MATEMATI^KI MODEL MIKROSEGREGACIJE U Al-Cu-Mg LEGURI SA PROMJENLJIVIM KONCENTRACIJAMA TOKOM O^VR[]AVANJA
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1 Igor Vušanovi} Ma{inski fakultet, Podgorica, Crna Gora, Jugoslavia Mat thew John M. Krane School of Ma te rials En gi neering, Purdue Uni ver sity West La fay ette, In di ana, USA MATEMATI^KI MODEL MIKROSEGREGACIJE U Al--Mg LEGURI SA PROMJENLJIVIM KONCENTRACIJAMA TOKOM O^VR[]AVANJA Originalni nau~ni rad UDC: '35'721:517.9 BIBLID: X, 27 (2001), 1-4, Uvod U radu e predstavlen matemati~ki model za opisivane fenomena mikrosegregacie tokom o~vr{}avana trone legure Al--Mg. Makroskopska redistribucia legirau}ih elemenata (bakra i magneziuma) po zapremini kalupa tokom o~vr{}avana ima utica na redistribuciu legirau}ih elemenata koe se odigravau na nivou kristala, kako na pro file koncentracia tako i na relativni odnos faza koe se izdvaau tokom o~vr{}avana. Predstavleni model omogu}ava istra`ivane uticaa promena makroskopskih koncentracia bakra i magneziuma na proces o~vr{}avana na mikro nivou. Prora~uni su sprovedeni za dvie trone Al--Mg legure za tri razli~ita slu~aa, kada se sredne koncentracie bakra i magneziuma ne mienau, kada opadau i kada rastu u zadatom vremenskom intervalu. Matemati~ki model predstavlen u ovom obliku mo`e biti kori{}en u sprezi sa poznatim makrosegregacionim modelima, i predstavla dobru osnovu za dalu nadgradnu u cilu dobiana precizniih mikrosegregacionih modela. Klu~ne rie~i:mikrosegregacia, Al--Mg legura, redistribucia, o~vr{}avane, koncentracie, q faza, S faza, makrosegregacia Mikrosegregacia predstavla proces razdvaana legirau}ih elemenata tokom procesa o~vr{}avana, uslied razli~ite rastvorlivosti elemenata u te~nosti odnosno ~vrsto fazi. Precizna kvantifikacia faza koe se izdvaau tokom procesa o~vr{}avana kao i koncentracie legirau}ih elemenata u fazama veoma e va`no sa stanovi{ta odre ivana 25
2 mehani~kih i fizi~kih karakteristika dobiene legure. Tako e, precizne informacie o sastavu i koncentracii elemenata u fazama e zna~ano sa stanovi{ta eventualnih termi~kih tretmana dobiene legure. Mikrosegregacioni modeli su tako e zna~ani kao konstitutivne korelacie kod makroskopskih modela, radi dobiana zatvorenog sistema edna~ina pogodnog za numeri~ku integraciu 1 5. Ve}ina danas postoe}ih modela za opisivane fenomena mikrosegregacie odnosi se na dvone legure, dok e pregled modela koi tretirau legure sa vi{e od dva elementa re i, a nihov pregled e dat u radu 6. Ne ra~unau}i neke izuzetke 7, 8, ve}ina mikrosegregacionih modela nie dizanirana tako da se mo`e sprezati sa postoe}im makroskopskim modelima 3 5 za opisivane fenomena makro redistribucie elemenata tokom o~vr{}avana. Kao spola{ni utica koi kontroli{e proces o~vr{}avana u ve}ini modela se obi~no zadae gradient entalpie ili tem per a ture, dok su sredne koncentracie legirau}ih elemenata konstantne tokom procesa o~vr{}avana. Makrosegregacioni modeli sra~unavau srednu entalpiu i koncentracie legirau}ih elemenata tokom o~vr{}avana, dok mikrosegregacioni model treba da na osnovu ova tri parametra izvr{i prora~un sastava faza i profila koncentracia na