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Table 6 The effective conductivity of the PCM and insert-composites [27,52–56]. H. Zhang et al./Progress in Energy and Combustion Science 53 (2016) 1–40 13 Ref. 52 53 54 55 56 τ = kt ρCpR2 R2 1 = k (inverse Bi hR keff ε (1− ε ) kPCM ∗kINS ⎡kINS +2kPCM −(1−ε)(kPCM −kINS )⎤ kPCM ⎢⎣kINS +2kPCM +(1−ε)(kPCM −kINS )⎥⎦ ⎡ ⎛kPCM⎞⎤ ⎢1+(1−ε)×⎜⎝F k −1⎟⎠⎥ with F=1 3 −1 ⎡1+⎛kPCM −1⎞f ⎤ fi are the semi-principal axes of ellipsoids, proposed as f1 =f2 =0.125f3 =0.75 ⎡1−3 (1−ε)+(1−ε)+kPCM ⎡3 (1−ε)−(1−ε)⎤⎤ k ⎢ kINS ⎣ ⎦⎥ kPCM ⎢ INS ⎢ 1−(1−ε)×(F −1) ⎥ ⎢⎥ ⎣⎦ ∑⎜⎟i ⎥ 3i=1⎢⎣ ⎝kINS ⎠ ⎥⎦ PCM⎢ 1−3(1−ε)+kPCM 3(1−ε) ⎥ ⎢k⎥ ⎣ INS ⎦ ⎡⎛3⎞⎛3⎞ ⎤ ⎢kINS +⎜⎝ψ −1⎟⎠kINS −⎜⎝ψ −1⎟⎠(1−ε)(kPCM −kINS )⎥ kPCM⎢ ⎛3⎞ ⎥ ⎢ kINS +⎜⎝ψ −1⎟⎠kINS +(1−ε)(kPCM −kINS ) ⎥ ⎣⎦ = α t (Fourier-number) (5) Fig. 13. Comparison of experimental temperature vs. time curves and predictions by solving the unsteady-state conduction equation and as predicted by Comsol Multiphysics [27]. thermal conductivity of the PCM, with sponge (porosity 0.9) having a slightly higher effect that the metallic foam. Fig. 13 illustrates the comparison of calculated and experimen- tal discharging rates of both liquid and solid PCM [27,49]: within the temperature range under consideration, cooling rates can be fairly well represented by a linear relationship of temperature and time. This results in respective cooling rates, ΔT/Δt (°C/s), which provide a tentative comparison as presented in Fig. 12. The theoretical calculation of the cooling rate of the liquid PCM underestimates the experimental temperature gradient, due to the fact that only conduction is taken into consideration, whilst natural convection within the liquid phase also contributes to the heat trans- fer. When calculating the Rayleigh-number within the liquid salt, values exceeding 107 are obtained, significantly above the thresh- old 1700 normally given for the onset of natural convection. Since it is difficult to predict the contribution of natural convection, Zalba et al. [36] propose to characterize the melt by an effective thermal conductivity, exceeding the thermal conductivity as such. An ex- pression of this effective thermal conductivity could not be obtained yet, but it should indeed be expressed as: ktotal = kmelt × f (Ra) with Ra being the Rayleigh number. To fit the experimental results, the value of the f(Ra) ranges from 1.2 to 1.4, but further refining is required. The Comsol predictions of the temperature evolution are very promising, again with an un- derestimation of the cooling rate in the liquid phase, but a close prediction of the periods of phase transformation and solid’s cooling. 3.4. SpecificPCMapplications 3.4.1. Latentheatstoragewithcryogenics Cryogenic fuels, such as liquefied natural gas, have been deemed to be promising alternative fuels due to their high energy density [57]. They are stored at extremely low temperatures to maintain the liquid state. At the consumers end, the cryogenic liquids need to be re-gasified to about ambient temperature, and plenty of cryo- genic cooling capacity would be released during this vapourization process. It is therefore quite important and imperative to maxi- mize the recovery of the cold energy either through the use of other cryogens or PCMs [55,56]. Cryogens normally refer to as liquid media at a temperature below approximately −150 °C and are physical energy carriers [58]. The use of cryogen as an energy carrier is dif- ferent from normal heat storage in that the energy is stored through Biot-number ) (6) with h = external heat transfer coefficient (W/m2 K) and n = r/R di- mensionless distance in the direction of the heat conduction. (ii) By using Comsol-Multiphysics, as previously used by Lopez et al. [49] and later extended by Pitié et al. [50] and Parrado et al. [51], presenting the confined melting in composite ma- terials made of graphite-nitrates, ceramic-nitrates, and copper- nitrates, respectively. Comsol-Multiphysics offers additional calculation procedures such as internal pressure build-up due to the thermal expansion of the PCM. The procedure “Thermal Stress contains Heat Transfer and Solid Mechanics phenom- ena” needs to be used. In order to provide results with a minimum error range, it uses an extra-fine mesh. The shell wall is considered to be homogeneous, isotropic and present at a known and uniform temperature. At the interfaces, an equality of phase temperature and heat flux continuity at the melting front, and an equality of temperature and pressure at the shell/salt interface are accepted. To predict the effective thermal conductivities of the PCM with inserts (foam, sponge), a series of empirical correlations was used, as listed in Table 6. The symbols keff, kPCM and kINS represent respec- tively the effective thermal conductivities of the composite mix, the thermal conductivity of the PCM itself, and the thermal conduc- tivity of the insert. The porosity, ε, represents the void space (PCM- filled) of the composites. The sphericity, Ψ, is estimated at 0.95 for the composites. The predicted values for the composite systems are given in Table 7, with ±10% deviation noticed between the differ- ent predicted and average values. The inserts clearly increase the Table 7 Calculated effective thermal conductivity of the PCM-insert composite. System Nitrate melt + foam Nitrate solid + foam Nitrate melt + sponge Nitrate solid + sponge Reference of Equation: keff (W/mK) 52 53 54 0.565 0.496 0.435 0.965 0.869 0.763 0.710 0.547 0.492 1.178 0.955 0.766 55 56 0.485 0.522 0.851 0.914 0.532 0.602 0.932 1.050 Average 0.501 0.872 0.567 0.978PDF Image | Thermal energy storage: Recent developments
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