Total solidification time of a liquid phase change material enclosed in cylindrical/spherical containers

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1 Appled Thermal Engneerng 25 (2005) Total soldfcaton tme of a lqud phase change materal enclosed n cylndrcal/sphercal contaners Levent Blr *, Zafer _ Ilken Department of Mechancal Engneerng, _Izmr Insttute of Technology, Gülbahçe-Urla, 35430, _Izmr, Turkey Receved 4 Aprl 2004; accepted 8 October 2004 Avalable onlne 6 December 2004 Abstract Ths study nvestgates the nward soldfcaton problem of a phase change materal (PCM) encapsulated n a cylndrcal/sphercal contaner wth a thrd knd of boundary condton. The governng dmensonless equatons of the problem and boundary condtons are formulated and solved numercally by usng enthalpy method wth control volume approach. The problem s solved many tmes for dfferent values of the affectng parameters and data sets are obtaned for dmensonless total soldfcaton tme of the PCM. These data sets are then used to derve correlatons whch express the dmensonless total soldfcaton tme of the PCM n terms of Stefan Number, Bot Number and Superheat Parameter. Ó 2004 Elsever Ltd. All rghts reserved. Keywords: Phase change; Latent heat energy storage; Inward soldfcaton n cylndrcal and sphercal regon. Introducton Cool storage systems remove heat from a thermal storage medum durng the perods of low coolng demand and use ths cool energy when t s needed. For that purpose, latent heat systems are more attractve than sensble ones due to ther large storage capactes and constant charge and dscharge temperatures. One of most popular latent heat storage systems s the encapsulated * Correspondng author. Tel.: ; fax: E-mal address: leventblr@yte.edu.tr (L. Blr) /$ - see front matter Ó 2004 Elsever Ltd. All rghts reserved. do:0.06/j.applthermaleng

2 L. Blr, Z. _Ilken / Appled Thermal Engneerng 25 (2005) Nomenclature B Bot number ¼ hr 0 k s c p specfc heat (j/kgk) C * dmensonless specfc heat ¼ c p l c ps C + dmensonless specfc heat ¼ c p l h convectve heat transfer coeffcent (W/m 2 K) H enthalpy (j/kg) H * H dmensonless enthalpy ¼ c ps ðt ntal T Þ nodal pont k thermal conductvty (W/mK) K * dmensonless thermal conductvty ¼ k l ; k s K + dmensonless thermal conductvty ¼ k l L latent heat of soldfcaton (j/kg) N total grd number nsde the contaner r 0 radus of the sphercal or cylndrcal contaner (m) R dmensonless radal poston ¼ r r 0 Ste Stefan number ¼ c p s ðt ntal T Þ L T ntal ntal temperature of PCM ( C) T m phase change temperature ( C) T coolant flud temperature ( C) V e dmensonless volume of the control volume V s dmensonless volume of the sold part n the control volume V l dmensonless volume of the lqud part n the control volume X dmensonless lnear nterpolaton factor Greek symbols a thermal dffusvty (m 2 /s) DR dmensonless radal dstance between grd ponts Ds dmensonless tme step h dmensonless temperature ¼ T T T ntal T h m superheat parameter ¼ T m T T ntal T c ps k s

