Hernandez, E. M., et al.: Non-Parabolc Interace Moton or the -D Stean Problem... THERMAL SCIENCE: Year 07, Vol., No. 6A, pp. 37-336 37 NON-PARABOLIC INTERFACE MOTION FOR THE -D STEFAN PROBLEM Drchlet Boundary Condtons by Ernesto M. HERNANDEZ, Jose A. OTERO *, Ruben D. SANTIAGO, Raul MARTINEZ, Francsco CASTILLO, and Joaqun E. OSEGUERA Monterrey Insttute o Technology, School o Engneerng and Scence, Atzapan de Zaragoza, State o Mexco, Mexco Orgnal scentc paper https://do.org/0.98/tsci54098h Over a nte -D specmen contanng two phases o a pure substance, t has been shown that the lqud-sold nterace moton exhbts parabolc behavor at small tme ntervals. We study the nterace behavor over a nte doman wth homogeneous Drchlet boundary condtons or large tme ntervals, where the nterace moton s not parabolc due to nte sze eects. Gven the physcal nature o the boundary condtons, we are able to predct exactly the nterace poston at large tme values. These predctons, whch to the best o our knowledge, are not ound n the lterature, were conrmed by usng the heat balance ntegral method o Goodman and a non-classcal nte derence scheme. Usng heat transport theory, t s shown as well, that the temperature prole wthn the specmen s exactly lnear and ndependent o the ntal prole n the asymptotc tme lmt. The physcs o heat transport provdes a powerul tool that s used to ne tune the numercal methods. We also ound that n order to capture the physcal behavor o the nterace, t was necessary to develop a new non-classcal nte derence scheme that approaches asymptotcally to the predcted nterace poston. We oer some numercal examples where the predcted eects are llustrated, and nally we test our predctons wth the heat balance ntegral method and the non-classcal nte derence scheme by studyng the lqud-sold phase transton n alumnum. Key words: Stean problem, heat balance, nte derence method Introducton The study and modelng o movng boundary problems such as lqud-sold phase transtons, mples the buldng o solutons as one o the most mportant tasks. Just or a ew -D problems on nnte or sem-nte regons s possble to nd exact analytcal solutons []. However, or the vast majorty o movng boundary problems, the search o solutons s done by means o derent strateges and approxmate methods. The nte derence method oers approxmate solutons and s requently used to buld numercal solutons o phase change problems wth tme ndependent boundary condtons [-7] and tme dependent boundary condtons [8-0]. Approxmate analytcal solutons can be ound as well or the class o problems that can not be solved exactly. One o these approaches s the Goodman s method or heat balance * Correspondng author, e-mal: j.a.otero@tesm.mx
Hernandez, E. M., et al.: Non-Parabolc Interace Moton or the -D Stean Problem... 38 THERMAL SCIENCE: Year 07, Vol., No. 6A, pp. 37-336 ntegral method (HBIM) [-5]. Ths method allows good results wth less numercal and computatonal resources than other approxmate methods. The lqud-sold phase transton on nte sze systems has been studed by several authors where the man concern s to develop derent numercal strateges to solve the same problem and compare wth the ew exact solutons avalable n the lterature. For example, the exact soluton on a sem-nnte regon s compared wth several sem-analytc and numercal methods or small tme ntervals [, 3, 7, 5-7] where the nterace moton s approxmately parabolc or boundary condtons are chosen n such a way that parabolc moton s observed throughout the entre process [7, 6]. However, lttle s mentoned about the behavor or tme values where the numercal solutons start to devate sgncantly rom the exact soluton. We wll study the physcal consequences o havng Drchlet boundary condtons on both sdes o the specmen, where nte sze eects become sgncant or large tmes. Thereore, the goal o ths work s to oer a physcal nterpretaton o -D heat transport wth Drchlet boundary condtons, where a lqud-sold phase transton s takng place on a pure substance. For the soldcaton process o the lqud phase, t wll be assumed that heat low s low enough, so a super coolng phase s not ormed. Gven that the nature o the