NUMERICAL SOLUTION OF STEFAN PROBLEM WITH TIME-DEPENDENT BOUNDARY CONDITIONS BY VARIABLE SPACE GRID METHOD. Svetislav SAVOVI] and James CALDWELL

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1 THERMAL SCIENCE: Vol. 13 (2009), No. 4, pp NUMERICAL SOLUTION OF STEFAN PROBLEM WITH TIME-DEPENDENT BOUNDARY CONDITIONS BY VARIABLE SPACE GRID METHOD by Sveislav SAVOVI] and James CALDWELL Orig i nal sci en ific pa per UDC: : DOI: /TSCI S The vari able space grid mehod based on fi nie dif fer ences is ap plied o he one-di - men sional Sefan prob lem wih ime-dependen boundary condiions describing he so lid i fi ca ion/mel ing pro cess. The emperaure disribuion, he posiion of he mov ing bound ary and is ve loc iy are eval u aed in erms of fi nie dif fer ences. I is found ha he com pu a ional re suls ob ained by he vari able space grid mehod ex - hibi good agree men wih he ex ac so lu ion. Also he pres en re suls for em per a - ure dis ri bu ion are found o be more ac cu rae com pared o hose ob ained pre vi - ously by he vari able ime sep mehod. Key words: Sefan problem, variable space grid mehod, finie differences Inroducion Mov ing bound ary prob lems known as Sefan prob lems in volv ing hea con duc ion in con junc ion wih change of phase are of grea in er es in nu mer ous im por an ar eas of sci ence, en gi neer ing, and in dus ry. Such a pro cess cov ers a wide range of ap pli ca ions in which phase changes from liq uid, solid, or vapour saes. The mov ing bound ary prob lems oc cur in many ar - eas such as he meal, glass, plas ic and oil in dus ries, space ve hi cle de sign, pres er va ion of food suffs, chem i cal and dif fu sion pro cesses, ec. The ma e rial is as sumed o un dergo a phase change wih a mov ing bound ary whose po si ion is un known and has o be de er mined as par of he anal y sis. Across he phase bound ary he hea flux is no con in u ous, and he hea equa ion is re placed by a flux con di ion which re laes he ve loc iy of he phase bound ary and he jump of hea flux across he phase fron. Since mov ing bound ary prob lems re quire solv ing he hea equa ion in an un known re - gion which has also o be de er mined as par of he so lu ion, hey are in her enly non-lin ear. Be - cause of non-lin ear iy of mov ing bound ary prob lems hey can be solved an a ly i cally for only a lim ied num ber of spe cial cases [1]. Due o dif fi cul ies in ob ain ing an a ly i cal so lu ion, nu mer i - cal ech niques are far more com mon [2-9]. Nu mer i cal ech niques are spe cially known o have dif fi cul ies wih ime-de pend en bound ary con di ions and very small ime seps are of en needed for ac cu rae so lu ions. So lu ions of such Sefan prob lems re pored in he li er a ure in clude lin - ear, ex po nen ial, and pe ri od i cal vari a ion of he sur face em per a ure or he flux wih ime [8, 10-13]. Com par i son of var i ous nu mer i cal meh ods has been made by Furzeland [11] and Caldwell e al. [14].