3 490 L. Blr, Z. _Ilken / Appled Thermal Engneerng 25 (2005) s dmensonless tme ¼ a st r 2 0 dmensonless total soldfcaton tme s total Subscrpts e control volume face between P and E E neghbour grd pont on the east sde l lqud phase P central grd pont under consderaton s sold phase, Interface w control volume face between P and W W neghbour grd pont on the west sde Superscrpt k tme level PCM systems, whch uses cylndrcal or sphercal contaners packed nto a storage tank. These geometres are commonly preferred due to ther favourable proporton of the energy storage volume to the heat transfer surface area. Tao [] presented a numercal method for the problem of freezng a saturated lqud nsde a cylndrcal or a sphercal contaner. The heat capacty and thermal conductvty of sold phase and the heat transfer coeffcent are assumed to be constants. Cho and Sunderland [2] obtaned an approxmate soluton for the nward and outward phase change of a sphercal body. The ntal temperature of the body s assumed constant at the fuson value and the surface temperature s assumed to change nstantaneously. The accuracy of the approxmaton s checked by a fnte dfference formulaton. Shh and Tsay [3] generated successve approxmate solutons for the freezng of a saturated lqud nsde a cylndrcal contaner under the constant heat transfer coeffcent boundary condton. The same method s also used for outward soldfcaton. Pedroso and Domoto [4] appled the method of straned coordnates to obtan a perturbaton soluton for nward soldfcaton of a saturated lqud n sphercal coordnates. The temperature at the boundary s assumed constant, but t s shown that the same technque s applcable to other types of boundary condtons. Rley et al. [5] presented an analytcal study of the freezng of a lqud nsde a sphere or a cylnder whch s ntally at the fuson temperature, when the outsde surface temperature s suddenly lowered to a value less than the fuson one. The perturbaton method s used n the soluton and the rato of the latent heat to the sensble heat s assumed to be large. Solomon [6] presented an analytcal expresson for the total meltng tme of a slab, a cylnder and a sphere, ntally at fuson temperature and subject to convectve boundary condton. Voller and Cross [7] developed an explct algorthm n order to obtan the soldfcaton and meltng tme of crcular regons wth frst knd of boundary condton. They derved a non-dmensonal expresson to predct the soldfcaton/meltng tme of a crcular cylnder. They also used ths expresson to determne upper and lower bounds of soldfcaton/meltng tme of symmetrc cylndrcally shaped regons. Hll and Kucera [8] developed a sem-analytcal method for the problem of freezng a saturated lqud nsde

4 L. Blr, Z. _Ilken / Appled Thermal Engneerng 25 (2005) a sphercal contaner. They ncluded the effect of radaton at the contaner surface besde convecton. They gave estmates of the tme for complete soldfcaton of the sphere. Mlanez [9] compared the exact solutons for the slab wth numercal solutons obtaned for the sphere under the same ntal and boundary condtons and developed a correcton factor whch when appled to the exact soluton of the sem-nfnte slab wll gve a soluton vald for the sphere. PrudÕhomme et al. [0] appled the method of straned coordnates to study the nward soldfcaton n slabs, cylnders and spheres for three dfferent types of boundary condtons. It s assumed that the lqud s ntally at the fuson temperature and the thermal propertes are constant throughout the process. Caldwell and Chan [] appled a numercal scheme based on the enthalpy method to sphercal soldfcaton. The materal nsde the sphere s assumed at fuson temperature ntally and constant temperature boundary condton s appled on the surface. The results are compared to the ones obtaned by heat balance ntegral method. Ismal and Henrquez [2] presented a numercal study of the soldfcaton of a materal enclosed n a sphercal contaner. The fnte dfference approxmaton and movng grd approach s used throughout the procedure. Constant temperature and convectve heat transfer boundary condtons are taken nto account. The effects of the sze, thckness and materal of the contaner and the external wall temperature on the soldfcaton rate are analysed. Ismal et al. [3] reported the results of a numercal study for the soldfcaton of water n sphercal contaner under convectve boundary condtons. The sze and the materal of the contaner, external temperature, ntal temperature of water are nvestgated and ther effects on the soldfcaton rate and complete soldfcaton tme are dscussed. Ths paper reports the results of a numercal study on the nward soldfcaton of a lqud PCM encapsulated n a cylndrcal/sphercal contaner wth an ntal temperature dfferent than the fuson value. Thrd knd of boundary condton s consdered at the contaner surface. The enthalpy method wth control volume approach s used n the formulaton. The governng dmensonless equatons are solved many tmes to obtan data sets for dfferent values of the affectng dmensonless parameters, namely, Stefan Number, Bot Number and Superheat Parameter. These data then correlated to gve total soldfcaton tme n terms of these parameters. 2. Mathematcal formulaton Consder a cylndrcal/sphercal contaner flled wth a lqud PCM at an ntal temperature T ntal, whch s hgher than ts fuson temperature T m.att = 0, the external boundary of the contaner s exposed to a flud wth a temperature of T, whch s lower than T m. Due to the temperature dfference, heat exchange takes place between the flud and PCM by convecton. PCM starts to soldfy from the outer boundary of the contaner after the temperature at that boundary reaches T m and phase change front moves toward the center. The tme at whch phase change front reaches the center of the contaner s defned as the total soldfcaton tme of the PCM. The followng assumptons are consdered n the formulaton: (a) The contaner wall s so thn and ts materal s so conductve that the thermal resstance through the wall s neglgble. (b) The temperature of the coolant flud T and the convectve heat transfer coeffcent h are constant.

