Dual Phase Lag Model of Melting Process in Domain of Metal Film Subjected to an External Heat Flux

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1 A C H I V E S o F O U N D Y E N G I N E E I N G Published uarerly as he organ o he Foundry Commission o he Polish Aademy o Sienes ISSN ( ) Volume 16 Issue / / Dual Phase Lag Model o Meling Proess in Domain o Meal Film Subeed o an Exernal Hea Flux Absra B. Mohnaki a, *, M. Ciesielski b a Universiy o Oupaional Saey Managemen in Kaowie, Bankowa 8, Kaowie, Poland b Czesohowa Universiy o ehnology, Dabrowskiego 69, -00 Czesohowa, Poland *Corresponding auhor: bmohnaki@wszop.edu.pl eeived ; aeped in revised orm Heaing proess in he domain o hin meal ilm subeed o a srong laser pulse are disussed. he mahemaial model o he proess onsidered is based on he dual-phase-lag euaion (DPLE) whih resuls rom he generalized orm o he Fourier law. his approah is, irs o all, used in he ase o miro-sale hea ranser problems (he exremely shor duraion, exreme emperaure gradiens and very small geomerial dimensions o he domain onsidered). he exernal heaing (a laser aion) is subsiued by he inroduion o inernal hea soure o he DPLE. o model he meling proess in domain o pure meal (hromium) he approah basing on he ariiial mushy zone inroduion is used and he main goal o invesigaion is he veriiaion o inluene o he ariiial mushy zone widh on he resuls o meling modeling. A he sage o numerial modeling he auhor s version o he Conrol Volume Mehod is used. In he inal par o he paper he examples o ompuaions and onlusions are presened. Keywords: heoreial basis o oundry proesses, Solidiiaion proess, Miro-sale hea ranser, Dual phase lag model, Conrol volume mehod 1. Inroduion he base or he marosopi model o alloys solidiiaion or meling is he well known Fourier euaion wih an addiional erm (he soure union) onrolling he evoluion o laen hea [1-3]. he assumpion ha he loal and emporary value o solid sae volumeri raion is he known emperaure-dependen union allows one o ransorm he basi energy euaion, and hen he parameer alled a subsiue hermal apaiy appears (e.g. []). Suh an approah is known as a one domain mehod or a ixed domain mehod. I urned ou ha he similar ransormaion an be done in he ase o dual-phase-lag euaion []. he one domain mehod desribes he meling or solidiiaion proesses in he ase o maerials or whih he phase ransiion proeeds in he inerval o emperaure (e.g. binary alloys). In he ase o pure meal solidiiaion he solidiiaion poin * has o be replaed by a erain emperaure inerval [*, * + ] and in his way he ariiial mushy zone is inrodued. Nex, or his inerval he subsiue hermal apaiy should be deined [1-3]. In lieraure he maro models o meling (solidiiaion) are widely disussed (e.g. [1-3], [5-7]), while he number o papers onerning he miro-sale models is raher small, among ohers [, 8, 9, 10]. In his paper he analysis o relaionships beween widh o he border emperaures inerval * and * + and he resuls o numerial soluion o he ask disussed is done. he similar researh onerning he maro models o meling (solidiiaion) proess has been presened by Szopa in [5]. A C H I V E S o F O U N D Y E N G I N E E I N G V o l u m e 1 6, I s s u e / 0 1 6,

