# Computational Model of Shell and Finned Tube Latent Thermal Energy Storage Developed as a New TRNSYS Type

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Physical Problem

## 3. Computational Model

^{th}segment with calculated heat fluxes, which form the energy conservation equation for the segment.

^{th}segment shown in Figure 3 thermally interacts with the neighboring upper and lower segments, the HTF and the environment. The energy conservation equation for the i

^{th}segment is defined as follows:

^{th}segment, H

_{i}[J/kgK] is PCM specific enthalpy in i

^{th}segment, t [s] is time, ${\dot{Q}}_{\mathrm{HTF}}$ [W] is exchanged heat flux with the HTF, ${\dot{Q}}_{\mathrm{PCM},i-1}$ [W] is exchanged heat flux with neighboring upper segment, ${\dot{Q}}_{\mathrm{HTF},i+1}$ [W] is exchanged heat flux with neighboring lower segment and ${\dot{Q}}_{\mathrm{loss},i}$ [W] are i

^{th}segment thermal losses through the outer wall of LTES. Equation (1) can further be written as:

_{hx,i}[W/K] is the reciprocal value of the total heat transfer resistance between HTF and PCM, ∆T

_{m,i}[K] is the mean logarithmic temperature difference between HTF temperature at inlet and outlet of tube i

^{th}segment and PCM temperature in the i

^{th}segment, k

_{PCM}[W/mK] is PCM thermal conductivity, A

_{IF}[m

^{2}] is interface area between neighboring segments, δx [m] is height of the i

^{th}segment, T

_{PCM,i−1}, T

_{PCM,i}and T

_{PCM,i+1}[K] are PCM temperatures at segments i − 1, i and i + 1, U

_{loss}[W/m

^{2}K] is the total heat transfer coefficient between PCM and environment, A

_{s,i}[m

^{2}] is the area of i

^{th}segment to environment and T

_{env}[K] is environment temperature.

_{PCM}[kJ/kgK] is the PCM specific heat capacity and γ [-] is the liquid fraction defined as:

_{solidus}and T

_{liquidus}are the temperatures at which the phase change begins or ends, depending on whether the charging or discharging process is occurring i.e., whether heat is being stored or released from the PCM. During the charging process, the PCM in the i

^{th}segment is in the solid state when its temperature is lower than T

_{solidus}; it consists of both solid and liquid phases when its temperature is T

_{solidus}< T

_{PCM,i}< T

_{liquidus}and it is liquid when its temperature is higher than T

_{liquidus}.

_{solidus}and T

_{liquidus}were the same for the LTES discharge.

_{hx}is derived as the reciprocal of the total thermal resistance between the HTF and PCM, which is given in Equation (5).

_{1}, R

_{2}, R

_{3}and R

_{4}represent thermal resistances due to convection heat transfer on the inside of tubes, conductive heat transfer through the tube wall, heat transfer from outside of the tube to the PCM and heat transfer from the fins to the PCM, respectively, as given in Equations (6)–(9).

_{HTF}[W/m

^{2}K] denotes the convective heat transfer coefficient inside the tubes, A

_{in}[m

^{2}] is the inside area of the tube, d

_{o}and d

_{i}[m] are the outside and inside tube diameters, L

_{i}[m] is tube and fin length in the i

^{th}segment, k

_{t}[W/mK] is tube thermal conductivity, D

_{PCM}[m] is the diameter of a PCM subdomain, which, considering all tubes are equidistant, is equal to the tube pitch, k

_{PCM}[W/mK] is PCM thermal conductivity, δ

_{PCM}[m] is average PCM thickness between two adjacent fins, A

_{f,i}[m

^{2}] is fin area and η

_{f}[-] is fin efficiency.

_{f}[W/mK] and t

_{f}[m] denote thermal conductivity of the fin and the fin thickness respectively. The convective heat transfer on the HTF side is defined by the Gnielinski correlation [41,42,43]:

_{HTF}is thermal conductivity of the HTF, Re and Pr are Reynolds and Prandtl numbers given as:

_{HTF}[kg/m

^{3}] is HTF density, μ

_{HTF}[Pa s] is HTF dynamic viscosity, c

_{HTF}[J/kgK] is HTF specific heat capacity and u

_{HTF}[m/s] is HTF inlet velocity.

_{PCM}[W/mK] is thermal conductivity of the PCM and Ra is the Rayleigh number:

^{2}] denotes gravitational acceleration, β [1/K] denotes the thermal expansion coefficient, T

_{HTF,in}[K] is HTF inlet temperature, T

_{PCM,m}[K] is melting temperature of the PCM, ν

_{PCM}[m

^{2}/s] is kinematic viscosity of the PCM and a

_{PCM}[m

^{2}/s] is thermal diffusivity of the PCM.

