# Compact Model of Latent Heat Thermal Storage for Its Integration in Multi-Energy Systems

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## Abstract

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## Featured Application

**Dynamic simulation of Latent Heat Thermal Storage at system level; optimization of control strategies in multi-energy systems; investigation of Demand Side Management strategies.**

## Abstract

## 1. Introduction

## 2. Materials and Methods

#### 2.1. D Detailed Model

_{W}) and LHTS aspect ratio (A) can be used to determine the relevance of the wall heat flux due to natural convection in the liquid PCM. This correlation asserts that the flow is dominated by natural convection only if $\frac{R{a}_{W}}{A}\ge 500$. Regarding the LHTS configuration proposed in this analysis, the above-mentioned correlation yields $\frac{R{a}_{W}}{A}=374$. Therefore, the heat flux due to natural convection is likely to be negligible both during charging and discharging phases if compared to conduction. Hence, the computational domain is reduced to the horizontal cross-section plane. Here, the temperature gradient is expected to be larger compared to the axial direction [19], and heat conduction is dominant. The numerical domain is also further reduced, taking advantage of the symmetry created by the fins. Then, the physical problem is modeled using the finite volume method implemented in the commercial code Ansys Fluent.

_{3}in Figure 1), while symmetry is considered on Γ

_{2}. The heat transfer with the water in the pipes is modeled with an implicit Robin boundary condition on Γ

_{1}, as shown by Equations (2)–(4). ${T}_{ref}$ represents the average temperature between the water inlet (${T}_{in}$) and outlet (${T}_{out}$) temperatures. However, ${T}_{out}$ depends on the efficacy of the heat transfer between the flowing HTF and the rest of the LHTS (i.e., the ensemble of fins and PCM). Therefore, if ${T}_{out}$ is expressed as a function of the heat flux at the wall ($\dot{{\mathrm{q}}_{\mathrm{wall}}}$), Fluent solves Equation (2) iteratively. The heat transfer coefficient ${h}_{conv}$ is computed using the Dittus–Boelter correlation (5) and the Nusselt number (6). In (5), n = 0.4 for discharge or n = 0.3 for charge. This correlation was preferred over the implicit one proposed by Sieder and Tate due to its simplicity and acceptable uncertainty. Substantially, the simulated domain represents an average cross-section of the whole LHTS unit.

^{−4}m. The selected mesh proved to be sufficiently fine not to influence the results. A Second Order Upwind scheme is used for the spatial discretization of the energy equation together with the Least Squares Cell Based method for gradient calculation. The convergence is reached when residuals are lower than 10

^{−8}for the energy equation. On the other hand, the transient nature of the problem is approached with a Second Order Implicit Euler method, with a time-step of 1 s. The selected value proved to be sufficiently fine not to influence the results.

#### 2.2. D Compact Model

_{n}is the normalized state of charge, and it is equal to $\left(\frac{SOC}{SO{C}_{0}}\right)$ during discharge and $\left(\frac{SOC-SO{C}_{0}}{1-SO{C}_{0}}\right)$ during charge, SOC

_{0}is the initial state of charge, $\dot{{Q}_{dis}}$ is the thermal power released by each tube during discharge, $\dot{{Q}_{chrg}}$ is the thermal power requested by each tube during the charging phase, and $\dot{{Q}_{idle}}$ is the thermal power when the LHTS is not operated. Therefore, depending on the operational phase of the LHTS, the value of the thermal power released or requested by each LHTS tube can be expressed as follows (11):

## 3. Case Study

#### 3.1. LHTS System Description

#### 3.2. Model Application to Thermal Energy Networks

#### 3.3. Model Application to Multi-Energy Systems

## 4. Results and Discussion

#### 4.1. Comparison between 0D and 2D Models

_{0}= 1 and SOC

_{0}= 0 (Figure 5 and Figure 6). The former fitting curve is characterized by a correlation coefficient R

^{2}= 0.995, while the latter has R

^{2}= 0.992. Overall, the minimum value is R

^{2}= 0.939, while the maximum standard deviation is 0.138 kW.

