# Numerical Simulation and Optimization of a Phase-Change Energy Storage Box in a Modular Mobile Thermal Energy Supply System

^{*}

## Abstract

**:**

## 1. Introduction

_{2}nanocomposite-based [6] energy storage materials. In M-TES systems, two types of heat storage vessels are commonly used, each catering to distinct heat transfer mechanisms. The first type involves the direct contact vessel [7], wherein the PCM is directly mixed with the heat transfer medium, typically heat transfer oil (HTO). Since PCMs and HTO are immiscible and possess different densities (usually with PCM being denser than HTO), they can be easily separated after mixing. The other container type is the indirect contact vessel, which utilizes a built-in heat exchanger [8,9] to facilitate heat transfer between the PCM and the heat transfer medium. Notably, the indirect contact container allows for enhanced heat transfer, as demonstrated by Kang et al. [10], who proposed the vertical mounting of Y-shaped fins to improve the heat charging efficiency of the M-TES container. On the other hand, the direct-contact structure reduces container weight and lowers transportation costs. Economic evaluations of M-TES primarily focus on various factors, such as the type of heat source, the distance from the heat source, the heat-carrying medium, the heat storage capacity, the number of daily cycles, and the charging and discharging time. Enhancing the thermal charging and discharging processes is crucial to achieving a low operating cost for M-TES, as highlighted by research studies [11]. For instance, Krönauer et al. [12] designed, proposed, and experimentally validated an M-TES system using a tubular heat exchanger in a filled tank with sodium acetate trihydrate as the PCM, aimed at recovering industrial waste heat. Similarly, Kuta et al. [13] constructed an experimental platform to verify the utilization of geothermal energy as a power source for M-TES. Nekoonam et al. [14] performed numerical simulations on a system comprising a solar collector and a PCM co-storage unit, showcasing stable system performance and improved heat storage efficiency between 15 °C and 90 °C. In another study, Elfeky et al. [15] conducted simulations with different phase-change materials and spherical capsules to optimize the performance of multilayer phase-change materials in the thermocline tank of a concentrating solar power plant. Additionally, Han et al. [16] prepared and validated the stability of chloride salt/nanoparticle composite phase-change materials (CPCMs) for high-temperature thermal energy storage. Fragnito et al. [17] explored the performance of heat exchangers with biological phase-change materials in chilled thermal energy systems through research experiments and numerical modelling, revealing that the design limits the thermal storage potential of the phase-change materials. Bianco et al. [18] conducted a numerical analysis of latent heat thermal energy storage based on microencapsulated phase-change materials (MEPCM) to enhance the efficiency of a chilled water system. They employed cylindrical MEPCM modules within a commercial water tank to cool a 150-square-meter residential space. Pourhemmati et al. [19] improved heat transfer efficiency and prevented boundary layer merging by adjusting the thickness of vertical finned heat sinks. COMSOL simulations indicated that increasing the number of fins could reduce thermal resistance by up to 45.5%.

## 2. Validation of Model Validity

#### 2.1. Physical Model

#### 2.2. Phase-Change Heat Transfer Mathematical Model

_{s}and c

_{l}being constants.

_{l}are the saturation enthalpies in the solid and liquid phase regions and T

_{m}is the phase-transition temperature of the PCM.

#### 2.3. Governing Equation

_{p}is the constant-pressure specific heat capacity of the material; S is the source term of the energy equation; μ is the kinetic viscosity of the material; P is the pressure; and S

_{u}and S

_{w}are the x-direction momentum source term and z-direction momentum source term, respectively. The source term is actually a damping term, which will have a large value as the melt fraction tends to 0, and disappears from the momentum equation as the melt fraction tends to 1.

_{i}, t

_{m}, and t

_{f}are the initial, final, and melting temperatures, respectively; m is the mass of the PCM; C

_{ps}and C

_{pl}are the specific heats of the solid and liquid phases; and ∆q is the latent heat of phase transition.

#### 2.4. Melting/Solidification Modelling

_{l}, is shown in Equation (11) [22].

_{l}< 1, at this time, the PCM can be regarded as a porous medium, and the porous part is the percentage of the total volume accounted for by the liquid phase. H

_{s}and H

_{l}: are the enthalpies of the solid and liquid phases of the PCM, respectively, where the viscous zone constant is taken to be 1 × 10

^{5}. For sufficiently fine grids, the numerical prediction of the isothermal phase-transition problem is independent of the permeability coefficient [23].

