The effects of radiative heat transfer during the melting process of a high temperature phase change material confined in a spherical shell☆
Introduction
The urgent need of sustainable and clean energy sources to secure the future energy production as well as expand the existing electricity generation market has motivated the rapid growth of commercial-scale central receiver concentrating solar power (CSP) plants during recent years [1], [2]. Among the major sub-components of the system namely, the heliostat field, the power conversion cycle and the receiver, the potential integration of thermal energy storage (TES) devices, during the operation of a CSP plant becomes extremely critical to dispatch power when solar irradiation is not available allowing the plant to operate in a cost-effective fashion and consequently reduce the plant levelized electricity cost. The current commercial TES option for CSP systems uses molten nitrate salts as the storage media while energy is stored via sensible heat [3], [4]. In these systems, large amounts of storage material are used and, therefore, large tanks are required, thus increasing the overall cost of the system. In addition to the economic drawback, technical barriers associated with the low exergetic efficiency and material compatibility issues [5], [6] are key factors to encourage the near-term development of advanced thermal energy storage concepts. Numerous research and development initiatives have been directed towards the single tank thermocline system for sensible heat storage [7], [8], [9] in an effort to reduce the cost of the TES system. However the primary challenge is the temperature degradation within the system, which results in lower exergetic efficiency [5]. In that context, the use of PCMs, for latent heat energy storage, seems to be one of the very promising methods because of its higher specific energy storage capacity which reduces the size and cost of the storage tank. The main drawbacks of PCMs for energy storage applications are low thermal conductivity, large volume expansion during melting [5] and chemical activity leading to corrosion. However, it is presumed that at elevated temperatures (>800 °C) radiative heat transfer becomes significant when compared to the others fundamental heat transfer modes. Consequently, the low thermal conductivity problem that penalizes the conduction energy transport during the melting and solidification processes can be overcome, particularly for transparent PCMs, by seeding the PCM with appropriate amounts of radiation absorbers. In this approach an infra-red transparent PCM with tailored absorptance for enhanced radiative transfer is encapsulated in several spherical shells with a highly emissive coating on the inner surface and use it in a packed bed heat exchanger. The objective of this study is to examine the energy exchange process during the melting of sodium chloride with enhanced radiative properties encapsulated in a spherical capsule at elevated temperatures. Among the different PCMs currently investigated as possible candidates to be used in thermal storage devices at elevated temperatures, NaCl has been selected in this investigation because of its high latent heat of fusion and relatively low cost. Also, its melting temperature (800–802 °C) matches with the typical operating range of central receiver power plants.
The problem of heat transfer by radiation in a gray medium confined between two concentric black spheres has been widely reported in the heat transfer literature. A simplified analytical method, based on a diffusion approximation, for calculating radiation heat transfer in the aforementioned system has been presented by Konakov [10]. Sparrow et al. [11] have discussed the contribution of the absorption coefficient and the size of the container on the medium temperature distribution. In the model, an absorbing-emitting, gray gas was assumed as a participating medium while uniform and equal temperatures have been imposed at the shell surfaces. A uniform internal heat generation rate per unit volume through the gas was also considered. Numerical predictions were reported and the study concluded that the gas temperature variations are greater when the absorption coefficient increases for a fixed geometry. An approximate solution of the previously described problem has been reported by Dennar and Sibulkin [12] and Chou and Tien [13] based on an extension of the Milne–Eddington approximation (moment method) originally developed for plane layers. Tong and Swathi [14] evaluated the accuracy of the spherical harmonic numerical method by solving the same problem. In their model the scattering of radiation within the medium was considered and the influence of the differential approximation technique on the wall heat fluxes was presented. Ryhming [15], Viskanta and Crosbie [16] and Jia et al. [17] extended the analysis of Sparrow et al. [11] to include a uniform but different temperature at the bounded surfaces. It should be mentioned that in the results reported by Ryhming [15] three values of inner to outer wall temperature ratios T1/T2 = 2, 5 and 25 were reported. Viskanta and Merriam [18] performed a parametric investigation on the steady state conduction and radiation heating and cooling processes of an absorbing, emitting, scattering and gray medium enclosed in the space between two black, isothermal and concentric spheres. The influences of the Planck number, optical thickness, surface emissivities, inner to outer radius ratio, internal heat generation per unit volume and inner to outer wall temperature ratios were investigated. The study highlighted the strong influence of the surface emissivities on the local radiation