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M.M. Kenisarin, et al. Journal of Energy Storage 27 (2020) 101082 Table 1 Experimental parameters and results obtained by Toksoy and İlken [60]. Experiment number 1 2 3 4 5 6 7 Tair ( °C) 39.9 40.0 40.3 39.9 40.3 39.7 45.0 PCM mass (gram) 424.85 424.85 434.11 434.11 82.81 82.81 82.81 Container radius (cm2) 3.93 3.93 3.93 3.93 2.34 2.34 2.34 Sphere surface area (m2) 0.0194 0.0194 0.0194 0.0194 0.0069 0.0069 0.0069 Stefan Complete melting time number (s) 0.0946 16080 0.0950 14488 0.0976 17317 0.0946 16223 0.0976 8391 0.0930 8981 0.1327 7800 Stored energy (kJ) 49.47 46.0 58.09 54.23 11.5 12.8 13.17 Heat flux (kW/m2) 0.1586 0.1614 0.1730 0.1723 0.1987 0.2605 0.2446 close-contact and natural convection melting in the ice storage sphe- rical container. The following assumptions were made for the analysis: the flow is laminar; customary Boussinesq approximation can be used; properties are constant and there is a constant volumetric expansion rate during the melting process. The fixed boundary method was used to evaluate the interface position. Numerical simulations were per- formed for melting ice, which initially was packed in a shell with an inner diameter of 32 mm. Calculations were made for two different temperatures of the spherical capsule: 3 and 10 °C. As it can be seen in Fig. 15, at the early stage of unconstrained melting, the calculated molten mass fraction was in a good agreement with data of Bahrami and Wang [56]. As the process progressed, the increasing discrepancy in time was observed. These differences were caused by intensification of the natural convection flows. Considering the configuration of unconstrained melting, Hu and Shi [64] developed a mathematical model describing the close-contact melting of PCM inside a spherical shell. The calculation was performed for two sets of experimental conditions, studied by Moore and Bayazi- toglu [55], namely 5 the first set: Ste = =0.05, Pr = =54.7, Ar = =7.36 × 10 ; the second set: Ste = =0.1, Pr = =41.8, Ar = =1.26 × 106. Fig. 17 shows data produced by Hu and Shi [64] in comparison with the results of Moore and Bayazitoglu [55], Bahrami and Wang [56]. It can be seen, that there is a good agreement between theoretical and experimental data. Fomin and Saitoh [65] investigated the close-contact (un- constrained) melting within a spherical capsule numerically and ana- lytically. Equations of their mathematical model were solved numeri- cally by using the fixed boundary method. For constructing an approximate analytical model of contact melting in a sphere with non- isothermal wall, the original approach was developed and applied by Bareiss and Beer [58]. The use of the perturbation technique allowed to obtain an analytical solution for the evaluation of interface position, s2 = [( 2 A)+( 2 A B)Ste/4]2 +[A+(A+B)Ste/4]s (5) and for the complete time of melting, =( 2+A)+( 2+A+B)Ste, m28 (6) where A = 2a3 B= 24 Here a is2 the coefficient in capsule's wall temperature equation: Tw = =a sin θ. In accordance with the experiments, a varies from 0 to 0.4 and for the most cases, a can be chosen as 0.2. Assuming that the capsule wall was isothermal (a = 0) and the heat transfer was dominated by heat conduction across the molten layer only, the comparison was conducted between the computed solid shift, as a function of relative time, and the results of Bahrami and Wang [56]. As it can be seen in Fig. 17, for these conditions the theoretical values completely coincide with experimental ones. The following conclusions were made by the authors: • • tical model. The discrepancy in results did not exceed 10 ± 15%; The assumption, which was used in the previous research works on the temperature in the wall of capsule being constant, can lead to results, which significantly differ from those obtained for the real conditions of melting, in which the wall of the capsule is non-iso- thermal. Eames and Adref [66, 67] carried out a comprehensive experimental study of freezing and melting of water inside a spherical shell. This type of shells is usually used in ice storage systems. The test apparatus is shown schematically in Fig. 18. The first sphere acted as the test object and was filled to 80% of its volume with de-ionized water. The air space in the test sphere was connected to the second sphere by a plastic tubing. The second sphere contained only air. During the freezing process, the air pressure in the two spheres would rise due to the in- crease in the volume of water as it freezes in the first sphere. The pressure was measured using a manometer. The purpose of the second sphere was to set the initial air volume contained by the two spheres. Once freezing had been completed the resulting air pressure would not exceed the strength of the glass spheres or the maximum pressure range of the manometer. The measurements of air pressure were used to es- timate the quantity of ice produced at any point in time during the freezing process. Using this date, the position of the water/ice interface could be estimated. The experiments showed that at least 90% of the energy was discharged in 70% of the time needed to complete the full discharge from the spherical storage element. For practical purposes, it is important to know the rate of melting or solidification. The Fig. 17. The solid PCM travel distance as a function of ratio between current time elapsed and the time required for complete melting (τ/τm) [65. 314 a 14 The approximate analytical solutions of close-contact melting, ob- tained by the perturbations technique, were found to be in a good agreement with the numerical solution of the complete mathema- 1 + a a + 1 + a 2ln(1 + a) ; (7) 1/4 6a 5a2(6+8a+2a2)ln(1+a) . a(2a + a2 2(1 + a)ln(1 + a))3/4 (8) 9PDF Image | Journal of Energy Storage 27
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