5 492 L. Blr, Z. _Ilken / Appled Thermal Engneerng 25 (2005) (c) The heat transfer process nsde the contaner s only by conducton n radal drecton. (d) The denstes of sold and lqud phases of PCM are equal. Under these assumptons the governng dmensonless enthalpy equaton and ntal and boundary equatons become, oh os ¼ o R n or K R n oh ðþ or for cylndrcal contaner where, n ¼ 2 for sphercal contaner Intal condton : hðr; 0Þ ¼ Boundary condton : K oh or ¼ BðhÞ R¼ The dmensonless enthalpy s defned as 8 < h h m H ðhþ ¼ : C ðh h m Þþ Ste for h < h m for h > h m The enthalpy temperature relaton s as follows: 8 H þ h m for H < 0 >< h h ¼ m for 0 6 H 6 Ste >: H Ste C þ h m for H > Ste The cylndrcal/sphercal soluton regon can be dvded nto small control volumes to apply numercal formulaton (Fg. ). The nterfaces of the control volumes are placed at the mdway between the nodes. If the control volume approach s appled to the dmensonless enthalpy ð2þ ð3þ ð4þ ð5þ Fg.. Control volumes and nodal ponts nsde the cylndrcal/sphercal contaner.

6 equaton (Eq. ()) and explct scheme s used then the followng equatons for dfferent nodes are obtaned. For = H kþ ¼ H k þ m:ds ðdrþ 2 ðk þ K 2 Þðhk 2 hk Þ ð6þ 2 for cylndrcal contaner where, m ¼ 3 for sphercal contaner When dervng Eq. (6), the thermal conductvty at the nterface w s taken as the arthmetc mean of the conductvtes of neghbour nodes (W, P) (Fg. 2a). For =2,3,...,n H kþ P ¼ H k P þ R P Kw R w h k W hk P DR DR K e R e h k P hk E DR Ds for cylnder ð7aþ K h k w H kþ P ¼ H k P þ 3Ds R2 W hk P w K e DR R2 e R 2 w þ R for sphere ð7bþ wr e þ R 2 DR e where K w and K e are the thermal conductvtes at the nterfaces w and e, respectvely (Fgs. 2b and 2c) and are the harmonc means of the conductvtes of neghbour nodes. h k P hk E DR for cylnder K K P K E ln e ¼ 2 ð8aþ K E ln þ K :5 P ln :5 2 K W K P ln K w ¼ K P ln L. Blr, Z. _Ilken / Appled Thermal Engneerng 25 (2005) :5 þ K W ln 0:5 ð8bþ Fg. 2. (a) Frst, (b) nternal, (c) last control volumes of the cylndrcal/sphercal contaner.