nivou kristala. U ovom radu prikazan e matemati~ki model mikrosegregacie koi e neravnote`nog karaktera i koi omogu}ava opisivane procesa mikrosegregacie sa promenlivim srednim koncentraciama tokom procesa o~vr{}avana. Jedna~ine matemati~kog modela Osnovne pretpostavke na koima se zasniva matemat~ki model predstavlen u ovom radu su sli~ne onima koe va`e u tzv. [elovom (Scheil) modelu o~vr{}avana 9, a to su beskona~no brza difuzia u te~no fazi i beskona~no spora difuzia u ~vrsto fazi. Na me ufazno granici te~nost ~vrsto vlada termodinami~ka ravnote`a tokom procesa o~vr- {}avana. Tokom procesa ponovnog otapana difuzia u ~vrsto fazi treba da e dovolno brza da se na me ufazno granici izme u te~nosti i ~vrste faze mo`e podrazumievati termodinami~ka ravnote`a. Kod ve}ine legura to se ne mo`e uzeti kao realna pretpostavka, er e koeficient difuzie u ~vrsto fazi za obi~no tri reda veli~ine mani od odgovarau}eg u te~nosti, pa e pretpostavka o ravnote`i na me ufazno granici neta~na 1. U izvedenim prora~unima brzina o~vr{}avana e dovolno velika da se pretpostavka o zanemarivanu difuzie u ~vrsto fazi mo`e uzeti kao prili~no dobra, pa e kod procesa ponovnog otapana koncentracia bakra i magneziuma u ~vrsto fazi fiksirana vriednostima sa prethodno uspostavlenih profila koncentracia. Za razliku od ~vrste faze koncentracia bakra i magneziuma u te~no fazi se uzima sa povr{i te~nosti, {to predstavla ravnote`nu vriednost koa e prihvatliva zbog znatno ve}e difuzie u te~nosti i zbog konvektivnog mie{ana te~ne faze. Mikrosegregacioni model podrazumieva ravansku geometriu izme u bo~nih dendrita. Rastoane izme u bo~nih dendrita nie parametar u modelu er se difuzia u ~vrsto fazi ne ra~una. Matemati~ki model na osnovu koga se vr{i sra~unavane profila koncentracia u razli~itim fazama tokom o~vr{}avana sastoi se od bilansnih edna~ina masa legirau}ih elemenata, entalpie kao i odgovarau}ih edna~ina koima e opisan troni fazni diagram legure Al--Mg: 26 0 h f c T ( 1 f )( c T h ) m s s s L L (1)
3 I. Vu{anovi}, M. X. M. Kren: Matemati~ki model mikrosegregacie u Al--Mg... TERMOTEHNIKA bro 1-4 godina XXVII, (2001) i i C m f C ( 1 fs ) C i L, ( i, Mg;,, S ) (2) gde su indeksom predstavlane faze koe se izdvaau tokom o~vr{}avana, dok maseni udio ~vrste faze u sme{i fs predstavla sumu masenih udela svih ~vrstih faza ponaosob ( fs f ), a konstanta h 0 L ( cs cl ) TEUT hf. Sredne koncentracie legirau}ih i elemenata ( C m ) i entalpia sme{e (hm) su vriednosti koe se mienau tokom procesa o~vr{}avana i kontrolisane su makroskopskim fenomenima transporta. U ovom radu entalpia se smanue konstantnim gradientom od 1500 kj/kg za svaki vremenski korak, dok su promene srednih masenih koncentracia bakra i magneziuma zadate u tablici 1. Tablica 1. Po~etne koncentracie legura i gradienti promena tokom o~vr{}avana Slu~a Legura C m te`.% C m Mg te`.% dc m / dt dc Mg m / dt te`.%/s te`.%/s 1 1 5,0 1, ,0 1,0-0,30-0, ,0 1,0 0,30 0, ,0 3, ,0 3,5 0,3 0, ,0 3,5-0,3-0, ,9 0, ,0 t po~ s t kra s Sredna koncentracia bakra i magneziuma u ~vrsto fazi ( Ci ) sra~unava se integralisanem profila koncentracia do trenutne vriednosti masenog udela ~vrste faze f s : i f C C df, ( i, Mg;,, S ) (3) s i fs 0 s Izdvaane odre enih ~vrstih faza iz rastopa legure definisano e faznim diagramom legure. Fazni diagram legure Al--Mg definisan e sa edna~inom povr{i te~nosti i tzv. parcialnim koeficientima uzetih prema 11 : T , 4C 538, 5C 438, 8C C Mg Mg LIQ L L L L (4) k ( C ) C L * 012, (5) 27
