Several Intensive Steel Quenching Models for Rectangular and Spherical Samples

Transcription

1 Recen Advances n Fud Mecancs and Hea & Mass ansfe Sevea Inensve See Quencng Modes fo Recangua and Speca Sampes SANDA BLOMKALNA MARGARIA BUIKE ANDRIS BUIKIS Unvesy of Lava Facuy of Pyscs and Maemacs Insue of Maemacs and Compue Scences Rana buv 9 Rga LV459 LAVIA sandabomkana@gmacom mbuke@anev buks@anev p://www zav/scenss/buksm Absac: - In s pape we deveop maemaca modes fo 3-D and -D ypeboc ea equaons and consuc e anayca souons fo e deemnaon of e na ea fux fo ecangua and speca sampes Some souons of me nvese pobems ae obaned n cosed anayca fom Some numeca esus ae gven fo a sve ba e nfuence of eaxaon me on souon neay of cassca and ypeboc ea equaon nea and non-nea bounday condons ae nvesgaed Key-Wods: Inensve quencng Hypeboc Hea equaon Invese pobem Exac souon Consevave aveagng meod Inoducon Conay o e adona meod e nensve quencng pocess uses envonmenay fendy gy agaed wae o ow concenaon of wae/mnea sa souons and vey fas coong aes ae apped []-[] We popose o use ypeboc ea equaon fo moe easc descpon of e nensve quencng (IQ) pocess (especay fo pocess na sage) Compee bbogapy on ypeboc ea conducon equaon can be found n [3] In ou pevous papes we ave consuced anayca exac and appoxmae [4] [5] souons fo IQ pocesses Hee we consde few oe modes and consuc souons fo dec and nvese pobems of ypeboc ea conducon equaon Hee ae bo appoxmae (on e bass of consevave aveagng meod see [6] [7]) and exac (on e bass of Geen funcon meod see [8]-[]) souons Maemaca Fomuaon of 3-D Pobem and Souons fo Paaeepped In s secon we gve e maemaca saemen fo dec and me nvese pobems Maemaca Saemen of Fu 3-D Pobem fo Paaeepped e non-dmensona empeaue fed fufs ypeboc ea equaon (eegap equaon): V V V V V a x y z x( ) y( b) z( w) () k ( ] a c Hee c s e specfc ea capacy k - e ea conducon coeffcen - e densy - e eaxaon me I s naua assumpon a panes x y z ae symmey sufaces of e sampe: V V V () x y z x y z On e a oe sdes of see pa we ave ea excange w envonmen Aoug e meod poposed ee s appcabe fo non-omogeneous envonmen empeaue fo smpcy we consde modes of consan envonmen empeaue s escon gves foowng omogeneous d ype bounday condons on e a ee oue sdes (ee s ea excange coeffcen): V V (3) x k x ISBN:

2 Recen Advances n Fud Mecancs and Hea & Mass ansfe V V V V (4) y z yb zw e na condons ae assumed n fom: V V ( x y z) (5) V W ( x y z) (6) Fom e pacca pon of vew e condon (6) s uneasc e na ea fux mus be deemned eoecay As addona condon we assume a e empeaue dsbuon and e ea fuxes dsbuon a e end of pocess ae gven (known): V V ( x y z) (7) V W ( x y z) (8) As e fs sep we use we known subsuon: V( x y z ) exp U( x y z ) (9) en e dffeena equaon () ansfoms no dffeena equaon wou fs me devave: U U U U a U x y z 4 x( ) y( b) z( w) ( ] () a a / e na and bounday condons ake e fom: U V ( x y z) () U V ( x y z) W ( x y z) () U U U x y z x y z (3) U U U U x y (4) x yb U U (5) z zw Addona condons (7) (8) ansfom as foow: U exp V ( ) x y z (6) U V ( x y z) exp W ( x y z) (7) Exac Souon of Dec -D Pobem We w sa w fomuaon of e maemaca mode fo n n yz decons of see pa (onedmensona mode): w b en n accodance w consevave aveagng meod [6] [7] we noduce foowng nega aveaged vaue: b ux ( ) bw dyuxyzdz ( ) w (8) Assumng e smpes appoxmaon by consan n e y z decons we oban -D dffeena equaon w e souce em: u u a ( ) ( cu x ] x (9) c b w 4 Ina condons () () fo e dffeena equaon () ae as foow: u u ( x) () u ( x) bw dy V ( x y z) dz u b w u ( x) v( x) v( x) w( x) b w ( ) ( ) w x bw dy W x y z dz () e bounday condons eman n e same fom: u u u () x x x x Souon of s one-dmensona dec pobem (9)-() s we known see [9]: ux ( ) u ( ) Gx ( d ) (3) v ( ) G( x ) d e Geen funcon as epesenaon [9]-[]: Gx ( ) m m ( x) ( )sn a c ( x) ( )sn a c a a c c (4) ISBN:

3 Recen Advances n Fud Mecancs and Hea & Mass ansfe ( x) cos x Hee e naua numbe m n e bo sums s gven by nequaes: a m b w 4 a m b w 4 e egenvaues ae oos of e anscendena equaon: an ( ) 3 Souon of me nvese -D Pobem As was od eae fom expemena pon of vew na condon (7) s uneazabe and e v ( x) mus be cacuaed eoecay e dffeenaon of souon (3) gves: ux ( ) u ( ) Gx ( d ) (5) v ( ) G( x ) d e addona condons (6) and (7) a e end of pocess egadng e funcon u x ae as foow: u u( x) u( x) exp v( x) (6) b w v ( x) bw dy V ( x y z) dz especvey u v ( x) v ( x) v( x) exp w( x) (7) b w ( x) bw dy W ( x y z) dz w ee s an neesng suaon f bo addona condons ae known In s case we noduce new me agumen by fomua (8) e man dffeena equaon (9) emans s fom: u u a cu x ( ) ( ] (9) x e bounday condons () eman e same Bo addona condons ansfom o na condons fo e equaon (8): u u u ( ) ( ) x v x (3) e souon of dec pobem (9) () and (3) s sma w e souon (3): ux ( ) u ( ) Gx ( d ) (3) v ( ) G( x ) d Fo e ea fux we ave an expesson: ux ( ) u ( ) Gx ( d ) (3) v ( ) G( x ) d Fom fomua (3) mmedaey foows a nce expc epesenaon fo e na ea fux: v ( x) u ( ) G( x ) d (33) v ( ) G( x ) d In pevous pape [8] we ave used e Geen funcon fo cassca (paaboc) ea equaon bu ee we used Geen funcon fo e wave (ypeboc) equaon 3 Appcaon of Consevave Aveagng Meod fo me Invese Hypeboc Hea Conducon Pobem In s pa we consde -D speca ypeboc ea equaon: V V a V We known smpe ansfomaon U V aows us o go o Caesan coodnaes We compae e souon of s equaon w cassca paaboc ea equaon 3 Ogna Pobem We sa w e fomuaon of e onedmensona maemaca mode fo nensve see quencng wou ea osses: ISBN:

4 Recen Advances n Fud Mecancs and Hea & Mass ansfe U U U a ( ) f x x (34) x( H) ( ) H U k U () x [ ] (35) x U x H x (36) U U ( x) x [ H] (37) e na ea fux U V ( x) x[ H] (38) can be measued expemenay and mus be cacuaed As addona condon we assume expemenay eazabe condon e empeaue dsbuon a e end of pocess s gven: U( x ) U ( x) x [ H] (39) e second sub-pobem as non-omogeneous man equaon and omogeneous na condons: dw dw w () f() (45) d d dw() w() d e souon of s pobem as a foowng fom: () ( ) ( ) f e w e q d () () () Hee q s souon of e dffeena equaon (45) w speca na condons: dq() q() d e q () sn( ) 3 e Appoxmae Souon by Hence: Consevave Aveagng Meod By appyng consevave aveagng meod o e w() e sn( )( e ( )) d pobem (9)-(4) we oban eavey e nega c aveage empeaue foowng bounday Consequeny we ave fnay obaned e souon pobem fo odnay dffeena equaon: of e pobem (4) (4) as: du du u () u () f() d d (4) v e [ ue ( u u )sn( )] u() u u( ) u (4) ( ) e sn( )( e ( )) d We ae neesed o deemne c du() v ( As ) e as sep we use e addona nfomaon d condon () e e known vaue a e end of e o sove s pobem we sp n wo subpobems: pocess s nfomaon aows us o expess unknown second na condon n cosed and u () () () smpe fom: Fs of em as omogeneous man equaon: v du du u sn( ) (4) d d ogee w non-omogeneous na condons: u ( u e u e ( )sn( )) du() u() u v (43) d s pobem can be soved n adona way and s souon s: sn( ) (46) u v u () e ( ue (( ( ) )sn( )) ( ) sn( ) ( ) f ( ) e d c Hee 4 (44) We can ncease e ode of e appoxmaon fo e souon of e ogna pobem (9)-(4) by e ISBN:

5 Recen Advances n Fud Mecancs and Hea & Mass ansfe epesenaon w poynoma of second degee and exponena appoxmaon Lnea appoxmaon educes o appoxmaon by consan Appoxmaon w second degee poynoma: x x U( x ) u( ) u( ) u ( ) R R We use bounday condons o deemne u () and u () e negaon ove neva x [ H ] of e man equaon paccay gves e same odnay dffeena equaon (4) e ony dffeence s n e same coeffcen a wo ems: du du u () f() d d R (47) k c( R k) e addona condons eman e same I means a we can use obaned above fomuae epacng e paamees by foowng expessons: 4 (48) R ( ) H k k Exponena appoxmaon: x x U( x ) u( ) ( e ) u( ) ( e ) u ( ) Dffeena equaon s n fom du du ku d d cr( k snr cos R ) k() f() c( ksnrcos R) Ake pevous case dffeence s ony n paamees and : 4 ( (sn R (cos R)) k (49) ( (sn R (cos R )) k We ave obaned souon of we posed pobem n cosed fom s souon can be used as na appoxmaon fo negaed ove x [ H ] equaon Consevave aveagng meod can be apped o pobems w non nea BC Condon fo nuceae bong( m [3;3 ]): 3 U m m k [ U B ( )] x R [ ] x 4 Resus We soved sevea pobems and obaned numeca esus usng Mape and COMSOL Mupyscs Modeng s done fo a sve ba =m empeaue a = s 6 C a = C Fgue Dependence on vaue If we compae souons of cassc paaboc and ypeboc ea conducon pobems usng nonnea bounday condon case we oban gapc n Fgue Fgue Nonnea BC We examned empeaue on e adus As you can see e empeaue on adus s no monoony I means a e fom of bounday condon on e suface can vay: v ISBN:

6 Recen Advances n Fud Mecancs and Hea & Mass ansfe Fgue 3 empeaue dsbuon on adus I s vey cea a a e begnnng of e pocess ypeboc em s exemey mpoan bu ae pocess s descbed by cassc ea equaon I s possbe o defne pecse pons wee empeaue s compued: Fgue 4 empeaue canges a =m and =m 5 Concusons We ave consuced some souons fo me nvese pobems fo ypeboc ea equaon w nea and nonnea bounday condons e souons fo deemnaon of na ea fux ae obaned n cosed anayca fom Numeca esus ae obaned fo speca sampe Acknowedgemens: Reseac was suppoed by Unvesy of Lava (Pojec No: 9/3/DP//9/APIA/VIAA/8) and Counc of Scences of Lava (Gan 957) Refeences: [] Kobasko N I Inensve See Quencng Meods Handbook eoy and ecnoogy of Quencng Spnge-Veag 99 [] Kobasko NI Sef-eguaed ema pocesses dung quencng of sees n qud meda Inenaona Jouna of Mcosucue and Maeas Popees Vo No 5 p - 5 [3] Lqu Wang Xuesang Zou Xaoao We Hea Conducon Spnge 8 [4] MBuke ABuks Hypeboc Seveae Inensve See Quencng Modes fo Recangua Sampes Poceedngs of 5 NAUN/WSEAS Inenaona Confeence on Fud Mecancs and Hea and Mass ansfe Cofu Isand Geece Juy -4 WSEAS Pess p88-93 [5] Buke M Buks A Appoxmae Souons of Hea Conducon Pobems n Mu- Dmensona Cynde ype Doman by Consevave Aveagng Meod Pa Poceedngs of e 5 IASME/WSEAS In Conf on Hea ansfe ema Engneeng and Envonmen Vouagmen Aens Augus p 5 [6] Buks A Consevave aveagng as an appoxmae meod fo souon of some dec and nvese ea ansfe pobems Advanced Compuaona Meods n Hea ansfe IX WI Pess 6 p 3-3 [7] Vums R Buks A Consevave aveagng meod fo paa dffeena equaons w dsconnuous coeffcens WSEAS ansacons on Hea and Mass ansfe Vo Issue 4 6 p [8] ABuks S Gusenov Souon of evese ypeboc ea equaon fo nensve cabuzed see quencng Poceedngs of 3 Inenaona Confeence on Compuaona and Expemena Engneeng and Scences Decembe -6 5 Madas Inda p [9] Poyann AD Handbook of Lnea Paa Dffeena Equaons fo Engnees and Scenss Capman&Ha/CRC (Russan edon ) [] Guene RB Lee JW Paa Dffeena Equaons of Maemaca Pyscs and Inega Equaons Dove Pubcaons Inc New Yok 996 [] Roac GF Geen s Funcons Cambdge Unvesy Pess 999 [] Coe K D Haj-Sek A Beck J V Lkou B Hea Conducon Usng Geen s Funcons CRC Pess ISBN: