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New effective thermal conductivity model for the analysis of whole thermal storage tank
International Journal of Heat and Mass Transfer 131 (2019) 1109–1116
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
New effective thermal conductivity model for the analysis of whole thermal storage tank Min Ho Kim 1, Yong Tae Lee 2, Jiwon Gim, Abhishek Awasthi, Jae Dong Chung ⇑ Department of Mechanical Engineering, Sejong University, Republic of Korea
a r t i c l e
i n f o
Article history: Received 12 March 2018 Received in revised form 25 September 2018 Accepted 25 November 2018
Keywords: Effective thermal conductivity Phase change Thermal storage tank Natural convection
a b s t r a c t In this paper, a new effective thermal conductivity model is proposed and used to numerically investigate a whole tank designed for latent-heat thermal energy storage (LHTES). The tank was filled with phasechange materials (PCM) inside 9 9 20 spherical capsules. Previous studies have focused on the performance of only one capsule under the assumption that one capsule could represent the performance of the whole tank. The analysis of phase change involves much complexity, even for one capsule; thus it is challenging to analyze a whole tank due to the tremendous amounts of calculation time and memory capacity required. The new effective thermal conductivity model includes the effect of the natural convection in molten PCM, and estimates charging/discharging performance in the full scale system with reduced calculation time. This new effective thermal conductivity model can appropriately predict dissimilar melting behavior depending on the unique position of each capsule in the tank. The model was validated by comparison with experimental data, and also by rigorous numerical analysis including natural convection. Ó 2018 Elsevier Ltd. All rights reserved.
1. Introduction Soaring demand for electric power has increased interest in energy storage systems. Latent-heat thermal energy storage (LHTES) systems have been highlighted as one of the more efficient energy storage systems for its high energy storage density. It is hard to accurately estimate the performance of LHTES systems because the phenomena involved with phase change is very complicated. Much effort has been devoted to numerical analysis of phase change phenomena. Chung et al. [1] analyzed flow transition during the melting process in a horizontal cylinder with extended Rayleigh number to demonstrate the effect of thermal instability on solid-liquid interfaces. Khodadadi and Zhang [2] conducted a computational study of constrained melting within spherical containers and showed that the Rayleigh number was more significant than the Stefan number, for natural convection. They also demonstrated the important role of the Prandtl number in melting patterns at a fixed Rayleigh number. Assis et al. [3] explored how a phase change process depends on thermal and geometric parameters. They suggested a correlation of molten fraction as a function of Fourier, Stefan, and Grashof numbers. Elghnam et al. [4] ⇑ Corresponding author. 1 2
E-mail address: [email protected] (J.D. Chung). Kia Motors Corporation, Seoul, Republic of Korea. LG Chem, Gyeonggi-do, Republic of Korea.
https://doi.org/10.1016/j.ijheatmasstransfer.2018.11.122 0017-9310/Ó 2018 Elsevier Ltd. All rights reserved.
performed an experimental study on the charging and discharging process inside a spherical capsule with variations in size, capsule material, and temperature of the heat transfer fluid (HTF). They discovered that the charging performance improved when using bigger size, metallic materials, at lower HTF temperature. Lee et al. [5] investigated thermal storage performance more practically by including the convection effect outside a spherical capsule. They confirmed that the inclusion of forced convection surrounding the capsule resulted in much lower discharging performance. In order to understand phase change phenomena better, many researchers conducted studies under diverse conditions. Sparrow and Geiger [6] provided an experimental and numerical comparison between constrained and unconstrained melting in a horizontal tube. They demonstrated that melting was faster in unconstrained mode where the movement of solid PCM was allowed, and that the melting became faster as the solid PCM sank closer to the tube wall. Tan [7] used spherical capsules experimentally to demonstrate that melting in unconstrained mode was faster than in constrained mode. These results correspond to the conclusions of Sparrow and Geiger [6]. The same author [8] also conducted numerical analysis to compare the results of the experiments and validated the conclusions. Hong et al. [9] analytically examined unconstrained melting inside a spherical capsule, varying the conditions of capsule size, material, and wall temperature. They discussed the limitation of analytical approach in the high Stefan number. There have been many studies on phase change
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Nomenclature CP d fl Gr g H h k L l _f m Pr Ra r T t V
specific heat at constant pressure [J/kg-K] diameter [mm] liquid fraction in each computational mesh Grashof number gravitational acceleration [m/s2] total enthalpy [J/kg] sensible enthalpy [J/kg] thermal conductivity [W/m-K] latent heat [J/kg] length [mm] molten fraction Prandtl number Rayleigh number radial distance [mm] temperature [°C] time [s] volume [mm3]
phenomena; however, there is still an ongoing need for the complexity of these phenomena to be considered, such as movement of solid phase-change material (PCM), the effect of natural convection in molten PCM, the thermal instability on solid-liquid interfaces, and the external conditions surrounding the capsule. Although it is important to analyze an entire tank to predict the proper performance of thermal energy storage systems, previous studies have focused on only one capsule or one coil under the assumption that it could represent the performance of the whole tank. Considering that phase change even for one capsule is very complex, it is not feasible to analyze a whole tank due to the tremendous amounts of calculation time and memory capacity required. Because of these difficulties, there have been few studies of whole tanks, and these have been subject to the following assumptions: (1) if two phases are regarded one single phase, which ignores the temperature differences between solid and liquid, we can call it a single-phase model, or (2) if the solid and liquid phases are treated separately, we can call it a two-phase model. There are two typical variations of the two-phase model: continuous solid phase and concentric dispersion. In the continuous solid phase model, the phase change material is assumed to be a porous medium. Ismail and Stuginsky [10] conducted a numerical comparison of four models, that is, the continuous solid phase model, Schumann’s model, a single phase model, and a thermal diffusion model. They compared the computation time required for each model to solve a test problem, and investigated the influence of various parameters, such as size, void fraction, material, flow rate, and working fluid inlet temperature. They discovered that the working fluid inlet temperature had a greater influence on the wall-heat-losses in all models and that the mass flow rate had the least effect. Arkar and Medved [11] investigated the influence of the PCM thermal properties on the thermal responses of LHTES-containing spheres filled with paraffin. Using a continuous solid phase model, they compared the numerical results with an experiment consisting of 35 rows of spheres and validated them in slow running processes. Erek and Dincer [12] developed a heat transfer coefficient correlation using 120 numerical simulations for an ice TES system. Their conclusions from using a concentric dispersion model emphasized the importance of taking into account the variable heat transfer coefficient for analyzing thermal energy storage (TES) systems. Peng et al. [13] conducted numerical analysis of thermal behavior for LHTES systems by investigating radial heat transfer and wall heat losses. Using the concentric dispersion model, their study showed that charging efficiency could be increased under the following conditions: decrease in