4 k Mg * ( C ) Mg 0, 32 0, 52C L 182, C Mg L (6) C L gde e( C i ) * ravnote`na koncentracia legirau}ih elemenata u ~vrsto fazi tokom procesa o~vr{}avana. Koncentracia bakra i magneziuma u ~vrstim fazama (a, q, S) i odgovarau}e koncentracie u te~nosti tokom sekundarnog o~vr{}avana (L a + q ili L a + S) definisane su pravilom zategnutog trougla kao na sl. 1, gde su prikazani za tem per a ture 507 C i 515 C. Tokom eutekti~kog o~vr{}avana kada se iz te~nosti simultano izdvaau tri ~vrste faze (L a + q + S), koncentracie bakra i magneziuma u ~vrstim fazama mogu se sra~unati iz trougla ~ia su temena spoena sa eutekti~kom ta~kom na sl. 1. Proces o~vr{}avana odigrava se na slede}i na~in: kompletno posmatrani domen Slika 1. Koncentracie bakra i magneziuma u te~nosti i ~vrstim fazama tokom o~vr{}avana du` a-q i a-s binarnih kanala (Lazare, Lesoult) koi e u te~no fazi sa temperaturom koa se ra~una prema (4), hladi se tako {to se entalpia smanue za 1500 kj/kg za svaki posmatrani vremenski korak integracie (Dt = 1 s). Entalpia sme{e se stalno upore ue sa vriednostima entalpia u karakteristi~nim ta~kama ( h LIQ za prelaz L a, zatim h BIN za prelaz L a + q ili L + S, h EUT za prelaz L + + S i kona~no h SOL koa defini{e kra procesa o~vr{}avana) i na ta na~in se defini{e trenutni re`im o~vr{}avana. Kompletan prikaz metodologie odre ivana grani~nih entalpia koe defini{u re`im o~vr{}avana dat e u radu 3. Na primer, kada e entalpia sme{e h LIQ < hm < h BIN legura se nalazi u fazi primarnog o~vr{}avana kada se iz te~nosti izdvaa primarna a faza L a. Sistem i i * i edna~ina (1 6) re{ava se simultano i odre uu se T, f L, f,c,( C ), C (i =, Mg; = = a) 28 L
5 I. Vu{anovi}, M. X. M. Kren: Matemati~ki model mikrosegregacie u Al--Mg... TERMOTEHNIKA bro 1-4 godina XXVII, (2001) na osnovu definisanih fizi~kih karakteristika legure (c s, c L, h f ) kao i entalpie sme{e i Mg srednih koncentracia bakra i magneziuma u sme{i (h m, C m, C m ) koe se odre uu iz edna~ina makro modela. Jedna~ina (3) u razvienom obliku za i Mg ima slede}i oblik: fa, old fs C C df ( f f, old )( C ) (7) Mg 0 fa, old Mg Mg fs C C df ( f f, old )( C ) (8) 0 gde e f a trenutna vriednost masenog u~e{}a ~vrste faze za posmatrani vremnski trenutak a f a, old vriednost masenog u~e{}a ~vrste faze iz prethodnog vremenskog trenutka. Vriednosti koncentracia C i su ravnote`ne vriednosti iz prethodnih vremenskih trenutaka i ~ine profil koncentracia bakra i magneziuma u primarno fazi tokom o~vr{}avana. S obzirom da e usvoena pretpostavka o zanemarlivosti difuzie u ~vrsto fazi kao i odsustva ponovnog otapana ove vriednosti ostau nepromienene tokom procesa o~vr{}avana. Poslie odre ivana tem per a ture u~e{}e poedinih faza u ~vrsto leguri se odre ue na osnovu pravila zategnutog trougla, a vriednost ravnote`nih koncentracia ( C ) * i se dodau u ve} formirani profil koncentracia. Novi korak se sra~unava sa novim vriednostima entalpie sme{e i srednim koncentraciama hm, Cm, C Mg m. Predstavleni model lako se mo`e modifikovati u cilu opisivana slu~aa kada se sredne koncentracie bakra i magneziuma mienau u vremenu. Promene srednih koncentracia legirau}ih elemenata tokom procesa o~vr{}avana de{avau se u glavnom uslied struana te~nosti kroz ka{astu (mushy) zonu, a koa e ili oboga}ena ili sa smanenom koncentraciom legirau}ih elemenata u odnosu na po~etni sastav. Smanivane srednih koncentracia dovodi do pomerana radne ta~ke ka Al uglu, ~ime se pove}ava i lokalna temperatura po~etka o~vr{}avana sa ~ime se ubrzava ve} uspostavleni re`im o~vr{}avana. Suprotno, kada sredne koncentracie legirau}ih elemenata rastu radna ta~ka se pomera ka eutekti~ko ta~ki, sa ~ime se proces o~vr{}avana usporava ~ak i sa poavom ponovnog otapana ve} formirane ~vrste faze bez obzira na negativni gradient entalpie. U slu~au ubrzanog o~vr{}avana edna~ine modela ostau iste kao {to su napried navedene, dok e za slu~a ponovnog otapana potrebno izvr{iti odre ene korekcie. Kao {to e ve} napried re~eno nie za o~ekivati da se na me ufazno granici te~nost ~vrsto formira lokalna termodinami~ka ravnote`a er difuzia u ~vrsto fazi nie dovolno brza da apsorbue vi{ak bakra i magneziuma iz te~nosti. Znatno realnia pretpostavka e da su vriednosti koncentracia legirau}ih elemenata u ~vrsto fazi na granici te~nost ~vrsto one ve} prethodno formirane tokom procesa o~vr{}avana. Na ova na~in u profilima koncentracia bakra i magneziuma u ~vrsto fazi mogu se formirati sna`ni diskontinuiteti ~ak i kada se o~vr{}avanem formira samo edna faza. Ova fenomen vidliv e na slikama 4(b) i 6(b) za vriednosti f s = 0,1. 29
6 Slika 2. Putane o~vr{}avana tokom o~vr{}avana legure 1 Slika 3. Temperaturni profili tokom o~vr{}avana legure 1 30
7 I. Vu{anovi}, M. X. M. Kren: Matemati~ki model mikrosegregacie u Al--Mg... TERMOTEHNIKA bro 1-4 godina XXVII, (2001) Rezultati i diskusia Prve dvie legure koe su odabrane za istra`ivane imau po~etni sastav koi obezbe ue dominantnu primarnu a fazu u o~vrslo leguri. Primarno o~vr{}avane se zavr{ava u ednom od dva binarna kanala (du` koih se izdvaau q ili S faza), i kona~no proces o~vr{}avana se zavr{ava sa eutekti~kom reakciom sa simultanim izdvaanem sve tri faze (a, q, S) iz te~nosti. Legura 1 ima sastav Al-5 te`.%-1 te`.%mg kada se tokom sekundarnog o~vr{}avana izdvaa q faza (L a + q), dok se kod legure 2 sastava Al-6 te`.%-3,5 te`.%mg tokom sekundarnog o~vr{}avana izdvaa S faza (L a + S). Za svaku leguru razmatrana su tri slu~aa: zatvoren sistem kada su sredne koncentracie bakra i magneziuma konstantne, otvoren sistem sa smanivanem srednih koncentracia bakra i magneziuma i otvoren sistem sa pove}avanem srednih koncentracia bakra i magneziuma. Promene srednih koncentracia bakra i magneziuma u vremenu su definisane sa odgovarau}im gradientima i predstavlau idealizovano upro{}ene onoga {to se de{ava uslied struana fluida u ka{asto zoni na makro nivou. Redistribucia te~nosti bogate ili osiroma{ene legirau}im elementima mo`e dovesti do