2 166 Savovi}, S. Caldwell, J.: Numerical Soluion of Sefan Problem wih... There are wo main ap proaches in he so lu ion of he Sefan prob lem. One is he fron-rack ing mehod, where he po si ion of he phase bound ary is con in u ously racked. An ex - am ple is he hea bal ance in e gral mehod [2], which ex plic ily racks he mo ion of iso herms (he phase bound ary be ing one of hem). An al er na ive ap proach, namely, vari able grid meh ods (vari - able space grid and vari able ime sep) pro vide a way o rack he phase fron ex plic ily [15]. An oher ap proach is o use a fixed-do main for mu la ion. An ex am ple is he iso herm mi gra ion mehod, which uses he em per a ure as he in de pend en vari able [16]. A more com - mon mehod is he enhalpy mehod which uses an enhalpy func ion o geher wih he em per a - ure as de pend en vari able [8, 17, 18]. Al er na ively, us ing a sui able co or di nae rans for ma ion, one may im mo bi lise he mov ing fron a he ex pense of solv ing a more com pli caed prob lem by a nu mer i cal scheme de scribed by Kuluay e al. [6]. The one-di men sional Sefan prob lem wih ime-de pend en bound ary con di ions de - scrib ing he so lid i fi ca ion/mel ing pro cess is con sid ered in his pa per. The vari able space grid (VSG) mehod is em ployed in or der o de er mine he evo lu ion of he em per a ure dis ri bu ion and phase bound ary dur ing he pro cess. The com pu a ional re suls are com pared wih he ex ac so lu ion and wih hose ob ained ear lier by Caldwell e al. [19] who used he vari able ime sep fi nie dif fer ence mehod. So lu ions re pored in he li er a ure us ing he VSG mehod for solv ing he mov ing bound ary prob lem in clude he one-di men sional Sefan prob lem de scrib ing he pro - cess of mel ing of ice [6] and he pro cess of evap o ra ion of drop les [20]. Formulaion of he problem Here we con sider he Sefan prob lem de scrib ing a one-di men sional sin gle phase mel ing pro cess where he em per a ure is in creased ex po nen ially wih ime a he fixed bound ary x = 0. The em per a ure hrough ou he solid is as sumed o re main a he mel ing poin. We are in er esed in he emperaure disribuion u(x, ) in he re gion 0 x s() and in he lo ca ion of he mov ing bound ary. Wihin he dimensionless mahemaical model, he funcion u(x, ) is gov erned by he hea equa ion: u u a 2, 0 x s( ), 0 (1) x 2 sub jec o he bound ary con di ions u( x, ) ea, x 0, 0 (2) u( x, ) 1, x s( ), 0 where a is a phys i cal pa ram e er com bin ing he den siy, spe cific hea, and he her mal con duc iv - iy. The lo ca ion of he mov ing bound ary is given by he hea bal ance equa ion known as he Sefan con di ion: 1 ds u, x s( ), 0 (3) a d x The ini ial con di ion is The ex ac so lu ion of his prob lem is given by: s(0) = 0 (4) u( x, ) ea x (5) s( ) a We use he ex ac so lu ion (5) o in iial ise our nu mer i cal schemes and o com pare i wih our com pu a ional re suls.