Greek symbols density (kg/m3) / liquid fraction in each capsule
q
Subscript c eff i init in l m o ref s w
characteristic effective inner initial tank inlet liquid melting outer reference solid wall
capsule size and fluid inlet velocity, or increase in the storage height. From a literature review, even though analysis of entire tanks have been extensive, we can see the limitations in their assumptions: the single-phase model has limited accuracy because it does not consider phase change behavior. The continuous solid-phase model also has shortcomings because the temperature differences in the two phases are ignored. The other concentric dispersion model, which takes temperature differences into account, improved the accuracy to some extent; however, it does not include the natural convective effect, and thus retains a limit on its accuracy. Furthermore, these models do not reflect the differences in thermal behavior of capsules regarding their position; thus we cannot estimate the true charging and discharging performance because the melting/solidification will be different depending on the capsule position: near the tank top or tank bottom, or near the tank center or tank edge. In this study, an efficient, accurate approach is proposed for estimating the melting/solidification behaviors in a full tank. By introducing effective thermal conductivity that reflects the effect of natural convection in the molten PCM, we are able to investigate the performance of the full-scale tank within a much shorter calculation time. First, we validated the proposed model for a single capsule by comparing experimental results. Second, we checked whether the model could be applied over much wider conditions in relation to different capsule sizes, wall temperatures, and initial temperatures. In addition, we created 12-layer capsules in a vertical column to show that the present model is superior to the previous effective thermal conductivity model. Finally, we accomplish our goal of analyzing a whole tank containing 1692 spherical capsules (9 9 20 array).
2. Numerical method 2.1. Mathematical model Hirata and Nishida [14] proposed an effective thermal conductivity model. The constrained melting model is shown in Fig. 1(a). This model ignores the movement of solid PCM due to the density difference between liquid PCM and solid PCM. Natural convection plays an important role and accelerates melting in the upper part of solid PCM. However, this process is the main cause of the tremendous computation time required in simulation of melting and solidification. Thus, a combination of the effective thermal
M.H. Kim et al. / International Journal of Heat and Mass Transfer 131 (2019) 1109–1116
conductivity model exclusion of convection in molten PCM (by treating the molten PCM as solid-like), with a way to properly include the effect of the accelerated melting, would be of great benefit for full tank simulation. As shown in Fig. 1, the shape of the phase interface is different from the real situation. However, considering that the molten fraction is predicted precisely in time for each capsule, the detailed phase interface of each capsule may not be important when the main interest is to estimate the charging and discharging performance of the full tank. Holman [15] proposed using the effective thermal conductivity model in Eq. (1) for internal natural convection in a rectangular domain.
m keff L ¼ CðGrPrÞn k d
ð1Þ
The parameters C, m, and n are constants and d is the horizontal distance of the domain. The effective thermal conductivity model used in the phase-change analysis employed an analogous formula that reflects the natural convection phenomenon that occurs in molten PCM.
keff ¼ CRaðr o Þm kl
ð2Þ
In Eq. (2), use of the Rayleigh number incorporated the effect of natural convection in molten PCM, and was expressed by the temperature difference T in T m and characteristic length of the capsule radius r o . Some of the drawbacks in the previous effective thermal conductivity model were: (1) the capsule radius was used for the characteristic length, which is constant and cannot reflect the time-wise variation of the strength of natural convection in molten PCM, and (2) the tank inlet temperature T in rather than the capsule wall temperature of each capsule in the tank, was used for the temperature difference. Thus, the previous approach of effective thermal conductivity keff cannot estimate the separate phase-change phenomena of each capsule in a tank. Instead, we propose the same form of keff but express it as a function of the molten fraction, which reflects the time-wise variation of the strength of the natural convection in molten PCM. This enables analysis of the different thermal behavior of each capsule in relation to the different positions in a tank.
K eff ¼ CRam l kl
ð3Þ
The Rayleigh number in Eq. (3) is expressed in terms of the characteristic length l ¼ ro r i and T w T m . The different surface temperature of each capsule is calculated by averaging the angular surface temperature of each capsule for every time step. The diameters of the solid bodies are also updated at each time step, after determining the phase interface and the amount of molten PCM using Eqs. (4) and (5).
R qf dV /¼ R l qdV ðl /Þ
1111
ð4Þ
4 3 4 3 pr ¼ pr 3 o 3 i
ð5Þ
In addition to the aforementioned limitations in the previous keff , the previous effective thermal conductivity model required a new fitting process to obtain keff for each different condition, such as changes in capsule size, wall temperature, or initial temperature in the tank. Practically, the effective thermal conductivity method has been adopted to facilitate calculation; however, in case of estimating the performance of whole tank, the previous effective thermal conductivity model will become impractical because repetitive processing is required to find the proper fitting parameters. Moreover, there is no reference for these fitting processes. If an experimental result exists for the fitting reference, indeed, there is no need to conduct numerical analysis. Meanwhile, with the proposed keff model, once fitting parameters are found for a specific condition, the parameters can be used to analyze capsules under different conditions to a fair extent. 2.2. Governing equation The enthalpy method (Swaminathan and Voller [16]) is incorporated into the commercial software STAR-CCM+ v9.04. The solution for mass and momentum conservation is not required in the effective thermal conductivity method. The energy conservation equation, excluding the convection term, is as follows.
q
@H @ @T @ @T þ ¼ keff keff @t @x @t @y @y
ð6Þ
Eq. (6) only includes conduction terms, which reduces calculation time and makes the use of a rough mesh available. Here, H ðH ¼ h þ DHÞ is expressed as the sum of sensible enthalpy RT (h ¼ href þ T ref cp dT) and latent heat (DH ¼ f l L), where f l is the liquid fraction of PCM defined as follows:
fl ¼
8 > < 0 > :
if
T < Ts
TT s T l T s
if
Ts < T < Tl
1
if
Tl < T
ð7Þ
3. Results and discussion 3.1. Validation by experimental results VS Natural Convection Model VS keff Model The proposed keff model was compared with the experimental results from Tan [7] as well as the constrained melting model
Fig. 1. (a) Constrained melting model with natural convection, (b) effective thermal conductivity model, and (c) full scale thermal energy storage filled with 1692 (9 9 20) spherical capsules.