## 4. Numerical Procedure and Experimental Validation

#### 4.1. Numerical Setup

#### 4.2. Independency Analysis of Number of Segments and Time Step Size

#### 4.3. Experimental Validation

## 5. Application of the Developed Model

#### 5.1. The Influence of HTF Inlet Temperature

#### 5.1.1. Influence of HTF Inlet Temperature on LTES Thermal Performance during Charging

#### 5.1.2. Influence of HTF Inlet Temperature on LTES Thermal Performance during Discharging

#### 5.2. The Influence of HTF Flow Rate

#### 5.2.1. Influence of HTF Flow Rate on LTES Thermal Performance during Charging

#### 5.2.2. Influence of HTF Flow Rate on LTES Thermal Performance during Discharging

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

A_{f} | Fin area in the i^{th} segment (m^{2}) |

A_{hx,i} | Heat transfer area between HTF and PCM in the i^{th} segment (m^{2}) |

A_{IF} | Interface area between neighboring segments (m^{2}) |

A_{in} | Inside tube heat transfer area (m^{2}) |

A_{s,i} | Area of the i^{th} segment to environment (m^{2}) |

a | Thermal diffusivity (m^{2}/s) |

B | Fin width (m) |

b | Coefficient (-) |

C | Coefficient (-) |

c | Specific heat capacity (J/kgK) |

D | LTES tank inside diameter (m) |

d_{i} | Tube inside diameter (m) |

d_{o} | Tube outside diameter (m) |

f_{t} | Fin thickness (m) |

g | Gravitational acceleration (m/s^{2}) |

H | Specific enthalpy (J/kg) |

h | Convective heat transfer coefficient (W/m^{2}K) |

k | Thermal conductivity coefficient (W/mK) |

k_{eq} | Equivalent thermal conductivity coefficient (W/mK) |

L | Fin length & LTES tank height (m) |

l | Latent heat (J/kg) |

${\dot{m}}_{\mathrm{HTF}}$ | Mass flow of the HTF (kg/s) |

m_{PCM,i} | Mass of the PCM in i^{th} segment (kg) |

n | Coefficient (-) |

n_{f} | Number of fins per tube (-) |

n_{t} | Number of tubes (-) |

Pr | Prandtl number (-) |

$\dot{Q}$ | Exchanged heat flux (W) |

${\dot{Q}}_{loss}$ | Thermal losses through the LTES outer wall (W) |

R_{1} | Thermal resistance on the inside of the tubes (K/W) |

R_{2} | Thermal resistance through tube wall (K/W) |

R_{3} | Thermal resistance on the outside of the tubes (K/W) |

R_{4} | Thermal resistance on the fin surface (K/W) |

Ra | Rayleigh number (-) |

Re | Reynolds number (-) |

T | Temperature (K) |

T_{m} | Melting temperature (K) |

t | Time (s) |

t_{p} | Tube pitch (m) |

U | Total heat transfer coefficient (W/m^{2}K) |

u_{HTF} | HTF inlet velocity (m/s) |

Greek symbols | |

β | Thermal expansion coefficient (1/K) |

γ | Liquid fraction (-) |

ΔT_{m} | Mean logarithmic temperature difference (K) |

δ_{PCM} | Average PCM thickness between two adjacent fins (m) |

δx | Height of the i^{th} segment (m) |

η_{f} | Fin efficiency (-) |

μ | Dynamic viscosity (Pa s) |

ν | Kinematic viscosity (m^{2}/s) |

ρ | Density (kg/m^{3}) |

Subscripts | |

env | Environment |

HTF | Heat transfer fluid |

i | i^{th} segment |

in | Inlet |

liquidus | PCM liquidus temperature |

loss | Thermal losses to the environment |

n | Number of LTES segments |

out | Outlet |

PCM | Phase change material |

solidus | PCM solidus temperature |

t | Tube |

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**Figure 3.**Schematic representation of a one-dimensional computational model of shell and finned tube LTES and the i

^{th}segment.

**Figure 4.**Schematic representation of a PCM annulus and the detail showing schematic representation of thermal resistances between HTF and PCM.

**Figure 5.**Transient PCM temperature variations at the middle axial position of LTES during melting for HTF flow rate 620 l/h and HTF inlet temperature 37 °C, obtained by computational model with different number of LTES tank isothermal segments.

**Figure 6.**Transient PCM temperature variations at the middle axial position of LTES during melting for HTF flow rate 620 l/h and HTF inlet temperature 37 °C, obtained by a computational model with different time step sizes.

**Figure 7.**Schematic representation of an experimental test system segment with LTES tank, HTF flow rate and inlet temperature regulation, cold HTF inlet, cold HTF outlet, hot HTF inlet, and hot HTF outlet.

**Figure 9.**Comparison of transient temperature variations of the HTF and PCM during melting obtained numerically and experimentally at positions T2, T3, T4 and T5 for HTF flow rate 620 l/h and HTF inlet temperature 42 °C.