soluton s hghly dependent on the boundary condtons mposed on the specmen. In ths work, we ocus on the physcal mplcatons o homogeneous Drchlet boundary condtons and very our predctons wth the HBIM and a new non-classcal nte derence scheme (). We ound that, n order to approach asymptotcally to the predcted poston o the nterace, t was necessary to develop ths. Statement o the problem Consder a lqud phase n contact wth a sold phase, both separated by an nterace wth poston ξ at some uson temperature, T, where the total heat low through the nterace causes ts dsplacement. Let us assume that the lqud and sold phases have a temperature prole T (,) xt and T(,) xt, respectvely, where the temperature at any pont wthn the lqud phase s above the T and wthn the sold phase, the temperature at any pont, s below T. The temperature proles have the ollowng homogeneous Drchlet boundary condtons: T (0,) t = T, T ( L,) t = T, and T ( ξ,) t = T ( ξ,) t = T () l s where the subndex and represents lqud and sold phase, respectvely. The let edge o the sample n contact wth the lqud s xed at some temperature, T l, and the rght edge n contact wth the sold phase s xed at some temperature, T s. We wll assume temperature proles that n general are unctons o the poston: ( ) ( ) T x, 0 = x, =, () where ( x ) can be obtaned n order to satsy the boundary condtons gven by eq. (), where ξ (0) = B, wth B > 0. The other equatons that model ths problem are the duson heat equatons n medums and : T T = α, wth ( ) ξ x ( ) ξ + ( ) L (3) t x where α = k/ ρ C s the heat duson coecent n phase. These duson constants depend on the specc heat capacty, C, densty, ρ, and thermal conductvty, k, at each phase. We
Hernandez, E. M., et al.: Non-Parabolc Interace Moton or the -D Stean Problem... THERMAL SCIENCE: Year 07, Vol., No. 6A, pp. 37-336 39 wll assume that these thermodynamc varables do not depend on the temperature and or the soldcaton case, we do not consder the ormaton o a super cooled lqud to drve the phase transton. Wth these assumptons, the velocty o the nterace d ξ /dt, between the two materals, s gven by the Stean condton (SC): dξ T T ρ L = k k (4) d t x + x= ξ x x= ξ where L s the latent heat o uson and ρ s the densty o the sold or lqud phase, dependng on whch drecton s the phase transton, takng place. The problem s completely dened by eqs. ()-(4). The well-posedness has been establshed by other authors [8, 9], thereore, eqs. ()-(4) has a unque classcal soluton, whch wll be approxmated usng the and the HBIM. Numercal solutons Fnte derence methods are one o the most popular numercal methods to nd approxmate solutons or boundary problems descrbed by PDE. In ths method, a contnuous regon s transormed nto a nte number o ponts (nodes) and an approxmate soluton s ound only at these ponts, whch consttute a grd or mesh. For ths reason, the derental operators are approxmated or dscretzed at the mesh ponts. Non-classcal nte derence scheme An mplct scheme was used to nd the solutons o the duson heat equaton. That s, the partal tme dervatve o the temperature s expressed as a rst order approxmaton o the backward derence n tme, gven by: mn, mn, T T T (5) t t where t represents the length o the step n t. The dscretzaton o the argument x, s represented by m, and the argument t s represented by n. Thereore, n ths notaton, T = T( xm, t mn, n). To m+, n m, n obtan the proposed, we start by addng the Taylor expansons or T and T up to ourth order n x, whch s the length o the step n x, and keepng n mnd that mn, ( ) T j = j (, )/ j T xm tn x we obtan: 4,,,, ( ), ( 4) + m + n m n mn mn x mn T T = T + x T + T + (6) Then, we apply the central derence denton to the ourth dervatve: m+, n ( ) mn, ( ) m, n ( ) mn, ( 4) T T + T T = + o x x ( ) and substtute ths expresson n eq. (6), to obtan the ollowng relaton: m+, n mn, m, n m+, n ( ) mn, ( ) m, n ( ) T T + T T +0 T + T x 4 ( x ) = + o Instead o substtutng the Taylor expansons n the heat equaton, n ths hgher order scheme, the dscretzed heat equaton s substtuted n eq. (8) to obtan the ollowng model o sx ponts or nodes, whch we have dened as : (7) (8)