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including natural convection of molten PCM. The latter was used as a reference when there was no experimental result. We also used the result of the previous keff for comparison. The validation was conducted in a spherical capsule of diameter 101.66 mm. We used n-octadecane as the PCM and its properties are shown in Table 1. The initial temperature was 1 °C below the melting temperature. As it was being isothermally heated (40 °C), the solid PCM started melting when it arrived at its melting temperature of 28.2 °C. The results of the time-wise variation in molten fraction mf are shown in Fig. 2. The proposed keff model shows close agreement with both the experimental results and the constrained melting model results over the whole time of the melting process. However, the previous keff model shows a significant deviation from the real physics in the intermediate stages of melting, even though it also agrees with the experimental results at mf ¼ 1. Note that it is not surprising to find close agreement at mf ¼ 1 because the parameters C and m in Eq. (2) were fitted to match the result at mf ¼ 1. The previous keff model assumes the same Ra number during the whole melting period (i.e., the same strength of natural convection); the deviation in the intermediate stages of melting is an inherent limitation of the model. In order to check the range of application of the proposed keff model, we conducted simulations for other conditions with the same fitting parameters obtained from specific conditions for different capsule wall temperatures, variations in capsule size and initial temperature. Because there were no experimental results for the conditions considered here, the results from the numerical model including natural convection of molten PCM, was used as a reference for comparison. We compared the results with capsule wall temperatures of 35, 40, and 45 °C. The time-wise variations in the molten fraction are shown in Fig. 3. The proposed keff model closely agrees with the reference result, regardless of the capsule wall temperature. The results of the different capsule diameters (40, 60, 80, and 101.66 mm) are shown in Fig. 4. The proposed keff model shows close agreement with the reference result, especially for small capsule sizes, because conduction is stronger than natural convection in capsule of smaller diameter. Even for large capsules, for example, do = 101.66 mm, the difference between the results of the proposed keff model and the reference is acceptable. Considering that capsule diameters are less than 100 mm in conventional applications of latent thermal energy storage systems, the proposed keff model could be used over a fair range of capsule diameters. The results with different initial temperature can be observed in Fig. 5. The initial temperatures are 1, 10, and 20 °C lower than the melting temperature. Again, the proposed keff model closely agrees with the reference result, regardless of the initial temperature. The effect of the initial temperature turned out to be very small. This is because the sensible heat, even for 20 °C, has less influence than that of the latent heat in the melting process. Contrary to the previous keff model, we have confirmed that the proposed keff model can be used for other melting conditions with
Fig. 2. Molten fraction comparison of the proposed keff model with the previous keff model, constrained melting model including natural convection in the molten PCM, and the experimental data from Tan [7].
Fig. 3. Molten fraction comparison between the proposed keff model and the referential numerical model with different capsule wall temperatures.
the same fitting parameters obtained from a specific condition, which will extend its application.
3.2. One column with twelve layers of capsules Table 1 Properties of n-octadecane. n-octadecane Melting temperature [°C] Latent heat [J/kg] Density [kg/m3] Specific heat [J/kg-K] Thermal conductivity [W/m-K] Viscosity [Pa-s] Thermal expansion coefficient [1/K] Kinematic viscosity [m2/s] Thermal diffusivity [m2/s]
28.2 243,500 772.0 2330.0 0.1505 0.00386 0.00091 5.0 106 8.3669 108
Before the application of a full scale tank, a feasibility study was conducted for a smaller system with one column of 12 layers of capsules. Contrary to the simulation of a full scale tank including natural convection of molten PCM, the simulation of this smaller system of one column of 12 layers of capsules is possible within an acceptable computation time. This enabled checking of the validity of the proposed keff model and also allowed examination of the ability of the model to capture the differences in melting behavior depending on the capsule positions. Twelve capsules were vertically arranged to have a simple cubic form. The capsule diameter was 100 mm. Three models were
M.H. Kim et al. / International Journal of Heat and Mass Transfer 131 (2019) 1109–1116
Fig. 4. Molten fraction comparison between the proposed keff model and the referential numerical model with different capsule diameters.
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Fig. 6. The average molten fraction comparison of 12 layers of capsules in three models of natural convection, the proposed keff model, and the previous keff model.
for the proposed model was about 50 times more efficient than that of the constrained melting model. This was achieved by adopting a rough mesh, a large time-step, and only solving the conduction equation. The proposed model can be used to estimate structures composed of hundreds of capsules thanks to the shorter computation time, allowing for the entire tank to be simulated. 3.3. Thermal energy storage tank with array of 9920 capsules A tank containing an array of 9 9 20 capsules in a simple cubic arrangement, as shown in Fig. 8, was the tank modeled. Thanks to its symmetric nature, only one fourth of the entire system was used as the domain of interest. The heat transfer fluid (HTF) comes in at 5.0 104 m/s, from top to bottom. The computational configuration and conditions are from Arkar and Medved [11], Nallusamy et al. [17], and Cho and Choi [18]. The molten fraction of the previous keff model and the proposed keff model are shown in Fig. 9. Overall, the melting is faster in the
Fig. 5. Molten fraction comparison between the proposed keff model and the referential numerical model with different initial temperatures.
compared: the constrained melting model including natural convection, the previous keff model, and the proposed keff model. Because there was no experimental data, the constrained melting model including natural convection of molten PCM, was used as a reference. The time-wise variation in molten fraction is shown in Fig. 6. The results demonstrate that the proposed keff model matches well with the behavior of the constrained melting model, while the previous keff model only corresponds when conditions are near mf ¼ 1. The discrepancy in the intermediate melting and solidification processes shows the limitation of the previous model because full charging/discharging is not normal in real applications. The merit of the proposed model was made clearer by comparing the melting behavior of capsules in different positions. We examined the molten fraction of capsules in the 2nd, 8th, and 12th layer, as shown in Fig. 7. The proposed keff model precisely predicted the melting behavior. In addition, the computation time
Fig. 7. Comparison of three models in different capsule positions.
