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Development and Validation of a Latent Thermal Energy Storage Model Using Modelica^{ †}

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Experiment Design

#### 3.1. Device Geometry

#### 3.2. Thermophysical Properties

#### 3.3. Transport Parameters

#### 3.4. Heat Transfer Coefficient

## 4. First-Principles Framework

## 5. Modelica Implementation

#### 5.1. Preliminary Modelica Model

#### 5.2. Full Modelica Model

#### 5.2.1. Metal Frame Capacitance and Conductance

#### 5.2.2. Flow Channel Discretization

#### 5.2.3. Working Fluid Residence Time

## 6. Results and Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

${A}_{c}$ | Cross-sectional area of the working fluid flow passage |

${c}_{p,s}$ | Effective specific heat of the storage matrix |

${c}_{p,w}$ | Specific heat of the working fluid |

${h}_{ls}$ | Latent heat of fusion of the PCM in the storage matrix |

${k}_{s}$ | Effective thermal conductivity of the storage matrix |

${k}_{w}$ | Thermal conductivity of the working fluid |

L | Length of the TES device |

$\dot{m}$ | Working fluid mass flow rate per passage |

${s}_{w}$ | Wetted perimeter of the working fluid flow passage |

${T}_{e}$ | Temperature of a discrete element in the storage matrix |

${T}_{liq}$ | Liquidus temperature of the PCM in the storage matrix |

${T}_{m}$ | Melt temperature of the PCM in the storage matrix |

${T}_{max}$ | Maximum temperature encountered in the TES device |

${T}_{min}$ | Minimum temperature encountered in the TES device |

${T}_{sol}$ | Solidus temperature of the PCM in the storage matrix |

${T}_{w}$ | Temperature of a discrete parcel of working fluid |

U | Overall heat transfer coefficient |

${x}_{e}$ | Melt fraction of a discrete element in the storage matrix |

$\Delta {t}^{*}$ | Non-dimensional temporal discretization |

$\Delta \widehat{z}$ | Non-dimensional spatial discretization |

${\epsilon}_{s}$ | Fraction of the storage matrix occupied by PCM |

$\theta $ | Dimensionless temperature of a discrete element in the storage matrix |

${\nu}^{\prime}$ | Storage matrix volume per unit length of the device |

${\rho}_{s}$ | Effective density of the storage matrix |

${\rho}_{w}$ | Density of the working fluid |

$\varphi $ | Dimensionless temperature of a discrete parcel of working fluid |

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**Figure 1.**Photograph and schematic (not to scale) illustrating heat flux (q″) from hot heat transfer fluid (HTF) flow passages into phase change material (PCM) storage sections during melting process.

Geometry Parameter, Variable | Value | Unit |
---|---|---|

Length of TES Device, L | 0.407 | $\mathrm{m}$ |

Wetted perimeter of flow passage, ${s}_{w}$ | $9.42\times {10}^{-2}$ | $\mathrm{m}$ |

Cross sectional area of flow passage, ${A}_{c}$ | $8.97\times {10}^{-5}$ | ${\mathrm{m}}^{2}$ |

Matrix volume per unit flow length, ${\nu}^{\prime}$ | $1.99\times {10}^{-4}$ | ${\mathrm{m}}^{2}$ |