7 494 L. Blr, Z. _Ilken / Appled Thermal Engneerng 25 (2005) for sphere K e ¼ ð :5ÞK E K P 0:5ð ÞK P þ 0:5ð 2ÞK E K w ¼ ð 0:5ÞK P K W 0:5 K W þ 0:5ð ÞK P For = n ð9aþ ð9bþ H kþ n ¼ H k n þ Ds ðn ÞðDRÞ 2 2BK n 2K n þ BDR hk n for cylnder ðn :5ÞK n K n lnðn =n 2Þ K n lnðn =n :5ÞþK n lnðn :5=n 2Þ ðhk n hk n Þ ð0aþ H kþ n ¼ H k n þ 3Ds ðdr þðn :5ÞðDRÞ 2 þðn :5Þ 2 ðdrþ 3 Þ " # 2BK n 2K n þ BDR hk n ðn :5Þ 3 K n K n 0:5ðn ÞK n þ 0:5ðn DRðh k 2ÞK n hk n Þ n for sphere ð0bþ The dmensonless total heat n the control volume e s defned as H V e where V e s the dmensonless volume of the control volume; H V e ¼ ðh h m ÞV s þ C ðh h m Þþ V l ðþ Ste Ths equaton can also be expressed as; H pðr2 w R2 e Þ¼ ðh h m ÞpðR 2 w R2 s Þþ C ðh h m Þþ pðr 2 s Ste R2 e Þ for cylndrcal geometry, and H 4 3 pðr3 w R3 e Þ¼ ðh h m Þ 4 3 pðr3 w R3 s Þþ C ðh h m Þþ 4 Ste 3 pðr3 s R3 e Þ ð2aþ ð2bþ for sphercal geometry. When freezng front s on the node, R s ¼ R wþr e and h 2 = h m, so the dmensonless enthalpy value at ths tme s; H ¼ R 2 w þ 2R wr e 3R 2 e 4Ste R 2 w ð3aþ R2 e

8 for cylndrcal geometry, and H ¼ R 3 w þ 3R2 w R e þ 3R w R 2 e 7R3 e 8Ste R 3 w ð3bþ R3 e for sphercal geometry. So, when H kþ s smaller and H k s greater than the quanttes on the rght sdes of the Eqs. (3a) and (3b) for cylndrcal and sphercal geometres, respectvely, ths means that the phase change front has just passed through the node. Furthermore, f t s assumed that the enthalpy changes lnearly n any tme nterval, then the tme at whch the phase change front s on the node can be found by; s ¼ðkþXÞds ð4þ where, X s estmated va lnear nterpolaton n tme doman (Fg. 3). X ¼ k A H H k H kþ as the result, R 2 w þ 2R wr e 3R 2 e 4Ste R 2 w X ¼ H k R2 e H k H kþ for cylndrcal geometry, and X ¼ L. Blr, Z. _Ilken / Appled Thermal Engneerng 25 (2005) R 3 w þ 3R2 w R e þ 3R w R 2 e 7R3 e 8Ste R 3 w H k R3 e H k H kþ ð5aþ ð5bþ for sphercal geometry. The followng algorthm summarzes the method used to fnd the dmensonless total soldfcaton tme of a cylndrcal/sphercal contaner; () Dmensonless enthalpy values of all nodes are calculated at every tme step from Eqs. (6), (7a) and (0a) for cylndrcal, from Eqs. (6), (7b) and (0b) for sphercal contaner. Fg. 3. Dmensonless enthalpy values at the tme nterval between k and k +.

9 496 L. Blr, Z. _Ilken / Appled Thermal Engneerng 25 (2005) (2) The poston of the phase change front s determned by comparng the dmensonless enthalpy values of all nodes wth the enthalpy value gven by Eq. (3a) for cylndrcal and Eq. (3b) for sphercal contaner. (3) The tme at whch the freezng front reaches the node s calculated by usng Eqs. (4) and (5a) for cylndrcal and Eqs. (4) and (5b) for sphercal contaner. When the freezng front reaches the frst node, whch s located at the center of the contaner, the dmensonless total soldfcaton tme of the PCM nsde the contaner s found. 3. Verfcaton of the method In order to valdate the numercal code wrtten for cylndrcal and sphercal contaners, comparsons are made wth the results of some studes avalable n the lterature. The dmensonless total soldfcaton tmes for dfferent Stefan and Bot numbers obtaned from ths study are compared wth the results of [] n Fgs. 4 and 5, for cylndrcal and sphercal geometres, respectvely, n whch the lqud PCM s ntally at the fuson temperature. The results of present study are also compared wth the results of another work [7], n whch the lqud s ntally superheated (Fg. 6). Actually, the methods used n both studes are the same and therefore a very good agreement s obtaned. The results of [7] are vald for a cylnder wth constant temperature boundary condton at the surface, therefore a very bg Bot number s chosen durng the analyss to make the comparson possble. The results are also compared wth of [3], whch nvestgates the soldfcaton of water n sphercal contaner under convectve boundary condton (Fg. 7). The results are generally n τ total.ste Ste= /B Results of [] Present results Fg. 4. Comparson of the total soldfcaton tme of a cylnder wth Ref. [].

