis thery. Numerical calculatins fr mst f the scalefrming minerals were dne with cmputer prgrams based n the presented mdel. The btained results shw that the mdel predicts well the peratinal cnditins fr the effective use f MWT devices, prviding the chemical cmpsitin f the supplied water is knwn. plmer delca gstta magnetnega plja mlarna kncentracija inv i v raztpini snvni elektrièni nabj (1, As) energija medmlekularnega privlaka energija elektrstatskega dbja energija magnetnega privlaka celtna energija interakcij Bltzmannva knstanta (1, /K) Hamakerjeva knstanta števil delcev Avgadrv števil delcev (6, /ml) faktr p enaèbi (8) razdalja med središèema delcev razmerje med razdalj R in plmerm a abslutna temperatura krdinati lege delca valenca ina magnetna susceptibilnst minerala dielektrièna knstanta vakuuma (8, As/Vm) relativna dielektrièna knstanta vde elektrkinetski ptencial na trdni pvršini Debye-Hückelv parameter magnetna permeabilnst vakuuma (4p Vs/Am) kt med magnetnim mmentm delca in uprabljenim magnetnim pljem nakljuèna vrednst med 0 in 1 kt delvanja zunanjega magnetnega plja 4 OZNAÈBE 4 SYMBOLS a m B Vs/m 2 c i ml/l e As E a E r E m E t k B /K k H N N A 1/ml P R m s T K x,y m z i c e 0 e r j d m As/Vm V k 1/m m 0 Vs/Am q x y rad rad radius f interacting spheres magnetic field density mlar cncentratin f ins i in the slutin electrn charge (1, As) inter-mlecular attractin energy electrstatic repulsin energy magnetic attractin energy ttal interactin energy) Bltzmann cnstant (1, /K) Hamaker cnstant number f particles Avgadr number f particles (6, /ml) factr by equatin (8) distance between centers f particles rati between distance R and radius a abslute temperature crdinates f particle s psitin valence f in i magnetic susceptibility f the mineral dielectric cnstant f vacuum (8, As/Vm) relative dielectric cnstant f water electrkinetic ptential at a slid surface Debye-Hûckel parameter magnetic permeability f vacuum (4p Vs/Am) angle between the magnetic mment f the particle and the applied magnetic field randm number frm 0 t 1 angle f the external magnetic field stran 526

8 5 LITERATURA 5 REFERENCES [1] Grier, D.G., S.H. Behrens (2001) Interactins in the cllidal suspensins, The ames Franck Institute, The University f Chicag. [2] Svbda,. (1981) A theretical apprach t the magnetic flcculatin f weakly magnetic minerals. Internatinal urnal f Mineral Prcessing, N. 8, [3] Svbda,. (1982) Magnetic flcculatin and treatment f fine weakly magnetic minerals. IEEE Transactins n Magnetics, N. 2, Vl. 18, [4] Metrplis, N., A.W. Rsenbluth, M.N. Rsenbluth, A.H. Teller, E. Teller (1953) Equatin f state calculatins by fast cmputing machines.. Chem. Phys. 21, [5] Chantrell, R. W., A. Bradbury,. Ppplewell, S.W. Charles (1980) Particle cluster cnfiguratin in magnetic fluids.. Phys. D: Appl. Phys. 13(3), UK, [6] Chantrell, R. W., A. Bradbury,. Ppplewell, S.W. Charles (1982) Agglmerate frmatin in a magnetic fluid.. Appl. Phys. 53(3), USA, [7] Miles,.., R.W. Chantrell, M.R. Parker (1985) Mdel f magnetic field induced rdering in dispersins f fine paramagnetic particles.. Appl. Phys. 57(1), USA, [8] Lipus, L.C.,. Krpe, L. Crepinsek, L. (2001) Dispersin destabilizatin in magnetic water treatment. urnal f Cllid and Interface Science 235, [9] Pristvnik, A. (2002) Cnfiguratin and frmatin f clusters frm dispersin particles during the magnetic water treatment. Dctral thesis, University f Maribr, Faculty f Chemistry and Chemical Engineering. [10] Ivchev, M. (1966) The effect f magnetic field n an irn-xygen water system. Khim. Tekhnl. Inst. 19: [11] Herzg, R. E., O. Shi,. Patil,.L. Katz (1989) Magnetic water treatment - the effect f irn n calcium carbnate nucleatin and grwth. Langmuir, Vl. 5, N. 3, [12] Kaufman, L.,. Peter (1990) Russeeuw finding grups in data - an intrductin t cluster analysis. A Wiley-Interscience Publicatin, New Yrk, Trnt. [13] Kulskii, L.A., V.Z. Kchmarsky, V.V. Krivtsv (1983) Intensifying and destabilizing factrs f magnetic antiscale treatment f water. Himija i tehnlgija vdy, Vl. 5, N. 4, [14] CSEP Intrductin t Mnte Carl Methds ( ) electrnic bk, Cpyright (C) by the Cmputatinal Science Educatin Prject, [15] Raèunalniški prgrami/cmputer prgrams: MCM, MCMr_z_a in MCM view, CpyrightÓ Andrej Pristvnik COMPAQ ARRAY VISUALIZER 1.1 CpyrightÓ Digital Equipment Crpratin Micrsft Ó Excel, versin 97, CpyrightÓ Micrsft Crpratin Micrcal TM Origin, versin 6.0, CpyrightÓ Micrcal Sftware, Inc. Naslva avtrjev: dr. Andrej Pristvnik dr. Lucija Èrepinšek-Lipuš Univerza v Maribru Fakulteta za strjništv Smetanva Maribr prf. dr. urij Krpe Univerza v Maribru Fakulteta za kemij in kemijsk tehnlgij Smetanva Maribr Authrs Addresses: Dr. Andrej Pristvnik Dr. Lucija Èrepinšek-Lipuš University f Maribr Faculty f Mechanical Eng. Smetanva Maribr, Slvenia Prf. Dr. urij Krpe University f Maribr Faculty f Chemistry and Chemical Engineering Smetanva Maribr, Slvenia Prejet: Received: Sprejet: Accepted: stran 527