Keywords

1 Introduction

Several countries ramped up their investments on renewable energy based electricity generation to combat the harmful environmental outcomes from power plants running on fossil fuels. Indeed the availability of fossil fuels is also a big question after few decades. Concentrated solar power (CSP) is one of the promising large-scale power generation technologies among the renewables which is being widely commercialized now. The major problem that CSP plants face is the intermittent solar radiation which limits the capacity and reliability of the plant. To alleviate this, integration of thermal energy storage (TES) systems to the CSP plants is essential [1]. Till date, two TES technologies were implemented commercially, viz., molten salt systems for parabolic trough or solar tower CSP plants and steam accumulators for direct steam generation CSP plants [2]. The major disadvantage of steam accumulators is the decrease in pressure during the discharging of steam. Incorporating a flash evaporator or an encapsulated heat storage capsule is preferred to avoid or reduce the pressure drop. Due to entropy generation by mixing in the flash evaporator system, encapsulated capsule is beneficial in steam accumulators [3]. Thermal energy can be stored in the form of sensible heat, latent heat and thermochemical heat. Sensible and latent heat storage devices are viable options for usage in steam accumulators in the form of encapsulated capsules. SHS devices store the sensible heat of the material during the rise in temperature. LHS devices store the latent heat of phase change material (PCM) at near constant melting temperature. But thermal resistance during charging/discharging of certain SHS material like concrete and PCM is high because of its low thermal conductivity. Several techniques were analyzed by researchers to increase the performance of the storage systems [4,5,6]. A detailed review of performance improvement techniques for was reported in the literature [7]. Encapsulating storage materials inside capsule increases the specific surface area. Also, direct contact of heat transfer fluid with the capsule increases the heat transfer coefficient [8]. Spherical and cylindrical capsules filled with PCM were tested for storage characteristics in a constant temperature bath at lab scale and a steam accumulator at industrial scale [9,10,11,12,13]. It is understood from the literature survey that no work was reported comparing the performance characteristics of different configurations of encapsulated capsules. In the present work, a numerical model is developed for evaluating the performances of different configurations of encapsulated capsules. Concrete and sodium nitrate packed in four different configurations, viz., spherical, cylindrical (H = D, H = 4D), and novel cylindrical models are compared.

2 Model Description

Figure 1 shows a view of the 2D axisymmetric encapsulated capsule containing concrete or sodium nitrate. The capsules are designed for a heat storage capacity of 320 kJ each. Melting point, latent heat, and thermal conductivity are the major parameters which are having a great influence in the design of any LHS systems. Similarly, specific heat and thermal conductivity of the storage material decides the charging time of SHS systems. The dimensions of the capsules are given in Table 1. Encapsulating material should withstand the high pressures (60–80 bar) that exists inside the pressure vessel. SS304 is selected as the encapsulating material for the present analysis. The thermophysical properties of concrete, sodium nitrate and SS304 are given in Table 2. Three physical processes are to be simulated to study the thermal storage behavior of encapsulated LHS capsules, i.e. conduction, convection and phase change. Conduction heat transfer is the only process involved in the charging simulation of SHS capsules. A 2D axisymmetric model is developed in view of the symmetry of flow and heat transfer around the vertical axis. Molten PCM movement within the capsules due to natural convection heat transfer is assumed to be laminar, Newtonian and incompressible. The major problem associated with the modeling of LHS capsule is the inclusion of latent heat necessary to melt/solidify the PCM. This problem is resolved by using the effective heat capacity method, which takes both specific heat and latent heat of the PCM in a single term called effective heat capacity. The discontinuous modified heat capacity is applied in the COMSOL Multiphysics software using a Heaviside function [14]. Boussinesq approximation and Darcy law’s source term are added in the momentum equation to include the buoyancy effect and nullify the solid PCM’s velocity. The corresponding governing equations for simulating the capsules are given in Eqs. (1)–(7). Initially, the capsules are at 291.8 °C. At any time (t > 0), the boundaries are given a temperature of 321.8 °C thereby making a temperature swing of 30 °C

Fig. 1
figure 1

2D axisymmetric view of capsules a spherical b cylindrical, H = D c cylindrical, H = 4D and d novel cylindrical

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Table 1 Dimensions of capsules
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Table 2 Thermo-physical properties of sodium nitrate, concrete and SS304
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$$C_{P} = \left\{ {\begin{array}{*{20}l} {C_{PS} } \hfill & {{\text{for}}\;\;T < T_{S} } \hfill \\ {C_{{P,{\text{EFF}}}} } \hfill & {{\text{for}}\;\;T_{S} \le T \le T_{L} } \hfill \\ {C_{PL} } \hfill & {{\text{for}}\;\;T > T_{L} } \hfill \\ \end{array} } \right.$$
(1)
$$C_{{P,{\text{EFF}}}} = \frac{{C_{PS} + C_{PL} }}{2} + \frac{{\Delta H_{F} }}{{2\Delta T_{M} }}$$
(2)
$$\nabla .\vec{V} = 0$$
(3)
$$\frac{{\partial \vec{V}}}{\partial t} + (\vec{V}.\nabla )\vec{V} = \frac{1}{\rho }\left( { - \nabla P + \mu \nabla^{2} \vec{V} + F + \vec{S}} \right)$$
(4)
$$\rho C_{p\,} \frac{{{\text{D}}T}}{{{\text{D}}t}} = k\nabla^{2} T$$
(5)
$$F = \rho \vec{g}\beta \left( {T - T_{M} } \right)$$
(6)
$$\theta = \frac{{T - T_{S} }}{{T_{L} - T_{S} }} = \frac{{T - T_{M} + \Delta T_{M} }}{{2\Delta T_{M} }} = \left\{ {\begin{array}{*{20}l} 0 \hfill & {{\text{for}}\;\;T < T_{S} } \hfill \\ {0 - 1} \hfill & {{\text{for}}\;\;T_{S} \le T \le T_{L} } \hfill \\ 1 \hfill & {{\text{for}}\;\;T > T_{L} } \hfill \\ \end{array} } \right.$$
(7)