Hernandez, E. M., et al.: Non-Parabolc Interace Moton or the -D Stean Problem... 330 THERMAL SCIENCE: Year 07, Vol., No. 6A, pp. 37-336 β T + β T + β T T 0 T T = 0 (9) () m, n () mn, () m +, n m +, n mn, m, n () () where β = ( λ), β = (0 + 4 λ ), and λ = α t/ x, or =,. The dervatves that appear n the SC, eq. (4), are also obtaned by usng a ourth order approxmaton: mn, () j mn, m+ jn, m+ jn, m+ 3 jn, m+ 4 jn, T = ( 5 T + 48 T 36 T + 6 T 3 T) (0) x where j = or = (medum ), and j = or = (medum ). Approxmate analytcal soluton An approxmate analytcal method was proposed by Goodman [], to seek solutons or movng boundary problems, where the equatons that govern transport phenomena are the heat duson equatons, and the dynamcs o the movng boundary s governed by the balance n the heat lux. Temperature proles wll not only be assumed constant along each medum, we wll also consder cases wth parabolc proles n the poston. However, other prole shapes can be consdered snce, as we wll show later, the physcal behavor o the system at large tme values s completely ndependent o the ntal prole. In ths ramework, the HBIM [] suggests representng the temperature proles: ( ) ( ) T( xt, ) = a ξ x + b ξ x, wth ( ) ξ x ( ) ξ + ( ) L () where a and b wth =, or medums and, are unctons o tme. Ths equaton obeys the Drchlet boundary condtons gven by eq. (). The constants a and b at t = 0 wth =,, are determned n order to satsy the boundary condtons. Some extra condtons are needed to determne the ntal values o a and b whch can be obtaned rom the ntal temperature prole. Ater applyng the Drchlet boundary condtons to these parabolc proles, the ollowng relatons between the unctons a and b are obtaned or the lqud and sold phases: ( ) ( ) aξ + bξ = T, and a L ξ + b L ξ = T () l Once the boundary condtons have been appled to the temperature proles, the dynamcs o the movng boundary can be obtaned by substtuton o eq. () n the SC, eq. (4). The resultng equaton s an ODE n tme, or the poston ξ o the nterace: dξ L = ka + ka (3) dt The key element o ths method s to average the duson equatons over the poston varable, by ntegratng the duson equaton n medum, rom x = 0 to x = ξ, and n a smlar manner, ntegratng the duson equaton n medum, rom x = ξ to x = L. Ater averagng n the lqud phase (medum ), t s obtaned an ODE n tme, gven by: db dξ da ξ + ( bξ + a) + ξ αb= 0 (4) 3dt dt dt Smlarly, or the sold phase, the ODE obtaned ater averagng over the poston s gven by: db dξ da dξ ( L ξ) b( L ξ) + ( L ξ) a α b= 0 (5) 3dt dt dt dt s
Hernandez, E. M., et al.: Non-Parabolc Interace Moton or the -D Stean Problem... THERMAL SCIENCE: Year 07, Vol., No. 6A, pp. 37-336 33 Solvng or a and a rom eq. () and substtutng n eqs. (3)-(5), a set o three ODE n tme s obtaned, and solved wth the ntal boundary condtons. Results and dscusson In ths part o the results we show the asymptotc lmts ound or ths problem and a ew numercal experments that valdate the and the HBIM. For these experments, we wll set the thermodynamc varables o densty and specc heat equal to one. Thereore, the dusvty s reduced to α( α ) = k( k) or the lqud (sold) phase. The heat equaton n each medum s smpled correspondngly, and the SC equaton s reduced to: d ξ ( t) T( xt, ) T( xt, ) L = k k (6) d t x + x x= ξ( t) x= ξ( t) snce we are consderng phases wth the same densty ρ =. The ollowng examples show a comparson between the and HBIM, ndcatng also, the maxmum devaton between both types o approaches n each example as ξh BIM ξnc- FDS max ( TH BIM TNC- FDS max) or the nterace (temperature) hstory. For the nte derence smulatons, the results presented n 5 gs. and, use a ne mesh and a value o t =.5 0. Fgure s an example where the moton o the nterace s ntally set at ξ (0) = 0.0. The gure shows the soluton obtaned wth the and the HBIM. A latent heat o L =.0 s used, and the temperatures at the boundares are T l =.0 and T s = 0.4. The dusvtes used are k =.5 and k =.0. The tme nterval where the moton o the nterace was 4 studed s t max =.0 wth N t = 8.0 0 tme parttons, and the spatal mesh n ths example had, N = 4 and N = 8 nodes. Fgure also shows the tme evoluton o the temperature at x = 0.40. All remanng gures ndcate the lmtng value or the nterace and temperature, and as wll be shown n the next secton, these values are obtaned by studyng the proper physcal behavor o the nterace and temperature prole or large values o t. Large tme lmt o the soluton In ths secton, we wll oer a physcal nsght to the obtaned solutons accordng to the boundary condtons consdered. The physcal phenomenom s descrbed by the phase tran- ξ 0.7 0.6 0.5 0.4 0.3 ξ lm = 0.657639 Maxmum devaton ξ HBIM ξ = 0.007 HBIM Asymptotc lmt 0.5 T 0.4 0.3 0. 