promena srednih koncentracia znatno vi{e od ±25% od nominalne vriednosti kori{}ene u ovom radu. Odabrani gradienti koi obezbe uu promene srednih koncentracia od ±25% uzete su na osnovu procena koe se dobiau makro modelima o~vr{}avana. Vremenski intervali zaedno sa odabranim gradientima promene srednih koncentracia dati su u tablici 1. Model e tako e iskori{}en za prora~un mikrosegregacie legura 3 i 4, koe su eksperimentalno istra`ivane u radu 12. Rezultati prora~una za leguru 1 (slu~aevi 1 3) su prikazani na slikama 2 4. Putana o~vr{}avana (sl. 2) za slu~a zatvorenog sistema kre}e se niz povr{inu te~nosti od tem per a ture po~etka o~vr{}avana (T LIQ = 637 C) dok linia ne presie~e binarni kanal na T = 512 C kada po~ine sekundarno o~vr{}avane sa izdvaanem q sekundarne faze simultano sa primarnom a fazom. Zavisnost tem per a ture od masenog u~e{}a ~vrste faze f s koa e prikazana na sl. 3 pokazue nelinearan karakter promene tokom primarnog o~vr{}avana, dok e tokom kratkog sekundarnog o~vr{}avana promena skoro linearna. Na krau eutekti~ka reakcia sa koom se zavr{ava o~vr{}avane e izotermskog karaktera, {to se na diagramu vidi kao horizontalna linia. Kao {to se mo`e videti u tablici 2, 88% o~vrsle legure e primarna a Tablica 2. Masena u~e{}a poedinih faza tokom o~vr{}avana legura 1 i 2 Slu~a Legura Primarna faza (a) Sekundarna faza (a + S ili a + q) Eutekti~na faza (a + S + q) 1 1 0,883 0,030 0, ,923 0,029 0, ,842 0,027 0, ,804 0,136 0, ,738 0,191 0, ,865 0,087 0,048 31
8 faza, dok e sekundarna faza svega 3%. Preostalih 9% e u formi eutekti~ke strukture koa se posledna izdvaa na krau o~vr{}avana. Profili koncentracia bakra i magneziuma prikazani su na sl. 4, sa koih se vidi o~ekivani porast koncentracie tokom primarnog o~vr{}avana. Slika 4. Sredni sastavi u o~vrslo leguri 1: (a) ; (b) Mg Sa po~etkom sekundarnog o~vr{}avana dolazi do skoka koncentracie bakra na 33% uslied izdvaana q faze bogate bakrom ( C 53 %), dok za magnezium dolazi do pada er e u q fazi magnezium veoma malo zastuplen, {to se vidi sa linie L-q na 507 C i 515 C na sl. 1. Smanivanem srednih koncentracia tokom o~vr{}avana legure 1 (slu~a 2 u tablici 1) dolazi do promena u strukturi dobiene legure. Efekat smanena koncentracia bakra i magneziuma vidliv e u donem lievom dielu na sl. 2. Pomeranem radne ta~ke bli`e ~istom aluminiumu dolazi do pove}ana tem per a ture na koo po~ine proces o~vr{}avana (edna~ina 4) pa samim tim i do ubrzavana procesa o~vr{}avana. Ubrzavanem procesa o~vr{}avana dolazi i do pove}avana u~e{}a primarne a faze u o~vrslo strukturi na ra~un smanena sekundarne i eutekti~ke faze, kao i do pove}ana latentne toplote faznog prelaza u pore enu sa zatvorenim sistemom. S obzirom da slu~aevi 1 i 2 imau isti zadati gradient entalpie pove}anem latentne toplote kod slu~aa 2 dolazi do pove}ana tem per a ture da bi bila zadovolena energiska edna~ina (1), {to se vidi skokom na diagramu 3. Temperatura raste do 642 C {to e za 5 C vi{e od odgovarau}e tem per a ture na povr{i te~nosti, ali e ipak ni`a od tem per a ture sa povr{i te~nosti koa odgovara trenutnim srednim koncentraciama bakra i magneziuma. Kada se sredne