3 THERMAL SCIENCE: Vol. 13 (2009), No. 4, pp Here we deal wih he fi nie dif fer ence so lu ion of he dimensionless model prob lem given by eqs. (1)-(4). Sev eral nu mer i cal ech niques based on fi nie dif fer ences and fi nie el e - mens have been suc cess fully ap plied o he rea men of he Sefan prob lem [4, 5, 21-23]. In his pa per, in or der o de er mine s() for > 0 and u(x, ) for 0 x s() and > 0, we em ploy a vari able space grid ech nique. A vari able space grid (VSG) mehod The num ber of space in er vals be ween a fixed bound ary x = 0 and a mov ing bound ary x = s() is kep con san and equal o N, and hus he mov ing bound ary al ways lies on he N-h grid. Be fore wri ing he fi nie dif fer ence form of eq. (1), i is nec es sary o ake ino ac coun he con in u ous change in he nodal po si ions due o he bound ary move men. The fol low ing ex pres - sion ap plies a he i-h grid poin: u u x u (6) x i i x and he node x i is moved ac cord ing o he ex pres sion: dx xi ds (7) d s ( ) d in which he suf fices, i, and x are o be kep con san dur ing he dif fer en i a ion pro cess and omi ed for clar iy be low. By sub si u ing eqs. (1) and (7) ino eq. (6), he fol low ing equa ion is ob ained: u xi ds u u a 2, 0 x s( ), 0 (8) s d x 2 sub jec o he bound ary con di ions (2). Equa ion (3), sub jec o ini ial con di ion (4), re mains un - changed. One should noe here ha he grid size Dx = s()/n var ies wih ime as he in er face moves, since he num ber N of grid poins is con san. The em per a ure gra di en a he mov ing in er face [x = s() = NDx] is given by he fol - low ing hree poin back ward scheme [11]: u 3u 4u u x 2Dx x s N N 1 N 2 2 O( Dx ) (9) Us ing a for ward dif fer ence ap prox i ma ion for he ime de riv a ive and a cen ral dif fer - ence ap prox i ma ion for he space de riv a ive, he discreizaion of eq. (8) can be ex pressed as: kx s i, m m ka ui, m 1 ui, m ( ui 1, m ui 1, m ) ( ui 2h s h2 1, m 2ui, m u i 1, m ) (10) m m where u i,m u(x i,m, m ), s m s( m ), s m = (s m+1 s m )/D, x i,m = ih m, m = 0 + mk, h m is he space grid size Dx a m h ime sep, k( D) he ime sep, and 0 he ime a which he nu mer i cal pro cess is in iial ised. A run ca ion er ror for his scheme is O(k) + O(h m2 ). The sche maic di a gram in fig. 1 dem on sraes he con sruc ion of he grids for he fi nie dif fer ence so lu ion. The em per a ure dis ri bu ion a he or i gin is eas ily ob ained us ing he bound ary con di - ion (2) a x = 0, which in discreized form is: m u e i, m a m, i 0, m 0, 1, 2,... (11) For he em per a ure dis ri bu ion a 0 < x < s() (i = 1, 2,, N 1, m = 0, 1, 2,,) eq. (10) is o be used. The bound ary con di ion (2) a x = s() is:

4 168 Savovi}, S. Caldwell, J.: Numerical Soluion of Sefan Problem wih... Fig ure 1. Sche maic di a gram o il lus rae he con sruc ion of he grids for he fi nie dif fer ence so lu ion u i,m = 1, i = N, m = 0, 1, 2,... (12) Us ing eq. (9), he Sefan con di ion (3) a x = s() (i = N) in erms of fi nie dif fer ences is: s ka sm ( 3uN, m 4uN 1, m un 2, m ), m 0, 1, 2,..., (13) 2h m 1 m and he ini ial con di ion (4) be comes: s 0 = 0 (14) On he ba sis of he up daed in er face lo ca ion s m+1, he up daed grid size h m+1 is cal cu - laed a each ime sep as h m+1 = s m+1 /N. Nu mer i cal re suls and dis cus sion In his sec ion we pres en he com pu a ional re suls ob ained by us ing he VSG mehod ap plied o he one-di men sional Sefan prob lem de scrib ing he mel ing pro cess of a solid. We com pare our com pu a ional re suls wih he ex ac so lu ion for a = 2 and 10. Also he pres en re suls for em per a ure dis ri bu ion are com pared wih hose ob ained ear lier by Caldwell e al. [19] for a = 10, who use he vari able ime sep fi nie dif fer ence mehod. In he VSG mehod used in he pres en sudy, he nu mer i cal pro cess is in iial ised us ing he ex ac so lu - ion (5) of he Sefan prob lem de fined by eqs. (1)-(4). The ini ial ime 0 = 0.01 which ac