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Fig. 8. Geometry of latent heat thermal storage tank with an array of 9 9 20 capsules.
Fig. 9. Molten fraction comparison between the proposed model and the previous model for a full tank with an array of 9 9 20 capsules.
previous keff model than in the proposed model. A constrained melting model with natural convection is not feasible due to the lack of memory capacity and the tremendous amounts of time required. Nevertheless, we would expect that the proposed keff model would show melting behavior very close to actual conditions, as was shown from the results for the 12 layers of capsules. The fitting parameters used for the each keff model, Eqs. (2) and (3), respectively, were obtained from the one-capsule model because there were no experimental or numerical references for a full scale tank. To examine the spatially variable melting behavior, we sampled positions at the top (20th layer), middle (10th layer), and bottom (1st layer); as well as at the center, middle, and edge, as shown in Fig. 8. On the top layer of the system, PCM melted almost uniformly in the center, middle, and edge positions in both models. This is because the flow inlet is located in the top layer, and the brine temperature is uniform regardless of the capsule positions. The results of the middle and bottom layers are shown in Fig. 10 (b) and (c), respectively. The lower the layer is,
Fig. 10. Time-wise variation of the molten fraction depending on the position in a tank with an array of 9 9 20 capsules: (a) at the top (20th layer), (b) in the middle (10th layer), and (c) at the bottom (1st layer).
M.H. Kim et al. / International Journal of Heat and Mass Transfer 131 (2019) 1109–1116
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Fig. 11. Effective thermal conductivity behavior of the previous keff model.
Fig. 12. Effective thermal conductivity behavior of the proposed keff model.
the more differently the melting progresses at the center, middle, and edge. This is because the brine temperature changes along the flow direction, and the capsule temperature is different depending on their locations. Contrary to the previous keff model, the proposed keff model properly incorporated the effect of different capsule wall temperatures. This resulted in different melting speeds and different strengths of natural convection of molten PCM in each capsule. Fig. 11 shows the behavior of keff in the previous model at the cross section of Fig. 8. The previous model shows the same thermal conductivity of liquid PCM for all capsules in a tank. It cannot reflect the effect of the natural convection that becomes intense as the melting progresses. Thus, this model cannot predict the proper performance of such systems. To deal with this problem, we present an enhanced effective thermal conductivity model which is a function of the molten fraction. For this reason, it can
properly incorporate the increased strength of natural convection in the molten PCM as the melting progresses. Fig. 12 shows the different effective thermal conductivities in each capsule in a tank, reflecting the different melting speeds according to different capsule locations. 4. Conclusions In this study, a new effective thermal conductivity model was proposed and used to investigate numerically the whole tank of a latent heat thermal energy storage (LHTES) system filled with phase-change materials (PCM) in an array of 9 9 20 spherical capsules. Most of previous approaches assumed the charging/discharging performance of the entire tank would be the same as that of one capsule inside the tank due to the tremendous computation time required to simulate the natural convection in the molten
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PCM. There were some studies of whole tanks, but due to oversimplified assumptions the detailed physics of phase change was not captured. The main innovations in the proposed method are (1) Tremendous reduction of calculation time was achieved by introducing effective thermal conductivity, for example, for the case of one column, with 12 layers of capsules; the computational time of the proposed model was about 50 times shorter than that of a rigorous computational model (i.e., the constrained melting model). (2) Use of the Rayleigh number incorporated the effect of natural convection in the molten PCM. (3) The Rayleigh number (i.e., the strength of natural convection in the molten PCM), changed as the melting proceeded and the volume of the molten PCM increased. Thus, dissimilar melting behavior depending on the different positions of the capsules in the tank could be observed. (4) Close agreement with both experimental and rigorous computational results was observed over the whole time of the melting process. The previous keff model showed a significant deviation from the real physics in the intermediate stages of melting. The discrepancy in the intermediate melting/solidification process would be a big limitation because full charging/discharging is not normal in real applications (5) The same keff can be used for other melting conditions: different capsule wall temperatures, variations in capsule size, and differences in initial temperature, which made it possible to apply the same keff to the whole tank. However, the previous keff requires a new keff to find the proper fitting parameters. Moreover, there is no reference for these fitting processes, and if experimental results existed, there would be no need to conduct numerical analysis. Conflict of interest The authors declared that there is no conflict of interest. Acknowledgement This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (No. 2017R1D1A1B05030422).