3 Results and Discussions

3.1 Validation

In order to validate the numerical model, the results obtained for the melt fraction of storage bed were compared with the melt fraction reported in the literature [10]. The physical model chosen for the numerical validation, thermophysical properties of the PCM and initial and boundary conditions of the model are taken from literature [10]. It is evident from Fig. 2 that the current numerical results showed good agreement with the results reported in the literature [10].

Fig. 2
figure 2

Validation of the numerical model

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3.2 Grid Independency Test

Free triangular mesh has been adapted in the numerical model. In order to test the dependency of numerical results on the mesh element size, a simulation is run with the cylindrical capsule of H = D. The LHS capsule is initially at 291.8 °C. At any time t > 0, the boundary of the capsule is at 321.8 °C. The average temperature of the capsule is compared for different element sizes and it is observed from Fig. 3 that 10,377 elements can be taken for the numerical model. Similarly, grid independency test carried has been carried out for other configurations also. Time step used in the analyses is 0.01 s throughout all the models.

Fig. 3
figure 3

Grid independency test

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3.3 Temperature Distribution

Figure 4a, b shows the comparison of the average temperature variation of the LHS/SHS capsules of different configurations. During charging, PCM/concrete kept in the capsules is initially in the solid state at 291.8 °C. When a high temperature of 321.8 °C is applied on the boundary of the capsule, heat is transferred from periphery of the capsules and thereby stored in the form of sensible and latent heat. It is inferred from Fig. 4 that the increase in average temperature is faster in cylindrical capsules than the spherical capsule. Also, it can be noted that the cylindrical capsule with a higher aspect ratio reaches the boundary condition in lesser time. Also, the increase in temperature of novel cylindrical configuration is much faster than all of the configurations. This is due to two reasons, viz., increased heat transfer area and reduced thickness of the capsule. Screenshots of temperature variation of LHS capsules during charging are given in Fig. 5. The effect of natural convection is seen from the asymmetric temperature variation in all the LHS capsules. Similarly, the conduction in the concrete capsules is seen from the symmetric variation of temperature.

Fig. 4
figure 4

Temperature variation a LHS and b SHS capsules

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Fig. 5
figure 5

Temperature variation of LHS and SHS capsules

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3.4 Total Energy Storage Rate

Figure 6a, b show the total energy storage rate of all the LHS/ SHS capsules. For making a comparison between the LHS and SHS capsules of different configurations, the time taken for 95% of total energy stored, i.e., 304 kJ can be taken as the charging time of the capsules. It took about 58, 48, 36, and 26 min to store about 320 kJ in the LHS capsules. Similarly, it took about 69, 62, 57, and 22 min to store about 320 kJ in the SHS capsules.

Fig. 6
figure 6

Total energy storage variation a LHS and b SHS capsules

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In the SHS capsules, minor difference in the charging time exists between the spherical, cylindrical (H = D), and cylindrical (H = 4D) configurations. But there exists a huge decrease in the charging time using the novel cylindrical configuration. In the LHS capsules, considerable difference in the charging time exists between the spherical, cylindrical (H = D), cylindrical (H = 4D), and novel cylindrical configurations. This is due to the presence of natural convection in the capsules during melting. The proposed novel cylindrical configuration yields 61 and 28% lesser charging times for SHS and LHS capsules when compared with the cylindrical capsule (H = 4D).

4 Conclusions

A thermal model was developed to compare the performance of LHS/SHS capsules of different geometrical configurations. Numerical results shown that for the same mass of storage material, cylindrical configuration yields lesser charging time than the spherical configuration. This is due to the fact that the distance between the center and periphery of the capsule is more in spherical capsule than that of cylindrical capsule. In the cylindrical models, configuration with a higher aspect ratio takes lesser time for charging. The time taken for charging of LHS capsule in spherical, cylindrical (H = D, H = 4D) and novel cylindrical configurations are 58, 48, 36, and 26 min, respectively. Similarly, the time taken for charging of SHS capsule in spherical, cylindrical (H = D, H = 4D), and novel cylindrical configurations are 69, 62, 57 and 22 min, respectively. The proposed novel cylindrical configuration saves 61 and 28% charging times for SHS and LHS capsules when compared with the cylindrical capsule (H = 4D). The results of the current study will be useful in the design and optimization of storage modules which are having wide usage in solar thermal power plants, clean transportation systems, thermal conditioning for buildings, etc.