koncentracie prestanu mienati i postanu konstantne temperatura po~ine da te`i eutekti~ko na isti na~in kao za slu~a zatvorenog sistema. Kada sredne koncentracie bakra i magneziuma rastu u definisanom vremenskom intervalu (slu~a 3 u tablici 1), temperatura po~ine da se sni`ava {to izaziva lokalno ponovno otapane ve} formirane ~vrste faze. Ponovno otapane izazvano pove}anem koncentracia legirau}ih elemenata e veoma poznat fenomen koi dovodi do formirana oblaka ~estica ~vrste faze u strui te~nog metala tokom o~vr{}avana u ingotima 13. Uslied ponovnog otapana dolazi do apsorbovana latentne toplote od sistema i pada tem per a ture za red veli~ine 10 C. 32
9 I. Vu{anovi}, M. X. M. Kren: Matemati~ki model mikrosegregacie u Al--Mg... TERMOTEHNIKA bro 1-4 godina XXVII, (2001) Iznenadne promene tem per a ture tokom varirana koncentracia koe su prikazane na sl. 3. nisu primie}ene kod realnih sistema o~vr{}avana bilo eksperimentalnih bilo modelskih analiza fenomena makrosegregacie 14. One se poavluu kod predlo`enog modela er e gradient entalpie fiksan, dok e kod realnih sistema on definisan lokalnim temperaturskim gradientom. U realnim procesima o~vr{}avana promene srednih koncentracia izazva}e lokalno otapane ili ubrzano o~vr{}avane, ali ne}e do}i do formirana zna~aniih temperaturskih gradienata. Ako se, na primer, na odre enom mestu desi lokalno otapane ve} formirane ~vrste faze, olsobo ena latentna toplota bi}e transportovana kondukciom u lokalno okru`ene. Visoki koeficienti provo ena toplote za sve metale su prevencia da se na tim mestima ne poave izraziti temperaturski gradienti. Pove}avane odnosno smanivane koncentracie legirau}ih elemenata u te~nosti ima znatan utica na pro file koncentracia legirau}ih elemenata u ~vrsto fazi (sl. 4). Za slu~a 2 kada dolazi do smanivana sredne koncentracie bakra i magneziuma, dolazi i do smanivana koncentracia i Mg u primarno a fazi. Smanivanem koncentracia u a fazi dovodi do ka{nena u poavlivanu sekundarne i terciarne ~vrste faze pa se kod slu~aa 2 u~e{}e primarne faze pove}ava u ukupno o~vrslo masi (porast na 92,3% sa 88,3% za slu~a zatvorenog sistema). Suprotan trend odigrava se kada koncentracie bakra i magneziuma rastu sa vremenom, tada su koncentracie bakra i magneziuma u ~vrsto fazi ve}e dok se u~e{}e primarne faze u o~vrslo leguri smanue (84,3%). Slika 5. Putane o~vr{}avana tokom o~vr{}avana legure 2 33
10 Rezultati prora~una za leguru 2 prikazani su na sl. 5. i 6. Na sl. 5 prikazane su putane o~vr{}avana za slu~aeve 4 6 kada se primarno o~vr{}avane zavr{ava na binarnom kanalu du` koga se S faza (Al 2 Mg) izdvaa simultano sa a fazom iz preostalog te~nog rastopa. Binarni kanal ima negativan trend {to se ti~e koncentracie magneziuma, dok koncentracia bakra u te~nosti nastavla da raste sve do eutekti~ke vriednosti od 33,2%. Efekti promene srednih koncentracia bakra i magneziuma sli~ni su kao kod legure 1. Smanivanem koncentracia i Mg u vremenu dolazi do pomerana radne ta~ke bli`e ~istom aluminiumu, {to pomera putanu primarnog o~vr{}avana na dole. Slika 6. Sredni sastavi u o~vrslo leguri 2: (a) ; (b) Mg Za slu~a ubrzanog o~vr{}avana putana