cord ing o eq. (5) cor re sponds o he ini ial po si ion of he mov ing bound ary s( 0 ) = 0.02 and 0.1 for a = = 2 and 10, re spec ively, is used. We in ves i gae he evo lu ion of he em per a ure dis ri bu ion, he po si ion of he mov ing bound ary and is ve loc iy in a ime in er val from = 0 = 0.01 o 0.5. Ap ply ing he VSG mehod a grid size h m ( Dx) = s()/n (N = 10 is also adoped) var ies be ween and 0.1 for a = 2 and be ween 0.01 and 0.5 for a = 10, since we are an a lyz ing he move - men of he phase bound ary po si ion s() be ween 0.02 and 1 for a = 2 and be ween 0.1 and 5 for a = 10. The ime seps k ( D)= and are used for a = 2 and 10, re spec ively. Such a choice of ime sep and grid size guar an ees sa bil iy of our dif fer ence schemes ap plied wihin he VSG mehod. We firs pres en he re suls ob ained for a = 10. The pres en com pu a ional re suls for he em per a ure dis ri bu ion u(x, ) o geher wih he ex ac so lu ion are shown in ab. 1. Good agree men be ween he pres en re suls and ex ac so lu ion is seen. Fur her more, he ac cu racy of he pres en re suls for he em per a ure dis ri bu ion u(x, ) is abou one or der of mag ni ude beer han he ac cu racy of he re suls ob ained ear lier in [19] (shown in ab. 2) us ing he vari able ime

5 THERMAL SCIENCE: Vol. 13 (2009), No. 4, pp Ta ble 1. Tem per a ure dis ri bu ion u(x, ) ob ained us ing he VSG mehod com pared wih he ex ac so lu ion for a = x/s VSG u(x, ) Ex ac soluion Er ror [%]

6 170 Savovi}, S. Caldwell, J.: Numerical Soluion of Sefan Problem wih... Ta ble 2. Tem per a ure dis ri bu ion u(x, ) as cal cu laed in [19] us ing he vari able ime sep mehod and ex - ac so lu ion for a = 10 sep mehod. In fig. 2 he com pu a ional re suls and ex ac so lu ion for mov ing bound ary po si ion ver sus ime are shown. In ab. 3 he com pu a ional and ex ac val ues for bound ary po si ion and is u(x, ) x = 0.1 x = 0.3 x = 0.5 Com pued Exac Error [%] Com pued Exac Error [%] Com pued Exac Error [%] Figure 2. Posiion of moving boundary vs. ime for a = 10 Figure 3. Posiion of moving boundary vs. ime for a = 2 ve loc iy are shown o geher wih per cen age er rors. Rea son ably good agree men be ween he pres en re suls and he ex ac so lu ion is seen. In ab. 4 is shown a com par i son of u(x, ) de er mined from he ex ac so lu ion and from he fi nie dif fer ence cal cu la ions for a = 2. Very good agree men be ween he pres en re suls and Ta ble 3. Po si ion of mov ing bound ary and is ve loc iy ob ained us ing VSG mehod com pared wih ex ac so lu ion for a = 10 s [] dx/d VSG Error [%] Exac value VSG Error [%] Exac value

7 THERMAL SCIENCE: Vol. 13 (2009), No. 4, pp Ta ble 4. Tem per a ure dis ri bu ion u(x, ) ob ained us ing VSG mehod com pared wih ex ac so lu ion for a = x/s VSG u(x, ) Ex ac soluion Er ror [%]

8 172 Savovi}, S. Caldwell, J.: Numerical Soluion of Sefan Problem wih... Ta ble 5. Po si ion of mov ing bound ary and is ve loc iy obained using he VSG mehod com pared wih he ex ac so lu ion for a = 2 he ex ac so lu ion is seen. The com pu a - ional re suls and ex ac so lu ion for mov ing bound ary po si ion vs. ime are plo ed in fig. 3. In ab. 5 he bound ary po si ion and is ve loc iy de er mined us - ing fi nie dif fer ences are com pared wih he ex ac so lu ion. Again, good agree - men be ween he pres en re suls and ex - ac so lu ion is ev i den. Clearly, our com pu a ional re suls for he worse