2 . Mahemaial model he dual phase lag euaion resuls rom he generalized orm o he Fourier law onaining wo lag imes, in pariular he relaxaion and o hermalizaion imes [11, 1, 13] r, z, λ r, z, (1) where τ and τ are he phase lags (relaxaion and hermalizaion imes), while is a hea lux, λ is a hermal onduiviy, r, z are he geomerial o-ordinaes (axially-symmerial problem is onsidered), is a ime. Using he aylor series expansions, he ollowing irs-order approximaion o euaion (1) is obained ( r, z, ) ( r, z, ) ( r, z, ) τ λ ( r, z, ) τ () Inroduing his ormula o he well known diusion euaion aer he mahemaial manipulaions one has ( r, z, ) ( r, z, ) τ λ ( r, z, ) λ ( r, z, ) Q ( r, z, ) τ Q( r, z, ) τ Here is a volumeri speii hea o maerial, Q is he apaiy o inernal hea soures. In he ase onsidered he union Q is he sum o wo omponens. he irs o hem is onneed wih he laser beam aion and aording o [11] he suiable soure erm is o he orm ln I r z QL ( r, z, ) (1 ) exp ln π δ r δ ( ) 0 p p D p (3) () where I 0 [J/m ] is a laser inensiy, p [s] is a haraerisi ime o laser pulse, δ [m] is an opial peneraion deph, is a releiviy o irradiaed surae, r D [m] is a laser beam radius. he derivaive o Q L wih respe o ime (euaion ()) an be ound analyially. he seond soure union is onneed wih he evoluion o laen hea. he well known orm o his omponen is he ollowing One an see ha ouside he inerval beween he border emperaures he union S akes he values 1 and 0 (solid and liuid saes). In he paper [] was shown ha he soure erm onrolling he evoluion o laen hea an be wrien as ollows Q ( r, z, ) τ S L QS ( r, z, ) d S ( ) ( r, z, ) ( r, z, ) τ d and inally one obains he DPLE in he orm ( r, z, ) ( r, z, ) C τ λ ( r, z, ) λ ( r, z, ) QL ( r, z, ) τ QL ( r, z, ) τ where L * L * * C M * S while L, S are he volumeri speii heas o liuid and solid sae, M is (or example) he arihmei mean o L and S. he no-lux boundary ondiion given on he exernal surae o he sysem is he ollowing [, 1] r, z, r, z, n n 0 (10) where r, z, n is he emperaure derivaive in he normal direion. he iniial ondiions (he iniial emperaure o domain and he iniial heaing rae) are also given r, z, 0 : r, z,0 0, 0 0 (7) (8) (9) (11) S( r, z, ) d S( ) ( r, z, ) QS ( r, z, ) L L d (5) where 0 is he iniial emperaure o domain. where L is a volumeri laen hea, S is a volumeri solid sae raion a he neighborhood o poin onsidered. Le us assume ha he union S in he inerval [*, * + ] is a linear one, his means ( r, z, ) S * r z (,, ) (6) 3. Conrol Volume Mehod A he sage o numerial ompuaions he Conrol Volume Mehod (CVM) in he version proposed by Ciesielski and Mohnaki [1] is applied. As is well known, he CVM algorihm allows one o ind he ransien emperaure ield a he se o nodes orresponding o he enral poins o onrol volumes. he nodal emperaures an be ound on he basis o energy balanes 86 A C H I V E S o F O U N D Y E N G I N E E I N G V o l u m e 1 6, I s s u e / 0 1 6,

3 or he suessive volumes. he energy balanes orresponding o hea exhange beween he analyzed onrol volume and adoining onrol volumes resuls rom he inegraion o euaion (8) wih respe o onrol volume and ime. In Figure 1 he ylindrial, axially symmerial domain and is disreizaion is shown. Fig. 1. he disreizaion o domain So, he energy euaion (8) should be inegraed over he onrol volume,, r, z, r z r, z, r, z, d Q r, z, Qr, z, d d Inegraion o he le hand side o euaion (1) gives,, r, z, r z d d d V d d (1) (13) he soure erm in E. (8) is reaed in a similar way. Applying he divergene heorem o he erm deermining hea onduion beween he volume bounded by he suraes A and is neighbourhoods one obains r, z, r, z, d r, z, n r, z, da A and nex his erm an be wrien in he orm (1) A r, z, r, z, da k A n k (15) k 1 where d i 0 0 i z d z d i i nr 0 i i nr i1, i1, r d r d i nz 0 i nz z d z d i i 0 0 i i 0 i1, i1, r d r (16) (17) (18) (19) and ( k ) are he harmoni mean hermal onduiviies beween wo enral poins o adoining onrol volumes. Aer he inroduion o all disree erms ino euaion (1) one obains d d i, V k A k d d k 1 dq Qi, Vi, d (0) he seond sage o CVM is he inegraion o euaion (0) wih respe o ime and hen he energy balanes wrien in he onvenion o explii sheme (or he ransiion 1 o, =,...,F) ake he orm V 1 1 k, k i k 1 Q Q A Q V (1) where denoes a ime sep. he hea luxes appearing in he las euaion (aer non omplex mahemaial manipulaions) an be wrien as ollows i, 1 i, 1 i, 1 i, () A C H I V E S o F O U N D Y E N G I N E E I N G V o l u m e 1 6, I s s u e / 0 1 6,