#### 2.5. Model-Related Assumptions

- (1)
- The heat storage material is isotropic;
- (2)
- The heat loss of the heat storage box is neglected;
- (3)
- Compared with the model size, the thickness of the copper tubes inside the heat accumulator is negligible and has little effect on heat transfer, so the tube wall thickness is ignored in the calculations;
- (4)
- The density of the heat storage material is considered to vary with temperature only in the buoyancy force, while the other parameters are used to vary linearly, i.e., the Boussinesq assumption is used;
- (5)
- The radiative heat exchange of the heat storage material is ignored and only the heat exchange due to convection and conduction is calculated.

#### 2.6. Calculation Conditions and Calculation Methods

#### 2.7. Mesh Segmentation and Irrelevance Verification

#### 2.8. Model Validation

#### 2.9. Analysis of Phase-Transition Processes

## 3. Modular Design

## 4. Modeling and Calculation Methods

#### 4.1. Model Building

#### 4.2. Calculation Method

## 5. Simulation Results and Discussion

#### 5.1. Simulation Analysis of Different Finning Conditions

#### 5.2. Comparative Analysis of Composite Materials

## 6. Conclusions

- (1)
- The proposed new mobile heating system thermal storage box addresses the issue of uneven temperature distribution in traditional thermal storage boxes. The modular design optimizes the arrangement of heat accumulators, reducing the problem of uncoordinated heat storage in the length direction. The modular thermal storage box can be easily installed and uninstalled using a crane, making heat distribution more flexible and efficient.
- (2)
- The original model was optimized by introducing fins and subjected to numerical simulations to obtain model parameters under various operating conditions. The optimal solution resulted in a 30.7% increase in heat storage over 62 h and an 11.2% reduction in exothermic heat release over 48 h when compared to the no-fin condition. In contrast to the significant improvement observed between condition 1 and condition 2, the enhancement seen in condition 3 over condition 2 is less pronounced, accounting for only about half of the former’s improvement. It is noteworthy that continuing to increase the number of fins or their height further diminishes the heat transfer enhancement. This effect is compounded by the fact that excessive fins complicate the design, disrupt the natural flow of the melted phase-change material, increase the nonproductive weight of the container, and reduce the overall efficiency of the box. All these factors contribute to increased operational costs.
- (3)
- The combination of expanded graphite and erythritol in different proportions as a composite heat storage material shows promising results. By adding 12.3% expanded graphite, the complete heat storage time is reduced from 62 h to 26 h, achieving a 50% improvement in heat charging efficiency.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Guo, S.; Liu, Q.; Zhao, J.; Jin, G.; Wu, W.; Yan, J.; Li, H.; Jin, H. Mobilized thermal energy storage: Materials, containers and economic evaluation. Energy Convers. Manag.
**2018**, 177, 315–329. [Google Scholar] [CrossRef] - Diarce, G.; Gandarias, I.; Campos-Celador, A.; García-Romero, A.; Griesser, U. Eutectic mixtures of sugar alcohols for thermal energy storage in the 50–90 C temperature range. Sol. Energy Mater. Sol. Cells
**2015**, 134, 215–226. [Google Scholar] [CrossRef] - Höhlein, S.; König-Haagen, A.; Brüggemann, D. Thermophysical characterization of MgCl
_{2}· 6H_{2}O, xylitol and erythritol as phase change materials (PCM) for latent heat thermal energy storage (LHTES). Materials**2017**, 10, 444. [Google Scholar] [CrossRef] - Peiró, G.; Gasia, J.; Miró, L.; Cabeza, L.F. Experimental evaluation at pilot plant scale of multiple PCMs (cascaded) vs. single PCM configuration for thermal energy storage. Renew. Energy
**2015**, 83, 729–736. [Google Scholar] [CrossRef] - Kumar, Y.A.; Kim, H.-J. Effect of time on a hierarchical corn skeleton-like composite of CoO@ ZnO as capacitive electrode material for high specific performance supercapacitors. Energies
**2018**, 11, 3285. [Google Scholar] [CrossRef] - Moniruzzaman, M.; Anil Kumar, Y.; Pallavolu, M.R.; Arbi, H.M.; Alzahmi, S.; Obaidat, I.M. Two-dimensional core-shell structure of cobalt-doped@ MnO
_{2}nanosheets grown on nickel foam as a binder-free battery-type electrode for supercapacitor application. Nanomaterials**2022**, 12, 3187. [Google Scholar] [CrossRef] - Kaizawa, A.; Kamano, H.; Kawai, A.; Jozuka, T.; Senda, T.; Maruoka, N.; Akiyama, T. Thermal and flow behaviors in heat transportation container using phase change material. Energy Convers. Manag.