case ab u laed (cor re spond ing o = 0.5) wih a = 10 are ap prox i maely wihin 4% or less of he ex ac val ues. Since he val ues of a for al mos all he maerials of pracical ineres are less han 5, he VSG mehod may be as sumed sufficienly accurae for mos pracical applicaions. Also he VSG mehod has been ear lier suc cess fully ap plied o he Sefan prob lem wih Neumann bound ary condiion a x = 0 by Caldwell e al. [20] describing he evaporaion of droples and a ime-de pend en bound ary con di - ion a x = 0 by Kuluay e al. [6] de scrib - ing he pro cess of mel ing of ice. On he ba sis of he re suls ob ained we can con clude ha he VSG mehod, which uses con san ime sep, can be suc cess fully em ployed o he Sefan prob lem de scrib ing a one-di men sional sin gle phase mel ing pro cess where he em per a ure is in creased ex po nen ially a he fixed bound ary x = 0. Al hough he ex po nen ially in creas ing em per a - ure a he fixed bound ary x = 0 makes his prob lem more dif fi cul han he prob lem wih ime-in - de pend en bound ary con di ions suc cess fully reaed ear lier [20] us ing he VSG mehod, his mehod again proves o be very ef fi cien and ac cu rae. Con clu sions s [] VSG Er ror [%] Ex ac value ds/d VSG Er ror [%] Ex ac value We re por on he im ple men a ion of he vari able space grid mehod for he so lu ion of he Sefan prob lem de scrib ing he mel ing pro cess of a solid. Very good agree men be ween he com pu a ional re suls ob ained us ing he VSG mehod wih he ex ac so lu ion is ev i den. We find ha he ac cu racy of he VSG re suls for a = 2 is much beer han he ac cu racy of he com pu - a ional re suls achieved for a = 10. Since he val ues of a for al mos all he ma e ri als of prac i cal in er es are less han 5, he VSG mehod may be as sumed suf fi cienly ac cu rae for mos prac i cal ap pli ca ions. One ben e fi of he VSG mehod is ha he com pu a ion ime is com par a ively shor and so i is pos si ble o achieve higher ac cu racy by re fin ing he mesh size. Fur her more, his

9 THERMAL SCIENCE: Vol. 13 (2009), No. 4, pp mehod is shown o pro vide more ac cu rae so lu ions of he Sefan prob lem reaed in he pres en work com pared o hose ob ained us ing he vari able ime sep mehod [19]. The good agree men achieved in com par i son wih he an a ly i cal so lu ion gives us con fi dence in he use of his vari - able space grid ap proach for oher Sefan prob lems wih ime-de pend en bound ary con di ions. This is im por an for hose cases where an a ly i cal so lu ions are no avail able. Acknowledgmens The au hors would like o hank Ciy Uni ver siy of Hong Kong for fund ing his re - search by a Sra e gic Re search Gran (Pro jec No ). References [1] Hill, J., One-Di men sional Sefan Prob lem: An In ro duc ion, Longman Sci en ific and Tech ni cal, Harlow, Essex, UK, 1987 [2] Good man, T. R., The Hea-Bal ance In e gral Mehod and is Ap pli ca ion o Prob lems In volv ing a Change of Phase, Trans. ASME, 80 (1958), 2, pp [3] Crank, J., Free and Mov ing Bound ary Prob lems, Clar en don Press, Ox ford, UK, 1984 [4] Asaihambi, N. S., On a Vari able Time Sep Mehod for he One-Di men sional Sefan Prob lem, Compu. Meh. Appl. Mech. Engg., 71 (1988), 1, pp [5] Asaihambi, N. S., A Vari able Time-Sep Galerkin Mehod for a One-Di men sional Sefan Prob lem, Appl. Mah. Compu., 81 (1997), 2-3, pp [6] Kuluay, S., Bahadir, A. R. Özdes, A., The Nu mer i cal So lu ion of One-Phase Clas si