4 i1, i1, i1, 1 1 i n 1 1 r 0 i n n 1 1 z r n i 1, i 1, i 1, i, 1 i i 0 z (3) () (5) he aepaion o he direional hermal onduiviies as he harmoni means o he nodal values auses ha in he ormulas (30) deermining hea luxes he hermal resisanes beween he neighbouring nodes appear [15], in pariular or he inernal nodes 0.5z 0.5z r 0.5r i 1, 0.5z 0.5z r 0.5r i 1, (6) (7) (8) (9) he oeiien W is eual o W From ineualiy (3) one an deermine he proper ime sep 1 Wmax 1 8W max 0, W max where W. 1 max W i, max,. Examples o ompuaions (3) (33) Numerial simulaion o a hermal proess proeeding in omain o hin meal ilm subeed o he shor laser pulse has been done or he ylindrial domain wih dimensions Z = 100 nm, = 100 nm Figure [1]. while or he boundary ones he hermal resisane in direion o he boundary an be assuned as a very big number, e.g he inal orm o he CVM euaions an be wrien in he orm 1 (30) k, k Q i 1 Q i, k 1 k A /, k V i i A he same ime,. he ormula (30) shows ha in he ase o hyperboli euaions hree-level CVM approximaion mus be used, in oher words, in order o deermine he emperaure (ime level ) he nodal emperaures orresponding o ime levels 1 and mus be known. Using he iniial ondiions (11), he ollowing saring poin should be 0 1 aeped:. 0 Beause he explii sheme has been used, hereore he sabiliy ondiions should be ulilled. he auhors ound ha his ondiion or he ask onsidered is he ollowing Fig.. Miro-domain onsidered hermophysial parameers o maerial (hromium) are eual o = 93 W/(mK), S = L = J/(m 3 K), = ps, = 7.86 ps [16], L = J/m 3, solidiiaion poin * = 1857 o C, = 3, 5, 8 K, respeively. he parameers o he bellype laser pulse are ollowing: r d = 50 nm, I 0 = 3000 J/m, = 0.93, = 15.3 nm, p = 100 s. he iniial emperaure o he meal is 0 = 0 o C. he mesh seps used in his example: z = 10-9 m, r = 10-9 m and he ime sep = s. he resuls o ompuaions presened in Figures 3-5 illusrae he emperaure hisories a hree seleed nodes in he domain or he dieren values o. In Figure 6 he emperaure ield or ime 0.3 ps is shown. 1 W 0 (31) 88 A C H I V E S o F O U N D Y E N G I N E E I N G V o l u m e 1 6, I s s u e / 0 1 6,

5 5. Final remarks Fig. 3. Heaing/ooling urves a he poin P 1 he auhors solved he problem o pure meal meling and resolidiiaion by he inroduion o ariiial mushy zone o he mahemaial model o he proess. A he sage o numerial ompuaions he dieren values o (a widh o mushy zone) have been aken ino aoun. he dierenes o numerial soluions are small, bu visible. he same example has been also solved using he Fourier euaion (boh and are eual o zero). I urned ou ha or his model he heaing proess proeeds more inensively and or I 0 = 3000 J/m he meal emperaure (lose o he poin P 1 ) exeeds he evaporaion poin. he resuls loser o he presened above have been obained or I 0 = 600 J/m. he ypial derease o heaing/ooling rae lose o he solidiiaion poin (easily visible in he ase o maro-sale solidiiaion models e.g. [5]) is here almos unnoieable. I resuls rom essenial dierenes beween apaiy o inernal hea soure resuling rom he laser aion and he apaiy o hea soure resuling rom he meling or solidiiaion proesses. Aknowledgemen he paper is a par o esearh Proe 015/19/B/S8/01101 sponsored by NCN. Fig.. Heaing/ooling urves a he poin P Fig. 5. Heaing/ooling urves a he poin P 3 Fig. 6. emperaure ield or ime 0.3 ps ( = 5[K]) eerenes [1] Mohnak B. & Mahrzak, E. (010). Numerial modeling o asing solidiiaion using generalized inie dierene mehod. Maerials Siene Forum , [] Mohnak B. (01). Deiniion o alloy subsiue hermal apaiy using he simple marosegregaion models. Arhives o Foundry Engineering. 19(), [3] Mohnak B. (011). Compuaional simulaions and appliaions. Numerial modeling o solidiiaion proess (Chaper ), Ed. Jianping Zhu, INECH, [] Mahrzak, E. & Mohnak B. (016). Modeling o meling and resolidiiaion in domain o meal ilm subeed o a laser pulse. Arhives o Foundry Engineering. 16(1), 1-. [5] Szopa,. (015). Numerial modeling o pure meal solidiiaion using he one domain approah. Journal o Applied Mahemais and Compuaional Mehanis. 1(3), 8-3. [6] Bondarenko, V.I., Bilousov, V.V., Nedopekin, F.V. & Shalapko, J.I. (015). he mahemaial model o hydrodynamis and hea and mass ranser a ormaion o seel ingos and asings. Arhives o Foundry Engineering. 15(1), [7] Ivanova, A.A. (01). Calulaion o phase hange boundary posiion in oninuous asing. Arhives o Foundry Engineering. 13(), [8] Mohnak B., Mahrzak, E. (016). Chaper 86: Numerial modeling o biologial issue reezing proess using he Dual- Phase-Lag Euaion, Advanes in Mehanis: heoreial, Compuaional and Inerdisiplinary Issues, Proeedings o he 3rd Polish Congress o Mehanis (PCM) and 1s A C H I V E S o F O U N D Y E N G I N E E I N G V o l u m e 1 6, I s s u e / 0 1 6,

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