**Figure 10.**Comparison of accumulated energy during melting obtained numerically and experimentally for HTF flow rate 620 l/h and HTF inlet temperature 42 °C.

**Figure 11.**Comparison of transient temperature variations of the HTF and PCM during solidification obtained numerically and experimentally for HTF flow rate 620 l/h and HTF inlet temperature 7 °C.

**Figure 12.**Comparison of accumulated energy during solidification obtained numerically and experimentally for HTF flow rate 620 l/h and HTF inlet temperature 7 °C.

**Figure 13.**Comparison of transient variations of PCM temperatures during melting obtained numerically for different HTF inlet temperatures, HTF inlet flow rate 800 l/h and initial PCM temperature 15 °C.

**Figure 14.**Comparison of accumulated energy during melting obtained numerically for different HTF inlet temperatures, HTF inlet flow rate 800 l/h and initial PCM temperature 15 °C.

**Figure 15.**Comparison of mean heat transfer rates during melting obtained numerically for different HTF temperatures, HTF inlet flow rate 800 l/h and initial PCM temperature 15 °C.

**Figure 16.**Comparison of transient variations of PCM temperatures during solidification obtained numerically for different HTF inlet temperatures, HTF inlet flow rate 800 l/h and initial PCM temperature 35 °C.

**Figure 17.**Comparison of released energy during solidification obtained numerically for different HTF inlet temperatures, HTF inlet flow rate 800 l/h and initial PCM temperature 35 °C.

**Figure 18.**Comparison of mean heat transfer rates during solidification obtained numerically for different HTF temperatures, HTF inlet flow rate 800 l/h and initial PCM temperature 35 °C.

**Figure 19.**Comparison of transient variations of PCM temperatures during melting obtained numerically for different HTF flow rates, HTF inlet temperature 40 °C and initial PCM temperature 15 °C.

**Figure 20.**Comparison of accumulated energy during melting obtained numerically for different HTF flow rates, HTF inlet temperature 40 °C and initial PCM temperature 15 °C.

**Figure 21.**Comparison of mean heat transfer rate during melting obtained numerically for different HTF flow rates, HTF inlet temperature 40 °C and initial PCM temperature 15 °C.

**Figure 22.**Comparison of transient variations of PCM temperatures during solidification obtained numerically for different HTF flow rates, HTF inlet temperature 10 °C and initial PCM temperature 35 °C.

**Figure 23.**Comparison of released energy during solidification obtained numerically for different HTF flow rates, HTF inlet temperature 10 °C and initial PCM temperature 35 °C.

**Figure 24.**Comparison of mean heat transfer rate during solidification obtained numerically for different HTF flow rates, HTF inlet temperature 10 °C and initial PCM temperature 35 °C.

LTES tank height, L [mm] | 1550 |

LTES tank diameter, D [mm] | 950 |

Tubes diameters, d_{o}/d_{i} [mm] | 30/25 |

Number of tubes, n_{t} [-] | 19 |

Number of fins per tube, n_{f} [-] | 8 |

Fin thickness, f_{t} [mm] | 1 |

Fin width, B [mm] | 66 |

Tube pitch, t_{p} [mm] | 180 |

**Table 2.**Thermo-physical properties of PCM paraffin RT25, water and aluminum used in numerical simulations.

Properties | PCM | Water | Aluminum | |
---|---|---|---|---|

Liquid | Solid | |||

Density [kg/m^{3}] | 760 | 880 | 998.2 | 2719 |

Thermal conductivity [W/mK] | 0.2 | 0.6 | 202.4 | |

Specific heat capacity [J/kgK] | 2000 | 4182 | 871 | |

Kinematic viscosity of PCM [mm^{2}/s] | 4.7 | 1.005 | - | |

Latent heat [J/kg] | 170,000 | - | - | |

Melting | Solidification | - | - | |

Liquidus temperature [°C] | 25 | 25 | - | - |

Solidus temperature [°C] | 18 | 25 | - | - |

Thermal expansion coefficient [1/K] | 0.001 | - | - |

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**MDPI and ACS Style**

Torbarina, F.; Lenic, K.; Trp, A.
Computational Model of Shell and Finned Tube Latent Thermal Energy Storage Developed as a New TRNSYS Type. *Energies* **2022**, *15*, 2434.
https://doi.org/10.3390/en15072434

**AMA Style**

Torbarina F, Lenic K, Trp A.
Computational Model of Shell and Finned Tube Latent Thermal Energy Storage Developed as a New TRNSYS Type. *Energies*. 2022; 15(7):2434.
https://doi.org/10.3390/en15072434

**Chicago/Turabian Style**

Torbarina, Fran, Kristian Lenic, and Anica Trp.
2022. "Computational Model of Shell and Finned Tube Latent Thermal Energy Storage Developed as a New TRNSYS Type" *Energies* 15, no. 7: 2434.
https://doi.org/10.3390/en15072434