cal Sefan prob lem, J. Compu. Appl. Mah., 81 (1997), 1, pp [7] Wood, A. S., A New Look a he Hea Bal ance In e gral Mehod, Appl. Mah. Mod el ling, 25 (2001), 10, pp [8] Esen, A., Kuluay, S., A Nu mer i cal So lu ion of he Sefan Prob lem wih a Neumann-Type Bound ary Con - di ion by Enhalpy Mehod, Appl. Mah. Compu., 148 (2004), 2, pp [9] Ang, W.T., A Nu mer i cal Mehod Based on Inegro-Dif fer en ial For mu la ion for Solv ing a One-Di men - sional Sefan Prob lem, Nu mer i cal Meh ods for Par ial Dif fer en ial Equa ions, 24 (2008), 3, pp [10] Mennig, J., Özisik, M. N., Cou pled In egral Equaion Approach for Solving Meling or Solidificaion, In. J. Mass Trans fer, 28 (1985), 8, pp [11] Furzeland, R. M., A Com par a ive Sudy of Nu mer i cal Meh ods for Mov ing Bound ary Prob lems, J. Ins. Mahs. Appl., 59 (1980), 26, pp [12] Rizwann-Uddin, One-Di men sional Phase Change wih Pe ri odic Bound ary Con di ions, Num. Hea Trans - fer, Par A, 35 (1999), 4, pp [13] Savovi}, S., Caldwell, J., Fi nie Dif fer ence So lu ion of One-Di men sional Sefan Prob lem wih Pe ri odic Bound ary Con di ions, In. J. Hea and Mass Trans fer, 46 (2003), 15, pp [14] Caldwell, J., Kwan, Y. Y., Nu mer i cal Meh ods for One-Di men sional Sefan Prob lems, Commun. Numer. Meh. Engng., 20 (2004), 7, pp [15] Mar shall, G., A Fron Track ing Mehod for One-Di men sional Mov ing Bound ary Prob lems, SIAM J. Sci. Sa. Compu., 7 (1986), 1, pp [16] Chur chill, S. W., Gupa, J. P., Ap prox i ma ions for Con duc ion wih Freez ing or Mel ing, In. J. Hea Mass Transfer, 20 (1977), 11, pp [17] Voller, V. R., Cross, M., Ap pli ca ions of Con rol Vol ume Enhalpy Meh ods in he So lu ion of Sefan Prob lems, in: Com pu a ional Tech niques in Hea Trans fer (Eds. R. W. Lewis, K. Mor gan, J. A. John son, W. R. Smih), Pineridge Press Ld., Mum bles, Swansea, UK, 1985 [18] Caldwell, J., Chan, C. C., Spher i cal So lid i fi ca ion by he Enhalpy Mehod and he Hea Bal ance In e gral Mehod, Appl. Mah. Model., 24 (2000), 1, pp [19] Caldwell, J., So, K. L., Nu mer i cal So lu ion of Sefan Prob lems Us ing he Vari able Time Sep Fi nie-dif - fer ence Mehod, J. Mah. Sci ences, 11 (2000), 2, pp [20] Caldwell, J., Savovi}, S., Nu mer i cal So lu ion of Sefan Prob lem by Vari able Space Grid Mehod and Bound ary Im mo bi li sa ion Mehod, J. Mah. Sci ences, 13 (2002), 1, pp [21] Gupa, R. S., Kumar, D., A Mod i fied Vari able Time Sep Mehod for he One-Di men sional Sefan Prob - lem, Compu. Meh. Appl. Mech. Engng., 23 (1980), 1, pp

10 174 Savovi}, S. Caldwell, J.: Numerical Soluion of Sefan Problem wih... [22] Murray, W. D., Landis, F., Nu mer i cal and Machine Soluions of Transien Hea Conducion Involving Mel ing or Freez ing, J. Hea Trans fer, 81 (1959), pp [23] Finn, W. D., Voroglu, E., Fi nie El e men So lu ion of he Sefan Prob lem, in: The Mah e ma ics of Fi nie El e mens and Ap pli ca ions, MAFELAP 1978 (Ed. J. R. Whie man), Ac a demic Press, New York, USA, 1979 Auhors' affiliaions: S. Savovi} (corresponding auhor) Deparmen of Mahemaics, Kowloon, Ciy Universiy of Hong Kong, Hong Kong, People s Republic of China Faculy of Science, Universiy of Kragujevac 12, R. Domanovi}a Sr., Kragujevac, Serbia savovic@kg.ac.rs J. Caldwell Deparmen of Mahemaics, Ciy Universiy of Hong Kong, Kowloon, Hong Kong, People s Republic of China Paper submied: Ocober 20, 2008 Paper revised: Ocober 25, 2008 Paper acceped Ocober 28, 2008