Gradient method of cast iron latent heat identification

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1 ARCHIVS o OUNDRY NGINRING Pulishd qurtrly s th orgn o th oundry Coission o th Polish Acdy o Scincs ISSN (897-) Volu 7 Issu 4/7 6 4/4 Grdint thod o cst iron ltnt ht idntiiction. Mjchrz,,*, J. Mndiwicz Dprtnt or Strngth o Mtrils nd Coputtionl Mchnics, Silsin Univrsity o Tchnology, Konrsigo 8, 44- Gliwic, Polnd Institut o Mthtics nd Coputr Scinc, Czstochow Univrsity o Tchnology, Drowsigo 7, 4- Czstochow, Polnd *Corrsponding uthor. -il ddrss: w.jchrz@polsl.pl Rcivd.7.7; ccptd in rvisd or Astrct In th ppr th cst iron ltnt ht in th or o thr coponnts corrsponding to solidiiction o ustnit nd utctic phss is idntiid. Th sic inortion concrning th or o dqut unctions pproxition hs n tn on th sis o cooling curv nd tprtur drivtiv courss ound y ns o th TDA tchniqu. On th stg o invrs prol solution th grdint thod hs n usd. Th nuricl coputtions hv n don using th init dirnc thod. In th inl prt o th ppr th xpl o ltnt ht idntiiction is shown. Kywords: Appliction o inortion tchnology to th oundry industry, Solidiiction procss, Nuricl tchniqus, Invrs prols, Grdint thods, Idntiiction o ltnt ht. Introduction Th invrs prols constitut vry ctiv tool or th nlysis o thrl procsss procding in th syst cstingould-nvironnt. In this ppr th prtric invrs prol is discussd, this ns th ltnt hts connctd with th cst iron solidiiction r idntiid. Th vlus o ths prtrs dtrin th cours o sustitut thrl cpcity o tl. Th sustitut thrl cpcity constituts vry ssntil prtr ppring in th govrning qution dtrining csting solidiiction, in prticulr whn th on doin pproch is pplid [,, ]. To idntiy th ltnt hts corrsponding to ustnit nd utctic phss th grdint thods hv n usd [4,, 6, 7, 8]. Additionlly, nowldg o cooling (hting) curvs t th points slctd ro csting (ould) doin is ncssry to solv th prol considrd nd such inortion (in this ppr) rsults ro th nuricl solution o dirct prol or rl vlus o cst iron prtrs.. Cst iron sustitut thrl cpcity To dtrin th cours o sustitut thrl cpcity o cst iron th xprintl rsrchs hv n rlizd. Th ht cst o hypo-utctic gry cst iron o Zl-Zl clss hs n prprd. Th chrg tril hs n choosn ccording to th ruls concrning th slting o cst iron in th induction urnc. In th cntrl prt o th spling csting th throcoupl PtRh-Pt hs n instlld. Th throcoupl hs n connctd to th rgistring pprtus. Th thrl nd drivtiv nlysis (TDA) hs n don in ordr to dtrin th chrctristic tprturs ssocitd with th chng trnsition. So, th ht procsss procding in th solidiying tl connctd with th ltnt ht ission o succssiv phss hv n rgistrd ting into ARCHIVS o OUNDRY NGINRING Volu 7, Issu 4/7, -6

2 ccount th cooling curv T d (t) T (x d, t) nd its ti drivtiv T d (t)/ t. Using th digrs o th thrl nd drivtiv nlysis th vlus o tprtur-dpndnt ltnt ht hv n rgistrd [9] (ig. ). Nxt, th sustitut thrl cpcity distriution or ushy zon contining th inortion out th ustnit nd utctic phss hs n dscrid ig.. O cours, th physicl condition in th or T CTdT cp T TS + () TS ust ullild. In qution () us + u is th cst iron ltnt ht, us us + us, u r th ltnt hts connctd with th ustnit nd utctic phss volution whr T, T A, T, T S corrspond to th ordr tprturs,,,,,, 4, r th coicints nd c A us cp + T T A whr c P.(c + c S ). Th coicints, hv n ound on th sis o conditions ssuring th continuity o C clss nd physicl corrctnss o pproxition, nly CT ( ) c CT ( A) ca d CT dt T T d CT dt T TA T CT dt c T T + T A nd P A us () (4) C ( T ) [ MJ / K ] T [ o C ] ig.. Distriution o cst iron ltnt ht CT ( ) ca CT ( S) cs d CT dt T d CT dt T TS T CT dt c ( T T ) + T S P S u () Atr th thticl nipultions on hs T S T T A T T [ o C ] ig.. Sustitut thrl cpcity o cst iron So, in th cs o cst iron solidiiction th ollowing pproxition o sustitut thrl cpcity cn tn into ccount (ig. ) c, T T 4 + T + T + 4T + T, TA T < T CT ca, T < T A () 4 + T + T + 4T + T, TS T < cs, T < TS T T A us ( c ca) TTA( T + TA) ( A) ca T c TA + + T TA T T ( T TA) 6( c ) 6 ca TT T TA ( T + TA ) A us ( T TA) ( T TA) ( )( + ) ( ) ( T TA) ( T TA) ( c ) 6 ca ( T + TA ) us ( T TA) ( T TA) A A A A us + 4 c c T T T T T T us ( T T A ) (6) ARCHIVS o OUNDRY NGINRING Volu 7, Issu 4/7, -6

