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M.M. Kenisarin, et al. Journal of Energy Storage 27 (2020) 101082 data presented in [71]. It would also appropriate to highlight the work of Bulunti and Arslantürk [70], who reported main results in the gra- phical form, and which could be further turned into a dimensionless correlation. Solidification. As presented in Section 3.3, the process of solidifica- tion was studied at least in 34 published papers. The analytical corre- lations for determination of the time of solidification were obtained in [85, 86, 88-93]. The minimum time required for complete solidification and determined in the above works for the sphere, immersed in the media with a constant temperature, was of the same magnitude and can be expressed by Eq. (25). According to Hill and Kucera [92], the maximum time of complete solidification can be evaluated using the correlation (35). The experimental studies were performed in [66, 67, 91, 105-113, 115, 117], however, the results obtained in these works do not always correlate with each other. All comments made above on the study of Eames and Adref [66, 67] of melting process, are also relevant and valid for the solidification stage of the process. The comprehensive study of Assis et al. [101] resulted in the correlation (45) for the calculation of the molten fraction during the solidification process. The correlations obtained in other experiments in [115, 117,118] are also used for practical applications. The change in the solidified mass fraction in time, reported in [106, 107], was obtained for two compositions based on water. The absence of measured thermophysical properties and presentation of data only in the graphical form makes results less at- tractive for practical purposes. It should be noted that the experiments performed by Chinese researchers in [112, 117] are completed with proposed dimensionless correlations (46), (47), (48), and (49). How- ever, most of numerical studies do not produce dimensionless correla- tions, which could expand the applicability of their investigations. An example of an exception in this list is [101]. Melting and solidification in finned and pinned spherical containers. To our knowledge, there are four works, [119, 121-123], in which the phase change characteristics of PCMs inside finned or pinned spherical enclosures were studied. Fan et al. [119] suggested Eq. (54) for eva- luation of the molten fraction as a function of the dimensionless time and height of circumferential fin. The second study of Govindaraj et al. [121] clearly demonstrated that the application of orthogonal fin is preferable compared to circumstance one. The results of [123] after revision by Jia et al. can be presented in the form of a dimensionless correlation. The effective thermal conductivity correlations of the PCM constrained melting in spherical envelopes. As it was shown, there are only three publications [124-126], in which the correlations of the effective thermal conduction have been proposed. The comparison of these correlations in [126] that results of predictions made with using cor- relations of Amin et al. [124] and Liao et al. [125] are significantly overestimate of melting rate in comparison with experiments as well as prediction calculated by the correlation of Gao et al. [126]. These sig- nificant discrepancies are mainly caused by inadequate choosing the characteristic size δ in the Rayleigh number that reflects the hydraulic gap between the phase change interface and the inner diameter of spherical shell. The range of applicability of the correlation (59) of Gao et al. [126] is restricted by the cases considered in [126] and cannot captured all potential PCMs. Table 13 summarises the brief data of all reviewed original papers. Results from nine publications have restricted applicability for solving practical issues. The results in most investigations are applicable only in the narrow range of conditions, applied in each case. Applicability of the proposed analytical equations requires rigorous validation by ex- perimental data. The most of experimental, analytical, and numerical studies were performed for the cases in which the spherical capsules were placed in the baths with a constant temperature, though in nu- merous real applications, a heat transfer fluid flows around the cap- sules. that the correlation of Raithby and Holland was obtained for single phase natural convection heat transfer between concentric spheres. The correlations of Amin et al. [124] and Liao et al. [125] are overestimate the molten fractions observed in experiments. 4. Discussion State of the art on investigations of phase change materials inside spherical shells was analyzed in this paper. Below we discuss the main results of reviewed works. Constrained melting. As it was shown in Section 3.1, the constrained melting studies were described in eight publications [47-54]. Compar- ison of the time fixed profiles of PCMs in Fig. 2–Fig. 9 clearly shows that the physics of melting process of the fixed solid PCM is very complex. From experiments, it follows that the sliding of solid takes place during its melting, and the rate of sliding depends on the design of the fixing elements and Stefan number. A comparison of the above figures reveals that not all mathematical models, used for the description of con- strained melting, adequately describe the real physical process of melting. As it follows from Fig. 6, that only 3-D numerical simulation, performed by Galione et al. [51], produces results close to experimental observations. Thus, available numerical simulation and experimental results are not fully sufficient to firmly recommend a certain correlation for ultimate evaluation of the complete time of melting and the rate of constrained melting. It essential to emphasize that Li et al. [52] were close to obtaining the correlation, derived from using numerical data (see Fig. 7), for calculation of the molten fraction as a function of di- mensionless time. The shortcoming was that generalization of results for unconstrained melting was not performed to the level as in Assis et al. [71], Archibold et al. [74, 78, 79]. Unconstrained melting. Unconstrained melting inside a sphere, as it follows from Section 3.2, was studied to a larger extent compared to the constrained melting. Thus, the unconstrained melting was studied in 26 papers [55-84] with experimental studies being described in eleven works [48, 60, 66-68, 71, 73, 75, 77, 80]. In all these works, except [60, 68, 81], studies on PCM melting were performed in water baths. All obtained data from these experimental investigations was used for verification and validation of the developed mathematical models. Toksoy and İlken [60], Ettouney et al. [68], and Hariharan [81] have studied the charging and discharging processes in the spherical con- tainer with the PCM in the airflow. Using produced experimental data, they proposed dimensionless correlations for calculation of the time for complete melting and solidification, as well as heat transfer between the spherical container and the ambient HTF (air). Taking into account the fact that water, salt compositions, and oil (as heat transfer media) are often used in many heat storage systems, it is reasonable to study heat transfer characteristics of PCMs in containers of the spherical form. The time of complete melting of PCMs can be evaluated by using analytical correlations, suggested by Bahrami and Wang [56], Fomin and Saitoh [65], and Veerappan et al. [72]. The accuracy of correlations derived in these works is quite high with errors, not exceeding 20%. The results of all experimental works, which are under discussion, are presented in graphical and tabulated forms for better comparison pur- poses. The absence of numerical data restricts the range of application of dimensionless correlations and their further usefulness. The appli- cation of experimental results for the verification and validation of mathematical models is of paramount importance. It is necessary to mention the experimental work of Eames and Adref [66, 67]. These works are the only works in which the rate of energy stored was found, and having information on such an important parameter is very bene- ficial for practical purposes. As far as numerical studies are concerned, it is worth pointing out that there is a trend to present results as di- mensionless correlations, relating main parameters in the melting of PCMs [71, 74, 77-80]. It necessary to mention that the first correlation (15) obtained by Assis et al. [71] was based on their own experiments. All other numerical simulation results were verified with experimental 31PDF Image | Journal of Energy Storage 27
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