0. 0.0 0. T lm = 0.38666667 Maxmum devaton T HBIM T = 0.049 HBIM Asymptotc lmt 0. 0.0 0.5.0.5.0 0. 0.0 0.5.0.5.0 (a) Tme (b) Tme Fgure. (a) Interace movement and (b) temperature hstory at x = 0.4
Hernandez, E. M., et al.: Non-Parabolc Interace Moton or the -D Stean Problem... 33 THERMAL SCIENCE: Year 07, Vol., No. 6A, pp. 37-336 ston between the two phases and heat transport wthn each phase, gven that the specmen s subject to Drchlet boundary condtons. These boundary condtons mply that the specmen s constantly heated rom the let edge by a heat source, keepng the temperature at ths boundary xed, at some value above the uson pont o the substance. On the other sde o the specmen, heat s beng draned by a cold reservor, keepng the rght edge at some temperature below the uson pont. The heat rom the hot reservor wll low to the nterace at constant rate k ( )/ Tl T ξ, and at the same tme, heat wll be removed rom the nterace by the cold reservor at a constant rate k ( )/( ) T Ts L ξ. Thereore, the net lux through the nterace, determned by these two quanttes, wll dctate how much sold melts or how much lqud s solded. At some tme, whch depends on the dusvtes at each phase and the latent heat o uson, the net lux through the nterace wll be equal to zero. When the nterace reaches a specc poston, where the net lux s zero, there s no energy let to sustan the phase transton, and the lqud-sold nterace wll stop movng. The exact value o the poston or the nterace, where ths happens, can be obtaned through ths analyss, thereore, ths poston s gven by: k( Tl T ) L ξlm = (7) k ( T T ) + k ( T T ) s l whch wll happen at some tme t 0, dependng on the startng poston o the nterace. Accordng to ths argument, any soluton or the nterace poston must approach asymptotcally to ths value. Ths lmt s shown n g., and s compared wth the value obtaned by usng the and HBIM. Durng the numercal smulatons t was observed that the nterace moton practcally stopped at t max =, so the exact value gven by eq. (7) s compared wth the correspondng numercal solutons, as shown n g.. For large values o t, the net lux through any pont x wthn the specmen s also zero. We can use the lux equaton at any poston, and obtan an asymptotc value or the temperature T lm at that poston as well. By settng the ncomng and outgong lux equal to each other at a gven poston x, the temperature o a pont wthn the lqud and sold phase or large values o t s gven by: ξlm x x L x x ξ lm T ( x) = T and lm l + T T ( x) = T lm + Ts (8) ξlm ξlm L ξlm L ξlm Here ξ lm s gven by eq. (7), L s the length o the specmen and x s the poston o any pont where the asymptotc value or the temperature s to be determned. In g., the asymptotc temperature at x = 0.4 s shown as well, and compared wth the temperatures obtaned rom the and HBIM at t max =. The predctng power o eq. (7) also les n the act that we are able to calculate the amount o lqud or sold that wll reman n the specmen, ndependently o the ntal poston o the nterace. In g., the nterace s ntally placed to the rght sde o the lmtng poston gven by eq. (7). In ths case, the lqud close to the nterace wll be transormed nto sold phase, approachng the asymptotc value predcted by eq. (7). The temperatures at the boundares and thermodynamc varables o latent heat and dusvtes rom the rst example were used n ths example as well, only that the nterace s ntentonally placed at ξ (0) = 0.80, so t moves to the let, as predcted by eq. (7). All ths s shown n g., where t s also evdent that the behavor o the nterace moton s not parabolc as t approaches the asymptotc poston.