3 nd T T S u ( ca cs) TS( + TS) ( S) ct S cats + + TS T T ( TS) 6( c ) 6 A cs T TS( + TS) S u ( )( + ) ( ) ( c ) 6 A cs ( + TS) u A S S S S u + 4 c c T T T T T T u ( TS) (7) or ti t th initil condition t : T( x, ) T ( x), T ( x, ) T ( x) () is lso nown. I th prtrs ppring in govrning qutions r nown thn th dirct prol is nlyzd, whil i prt o th is unnown thn th invrs prol should considrd [4, 6, 7, 8]. In th ppr th cst iron ltnt hts us, us nd u r idntiid. To solv th invrs prol th snsitivity coicints should dtrind [, ]. So, th dditionl oundry initil prols rsulting ro th dirntition o sic qutions with rspct to th unnown prtrs ust orultd. 4. Snsitivity coicints To dtrin th snsitivity coicints th govrning qutions (8) () r dirntitd with rspct to p us, p us nd p u. So, th ollowing dditionl prols should solvd. Govrning qutions Th nrgy qution dscriing th csting solidiiction hs th ollowing or [, 8, 9, ] T x Ω CT T x t t : λ (, ) whr C (T ) is th sustitut thrl cpcity o cst iron (c.. qution ()), λ is th thrl conductivity, T, x, t dnot th tprtur, gotricl co-ordints nd ti. Th considrd qution is supplntd y th qution concrning ould su-doin T x Ω t : c λ T whr c is th ould volutric spciic ht, λ is th ould thrl conductivity. In th cs o typicl snd oulds on th contct surc twn csting nd ould th continuity condition in th or x Γ c : λ n T λ n T T T cn ccptd. On th xtrnl surc o th syst th Roin condition (8) (9) () x Γ : λ (, ) α (, ) n T x t T x t T () is givn (α is th ht trnsr coicint, T is th int tprtur). CT T T( xt, ) x Ω : + C( T) t t λ T T x Ω : c λ T t p λ n (, ) λ (, ) x Γc : T T x Γ : λ n [ T ] T( x, ) T ( x, ) t :, or [ T x t ] n [ T x t ] Z ( xt, ) CT T( xt, ) x Ω : C( T) λ Z t t Z x Ω c Z x t t : λ (, ) λ n Z λ n Z x Γc : Z Z x Γ : λ n Z t : Z ( x, ), Z ( x, ) whr () (4) ARCHIVS o OUNDRY NGINRING Volu 7, Issu 4/7, -6

4 T Z Z T () nd,,. Dirntition o sustitut thrl cpcity with rspct to th prtrs p, p, p lds to th ollowing oruls, T + + p p 4 4 CT T + T + T + TA T < T p p p + T + 4T + 4 T Z, T A, S < T T (6), T < TA, TS T <, T < TS nd, T T, TA T < T CT, T T < TA (7) whil, T T T T < T, T T, TA T < T, T < T CT + T T + T + T + TS T < T ( + T + 4T + 4 T ) Z,, T < TS S A (8) Ting into ccount th dpndncs (6), (7), th clcultions o / p nd / p r vry sipl. Th oundry initil prols (4) r coupld with th sic on (8) (), cus in ordr to ind thir solutions, th ti drivtiv T (x, t)/ t should nown. Th sic prol or th ssud vlus o p, p, p nd th dditionl ons connctd with th snsitivity unctions Z coputtions hv n solvd using th xplicit sch o init dirnc thod [].. Grdint thod In ordr to solv th invrs prol th lst squrs critrion is pplid [, 4] S p p p T T (9) M (,, ) ( i di) M i whr T di nd T i T (x i, t ) r th surd nd stitd tprturs, rspctivly. Th stitd tprturs r otind ro th solution o th dirct prol (c.. chptr ) y using th currnt vill stit or th unnown prtrs. In th cs o typicl grdint thod ppliction [, 4, 6, 8] th critrion (9) is dirntitd with rspct to th unnown prtrs p,,, nd nxt th ncssry condition o optiu is usd. inlly on otins th ollowing syst o qutions M S ( Ti Tdi) ( Z i) () M i whr ( Z i ) Ti () p p r th snsitivity coicints, is th nur o itrtion, p r th ritrry ssud vlus o p, whil p or > rsult ro th prvious itrtion. Th coicints () cn collctd in th ollowing trix Z ( Z) ( Z) ( Z) ( Z) ( Z) ( Z) ( Z) ( Z) ( Z) ( Z) ( Z) ( Z) ( ZM ) ( ZM ) ( ZM ) ( ZM ) ( ZM ) ( ZM ) () unction T i is xpndd in Tylor sris out nown vlus o p l, this ns ( + ) T T + Z p p i i il l l l () 4 ARCHIVS o OUNDRY NGINRING Volu 7, Issu 4/7, -6