_{0}). However, SOC

_{0}affects the results only in case a new charging or discharging phase starts. For instance, if there are two consecutive charging or discharging phases (only interrupted by an idle period), the value of SOC

_{0}should not be updated at the beginning of the second charge/discharge simulation. Instead, the previous value must be retained. This issue is due to the fact that when a partial charge is performed, only the PCM close to the fins becomes liquid, while the PCM far from the fins remains solid. If the subsequent operational phase is a discharge, a peak of thermal power occurs at the beginning of this latter process because the first layer of PCM encountered in the heat propagation is liquid (Figure 5).

_{0}is much smaller than 1 (i.e., when a discharge is preceded by a very short charging phase). However, it is rather unlikely to discharge a thermal storage starting from such a low content of energy. Similarly, the 0D model for the LHTS charge thermal power (9) is less accurate when the SOC

_{0}is much larger than 0 (i.e., when a charge is preceded by a very short discharge phase). As a matter of fact, the 2D curves in Figure 5 are both shifted and contracted with respect to one other, while the 2D curves in Figure 6 are mainly shifted. This dissimilarity might be due to the fact that the HTF inlet temperature during the charging phase (75 °C) is much closer to the PCM average phase change temperature (70 °C) compared to the HTF inlet temperature during discharge (48 °C). In fact, as indicated by [24], the HTF inlet temperature affects the LHTS thermal power.

#### 4.2. Distributed LHTS in DH Networks

- The charging phase of each LHTS unit lasts 3 h; thus, the whole LHTS bundle is charged between 1.20 and 5.40 am (due to the imposed delay between the activation of each unit);
- The discharging phase of each LHTS unit lasts 1.5 h; thus, the whole LHTS bundle is discharged between 6.30 am and 9.20 am (due to the imposed delay between the activation of each unit).

^{3}, while a sensible water storage tank would require 7 m

^{3}for the same energy content (assuming also the same temperature difference 75–48 °C). Therefore, this feature constitutes a great advantage when the thermal energy storage is located in the small technical rooms available in the buildings.

#### 4.3. LHTS in Multi-Energy Systems

## 5. Conclusions

^{2}) is 0.939, and the maximum standard deviation is 0.138 kW.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Three-dimensional (3D) Latent Heat Thermal Storage (LHTS) shell-and-tube concept; (

**b**) 2D computational domain.

**Figure 5.**Latent Heat Thermal Storage (LHTS) discharge starting from different initial states of charge (after a partial/complete charge).

**Figure 6.**LHTS charge starting from different initial states of charge (after a partial/complete discharge).

**Figure 10.**Production/consumption evolution for the energy vectors: (

**a**) hot production; (

**b**) hot consumption.

PCM Physical Property | Value |
---|---|

Density (ρ) [kg/m^{3}] | 880 (constant) |

Specific Heat [J/(kg*K)] | 2000 |

Thermal Conductivity (k) [W/(m*K)] | 0.2 |

Latent Heat [kJ/kg] | 214 |

Solidus Temperature [°C] | 69 |

Liquidus Temperature [°C] | 71 |

Coefficient | Discharge Phase | Charge Phase |
---|---|---|

A | 0.1752 [kW] | 3.353 [kW] |

B | 3.112 [-] | −45.93 [-] |

C | −0.2078 [kW] | 1.337 [kW] |

D | −0.9345 [-] | −3.606 [-] |

K | 1.758 [kW] | 0.3296 [kW] |

E | 0.5518 [-] | 0.5908 [-] |

F | 0.3442 [-] | 0.4197 [-] |

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**MDPI and ACS Style**

Colangelo, A.; Guelpa, E.; Lanzini, A.; Mancò, G.; Verda, V. Compact Model of Latent Heat Thermal Storage for Its Integration in Multi-Energy Systems. *Appl. Sci.* **2020**, *10*, 8970.
https://doi.org/10.3390/app10248970

**AMA Style**

Colangelo A, Guelpa E, Lanzini A, Mancò G, Verda V. Compact Model of Latent Heat Thermal Storage for Its Integration in Multi-Energy Systems. *Applied Sciences*. 2020; 10(24):8970.
https://doi.org/10.3390/app10248970

**Chicago/Turabian Style**

Colangelo, Alessandro, Elisa Guelpa, Andrea Lanzini, Giulia Mancò, and Vittorio Verda. 2020. "Compact Model of Latent Heat Thermal Storage for Its Integration in Multi-Energy Systems" *Applied Sciences* 10, no. 24: 8970.
https://doi.org/10.3390/app10248970