# Numerical Analysis of Shell-and-Tube Type Latent Thermal Energy Storage Performance with Different Arrangements of Circular Fins

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## Abstract

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## 1. Introduction

## 2. Description of the Simulation Model

#### 2.1. Storage Setup

- Division of the storage into three sections of the same length, with a different fin density in each section—the density ratio ${F}_{3}$ is calculated according to Equation (1):$${F}_{3}=\frac{{N}_{\mathrm{sec}\text{}\mathrm{II}}}{{N}_{\mathrm{sec}\text{}\mathrm{I}}}=\frac{{N}_{\mathrm{sec}\text{}\mathrm{III}}}{{N}_{\mathrm{sec}\text{}\mathrm{II}}}$$
- Division of the storage into two sections of the same length, with a different fin density in each section—the density ratio ${F}_{2}$ is calculated according to Equation (2):$${F}_{2}=\frac{{N}_{\mathrm{sec}\text{}\mathrm{II}}}{{N}_{\mathrm{sec}\text{}\mathrm{I}}}$$
- Linear increase of distances between the fins towards the storage element inlet with a minimum distance $\Delta {x}_{0}$ between two fins—the distance $\Delta {x}_{n}$ between the fins n and n + 1 is calculated in consideration of the factor for linear increasing distances ${F}_{\mathrm{L}}$:$${F}_{L}=\frac{\Delta {x}_{n}-\Delta {x}_{0}}{n\xb7\left(\Delta {x}_{1}-\Delta {x}_{0}\right)}$$
- Exponential increase of distances between the fins towards the storage element inlet—the factor for exponential increasing distances ${F}_{E}$ is calculated according to Equation (4):$${F}_{E}=\frac{\Delta {x}_{n+1}-\Delta {x}_{0}}{\Delta {x}_{n}-\Delta {x}_{0}}$$
- Homogeneous arrangement

#### 2.2. Numerical Model

- Neglect of convectional effects in the liquid PCM;
- Neglect of the temperature dependency of material properties within one phase;
- Application of an enthalpy method with apparent heat capacity;
- Integration of the phase change enthalpy according to Rösler and Brüggemann [30] by applying an apparent heat capacity.$${c}_{app}={c}_{sen}+{c}_{L}$$$${c}_{L}=4L\frac{\mathrm{exp}\left(-\left\{{\left[\frac{4\left(T-{T}_{m}\right)}{{T}_{L,l}-{T}_{L,s}}\right]}^{2}\right\}\right)}{\left({T}_{L,l}-{T}_{L,s}\right)\xb7\sqrt{\mathsf{\pi}}}$$$$\mathsf{\rho}{c}_{\mathrm{eff}}\frac{\partial T}{\partial t}=\left(\frac{1}{r}\right)\frac{\partial}{\partial r}\left(\mathsf{\lambda}r\frac{\partial T}{\partial r}\right)+\frac{\partial}{\partial x}\left(\mathsf{\lambda}\frac{\partial T}{\partial r}\right).$$
- Incompressible fluid;
- One-dimensional convection (axial);
- Constant predefined velocity.

- ${T}_{0}=62\text{}\xb0\mathrm{C}$ for every element
- $\dot{q}=0$ for $r=0$
- $\dot{q}={\alpha}_{amb}\xb7\frac{\left({T}_{amb}-T\left({r}_{su}\right)\right)}{{N}_{su}}$ for $r={r}_{su}$
- $\dot{q}=0$ for $x=0$ and $r>{r}_{wall}$
- $\dot{q}=0$ for $x={x}_{su}$ and $r>{r}_{wall}$
- $\dot{q}=0$ for $x=0$ and ${r}_{HTF}<r\le {r}_{wall}$
- $\dot{q}=0$ for $x={x}_{su}$ and ${r}_{HTF}<r\le {r}_{wall}$
- $T={T}_{HTF,in}$ for $x=0$ and $r\le {r}_{HTF}$
- $\dot{q}=0$ for $x={x}_{su}$ and $r\le {r}_{HTF}$
- Simulation domain length of 1 m
- Simulation domain radius of 20 mm