primarnog o~vr{}avana presieca binarni kanal na mestu koe e bli`e eutekti~ko ta~ki u odnosu na slu~a zatvorenog sistema. Istovremeno, koncentracie bakra i magneziuma u ~vrsto fazi su mane (sl. 6) sli~no kao kod legure 1. Distribucia sastava tokom sekundarnog o~vr{}avana direktno e kontrolisana polo`aem i promenama koe diktira binarni kanal (smanivane koncentracie Mg u te~nosti i rast koncentracie ). U tablici 3. prikazane su uporedne vriednosti masenih u~e{}a poedinih faza za dvie legure koe su eksperimentalno istra`ivane 12, zaedno sa rezultatima dobienim predlo`enim modelom. Potrebno e naglasiti da maseni udeli primarne a faze koi su dobieni eksperimentom 12 i koi su prikazani u tablici sadr`e u sebi primarnu fazu koa se izdvaa tokom sekundarnog o~vr{}avana (L a+q). Na osnovu rezultata prikazanih u tablici 3. vidliva su odre ena gruba slagana, ali o{ uviek nedovolno precizna. Xie i koautori 12 sproveli su eksperimente sa nekoliko razli~itih brzina hla ena. U ovom radu kori{}eni su rezultati za slu~a nabr`eg hla ena u cilu minimizirana efekata difuzie u ~vrsto fazi. Za slu~aeve brzina hla ena koi su kori{}eni (0,039 0,78 K/s), efekat difuzie e vidliv u masenim udelima primarne a faze (mane vriednosti a faze za ve}e brzine hla ena). Drugi razlog koi mo`e biti uzrok neslagana rezultata modela i eksperimenta su nepreciznosti vezane za opisivane faznog diagrama legure. Za suvi{e velike vriednosti parcialnih koeficienata modelom }e se dobiti mane vriednosti koncentracia i Mg u ~vrsto fazi, kao i mane u~e{}e primarne a faze. Efekti koi su doveli 34
11 I. Vu{anovi}, M. X. M. Kren: Matemati~ki model mikrosegregacie u Al--Mg... TERMOTEHNIKA bro 1-4 godina XXVII, (2001) do nepreciznosti modela kao i difuzia u ~vrsto fazi koa se ne mo`e zanemarivati, su predmeti ve} sprovedenih i budu}ih istra`ivana koa }e biti uskoro obavlena na osnovu postoe}ih rezultata 15, 16. Tablica 3. Uporedni rezultati modela i eksperimenta prema 12 Legura a faza q faza Eutekti~ka faza (a + S + q) Modelska analiza 3 0,925 0,008 0,067 Mereni rezultati ,958 0,019 0,023 Modelska analiza 4 0,713 0,175 0,112 Mereni rezultati ,792 0,125 0,083 Zaklu~ak U radu e prikazano preliminarno istra`ivane efekata makroskopske redistribucie legirau}ih elemenata na proces mikrosegregacie u Al--Mg leguri. U tom cilu razvien e odgovarau}i matemati~ki model koim se mogu predvi ati razli~iti slu~aevi o~vr{}avana na nivou kristala, kao i mogu}nost opisivana fenomena mikrosegregacie kada su sredne koncentracie bakra i magneziuma promenlive {to e na~e{}i slu~a u procesima realnog o~vr{}avana u ingotima. Sprovedeni su prora~uni za dvie razli~ite legure za tri razli~ita slu~aa: kada nema promena srednih koncentracia i Mg, kada opadau i kada rastu sa definisanim gradientom u definisanom vremenskom intervalu. Izvr{eno e uporedno prikazivane rezultata za sva tri slu~aa o~vr{}avana na diagramima na koima su prikazani temperaturski profili tokom o~vr{}avana i profili koncentracia i Mg. Slu~aevi otvorenog sistema ( 2 i 3 ) dovode do ubrzanog o~vr{}avana i ponovnog otapana ve} formirane ~vrste faze redom. Ove promene dovode do promene sastava dobiene