References [1] J.D. Chung, J.S. Lee, H. Yoo, Thermal instability during the melting process in an isothermally heated horizontal cylinder, Int. Commun. Heat Mass Transf. (1997) 3899–3907. [2] J.M. Khodadadi, Y. Zhang, Effects of buoyancy-driven convection on melting within spherical containers, Int. Commun. Heat Mass Transf. 44 (2001) 1605– 1618. [3] E. Assis, L. Katsman, G. Ziskind, R. Letan, Numerical and experimental study of melting in a spherical shell, Int. J. Heat Mass Transf. 50 (2007) 1790–1804. [4] R.I. ElGhnam, R.A. Abdelaziz, M.H. Sakr, H.E. Abdelrhman, An experimental study of freezing and melting of water inside spherical capsules used in thermal energy storage systems, Ain Shams Eng. J. 3 (2012) 33–48. [5] Y.T. Lee, S.W. Hong, J.D. Chung, Effects of capsule conduction and capsule outside convection on the thermal storage performance of encapsulated thermal storage tanks, Sol. Energy 110 (2014) 56–63. [6] E.M. Sparrow, G.T. Geiger, Melting in a horizontal tube with the solid either constrained or free to fall under gravity, Int. Commun. Heat Mass Transf. 29 (1986) 1007–1019. [7] F.L. Tan, Constrained and unconstrained melting inside a sphere, Int. Commun. Heat Mass Transf. 35 (2008) 466–475. [8] F.L. Tan, S.F. Hosseinizadeh, J.M. Khodadadi, L. Fan, Experimental and computational study of constrained melting of phase change materials (PCM) inside a spherical capsule, Int. J. Heat Mass Transf. 52 (2009) 3464– 3472. [9] S.W. Hong, Y.T. Lee, J.D. Chung, Restrictions on the analytic approach of unconstrained melting inside a spherical capsule, J. Mech. Sci. Technol. 29 (2015) 5035–5042. [10] K.A.R. Ismail, R. Stuginsky Jr., A parametric study on possible fixed bed models for pcm and sensible heat storage, Appl. Therm. Eng. 19 (1999) 757–788. [11] C. Arkar, S. Medved, Influence of accuracy of thermal property data of a phase change material on the result of a numerical model of a packed bed latent heat storage with spheres, Thermochimica Acta 438 (2005) 192–201. [12] A. Erek, I. Dincer, Numerical heat transfer analysis of encapsulated ice thermal energy storage system with variable heat transfer coefficient in downstream, Int. J. Heat Mass Transf. 52 (2009) 851–859. [13] H. Peng, H. Dong, X. Ling, Thermal investigation of PCM-based high temperature thermal energy storage in packed bed, Energy Convers. Manage. 81 (2014) 420–427. [14] T. Hirata, K. Nishida, An analysis of heat transfer using equivalent thermal conductivity of liquid phase during melting inside an isothermally heated horizontal cylinder, Int. J. Heat Mass Transf. 32 (1989) 1663–1670. [15] J.P. Holman, Heat Transfer, sixth ed., McGraw-Hill, 1986, p. 350. [16] C.R. Swaminathan, V.R. Voller, A general enthalpy method for modeling solidification processes, Metall. Trans. B 23B (1992) 651–664. [17] N. Nallusamy, S. Sampath, R. Velraj, Experimental investigation on a combined sensible and latent heat storage system integrated with constant/varying (solar) heat sources, Renewable Energy 32 (2007) 1206–1227. [18] K. Cho, S.H. Choi, Thermal characteristics of paraffin in a spherical capsule during freezing and melting processes, Int. Commun. Heat Mass Transf. 43 (2000) 3183–3196.
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
New effective thermal conductivity model for the analysis of whole thermal storage tank Min Ho Kim 1, Yong Tae Lee 2, Jiwon Gim, Abhishek Awasthi, Jae Dong Chung ⇑ Department of Mechanical Engineering, Sejong University, Republic of Korea
a r t i c l e
i n f o
Article history: Received 12 March 2018 Received in revised form 25 September 2018 Accepted 25 November 2018
Keywords: Effective thermal conductivity Phase change Thermal storage tank Natural convection
a b s t r a c t In this paper, a new effective thermal conductivity model is proposed and used to numerically investigate a whole tank designed for latent-heat thermal energy storage (LHTES). The tank was filled with phasechange materials (PCM) inside 9 9 20 spherical capsules. Previous studies have focused on the performance of only one capsule under the assumption that one capsule could represent the performance of the whole tank. The analysis of phase change involves much complexity, even for one capsule; thus it is challenging to analyze a whole tank due to the tremendous amounts of calculation time and memory capacity required. The new effective thermal conductivity model includes the effect of the natural convection in molten PCM, and estimates charging/discharging performance in the full scale system with reduced calculation time. This new effective thermal conductivity model can appropriately predict dissimilar melting behavior depending on the unique position of each capsule in the tank. The model was validated by comparison with experimental data, and also by rigorous numerical analysis including natural convection. Ó 2018 Elsevier Ltd. All rights reserved.
1. Introduction Soaring demand for electric power has increased interest in energy storage systems. Latent-heat thermal energy storage (LHTES) systems have been highlighted as one of the more efficient energy storage systems for its high energy storage density. It is hard to accurately estimate the performance of LHTES systems because the phenomena involved with phase change is very complicated. Much effort has been devoted to numerical analysis of phase change phenomena. Chung et al. [1] analyzed flow transition during the melting process in a horizontal cylinder with extended Rayleigh number to demonstrate the effect of thermal instability on solid-liquid interfaces. Khodadadi and Zhang [2] conducted a computational study of constrained melting within spherical containers and showed that the Rayleigh number was more significant than the Stefan number, for natural convection. They also demonstrated the important role of the Prandtl number in melting patterns at a fixed Rayleigh number. Assis et al. [3] explored how a phase change process depends on thermal and geometric parameters. They suggested a correlation of molten fraction as a function of Fourier, Stefan, and Grashof numbers. Elghnam et al. [4] ⇑ Corresponding author. 1 2
E-mail address: [email protected] (J.D. Chung). Kia Motors Corporation, Seoul, Republic of Korea. LG Chem, Gyeonggi-do, Republic of Korea.
https://doi.org/10.1016/j.ijheatmasstransfer.2018.11.122 0017-9310/Ó 2018 Elsevier Ltd. All rights reserved.