#### 2.3. Validation

## 3. Results and Discussion

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Nomenclature

Latin symbols | |

a | Temperature conductivity (m^{2}·s^{−1}) |

A | Area (m^{2}) |

c | Specific heat capacity (J·kg^{−1}·K^{−1}) |

E | Element (-) |

F | Factor for fin concentration |

l | Length of the storage (m) |

L | Latent heat (J·kg^{−1}) |

$\dot{m}$ | Mass flow (kg·s^{−1}) |

N | Number of fins |

r | Radius (m)/radial coordinate (m) |

$\dot{q}$ | Heat flux (J·s^{−1}) |

S | Heat source term |

t | Time (s) |

T | Temperature (K) |

u | Flow velocity (m·s^{−1}) |

$\overrightarrow{u}$ | Flow velocity vector |

V | Volume (m^{3}) |

x | Axial coordinate (m) |

Greek symbols | |

α | Heat transfer coefficient (W·m^{−2}·K^{−1}) |

$\Delta $ | Difference (-) |

ρ | Density (kg·m^{−3}) |

ϕ | General variable |

Γ | Diffusion coefficient (m^{2}·s^{−1}) |

λ | Heat conductivity (W·m^{−1}·K^{−1}) |

Subscripts | |

amb | Ambience |

app | Apparent value |

E | Concerning the element left of the calculation element/exponential |

edge | Concerning the edge of the calculation element, that borders the ambience |

EP | Concerning the element east of the calculation element (CE) and the CE |

F | Heat transfer fluid |

h | Concerning the specific enthalpy |

HTF | Concerning the outer radius of the HTF |

in | Inlet |

I | Concerning the first section |

II | Concerning the second section |

III | Concerning the third section |

j | Position indicator in radial direction |

l | Liquid phase |

L | Linear/latent |

m | Thermodynamic mean |

n | Counting variable |

N | Concerning the element above the calculation element |

NP | Concerning the element north of the calculation element (CE) and the CE |

P | Concerning the calculation element |

s | Solid phase |

S | Concerning the element below the calculation element |

sec | Concerning one section of the storage element |

sen | Concerning the sensible heat capacity |

SP | Concerning the element south of the calculation element (CE) and the CE |

su | Storage unit |

W | Concerning the element right of the calculation element |

wall | Pipe wall |

WP | Concerning the element west of the calculation element (CE) and the CE |

0 | Concerning minimum distance |

1 | First element |

2 | Storage element divided into two parts |

3 | Storage element divided into three parts |

α | Convective heat transfer |

ϕ | Concerning the general variable |

Superscripts | |

i | Position indicator in axial direction |

0 | Concerning the last time step |

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**Figure 1.**(

**a**) shell-and-tube latent thermal energy storage (LTS) and (

**b**) single storage element. HTF: heat transfer fluid; and PCM: phase change material.

**Figure 2.**Single storage element with (

**a**) homogeneous; and (

**b**) uneven distribution of circular fins.

**Figure 3.**Section of the discretized two-dimensional model with exemplary declaration of elements, interfaces and radii.

**Figure 5.**Absolute average power until total storage discharge of all examined arrangements of fins.

**Figure 7.**Comparison of absolute output power and level of discharge at 5000 s and 6300 s for all allocations.

**Figure 8.**Heat transfer fluid (HTF) and wall element center temperatures of two distributions at 5000 s physical time.

**Figure 9.**Heat transfer fluid (HTF) and wall element center temperatures of two distributions at 6300 s physical time.

Material | Application | Phase Change Temperature Solid-Liquid (°C) | Latent Heat of Fusion (kJ/kg) | Liquid Heat Capacity (kJ/kg·K) | Liquid Density (kg/m^{3}) |
---|---|---|---|---|---|

RT42 | PCM | 40–44 | 176 | 2.0 | 760 |

Water | HTF | 0 | 334 | 4.18 | 998 |

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**MDPI and ACS Style**

Kuboth, S.; König-Haagen, A.; Brüggemann, D. Numerical Analysis of Shell-and-Tube Type Latent Thermal Energy Storage Performance with Different Arrangements of Circular Fins. *Energies* **2017**, *10*, 274.
https://doi.org/10.3390/en10030274

**AMA Style**

Kuboth S, König-Haagen A, Brüggemann D. Numerical Analysis of Shell-and-Tube Type Latent Thermal Energy Storage Performance with Different Arrangements of Circular Fins. *Energies*. 2017; 10(3):274.
https://doi.org/10.3390/en10030274

**Chicago/Turabian Style**

Kuboth, Sebastian, Andreas König-Haagen, and Dieter Brüggemann. 2017. "Numerical Analysis of Shell-and-Tube Type Latent Thermal Energy Storage Performance with Different Arrangements of Circular Fins" *Energies* 10, no. 3: 274.
https://doi.org/10.3390/en10030274