heat transfer flux and reported an increase of the total heat flux at the inner surface of the system when the Planck number and the optical thickness decreases and increases respectively. The transient solution of the model reported by Viskanta and Merriam [18] has been obtained by Chu and Weng [19]. In the study, the equation of radiative transfer has been numerically solved by the spherical harmonic method. The transient cooling process by radiation of a stationary spherical gas mass has been analyzed by Viskanta and Lall [20]. The energy transfer by conduction and convection was neglected and a numerical solution of the energy equation has been obtained by the method of moments. The effect of the optical thickness of the gas on the local heat flux and temperature distribution has been presented. The Galerkin method was used by Thynell and Özisik [21] to solve the steady state condition of the problem reported by Viskanta and Lall [20]. Tsai and Özisik [22] analyzed the transient combined conduction and radiation problem in an absorbing, emitting and isotropically scattering solid sphere subjected to a uniform temperature at the black boundary surface. The effects of the single scattering albedo and the radiation to conduction parameter on the temperature distribution and heat fluxes have been discussed. Siewert and Thomas [23] reported the steady state solution of the problem formulated by Tsai and Özisik [22]. In the study the mathematical model was reduced based on a sphere to plane transformation technique and the spherical harmonics method was employed to numerically solve the reduced system. Bayazitoglu and Suryanarayana [24] numerically analyzed the influence of the Planck number and the optical thickness on the transient cooling process of a solid sphere surrounded by a gray, emitting, absorbing and isotropically scattering gas. The influence of anisotropic scattering on the combined conduction and radiation heat transfer on a spherical medium has been considered by Thynell [25]. Reported studies on the influence of thermal radiation in non-opaque media during phase change processes have been largely limited to the treatment of solidification in planar systems [26], [27], [28], because of its simplicity and the fact that the contribution of natural convection can be justifiably ignored in this process [29], [30]. In the present study, a concise description of the energy exchange interactions and induced fluid flow during the melting process of a PCM with enhanced radiative properties and encapsulated in a spherical shell is reported. The role played by all the fundamental heat transfer modes in the process is highlighted and analyzed. Except for the studies reported by [31], [32] for the combined mode heat transfer during melting in a rectangular and a square cavity respectively, no similar work has been reported in this field especially in curved media. The primary motivation of this investigation is to enlarge the limited literature available in the thermal characterization of PCM-based latent heat storage systems for concentrating solar power plants.
Section snippets
Problem formulation and mathematical model
The PCM capsule is schematically shown in a cross sectional view in Fig. 1. It consists of an opaque, gray and diffuse spherical shell of inner radius Ri and wall thickness δ. The space inside the shell is completely filled with a phase change material (PCM) which can conduct heat as well as absorb, emit and scatter radiant energy. Initially the PCM is in its solid state at a temperature To and at time t > 0, the outer boundary surface is subjected to a uniform temperature Tw which is higher than
Numerical approach
In order to solve the governing equations, the control volume technique, which employs an implicit scheme based on the SIMPLE (Semi-Implicit Method for Pressure-Linked Equations) algorithm [38], was used. Commercially available software Ansys/Fluent v12 was employed for the calculations. The second order upwind scheme was employed to discretize the momentum equations while the PRESTO (Pressure Staggering Option) scheme was adopted for pressure correction via the continuity equation. The energy
Results and discussion
Ten different cases were analyzed in this study as summarized in Table 3. The effect of the optical thickness (τ) by changing the absorption coefficient (κa), on the melting process for a constant capsule size is considered in cases 1–4. It should be pointed out that the first case was used as a limiting case in which κa = 0 and also zero emissivity (ɛ) were assumed at the inner surface of the shell. It may be noticed that the Planck number also varies in these study cases.
Summary and conclusions
A comprehensive analysis of the energy transport due to the simultaneous contribution of convection, radiation and conduction during the melting process of a high temperature phase change material encapsulated in a spherical shell has been presented. The discrete ordinate method was used to solve the equation of radiative transfer and the finite volume scheme was used to solve the equations for mass, momentum and energy conservation. Based on the numerical analysis of the governing equations,
Acknowledgements
The work presented herewith has been funded by the United States Department of Energy through the Advanced Research Projects Agency - Energy (ARPA-E) under award number: DE-AR0000179. The authors also would like to acknowledge the use of the services provided by Research Computing at the University of South Florida.
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This paper is included in the Special Issue of Energy Storage edited by Prof. Anthony Roskilly, Prof. Phil Taylor and Prof. Yan.