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M.M. Kenisarin, et al. Journal of Energy Storage 27 (2020) 101082 Table 4 Comparison of numerical values of the approximate time of complete solidifi- cation obtained by using Eqs. (34) and (35) for Bi = =0.2 and various values of the Ste number (Hill and Kucera [92]). t*= (y 1)+(1+2 )(y 1)2+(2+3 1 ) 2 (y 1)3 (y 1)4 322+(4 2 )322 Ste 1st order approximation 2 1.55 1 2.53 0.5 4.40 0.2 9.92 0.1 19.09 0.01 184.1 2nd order approximation 1.42 2.55 4.51 10.09 19.28 184.3 3rd order approximation t*fmin 0.92 1.83 3.67 9.17 18.33 183.3 t *f m a x 2.75 3.67 5.550 11.00 20.17 185.2 (29) (30) (53 522+28 5 52 2+25 15 4 4 + ... , where =1; =1; = ( 1); =2 ;y=R(t);t==Fo. 1.17 2.38 4.47 10.17 19.41 184.5 6Bi2 6Bi × cos 1 6Bi2 (34) The comparison of Eq. (34) with results of Tao [86] and Shih and Ste Bi Fo = F (Ste, Bi) + F (Bi) + C 12 where F1=1 1+0.122, Ste 3Bi Chou [87] demonstrated a good agreement between these two sets of data. The approximate time of complete solidification tf* was found from Eq. (34) by setting y value to 0. Applying the further simplifica- tions, the minimal and maximum of the first order estimates of tf were suggested as F2(Bi<1+ 2)= 3Bi 4Bi4 10Bi3 3Bi 2 3Bi Bi+1 2 0.146 4Bi3 1 Bi + 0.5 (31) 1ln(Bi+0.5) Table 4 presents the comparison of numerical values of estimates of complete solidification time. It can be seen that the difference between 1st and 3rd order approximations for Stefan number less than 0.5 does not exceed 2%. Consequently, the 1st order approximation can be used for many practical cases. Milanez and Ismail [93] suggested a simplified solution for calcu- lation of the solidification time during freezing process inside the spherical shell assuming that the temperature profile in the solidified layer is linear: =FoSte= (1 r*)3 1 1 (1 r*)2 1 1 +(1 r*) . 3Bi Bi2Bi (36) t*=Fo =2+1/6Steandt*=Fo =1+12+1 fmin fmin Bi fmax fmax /6 Ste Bi (35) + 2 3Bi Bi Bi2 +4Bi+1 (Bi 1)2 0.146 ln(Bi+0.5) 4Bi3 2 3Bi6Bi F (Bi>1+ 2)= 2Bi 4Bi4 0.439. 10Bi3 Bi2+4Bi+1×ln 1 6Bi (Bi 1)2 2 2 Bi+1 +1 , (32) (Bi 1)2 2 C = Assuming r* = =0 in Eq. (36), the complete solidification time takes the same form as in Eq. (25) A more precise solution was obtained using a second-degree poly- nomial representation for the temperature profile. In order to solve the solidification problem of PCM inside a spherical shell, Caldwell and Chan [94] used the enthalpy method. In considering this problem, the same assumptions were applied which were used previously in other reviewed studies. Data obtained using the enthalpy method was compared to calculation results determined by using the heat balance integral method (HBIM). The comparison results are shown in Fig. 35. It can be seen that the applied enthalpy method agrees very well with the HBIM, except for very small Stefan numbers. Ismail and Henriquez [95, 96] performed a numerical study of the It should be noted that Eq. (29) is valid for dimensionless thickness of the solidified layer (ro-r)/ro < 1. The complete time for solidification was found from quasi-stationary solution Fof(Ste 0)=1 1+1 Ste 3Bi 6 (33) Hill and Kucera [92] proposed a new semi-analytical method for description of the solidification of PCM inside a spherical capsule. The approximation expression suggested is Fig. 35. Position of the solid-liquid interface versus non-dimensional time (here τ/α = =FoSte for a: α = =1/Ste = =1; b: α = =1/Ste = =10) [94]. 18PDF Image | Journal of Energy Storage 27
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