performed an experimental study on the charging and discharging process inside a spherical capsule with variations in size, capsule material, and temperature of the heat transfer fluid (HTF). They discovered that the charging performance improved when using bigger size, metallic materials, at lower HTF temperature. Lee et al. [5] investigated thermal storage performance more practically by including the convection effect outside a spherical capsule. They confirmed that the inclusion of forced convection surrounding the capsule resulted in much lower discharging performance. In order to understand phase change phenomena better, many researchers conducted studies under diverse conditions. Sparrow and Geiger [6] provided an experimental and numerical comparison between constrained and unconstrained melting in a horizontal tube. They demonstrated that melting was faster in unconstrained mode where the movement of solid PCM was allowed, and that the melting became faster as the solid PCM sank closer to the tube wall. Tan [7] used spherical capsules experimentally to demonstrate that melting in unconstrained mode was faster than in constrained mode. These results correspond to the conclusions of Sparrow and Geiger [6]. The same author [8] also conducted numerical analysis to compare the results of the experiments and validated the conclusions. Hong et al. [9] analytically examined unconstrained melting inside a spherical capsule, varying the conditions of capsule size, material, and wall temperature. They discussed the limitation of analytical approach in the high Stefan number. There have been many studies on phase change
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Nomenclature CP d fl Gr g H h k L l _f m Pr Ra r T t V
specific heat at constant pressure [J/kg-K] diameter [mm] liquid fraction in each computational mesh Grashof number gravitational acceleration [m/s2] total enthalpy [J/kg] sensible enthalpy [J/kg] thermal conductivity [W/m-K] latent heat [J/kg] length [mm] molten fraction Prandtl number Rayleigh number radial distance [mm] temperature [°C] time [s] volume [mm3]
phenomena; however, there is still an ongoing need for the complexity of these phenomena to be considered, such as movement of solid phase-change material (PCM), the effect of natural convection in molten PCM, the thermal instability on solid-liquid interfaces, and the external conditions surrounding the capsule. Although it is important to analyze an entire tank to predict the proper performance of thermal energy storage systems, previous studies have focused on only one capsule or one coil under the assumption that it could represent the performance of the whole tank. Considering that phase change even for one capsule is very complex, it is not feasible to analyze a whole tank due to the tremendous amounts of calculation time and memory capacity required. Because of these difficulties, there have been few studies of whole tanks, and these have been subject to the following assumptions: (1) if two phases are regarded one single phase, which ignores the temperature differences between solid and liquid, we can call it a single-phase model, or (2) if the solid and liquid phases are treated separately, we can call it a two-phase model. There are two typical variations of the two-phase model: continuous solid phase and concentric dispersion. In the continuous solid phase model, the phase change material is assumed to be a porous medium. Ismail and Stuginsky [10] conducted a numerical comparison of four models, that is, the continuous solid phase model, Schumann’s model, a single phase model, and a thermal diffusion model. They compared the computation time required for each model to solve a test problem, and investigated the influence of various parameters, such as size, void fraction, material, flow rate, and working fluid inlet temperature. They discovered that the working fluid inlet temperature had a greater influence on the wall-heat-losses in all models and that the mass flow rate had the least effect. Arkar and Medved [11] investigated the influence of the PCM thermal properties on the thermal responses of LHTES-containing spheres filled with paraffin. Using a continuous solid phase model, they compared the numerical results with an experiment consisting of 35 rows of spheres and validated them in slow running processes. Erek and Dincer [12] developed a heat transfer coefficient correlation using 120 numerical simulations for an ice TES system. Their conclusions from using a concentric dispersion model emphasized the importance of taking into account the variable heat transfer coefficient for analyzing thermal energy storage (TES) systems. Peng et al. [13] conducted numerical analysis of thermal behavior for LHTES systems by investigating radial heat transfer and wall heat losses. Using the concentric dispersion model, their study showed that charging efficiency could be increased under the following conditions: decrease in
Greek symbols density (kg/m3) / liquid fraction in each capsule
q
Subscript c eff i init in l m o ref s w
characteristic effective inner initial tank inlet liquid melting outer reference solid wall
capsule size and fluid inlet velocity, or increase in the storage height. From a literature review, even though analysis of entire tanks have been extensive, we can see the limitations in their assumptions: the single-phase model has limited accuracy because it does not consider phase change behavior. The continuous solid-phase model also has shortcomings because the temperature differences in the two phases are ignored. The other concentric dispersion model, which takes temperature differences into account, improved the accuracy to some extent; however, it does not include the natural convective effect, and thus retains a limit on its accuracy. Furthermore, these models do not reflect the differences in thermal behavior of capsules regarding their position; thus we cannot estimate the true charging and discharging performance because the melting/solidification will be different depending on the capsule position: near the tank top or tank bottom, or near the tank center or tank edge. In this study, an efficient, accurate approach is proposed for estimating the melting/solidification behaviors in a full tank. By introducing effective thermal conductivity that reflects the effect of natural convection in the molten PCM, we are able to investigate the performance of the full-scale tank within a much shorter calculation time. First, we validated the proposed model for a single capsule by comparing experimental results. Second, we checked whether the model could be applied over much wider conditions in relation to different capsule sizes, wall temperatures, and initial temperatures. In addition, we created 12-layer capsules in a vertical column to show that the present model is superior to the previous effective thermal conductivity model. Finally, we accomplish our goal of analyzing a whole tank containing 1692 spherical capsules (9 9 20 array).
2. Numerical method 2.1. Mathematical model Hirata and Nishida [14] proposed an effective thermal conductivity model. The constrained melting model is shown in Fig. 1(a). This model ignores the movement of solid PCM due to the density difference between liquid PCM and solid PCM. Natural convection plays an important role and accelerates melting in the upper part of solid PCM. However, this process is the main cause of the tremendous computation time required in simulation of melting and solidification. Thus, a combination of the effective thermal
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conductivity model exclusion of convection in molten PCM (by treating the molten PCM as solid-like), with a way to properly include the effect of the accelerated melting, would be of great benefit for full tank simulation. As shown in Fig. 1, the shape of the phase interface is different from the real situation. However, considering that the molten fraction is predicted precisely in time for each capsule, the detailed phase interface of each capsule may not be important when the main interest is to estimate the charging and discharging performance of the full tank. Holman [15] proposed using the effective thermal conductivity model in Eq. (1) for internal natural convection in a rectangular domain.
m keff L ¼ CðGrPrÞn k d
ð1Þ
The parameters C, m, and n are constants and d is the horizontal distance of the domain. The effective thermal conductivity model used in the phase-change analysis employed an analogous formula that reflects the natural convection phenomenon that occurs in molten PCM.
keff ¼ CRaðr o Þm kl
ð2Þ
In Eq. (2), use of the Rayleigh number incorporated the effect of natural convection in molten PCM, and was expressed by the temperature difference T in T m and characteristic length of the capsule radius r o . Some of the drawbacks in the previous effective thermal conductivity model were: (1) the capsule radius was used for the characteristic length, which is constant and cannot reflect the time-wise variation of the strength of natural convection in molten PCM, and (2) the tank inlet temperature T in rather than the capsule wall temperature of each capsule in the tank, was used for the temperature difference. Thus, the previous approach of effective thermal conductivity keff cannot estimate the separate phase-change phenomena of each capsule in a tank. Instead, we propose the same form of keff but express it as a function of the molten fraction, which reflects the time-wise variation of the strength of the natural convection in molten PCM. This enables analysis of the different thermal behavior of each capsule in relation to the different positions in a tank.