5 Putting () into () on otins (,, ) M + ( Zil ) ( Z i) ( pl pl ) i l M i T T Z di i i (4) Th syst o qutions (4) cn writtn in th trix or T + T T + ( Z Z p Z Z p Z Td T ) () whr ( T ) T d ( T ) T d ( T T ) d Td, T Td ( T ) T dm ( TM ) T dm ( TM ) nd (6) + p p + + p p, p p (7) + p p This syst o qutions llows to ind th vlus o p + or,,. Th itrtion procss is stoppd whn th ssud nur o itrtions K is chivd. ig.. Csting-ould syst In igur 4 th cooling curvs t th control points,, ro csting su-doin (c.. ig. ) r shown, whil igur illustrts th courss o snsitivity unctions Z t th point. T [ o C ],8 [ K /MJ ], t [ s ] ig. 4. Cooling curvs t th points,, 6. Rsults o coputtions Th csting-ould syst shown in igur hs n considrd. At irst, th dirct prol hs n solvd. Th ollowing input dt hv n introducd: λ [W/(K)], λ [W/(K)], c.88 [MJ/( K)], c S.4 [MJ/( K)], us 97. [MJ/ ], us 97.6 [MJ/ ], u 8. [MJ/ ], c.7 [MJ/( K)], pouring tprtur T o C, liquidus tprtur T º C, ordr tprturs T A º C, T º C, solidus tprtur T S º C nd initil ould tprtur T º C. Th dirct prol hs n solvd using th xplicit sch o DM []. Th rgulr sh crtd y nods with constnt stp h. [] hs n introducd, ti stp Δt. [s].,,9 Z,6, Z Z, t [ s ] ig.. Snsitivity unctions t th point ARCHIVS o OUNDRY NGINRING Volu 7, Issu 4/7, -6

6 On th sis o nowldg o cooling curvs shown in igur 4 th unnown prtrs hv n idntiid undr th ssuption tht us us u ig. 6. It is visil tht th itrtion procss or th ssud initil vlus is convrgnt nd th xct solution is otind tr twnty itrtions. igur 7 illustrts th cours o itrtion procss or on snsor (point in ig. ) nd th s initil vlus o prtrs. 7. Conclusions Th tsting coputtions show tht it is possil to idntiy th unnown prtrs only on th sis o on cooling curv nowldg ut it should loctd t th cntrl prt o th csting doin. In this cs or zro initil vlus o idntiid prtrs th itrtiv procss is convrgnt. [ MJ/ ] 8 us This ppr is prt o th projct Progrss nd ppliction o idntiiction thods in oving oundry prols. Rrncs u 6 us 4 ig. 6. Rsult o idntiiction or thr cooling curvs [ MJ/ ] us 8 u 6 us 4 ig. 7. Rsult o idntiiction or on cooling curv [] B.Mochnci, J.S.Suchy, Nuricl thods in coputtions o oundry procsss, PTA, Crcow, 99. [].Mjchrz, B.Mochnci, Idntiiction o thrl proprtis o th syst csting - ould, Mtrils Scinc oru 9-4 (7) [] S.r, R.Szop, Modl o cst iron solidiiction using th rtiicil ushy zon pproch, Archivs o oundry, 6, (6) [4] K.Kurpisz, A.J.Now, Invrs Thrl Prols, Coputtionl Mchnics Pulictions, Southpton- Boston, 99. [] R.Szop, Snsitivity nlysis nd invrs prols in th thrl thory o oundry, Pul. o th Czst. Univ. o Tchn., Czstochow, 6 (in Polish). [6] B.Mochnci, J.S.Suchy, Idntiiction o lloy ltnt ht on th sis o ould tprtur (Prt ), Archivs o oundry 6, (6) 4-. [7] B.Mochnci,.Pwl, J.S.Suchy, Idntiiction o lloy ltnt ht on th sis o ould tprtur (Prt ), Archivs o oundry, 6, (6) -7. [8].Mjchrz, J.Mndiwicz, Idntiiction o cst iron sustitut thrl cpcity, Archivs o oundry,6, (6) -. [9] J.Mndiwicz, Inlunc o choic o cst iron sustitut thrl cpcity on th rsults o nuricl coputtions, Archivs o oundry, 6, (6) - (in Polish). [] R.Szop, Th prtric snsitivity nlysis o solidiiction procss, Archivs o oundry, () [] M.Klir, Prtr snsitivity, J.Wily & Sons td., Chichstr, 997. [] K.Ds, B.Rousslt, Snsitivity nlysis or trnsint ht conduction in solid ody, Structurl Optiistion, 7 (999), ARCHIVS o OUNDRY NGINRING Volu 7, Issu 4/7, -6