Hernandez, E. M., et al.: Non-Parabolc Interace Moton or the -D Stean Problem... THERMAL SCIENCE: Year 07, Vol., No. 6A, pp. 37-336 333 0.80 ξ HBIM 0. Asymptotc lmt T 0.75 0. 0.70 Maxmum devaton ξ HBIM ξ = 0.004 0.0 HBIM Asymptotc lmt Maxmum devaton T HBIM T = 0.005 0.65 ξ lm = 0.65739 0. T lm = 0.055 0.0 0.5.0.5.0 0.0 0.5.0.5.0 (a) Tme (b) Tme Fgure. (a) Interace movement and (b) temperature hstory at x = 0.70 A ne mesh wth N = 8 and N = 4 was used, or medums and, respectvely. 4 The whole tme nterval studed was t max =.0, and N t = 8 0 tme parttons were used as n the prevous example. As expected, the and HBIM capture the moton o the nterace wth a small observable derence between them. Also n ths case, both solutons approach asymptotcally to the lmtng value, whch accordng to eq. (7) s: ξ lm = 0.65739. Fgure also shows the temperature hstory at x = 0.70, where both solutons approach the asymptotc value or the temperature at ths poston T lm = 0.055, accordng to eq. (8). Interace moton n alumnum In ths part o the results we wll dscuss the consequences o eqs. (7) and (8) or the phase transton n pure alumnum. The thermodynamc varables are obtaned rom [7], and assumed to be constant. For ths part o the dscusson, we use the SC as shown n eq. (4), and the duson equatons must take nto account the densty and specc heat capacty o each phase n the dusvtes α = k/ ρ C and α = k/ ρ C. The thermodynamc varables taken rom [7] are: ρ = 380 kg/m 3, ρ = 545 kg/m 3, C = 30 J/kgK, C = 06 J/kgK, k = 5 W/mK, 3 k = 5.5 W/mK, and L = 396 0 J/kg, and we use constant temperatures at the boundares, T l = 073 K and T s = 573 K. As n the prevous secton, we wll consder two cases: (a) one, where the nterace wll be placed ntally at ξ (0 s) = 0.0 m, and (b) when ξ (0 s) = 0.90 m. In g. 3, we show the nterace poston as a uncton o tme and obtaned rom the and HBIM, by assumng ntally, a temperature prole o the orm dscussed n the secton Approxmate analytcal soluton. In case (a), the nterace moves to the rght o ts ntal poston accordng to eq. (7), so n order to obtan a physcally vable soluton, we must use the densty o the sold, ρ, n the SC, eq. (4). In case (b), we must use the densty o the lqud phase, ρ, n the SC, snce ormaton o sold s expected accordng to eq. (7). Substtutng the thermal conductvtes o alumnum and temperatures at the edges o the specmen n eq. (7), we can predct the asymptotc poston o the nterace ξ lm = 0.6947 m. In order to obtan the proper behavor wth the nte derence method, we ound that a second order approxmaton n the space varable, overestmated the predcted value, and we developed a n order to approach asymptotcally to the expected value. Even though, the uses a ourth order approxmaton, a very ne mesh was needed. In the example shown n g. 3, we used a mesh