K eff ¼ CRam l kl
ð3Þ
The Rayleigh number in Eq. (3) is expressed in terms of the characteristic length l ¼ ro r i and T w T m . The different surface temperature of each capsule is calculated by averaging the angular surface temperature of each capsule for every time step. The diameters of the solid bodies are also updated at each time step, after determining the phase interface and the amount of molten PCM using Eqs. (4) and (5).
R qf dV /¼ R l qdV ðl /Þ
1111
ð4Þ
4 3 4 3 pr ¼ pr 3 o 3 i
ð5Þ
In addition to the aforementioned limitations in the previous keff , the previous effective thermal conductivity model required a new fitting process to obtain keff for each different condition, such as changes in capsule size, wall temperature, or initial temperature in the tank. Practically, the effective thermal conductivity method has been adopted to facilitate calculation; however, in case of estimating the performance of whole tank, the previous effective thermal conductivity model will become impractical because repetitive processing is required to find the proper fitting parameters. Moreover, there is no reference for these fitting processes. If an experimental result exists for the fitting reference, indeed, there is no need to conduct numerical analysis. Meanwhile, with the proposed keff model, once fitting parameters are found for a specific condition, the parameters can be used to analyze capsules under different conditions to a fair extent. 2.2. Governing equation The enthalpy method (Swaminathan and Voller [16]) is incorporated into the commercial software STAR-CCM+ v9.04. The solution for mass and momentum conservation is not required in the effective thermal conductivity method. The energy conservation equation, excluding the convection term, is as follows.
q
@H @ @T @ @T þ ¼ keff keff @t @x @t @y @y
ð6Þ
Eq. (6) only includes conduction terms, which reduces calculation time and makes the use of a rough mesh available. Here, H ðH ¼ h þ DHÞ is expressed as the sum of sensible enthalpy RT (h ¼ href þ T ref cp dT) and latent heat (DH ¼ f l L), where f l is the liquid fraction of PCM defined as follows:
fl ¼
8 > < 0 > :
if
T < Ts
TT s T l T s
if
Ts < T < Tl
1
if
Tl < T
ð7Þ
3. Results and discussion 3.1. Validation by experimental results VS Natural Convection Model VS keff Model The proposed keff model was compared with the experimental results from Tan [7] as well as the constrained melting model
Fig. 1. (a) Constrained melting model with natural convection, (b) effective thermal conductivity model, and (c) full scale thermal energy storage filled with 1692 (9 9 20) spherical capsules.
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including natural convection of molten PCM. The latter was used as a reference when there was no experimental result. We also used the result of the previous keff for comparison. The validation was conducted in a spherical capsule of diameter 101.66 mm. We used n-octadecane as the PCM and its properties are shown in Table 1. The initial temperature was 1 °C below the melting temperature. As it was being isothermally heated (40 °C), the solid PCM started melting when it arrived at its melting temperature of 28.2 °C. The results of the time-wise variation in molten fraction mf are shown in Fig. 2. The proposed keff model shows close agreement with both the experimental results and the constrained melting model results over the whole time of the melting process. However, the previous keff model shows a significant deviation from the real physics in the intermediate stages of melting, even though it also agrees with the experimental results at mf ¼ 1. Note that it is not surprising to find close agreement at mf ¼ 1 because the parameters C and m in Eq. (2) were fitted to match the result at mf ¼ 1. The previous keff model assumes the same Ra number during the whole melting period (i.e., the same strength of natural convection); the deviation in the intermediate stages of melting is an inherent limitation of the model. In order to check the range of application of the proposed keff model, we conducted simulations for other conditions with the same fitting parameters obtained from specific conditions for different capsule wall temperatures, variations in capsule size and initial temperature. Because there were no experimental results for the conditions considered here, the results from the numerical model including natural convection of molten PCM, was used as a reference for comparison. We compared the results with capsule wall temperatures of 35, 40, and 45 °C. The time-wise variations in the molten fraction are shown in Fig. 3. The proposed keff model closely agrees with the reference result, regardless of the capsule wall temperature. The results of the different capsule diameters (40, 60, 80, and 101.66 mm) are shown in Fig. 4. The proposed keff model shows close agreement with the reference result, especially for small capsule sizes, because conduction is stronger than natural convection in capsule of smaller diameter. Even for large capsules, for example, do = 101.66 mm, the difference between the results of the proposed keff model and the reference is acceptable. Considering that capsule diameters are less than 100 mm in conventional applications of latent thermal energy storage systems, the proposed keff model could be used over a fair range of capsule diameters. The results with different initial temperature can be observed in Fig. 5. The initial temperatures are 1, 10, and 20 °C lower than the melting temperature. Again, the proposed keff model closely agrees with the reference result, regardless of the initial temperature. The effect of the initial temperature turned out to be very small. This is because the sensible heat, even for 20 °C, has less influence than that of the latent heat in the melting process. Contrary to the previous keff model, we have confirmed that the proposed keff model can be used for other melting conditions with
Fig. 2. Molten fraction comparison of the proposed keff model with the previous keff model, constrained melting model including natural convection in the molten PCM, and the experimental data from Tan [7].
Fig. 3. Molten fraction comparison between the proposed keff model and the referential numerical model with different capsule wall temperatures.
the same fitting parameters obtained from a specific condition, which will extend its application.
3.2. One column with twelve layers of capsules Table 1 Properties of n-octadecane. n-octadecane Melting temperature [°C] Latent heat [J/kg] Density [kg/m3] Specific heat [J/kg-K] Thermal conductivity [W/m-K] Viscosity [Pa-s] Thermal expansion coefficient [1/K] Kinematic viscosity [m2/s] Thermal diffusivity [m2/s]
28.2 243,500 772.0 2330.0 0.1505 0.00386 0.00091 5.0 106 8.3669 108
Before the application of a full scale tank, a feasibility study was conducted for a smaller system with one column of 12 layers of capsules. Contrary to the simulation of a full scale tank including natural convection of molten PCM, the simulation of this smaller system of one column of 12 layers of capsules is possible within an acceptable computation time. This enabled checking of the validity of the proposed keff model and also allowed examination of the ability of the model to capture the differences in melting behavior depending on the capsule positions. Twelve capsules were vertically arranged to have a simple cubic form. The capsule diameter was 100 mm. Three models were
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Fig. 4. Molten fraction comparison between the proposed keff model and the referential numerical model with different capsule diameters.