**2008**, 49, 698–706. [Google Scholar] [CrossRef] - Wang, W.; Guo, S.; Li, H.; Yan, J.; Zhao, J.; Li, X.; Ding, J. Experimental study on the direct/indirect contact energy storage container in mobilized thermal energy system (M-TES). Appl. Energy
**2014**, 119, 181–189. [Google Scholar] [CrossRef] - Wang, W.; Li, H.; Guo, S.; He, S.; Ding, J.; Yan, J.; Yang, J. Numerical simulation study on discharging process of the direct-contact phase change energy storage system. Appl. Energy
**2015**, 150, 61–68. [Google Scholar] [CrossRef] - Kang, Z.; Zhou, W.; Qiu, K.; Wang, C.; Qin, Z.; Zhang, B.; Yao, Q. Numerical Simulation of an Indirect Contact Mobilized Thermal Energy Storage Container with Different Tube Bundle Layout and Fin Structure. Sustainability
**2023**, 15, 5511. [Google Scholar] [CrossRef] - Li, H.; Wang, W.; Yan, J.; Dahlquist, E. Economic assessment of the mobilized thermal energy storage (M-TES) system for distributed heat supply. Appl. Energy
**2013**, 104, 178–186. [Google Scholar] [CrossRef] - Krönauer, A.; Lävemann, E.; Brückner, S.; Hauer, A. Mobile sorption heat storage in industrial waste heat recovery. Energy Procedia
**2015**, 73, 272–280. [Google Scholar] [CrossRef] - Kuta, M. Mobilized thermal energy storage (M-TES) system design for cooperation with geothermal energy sources. Appl. Energy
**2023**, 332, 120567. [Google Scholar] [CrossRef] - Nekoonam, S.; Roshandel, R. Modeling and optimization of a multiple (cascading) phase change material solar storage system. Therm. Sci. Eng. Prog.
**2021**, 23, 100873. [Google Scholar] [CrossRef] - Elfeky, K.; Li, X.; Ahmed, N.; Lu, L.; Wang, Q. Optimization of thermal performance in thermocline tank thermal energy storage system with the multilayered PCM (s) for CSP tower plants. Appl. Energy
**2019**, 243, 175–190. [Google Scholar] [CrossRef] - Han, D.; Lougou, B.G.; Xu, Y.; Shuai, Y.; Huang, X. Thermal properties characterization of chloride salts/nanoparticles composite phase change material for high-temperature thermal energy storage. Appl. Energy
**2020**, 264, 114674. [Google Scholar] [CrossRef] - Fragnito, A.; Bianco, N.; Iasiello, M.; Mauro, G.M.; Mongibello, L. Experimental and numerical analysis of a phase change material-based shell-and-tube heat exchanger for cold thermal energy storage. J. Energy Storage
**2022**, 56, 105975. [Google Scholar] [CrossRef] - Bianco, N.; Caliano, M.; Fragnito, A.; Iasiello, M.; Mauro, G.M.; Mongibello, L. Thermal analysis of micro-encapsulated phase change material (MEPCM)-based units integrated into a commercial water tank for cold thermal energy storage. Energy
**2023**, 266, 126479. [Google Scholar] [CrossRef] - Pourhemmati, S.; Hossainpour, S. Thermal improvement of the vertical plate-fin heat sink by variable fin thickness pattern and utilizing phase change material: A numerical investigation. J. Energy Storage
**2023**, 59, 106480. [Google Scholar] [CrossRef] - Kheirabadi, A.C.; Groulx, D. Simulating phase change heat transfer using comsol and fluent: Effect of the mushy-zone constant. Comput. Therm. Sci. Int. J.
**2015**, 7, 427–440. [Google Scholar] [CrossRef] - Shamsundar, N.; Sparrow, E. Analysis of multidimensional conduction phase change via the enthalpy model. J. Heat Trans. Aug.
**1975**, 97, 333–340. [Google Scholar] [CrossRef] - Huang, R.; Wu, H.; Cheng, P. A new lattice Boltzmann model for solid–liquid phase change. Int. J. Heat Mass Transf.
**2013**, 59, 295–301. [Google Scholar] [CrossRef] - Ebrahimi, A.; Kleijn, C.R.; Richardson, I.M. Sensitivity of numerical predictions to the permeability coefficient in simulations of melting and solidification using the enthalpy-porosity method. Energies
**2019**, 12, 4360. [Google Scholar] [CrossRef] - Guo, S.; Zhao, J.; Li, X.; Wang, W.; Yan, J. Experimental study on waste heat recovery with an indirect mobilized thermal energy storage system. In Proceedings of the International Conference on Applied Energy, Perugia, Italy, 16–18 May 2011. [Google Scholar]
- Guo, S.; Zhao, J.; Wang, W.; Yan, J.; Jin, G.; Zhang, Z.; Gu, J.; Niu, Y. Numerical study of the improvement of an indirect contact mobilized thermal energy storage container. Appl. Energy
**2016**, 161, 476–486. [Google Scholar] [CrossRef] - Kenisarin, M.M. High-temperature phase change materials for thermal energy storage. Renew. Sustain. Energy Rev.
**2010**, 14, 955–970. [Google Scholar] [CrossRef] - Oya, T.; Nomura, T.; Okinaka, N.; Akiyama, T. Phase change composite based on porous nickel and erythritol. Appl. Therm. Eng.
**2012**, 40, 373–377. [Google Scholar] [CrossRef] - Oya, T.; Nomura, T.; Tsubota, M.; Okinaka, N.; Akiyama, T. Thermal conductivity enhancement of erythritol as PCM by using graphite and nickel particles. Appl. Therm. Eng.
**2013**, 61, 825–828. [Google Scholar] [CrossRef] - Gao, L.; Zhao, J.; An, Q.; Zhao, D.; Meng, F.; Liu, X. Experiments on thermal performance of erythritol/expanded graphite in a direct contact thermal energy storage container. Appl. Therm. Eng.
**2017**, 113, 858–866. [Google Scholar] [CrossRef]