Hernandez, E. M., et al.: Non-Parabolc Interace Moton or the -D Stean Problem... 334 THERMAL SCIENCE: Year 07, Vol., No. 6A, pp. 37-336 ξ [m] 0.8 0.4 0.0 0.6 0. ξ lm = 0.6947 m Maxmum devaton ξ HBIM ξ = 0.0047 HBIM Asymptotc lmt ξ [m] ξ lm = 0.6947 m HBIM Asymptotc lmt Maxmum devaton ξ HBIM ξ = 0.053 0.4 0 3 4 5 0 3 4 5 (a) t 0 4 [s] (b) t 0 4 [s] Fgure 3. Interace moton or alumnum wth N = 60 and N = 360 nodes n (a) and a mesh wth N = 360 and N = 60 nodes n (b). The tme step t = /3 seconds or t max = 50 0 3 second o smulaton. By usng eq. (8), we can also predct the temperature prole wthn the specmen n ths lmt. Ths equaton predcts a derent lnear temperature prole wthn the lqud and sold at large tme values. These asymptotc proles only depend on the boundary condtons, uson temperature, specmen s sze, and thermal conductvtes. Snce they must not depend on the ntal nterace poston and ntal temperature prole, we also test ths predcton n g. 4, where the tme evoluton o the prole wthn the specmen s shown or each case. The soluton was obtaned wth the, and compared wth the exact value n the asymptotc lmt. We also use two derent ntal proles n each case, n order to llustrate the generalty o the result predcted by eq. (8). The tme evoluton o the prole s shown or an ntal parabolc (IPP) and step uncton-lke proles (ISFP), reachng the same asymptotc temperature n both stuatons, ndependently o the ntal prole or nterace poston. 0.84 0.70 0.56 0.4 0.8 Fgure 4. Tme evoluton o two derent temperature proles n alumnum obtaned wth the ; these solutons correspond to the boundary condtons and ntal nterace postons used n g. 3; whte squares and whte crcles belong to the exact soluton predcted by eq. (8) IPP ξipp ISFP ξisfp t = 0000 s t = 0000 s t = 5000 s t = 5000 s t = 000 s t = 000 s t = 500 s t = 500 s t = 0 s t = 0 s 0.0 0. 0.4 0.6 0.8.0 0.0 0. 0.4 0.6 0.8.0 (a) x [m] (b) x [m]
Hernandez, E. M., et al.: Non-Parabolc Interace Moton or the -D Stean Problem... THERMAL SCIENCE: Year 07, Vol., No. 6A, pp. 37-336 335 Conclusons Under homogeneous Drchlet boundary condtons appled over a -D sample wth a lqud-sold phase transton takng place, we have ound several results that, to the best o our knowledge, are not mentoned anywhere n the lterature. y Non-parabolc moton o the nterace could be explaned by usng heat transport theory. y The nature o the boundary condtons mposed on the specmen, mply an asymptotc behavor that can be predcted, and enables to nd the amount o lqud and sold that wll reman on the sample. y The net lux o energy through the nterace wll determne ts poston at large tme values, accordng to eq. (7), ndependently o the ntal amount o lqud or sold. y For large tme values, the temperature prole n each phase s exactly lnear accordng to eq. (8). y Equaton (8), predcts an asymptotc temperature prole wthn the lqud and sold phase, that s ndependent o the ntal prole and ntal poston o the nterace. y The and HBIM capture the physcs predcted by eqs. (7) and (8). y Gven that a second order nte derence scheme overestmates the poston o the nterace at large tme values, a new was developed, n order to obtan the proper asymptotc behavor. y The general results presented n ths work provde a deeper understandng o heat transport phenomena n pure substances, and experments ought to be planned n order to valdate these results. Reerences [] Tarza, D. A., Explct and Approxmated Solutons or Heat and Mass Transer Problems wth a Movng Boundary, n: Advanced Topcs n Mass Transer (Ed. M. El-Amn), Rjeka, Croata, 0, pp. 439-484 [] Javerre-Perez, E., Lterature Study: Numercal Problems or Solvng Stean Problems. Report No. 03-6, Del Unversty o Technology, Delt, The Netherland, 003 [3] Javerre, E., et. al., Comparson o Numercal Models or One-Dmensonal Stean Problems, J. Comput. Appl. Math., 9 (006),, pp. 445-459 [4] Mtchell, S. L., Vynnycky M., On the Numercal Soluton o Two-Phase Stean Problems wth Heat-Flux Boundary Condtons, J. Comput. Appl. Math., 64 (04), July, pp. 49-64 [5] Mtchell, S. L., Vynnycky M., Fnte-Derence Methods wth Increased Accuracy and Correct Intalzaton or One-Dmensonal Stean Problems, Appl. Math. Comput., 5 (009), 4, pp. 609-6 [6] Tad, M., A Four-Step Fxed-Grd Method or D Stean Problems, J. Heat Trans., 3 (00),, pp. 450-4505 [7] Wu, Z.-C., Wand, Q.-C., Numercal Approach to Stean Problem n a Two-Regon and Lmted Space, Thermal Scence, 6 (0), 5, pp. 35-330 [8] Savovc, S., Caldwell, J., Numercal Soluton o Stean Problem wth Tme-Dependent Boundary Condtons by Varable Space Grd Method, Thermal Scence, 3 (009), 4, pp. 65-74 [9] Savovc, S., Caldwell, J., Fnte Derence Soluton o One-Dmensonal Stean Problem wth Perodc Boundary Condtons, Int. J. Heat Mass Tran., 46 (003), 5, pp. 9-96 [0] Caldwell, J., et. al., Nodal Integral and Fnte Derence Soluton o One-Dmensonal Stean Problem, J. Heat Trans-T. ASME, 5 (003), 3, pp. 53-57 [] Goodman, T. R., Applcaton o Integral Methods to Transent Nonlnear Heat Transer, Advances n Heat Transer, Academc Press, New York, USA, 964 [] Fraguela, A., et. al., An Aapproach or the Identcaton o Duson Coecents n the Quas-Steady State o a Post-Dscharge Ntrdng Process, Math. Comput. Smulat., 79 (009), 6, pp. 878-894 [3] Mtchell, S. L., Myers, T. G., Applcaton o Standard and Rened Heat Balance Integral Methods to One Dmensonal Stean Problems, SIAM Rev., 5 (00),, pp. 57-86 [4] Sadoun, N., et. al., On the Goodman Heat Balance Integral Method or Stean Lke Problems: Further Consderatons and Renements, Thermal Scence, 3 (009),, pp. 8-96
Hernandez, E. M., et al.: Non-Parabolc Interace Moton or the -D Stean Problem... 336 THERMAL SCIENCE: Year 07, Vol., No. 6A, pp. 37-336 [5] Sadoun, N., et. al., On Heat Conducton wth Phase Change: Accurate Explct Numercal Method, J. Appl. Fud Mech., 5 (0),, pp. 05- [6] Wu, Z., et. al., A Novel Algorthm or Solvng the Classcal Stean Problem, Thermal Scence, 5 (0), Suppl., pp. S39-S44 [7] Yvonnet, J., et. al., The Constraned Natural Element Method (C-NEM) or Treatng Thermal Models Involvng Movng Interaces, Int. J. Therm. Sc., 44 (005), 6, pp. 559-569 [8] Fasano, A., Prmcero, M., General Free Boundary Problems or the Heat Equaton, I, J. Math. Anal. Appl., 57 (977), 3, pp. 694-73 [9] Fasano, A., Prmcero, M., General Free Boundary Problems or the Heat Equaton, II, J. Math. Anal. Appl., 58 (977),, pp. 0-3 Paper submtted: November 4, 05 Paper revsed: Aprl 7, 06 Paper accepted: Aprl 6, 06 07 Socety o Thermal Engneers o Serba Publshed by the Vnča Insttute o Nuclear Scences, Belgrade, Serba. Ths s an open access artcle dstrbuted under the CC BY-NC-ND 4.0 terms and condtons