Number of flow passages, ${n}_{w}$ | 5 | |

Number of storage matrix sections, ${n}_{s}$ | 4 | |

Void fraction in storage matrix, ${\epsilon}_{s}$ | 0.729 |

Property Parameter, Variable | Value | Unit |
---|---|---|

Thermal Conductivity, ${k}_{PCM}$ | 0.584 | W/mK |

density, ${\rho}_{PCM}$ | 1500 | $\mathrm{kg}/{\mathrm{m}}^{3}$ |

specific heat, ${c}_{p,PCM}$ | 2910 | $\mathrm{J}/\mathrm{kgK}$ |

latent heat of fusion, ${h}_{ls}$ | 278 | $\mathrm{kJ}/\mathrm{kg}$ |

melt temperature, ${T}_{m}$ | 30 | ${}^{\circ}\mathrm{C}$ |

Property Parameter, Variable | Value | Unit |
---|---|---|

Thermal Conductivity, ${k}_{m}$ | 117 | W/mK |

density, ${\rho}_{m}$ | 2640 | $\mathrm{kg}/{\mathrm{m}}^{3}$ |

specific heat, ${c}_{p,m}$ | 910 | $\mathrm{J}/\mathrm{kgK}$ |

Property Parameter, Variable | Value | Unit |
---|---|---|

Effective Thermal Conductivity, ${k}_{s}$ | 32.1 | W/mK |

effective density, ${\rho}_{s}$ | 1810 | $\mathrm{kg}/{\mathrm{m}}^{3}$ |

effective specific heat, ${c}_{p,s}$ | 2370 | $\mathrm{J}/\mathrm{kgK}$ |

Property Parameter, Variable | Value | Unit |
---|---|---|

Thermal Conductivity, ${k}_{w}$ | 0.608–0.623 | W/mK |

density, ${\rho}_{w}$ | 994–997 | $\mathrm{kg}/{\mathrm{m}}^{3}$ |

specific heat, ${c}_{p,w}$ | 4090–4130 | $\mathrm{J}/\mathrm{kgK}$ |

Operating Parameter, Variable | Value | Unit |
---|---|---|

Mass flow rate for melting, ${\dot{m}}_{ext}$ | $3.44\times {10}^{-3}$ | $\mathrm{kg}/\mathrm{s}$ |

Mass flow rate for freezing, ${\dot{m}}_{char}$ | $3.56\times {10}^{-3}$ | $\mathrm{kg}/\mathrm{s}$ |

Inlet temperature for melting, ${T}_{wi,ext}$ | 36 | ${}^{\circ}\mathrm{C}$ |

Inlet temperature for freezing, ${T}_{wi,char}$ | 26 | ${}^{\circ}\mathrm{C}$ |

Process | $\overline{\mathit{U}}$ Value | Unit |
---|---|---|

Melting | 2980 | $\mathrm{W}/{\mathrm{m}}^{2}\mathrm{K}$ |

Freezing | 2930 | $\mathrm{W}/{\mathrm{m}}^{2}\mathrm{K}$ |

Process | ${\mathbf{N}}_{\mathbf{tu}}$ | ${\mathbf{R}}_{\mathbf{we}}$ | ${\mathbf{St}}_{\mathbf{io}}$ |
---|---|---|---|

Melting | 32.5 | 0.534 | 0.234 |

Freezing | 30.4 | 0.541 | 0.234 |

Energy Balance | ${\mathbf{E}}_{\mathbf{in}}$ | ${\mathbf{E}}_{\mathbf{out}}$ | ${\mathbf{E}}_{\mathbf{diff}}$ | Unit |
---|---|---|---|---|

Experimental | 171.5 | 179.1 | −7.6 | $\mathrm{kJ}$ |

First-principles | 155.0 | 148.4 | 6.6 | $\mathrm{kJ}$ |

Modelica | 142.2 | 142.3 | −0.1 | $\mathrm{kJ}$ |

Process Completed | Value | Unit |
---|---|---|

Experimental heating | 1590 | s |

First-principles heating | 1530 | s |

Modelica heating | 1908 | s |

Experimental cooling | 5370 | s |

First-principles cooling | 4488 | s |

Modelica cooling | 5160 | s |

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**MDPI and ACS Style**

Helmns, D.; Blum, D.H.; Dutton, S.M.; Carey, V.P.
Development and Validation of a Latent Thermal Energy Storage Model Using Modelica. *Energies* **2021**, *14*, 194.
https://doi.org/10.3390/en14010194

**AMA Style**

Helmns D, Blum DH, Dutton SM, Carey VP.
Development and Validation of a Latent Thermal Energy Storage Model Using Modelica. *Energies*. 2021; 14(1):194.
https://doi.org/10.3390/en14010194

**Chicago/Turabian Style**

Helmns, Dre, David H. Blum, Spencer M. Dutton, and Van P. Carey.
2021. "Development and Validation of a Latent Thermal Energy Storage Model Using Modelica" *Energies* 14, no. 1: 194.
https://doi.org/10.3390/en14010194