**Figure 2.**Heat accumulator main view [10].

**Figure 12.**Two-dimensional model after adding fins. (

**a**) Condition 1; (

**b**) Condition 2; (

**c**) Condition 3.

**Figure 14.**(

**a**) Phase-transition diagrams for 62 h of heat storage (left) and 48 h of exothermic heat release (right) for Condition 1; (

**b**) Phase-transition diagrams for 62 h of heat storage (left) and 48 h of exothermic heat release (right) for Condition 2; (

**c**) Phase-transition diagrams for 62 h of heat storage (left) and 48 h of exothermic heat release (right) for Condition 3.

Thermophysical Property Parameters | Numerical Value |
---|---|

Density $(\mathrm{k}\mathrm{g}/{\mathrm{m}}^{3})$ | 1480 (20 °C), 1300 (140 °C) |

Specific heat $(\mathrm{k}\mathrm{J}/\mathrm{k}\mathrm{g}\xb7\xb0\mathrm{C})$ | 1.35 (20 °C), 2.74 (140 °C) |

Latent heat $(\mathrm{k}\mathrm{J}/\mathrm{k}\mathrm{g})$ | 339 |

Phase-transition temperature $(\xb0\mathrm{C})$ | 117.7 |

Viscosity $(\mathrm{k}\mathrm{g}/\mathrm{m}\xb7\mathrm{s})$ | 0.02895 (20 °C), 0.01602 (140 °C) |

Thermal conductivity $(\mathrm{W}/\mathrm{m}\xb7\xb0\mathrm{C})$ | 0.732 (20 °C), 0.326 (140 °C) |

Condition | Height of Fins (cm) | Distance between Fins (cm) | Number of Fins |
---|---|---|---|

1 | 8 | 10 | 83 |

2 | 12 | 10 | 83 |

3 | 12 | 8 | 106 |

PCM | Melting Enthalpy (kJ/kg) | Phase-Transition Temperature (°C) | Thermal Conductivity (W/m·K) | Density (kg/m^{3}) | Specific Heat (KJ/kg) |
---|---|---|---|---|---|

Erythritol | 339 | 117.7 | 0.326 | 120 | 2.71 [29] |

Expanded Graphite Volume Share | Heat Storage 62 h Liquid Phase Ratio | Complete Heat Storage Time | 48-h Liquid Phase Ratio | Liquid Phase Ratio Reduced by 75% of the Time |
---|---|---|---|---|

7.1 | 1 | 33.5 | 14.2 | 21.25 |

12.3 | 1 | 26 | 8.2 | 14.5 |

15.1 | 1 | 22 | 4.6 | 11.5 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Kang, Z.; Tan, R.; Zhou, W.; Qin, Z.; Liu, S.
Numerical Simulation and Optimization of a Phase-Change Energy Storage Box in a Modular Mobile Thermal Energy Supply System. *Sustainability* **2023**, *15*, 13886.
https://doi.org/10.3390/su151813886

**AMA Style**

Kang Z, Tan R, Zhou W, Qin Z, Liu S.
Numerical Simulation and Optimization of a Phase-Change Energy Storage Box in a Modular Mobile Thermal Energy Supply System. *Sustainability*. 2023; 15(18):13886.
https://doi.org/10.3390/su151813886

**Chicago/Turabian Style**

Kang, Zhangyang, Rufei Tan, Wu Zhou, Zhaolong Qin, and Sen Liu.
2023. "Numerical Simulation and Optimization of a Phase-Change Energy Storage Box in a Modular Mobile Thermal Energy Supply System" *Sustainability* 15, no. 18: 13886.
https://doi.org/10.3390/su151813886