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M.M. Kenisarin, et al. Journal of Energy Storage 27 (2020) 101082 Table 3 Operating conditions and experimental parameters [84]. Parameters Temperature (during the charging process) Temperature (during the discharging process) Volume flow rate Diameter of spherical container Spherical shell material Thickness of spherical shell Density of spherical shell Thermal conductivity of spherical shell Specific heat of spherical shell Thermal capacity of spherical shell Value −5 °C, −10 °C, −12 °C, −15 °C, −18 °C, −20 °C, −25 °C 10°C,14°C,18°C,22°C,25°C 3 0.003 m /minute 0.131, 0.106, 0.076, 0.035 m Plastic 0.002 m 3 2.16 g/cm 0.35 W/mK 0.960–1.03 J/g °C 46.74–1393.14 J/.K 0 ≤ Fo ≤ 0.491. Just recently, Lago at al. [84] reported on results of comprehensive experimental investigation of melting and solidification duration of water and water-polyethylene glycol mixtures inside a spherical con- tainer. Phase change experiments are conducted under operating con- ditions and experimental parameters presented in Table 3. Refrigerant fluids R-22 and ethanol were used as heat transfer fluids. The statistical processing of empirical data on the complete melting time gave the following correlations t =4.7×10D (1 W) T T , complete melting 5 0.99 2.34 ini 0.033 bath 0.53 (24b) where D is diameter (m), W – percentage of polyethylene glycol (dec- imal number), Tini – initial temperature ( °C), Tbath – thermal bath temperature ( °C), t – time for complete melting (s). This correlation is applicable in the range of variables: 0.076 m ≤ D ≤ 0.131 m, 0.1 (10%) ≤ W ≤ 0.5 (50%), −20°C≤Tini ≤−5°C,10°C≤Tbath ≤25°C. The comparison of predicted values with experimental fixed time of complete melting of water or water-polyethylene glycol mixtures showed that the error does not exceed 7.2%. 3.3. Solidification inside a spherical container The publication in 1943 of the paper by London and Seban [85] stimulated further investigations into the Stefan problem. Considering the idealized system, they suggested a general approximation method for the solution of freezing problems with applications to ice formation on spherical, cylindrical, and plane boundaries. Particularly, for the spherical container the resulting solution was presented as Fig. 34. The time required to freeze the center [86]. liquid inside or outside spherical shells using an analytical iteration technique. Comparison of the solution found with the numerical solu- tions of Tao [86] showed good agreement only for very low Stefan numbers when the third-order approximation was applied. For the higher values of Stefan number, the deviation was significant. Pedroso and Domoto [88, 89] applied the method of strained co- ordinates to obtain a perturbation solution for solidification inside a spherical shell. It was assumed that the wall temperature is constant. The comparison of the solution with the numerical solution of Tao [86] demonstrated a very good agreement. This solution has the following form 3(r* 1)2 + 2(r* 1)3 (r* 1)3Ste 1 (r* 1)2Ste2 =FoSte=11+1(r*3 1) 1(r*2 1), 3 Bi 2 (24c) =FoSte= 6 + 6 45 r* (27) where r* = =r/ro is dimensionless radius. Assuming r* = =0, the time for complete solidification takes the form f = FoSte = 1 + 1 . (25) 6 3Bi Considering the problem of solidification of saturated liquid with a constant thermal conductivity and density inside cylinder and sphere, Tao [86] produced the numerical solution and such the generalized solution obtained for the dimensionless solidification time t* = =τ = =FoSte is shown in Fig. 34. For Ste →0, the solidification time can be calculated analytically using a steady state transport equation t*= =(1 r*2)12+(1 r*3)( 1)/3 (26) Assuming r* = =0, the time of complete solidification takes the same form as in Eq. (25) Shih and Chou [87] suggested the solution for freezing of saturated Riley et al. [90] proposed an analytical solution of the inward so- lidification in a sphere or circular cylinder using the perturbation method. It was assumed that at the initial stage the PCM was fully melted and maintained at the melting temperature, when the outside surface was suddenly cooled. The complete solidification time τ of the PCM inside the spherical shell was presented as 17 f=FoSte= + 6 1+... 1 S t e 6 S t e 32 3(2 )2 (28) Kern and Wells [91] suggested a physical model, which can be solved analytically for the most common types of boundary conditions. Assumptions were that the temperature of the wall was constant or there was the finite rate heat transfer to the coolant. It was also as- sumed that there was a linear temperature profile in the shell with a corresponding differential removal of internal energy. As a result, the solution for the complete inward spherical solidification time was proposed asPDF Image | Journal of Energy Storage 27
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