10 L. Blr, Z. _Ilken / Appled Thermal Engneerng 25 (2005) τ total.ste Ste= /B Results of [] Present results Fg. 5. Comparson of the total soldfcaton tme of a sphere wth Ref. []. τ total θ m = θ m = Stefan Number Results of [7] Present Results Results of [7] Present Results Fg. 6. Comparson of the total soldfcaton tme of a cylnder wth Ref. [7]. good agreement. The small dfferences are due to the excepton of the thermal resstance of the contaner and the tme dependency of the convecton coeffcent n the present study. It can be observed from Fg. 7 that whle the results of both studes are very close for smaller contaners, as the dameter of the contaner ncreases, the dscrepancy between the results s gettng larger.

11 498 L. Blr, Z. _Ilken / Appled Thermal Engneerng 25 (2005) Dameter of the sphere (m) Ste= θ m = PCM: water Intal PCM temp: 25( o C) Coolant flud temp: -0( o C) Coolant flud: ethanol Complete soldfcaton tme (s) Results of [3] Present results Fg. 7. Comparson of the total soldfcaton tme of a sphere wth Ref. [3]. The reason of ths ncrease n the dfference s the constant convecton coeffcent assumpton made n the present study. When the sphercal contaner dameter s small, ths assumpton does not lead to a great dfference n total soldfcaton tme compared to the case of great sphere dameter n whch the dfference s larger. When the contaner s small, t does not take much tme to soldfy completely, so the change n convecton coeffcent dependng on the surface temperature of the contaner does not affect the results too much. But the stuaton s dfferent for a bg contaner; n that case soldfcaton takes a longer tme and therefore, the dfference between two cases becomes larger. Some oscllatons n nodal temperature values are encountered durng the soldfcaton process. The reason of these numercal oscllatons s the use of harmonc mean approxmaton for the calculaton of thermal conductvty values at the nterfaces of control volumes. Voller and Swamnathan [4] also ndcated that temperature predctons obtaned wth the harmonc and arthmetc mean approxmatons qute often show such oscllatons. However, when the total grd number nsde the contaner (N) s ncreased, t s observed that there s a reducton n numercal fluctuatons n temperature values. Ths effect can be seen from Fg. 8 whch shows the change of surface temperature of a sphercal contaner wth tme. In addton, t s found that the numercal oscllatons are not affected when smaller tme step values are used. Snce explct fnte dfference technque s used n numercal calculatons, t s needed to perform a stablty analyss. To assure numercal stablty, the coeffcent of the dmensonless temperature value of the nvestgated node at the k th tme level ðh k Þ should be equal to or greater than 0. If the frst node of a sphercal contaner s consdered as an example, the numercal equaton for ths node can be wrtten as follows:

12 L. Blr, Z. _Ilken / Appled Thermal Engneerng 25 (2005) Suface Temperature ( o C) Dameter of the sphercal contaner = 0,06(m) PCM: water Intal PCM temperature = 25( o C) Coolant flud temperature = -0( o C) h = 28,7(W/m 2 K), t = 0,2(s) -0 Tme (s) N=4 N=3 N= Fg. 8. Surface temperature of a sphere for dfferent total grd numbers. H kþ ¼ H k þ 3Ds ðdrþ 2 ðk þ K 2 Þðhk 2 hk Þ ð6þ Here the dmensonless enthalpy value of the frst node at tme level kðh k Þ can be wrtten as 8 < h k H k ¼ h m for h < h m ðsoldþ C ðh k h mþþ ð7þ : for h > h m ðlqudþ Ste The frst node s always lqud, because when the phase change front reaches the frst node, the program s ended. So, by substtutng the dmensonless enthalpy value taken from Eq. (7) nto Eq. (6), the rght hand sde of Eq. (6) becomes: C ðh k h mþþ Ste þ 3Ds ðdrþ 2 ðk þ K 2 Þðhk 2 hk Þ ð8þ The coeffcent of the dmensonless temperature value of the frst node s obtaned from Eq. (8) as; C 3Ds ðdrþ 2 ðk þ K 2 Þ ð9þ Ths value should be equal to or greater than 0 to satsfy the stablty. As a result, the upper lmt for dmensonless tme step can be calculated from Eq. (20): Ds 6 C ðdrþ 2 3ðK þ K 2 Þ ð20þ In the present study, two cases are nvestgated. In the frst case, water s selected as PCM (C * = C + = ,K + = 0.306), whle n the second case the specfc heat and thermal conductvty values for lqud and sold phases of PCM are taken equal (C * = C + =,K * = K + = ). To calculate

13 500 L. Blr, Z. _Ilken / Appled Thermal Engneerng 25 (2005) the dmensonless tme step value to be used n the numercal calculatons, the second case should be taken nto consderaton, because n that case a smaller Ds value should be used. A total of 4nodal ponts are used for all cases n the study. The correspondng dmensonless radal dstance between grd ponts (DR) s calculated as As a result, the upper lmt of the dmensonless tme step for the frst node s determned from Eq. (20) as Smlar nvestgatons are made for the nner and last nodes by takng nto consderaton the K * and C * values whch gve mnmum value for Ds. It s found that Ds should be smaller than for the nner nodes and for the last node of a sphercal contaner. The result of smlar calculatons made for a cylndrcal contaner shows that Ds should be smaller than for the frst node, for the nner nodes and for the last node. Fnally, t s decded to use the value of 0.000as the dmensonless tme step (Ds) for all cases, because ths value satsfes the stablty crteron of all cases. 4. Results and concluson A phase change problem s expressed by heat conducton equatons for lqud and sold phases and energy balance equaton for the sold lqud nterface. If these equatons are wrtten n dmensonless form, t can be seen that dmensonless phase change front and as the result, the dmensonless total soldfcaton tme of a PCM held n a cylndrcal or sphercal contaner has a functonal relaton expressed as below; s total ¼ f ðb; Ste; h m ; K þ ; C þ Þ In ths study, t s amed to fnd correlatons gvng the dmensonless total soldfcaton tme n terms of affectng parameters ndcated n Eq. (2). The correlatons are obtaned for the case of equal specfc heat and thermal conductvty values for lqud and sold phases (K + =, C + = ) and for water, whch s one of the most common phase change materal (K + = 0.306, C + = ). The correlatons to be found are n the form of, s total ¼ a ðsteþ b ðbþ c ðh m Þ d ð22þ To fnd the coeffcents a,b,c and d, the codes wrtten for cylndrcal and sphercal contaners are run for many tmes for dfferent values of Stefan Number, Bot Number and Superheat Parameter values and data sets are obtaned for the dmensonless total soldfcaton tme dependng on these parameters, then the correlatons n the form gven by Eq. (22) are obtaned by applyng multple regresson analyss to these data. The values of the parameters whle the data sets are created are; h m ¼ 0:2; 0:3; 0:4; 0:5; 0:6; 0:7; 0:8; 0:9 and fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} 9 values ð2þ

14 L. Blr, Z. _Ilken / Appled Thermal Engneerng 25 (2005) Ste ¼ 0:0; 0:025; 0:05; 0:075; 0:; 0:25; 0:5;...; 0:45; 0:475 and 0:5 fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} 2values B ¼ ; 2; 3; 4; 5;...; 47; 48; 49 and 50 fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} 50 values So, a total of 9450 values are calculated for the dmensonless total soldfcaton tme, and then the data s correlated to gve the followng correlatons: (a) For the case of equal thermal conductvty and specfc heat values of sold and lqud phases of PCM () For a cylndrcal contaner s total ¼ 0: ðsteþ 0: ðbþ 0:94324 ðh m Þ 0: () For a sphercal contaner s total ¼ 0: ðsteþ 0: ðbþ 0:94888 ðh m Þ 0: (b) For water () For a cylndrcal contaner s total ¼ 0: ðsteþ 0: ðbþ 0:88035 ðh m Þ :00093 () For a sphercal contaner s total ¼ 0:5028 ðsteþ 0: ðbþ 0: ðh m Þ 0: The correlaton coeffcents for each of the correlatons are hgher than 0.996, whch means that these correlatons found as the result of multple regresson analyss represent the dmensonless total soldfcaton tme found from the solutons very well and can be used n engneerng problems wth hgh relablty. References [] L.C. Tao, Generalzed numercal solutons of freezng a saturated lqud n cylnders and spheres, AIChE Journal 3 () (967) [2] S.H. Cho, J.E. Sunderland, Phase change of sphercal bodes, Internatonal Journal of Heat and Mass Transfer 3 (970) [3] Y.P. Shh, S.Y. Tsay, Analytcal solutons for freezng a saturated lqud nsde and outsde cylnders, Chemcal Engneerng Scence 26 (97) [4] R.I. Pedroso, G.A. Domoto, Inward sphercal soldfcaton soluton by the method of straned coordnates, Internatonal Journal of Heat and Mass Transfer 6 (973) [5] D.S. Rley, F.T. Smth, G. Poots, The nward soldfcaton of spheres and crcular cylnders, Internatonal Journal of Heat and Mass Transfer 7 (974) [6] A.D. Solomon, On the meltng tme of a smple body wth a convecton boundary condton, Letters n Heat and Mass Transfer 7 (980)

15 502 L. Blr, Z. _Ilken / Appled Thermal Engneerng 25 (2005) [7] V.R. Voller, M. Cross, Estmatng the soldfcaton/meltng tmes of cylndrcally symmetrc regons, Internatonal Journal of Heat and Mass Transfer 24 (9) (98) [8] J.M. Hll, A. Kucera, Freezng a saturated lqud nsde a sphere, Internatonal Journal of Heat and Mass Transfer 26 () (983) [9] L.F. Mlanez, Smplfed relatons for the phase change process n sphercal geometry, Internatonal Journal of Heat and Mass Transfer 28 (4) (985) [0] M. PrudÕhomme, T.H. Nguyen, D.L. Nguyen, A heat transfer analyss for soldfcaton of slabs, cylnders and spheres, Journal of Heat Transfer (989) [] J.D. Caldwell, C.C. Chan, Sphercal soldfcaton by the enthalpy method and heat balance ntegral method, Appled Mathematcal Modellng 24 (2000) [2] K.A.R. Ismal, J.R. Henrquez, Soldfcaton of PCM nsde a sphercal capsule, Energy Converson and Management 4 (2000) [3] K.A.R. Ismal, J.R. Henrquez, T.M. da Slva, A parametrc study on ce formaton nsde a sphercal capsule, Internatonal Journal of Thermal Scences 42 (2003) [4] V.R. Voller, C.R. Swamnathan, Treatment of dscontnuous thermal conductvty n control volume solutons of phase change problems, Numercal Heat Transfer Part B 24 (993) Levent B _ IL _ IR was awarded a B.Sc. degree at Dokuz Eylül Unversty, _ Izmr, Turkey, and a M.Sc. degree at _ Izmr Insttute of Technology, Turkey. He s currently a Ph.D student and research assstant n mechancal engneerng department at _ Izmr Insttute of Technology. Zafer _ ILKEN ganed hs B.Sc. at Mddle East Techncal Unversty, Ankara, Turkey, M.Sc. and Ph.D. degrees from Dokuz Eylül Unversty, _ Izmr, Turkey. He s employed as a full professor at _ Izmr Insttute of Technology n 999. Hs research areas are thermal energy storage and renewable energy sources.