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Fig. 6. The average molten fraction comparison of 12 layers of capsules in three models of natural convection, the proposed keff model, and the previous keff model.
for the proposed model was about 50 times more efficient than that of the constrained melting model. This was achieved by adopting a rough mesh, a large time-step, and only solving the conduction equation. The proposed model can be used to estimate structures composed of hundreds of capsules thanks to the shorter computation time, allowing for the entire tank to be simulated. 3.3. Thermal energy storage tank with array of 9920 capsules A tank containing an array of 9 9 20 capsules in a simple cubic arrangement, as shown in Fig. 8, was the tank modeled. Thanks to its symmetric nature, only one fourth of the entire system was used as the domain of interest. The heat transfer fluid (HTF) comes in at 5.0 104 m/s, from top to bottom. The computational configuration and conditions are from Arkar and Medved [11], Nallusamy et al. [17], and Cho and Choi [18]. The molten fraction of the previous keff model and the proposed keff model are shown in Fig. 9. Overall, the melting is faster in the
Fig. 5. Molten fraction comparison between the proposed keff model and the referential numerical model with different initial temperatures.
compared: the constrained melting model including natural convection, the previous keff model, and the proposed keff model. Because there was no experimental data, the constrained melting model including natural convection of molten PCM, was used as a reference. The time-wise variation in molten fraction is shown in Fig. 6. The results demonstrate that the proposed keff model matches well with the behavior of the constrained melting model, while the previous keff model only corresponds when conditions are near mf ¼ 1. The discrepancy in the intermediate melting and solidification processes shows the limitation of the previous model because full charging/discharging is not normal in real applications. The merit of the proposed model was made clearer by comparing the melting behavior of capsules in different positions. We examined the molten fraction of capsules in the 2nd, 8th, and 12th layer, as shown in Fig. 7. The proposed keff model precisely predicted the melting behavior. In addition, the computation time
Fig. 7. Comparison of three models in different capsule positions.
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Fig. 8. Geometry of latent heat thermal storage tank with an array of 9 9 20 capsules.
Fig. 9. Molten fraction comparison between the proposed model and the previous model for a full tank with an array of 9 9 20 capsules.
previous keff model than in the proposed model. A constrained melting model with natural convection is not feasible due to the lack of memory capacity and the tremendous amounts of time required. Nevertheless, we would expect that the proposed keff model would show melting behavior very close to actual conditions, as was shown from the results for the 12 layers of capsules. The fitting parameters used for the each keff model, Eqs. (2) and (3), respectively, were obtained from the one-capsule model because there were no experimental or numerical references for a full scale tank. To examine the spatially variable melting behavior, we sampled positions at the top (20th layer), middle (10th layer), and bottom (1st layer); as well as at the center, middle, and edge, as shown in Fig. 8. On the top layer of the system, PCM melted almost uniformly in the center, middle, and edge positions in both models. This is because the flow inlet is located in the top layer, and the brine temperature is uniform regardless of the capsule positions. The results of the middle and bottom layers are shown in Fig. 10 (b) and (c), respectively. The lower the layer is,
Fig. 10. Time-wise variation of the molten fraction depending on the position in a tank with an array of 9 9 20 capsules: (a) at the top (20th layer), (b) in the middle (10th layer), and (c) at the bottom (1st layer).
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Fig. 11. Effective thermal conductivity behavior of the previous keff model.
Fig. 12. Effective thermal conductivity behavior of the proposed keff model.
the more differently the melting progresses at the center, middle, and edge. This is because the brine temperature changes along the flow direction, and the capsule temperature is different depending on their locations. Contrary to the previous keff model, the proposed keff model properly incorporated the effect of different capsule wall temperatures. This resulted in different melting speeds and different strengths of natural convection of molten PCM in each capsule. Fig. 11 shows the behavior of keff in the previous model at the cross section of Fig. 8. The previous model shows the same thermal conductivity of liquid PCM for all capsules in a tank. It cannot reflect the effect of the natural convection that becomes intense as the melting progresses. Thus, this model cannot predict the proper performance of such systems. To deal with this problem, we present an enhanced effective thermal conductivity model which is a function of the molten fraction. For this reason, it can
properly incorporate the increased strength of natural convection in the molten PCM as the melting progresses. Fig. 12 shows the different effective thermal conductivities in each capsule in a tank, reflecting the different melting speeds according to different capsule locations. 4. Conclusions In this study, a new effective thermal conductivity model was proposed and used to investigate numerically the whole tank of a latent heat thermal energy storage (LHTES) system filled with phase-change materials (PCM) in an array of 9 9 20 spherical capsules. Most of previous approaches assumed the charging/discharging performance of the entire tank would be the same as that of one capsule inside the tank due to the tremendous computation time required to simulate the natural convection in the molten
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PCM. There were some studies of whole tanks, but due to oversimplified assumptions the detailed physics of phase change was not captured. The main innovations in the proposed method are (1) Tremendous reduction of calculation time was achieved by introducing effective thermal conductivity, for example, for the case of one column, with 12 layers of capsules; the computational time of the proposed model was about 50 times shorter than that of a rigorous computational model (i.e., the constrained melting model). (2) Use of the Rayleigh number incorporated the effect of natural convection in the molten PCM. (3) The Rayleigh number (i.e., the strength of natural convection in the molten PCM), changed as the melting proceeded and the volume of the molten PCM increased. Thus, dissimilar melting behavior depending on the different positions of the capsules in the tank could be observed. (4) Close agreement with both experimental and rigorous computational results was observed over the whole time of the melting process. The previous keff model showed a significant deviation from the real physics in the intermediate stages of melting. The discrepancy in the intermediate melting/solidification process would be a big limitation because full charging/discharging is not normal in real applications (5) The same keff can be used for other melting conditions: different capsule wall temperatures, variations in capsule size, and differences in initial temperature, which made it possible to apply the same keff to the whole tank. However, the previous keff requires a new keff to find the proper fitting parameters. Moreover, there is no reference for these fitting processes, and if experimental results existed, there would be no need to conduct numerical analysis. Conflict of interest The authors declared that there is no conflict of interest. Acknowledgement This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (No. 2017R1D1A1B05030422).
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