legure, posebno u~e{}a sekundarne i eutekti~ke faze u o~vrslo leguri. Temperaturske promene koe su uzrokovane ubrzanim o~vr{}avanem i ponovnim otapanem ne de{avau se kada se mikrosegregacioni model spre`e sa makroskopskim transportnim modelima, ili kada e proces o~vr{}avana kontrolisan promenama tem - per a ture, a ne entalpie. Preliminarni rezultati koi su prikazani u ovom radu predstavlau uvod za istra`ivana koa su u toku ili koa }e biti sprovedena 15, 16 u naskorio budu}nosti, a vezana su za razmatrane procesa difuzie u ~vrsto fazi i pobol{ana u opisivanu faznog diagrama legure. Literatura 1 Bennon, W. D., Incropera, F. P., Int. J. Heat Mass Trans., 30 (1987), Schnei der, M. C., Beckermann, C., Int. J. Heat Mass Trans., 38 (1995), Krane, M. J. M., Incropera, F. P., Gaskell, D. R., Int. J. Heat Mass Trans., 40 (1997),
12 4 Schnei der, M. C., Beckermann, C., Metall. Ma ter. Trans. A, 26A (1995), Swaminathan, C. R., Voller, V. R., Int. J. Heat Mass Trans., 40 (1997), Kraft, T., Chang, Y. A., JOM, 49:7, (1997), 20 7 Voller, V. R., in: So lid i fi ca tion 1998 (Eds. S. P. Marsh et al.), TMS, 1998, 27 8 Voller, V. R., in: Modeling of Cast ing, Welding and Ad vanced So lid i fi ca tion Pro cesses, IX, 9 (Eds. P. R. Sahm et al.), Shaker,Verlag, 2000, 607 Kurtz, W., & Fisher, D. J., Fun da men tals of So lid i fi ca tion, 4th ed., TransTech Pub li ca tions, Rettenmayr, M., Warkentin, O., Exner, H., Zeit. für Metallkunde, 88 (1997), Lacaze, J., Lesoult, G., ISIJ In ternational, 35 (1995), Xie, F. Y., Kraft, T., Zuo, Y., Moon, C.-H., Chang, Y. A., Acta. Ma ter., 47 (1999), Beckermann, C., Gu, J. P., Boettinger, W. J., Metall. Ma ter. Trans. A, 31A (2000), Krane, M. J. M., Incropera, F. P., Gaskell, D. R., Metall. Ma ter. Trans. A, 29A (1998), Cefalu, S., M. S. the sis, School of Ma te rials En gi neering, Purdue Uni ver sity, Vušanovi}, I., Ph. D. dissertation, Faculty of Mechanical Engineering, University of Montenegro, Podgorica, Montenegro, Yugoslavia2002 Ab stract MICROSEGREGATION DURING SO LID I FI CA TION OF Al--Mg AL LOYS WITH VARYING COM PO SI TION Igor VU[ANOVI] and Mat thew John M. KRANE A model is pre sented to ex am ine the ef fect of mac ro scopic re dis tri bu tion on lo - cal microsegregation dur ing so lid i fi ca tion of in multicomponent al loys. The mac ro scopic ef fects are felt through lo cal changes in mix ture com po si tion that can af fect the so lid i fi ca - tion be hav ior and com po si tion pro files of the al loys. Cal cu la tions were made for the so - lid i fi ca tion of two Al--Mg al loys for three cases in which the com po si tion re mains the same, de creases, or in creases. Vari a tions in com po si tion lead to changes in com po si tion pro files and rel a tive amounts of dif fer ent solid phases.the model in its pres ent form can be used in con unc tion with mac ro scopic so lid i fi ca tion mod els and is the ba sis for more so phis ti cated microsegregation mod els. Key words: microsegregation, Al--Mg alloy, redistribution, solidification, compositions, q phase, S phase, macrosegregation 36
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