# Numerical Analysis of Melting Process in a Rectangular Enclosure with Different Fin Locations

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## Abstract

**:**

## 1. Introduction

## 2. Model Construction

#### 2.1. Physical Model

_{c}, of 50 mm and height, H

_{c}, of 120 mm is used, as shown in Figure 1. A horizontal fin with thickness, t

_{f}, of 3 mm and length, L

_{f}, of 38 mm is mounted on the right hot surface. The distance between the lower surface of fin and the bottom surface of enclosure is denoted by d. A parameter, d

_{f}, named as fin location, is employed to characterize the dimensionless location of fin. d

_{f}varies from 0.05 to 0.95 in this study.

_{w}, is imposed on the right surface, while the adiabatic condition is used for other surfaces, as presented in Figure 1. Lauric acid and aluminum are respectively selected as the phase change medium and fin material, whose thermo-physical properties are shown in Table 1 and Table 2, respectively.

#### 2.2. Mathematical Model

- The motion of liquid is viewed as a 2D, laminar and incompressible flow.
- Both PCM volume change and radiation are ignored.
- The thermo-physical properties of fin material and PCM are constant, except for PCM density, ρ, in the buoyancy force term of momentum equations where the Boussinesq approximation is used, i.e., ρ varies linearly with temperature, T, as presented by:$$\rho -{\rho}_{ref}=-\rho \beta (T-{T}_{ref})$$
_{ref}and T_{ref}are the reference density and reference temperature which are set as liquid density, ρ_{l}, and melting temperature, T_{l}[22,39].

_{mush}is the mushy zone constant. ε is a small number, taking 10

^{−3}to avoid the division by zero. The formula of γ is:

_{s}:

_{ref}is the reference enthalpy at T

_{ref}. Only the heat conduction is taken into account for fins. The corresponding governing equation is:

- Boundary condition:$${\frac{\partial T}{\partial y}|}_{Bottom\text{}surface}=0,\text{}{\frac{\partial T}{\partial y}|}_{Top\text{}surface}=0,\text{}{\frac{\partial T}{\partial x}|}_{Left\text{}surface}=0,\text{}{{T}_{w}|}_{Right\text{}surface}=70\mathbb{C}$$

- Initial condition:$$t=0,\hspace{1em}T(0)={T}_{0}=25\xb0C\hspace{1em}\overrightarrow{u}(0)=0$$

^{−3}for velocity and momentum and 10

^{−6}for energy. The computation continues until the PCM melts completely, i.e., γ = 1.

#### 2.3. Independence Tests

_{f}= 0.20 is presented in Figure 2a, where the grids with 3840, 7448 and 14,060 quadrilateral cells are examined. As the cell number increases, the deviation between adjacent grids reduces. For the compromise of computation cost and accuracy, the grid with 7448 cells is picked up. Similar tests are performed for other d

_{f}and no fin cases, and the grids with 7448 quadrilateral cells are also selected. Figure 2b shows the schematic view of the grid which shares high quality with the maximum aspect ratio of 1.55 and minimum orthogonal quality of 0.9997.

^{−3}in some of time steps. As a result, Δt = 0.1 s is picked up for the better convergence and low computation cost [44].

#### 2.4. Model Validation

_{w}, from the initial value, T

_{0}, of 25 °C to the target value of 70 °C, occupying approximately 4 min, is not considered in the simulation. In the middle and late stages, as in the Abdi et al. model, the present model matches well with experiments. A difference is presented between the present model and Ji et al. model, because of the adopted A

_{mush}, 10

^{6}for the present model and 10

^{5}for the Ji et al. model.

## 3. Results and Discussion

#### 3.1. Patterns of Melting

_{f}, and no fin case at the flow time, t, of 300 s, 1800 s, 4800 s, and 8400 s are presented in Figure 5. Velocity vectors with high resolution are also provided in the Supplementary Materials. The images of d

_{f}= 0.05, 0.20 and 0.35 at t = 8400 s are not provided because the melting time, t

_{m}, of them is less than 8400 s.

_{f}and notably larger than the no fin case, which indicates that the effect of d

_{f}is insignificant and heat conduction is enhanced significantly by the fin at the beginning.

_{f}= 0.05, the lower area nearly melts totally and considerable portion of solid phase melts at the upper part with the interface shape similar to the no fin case. The behavior reverses for d

_{f}= 0.95 where the upper part melts completely and only a small region at the lower part, less than the no fin case, becomes liquid. The result signifies that the effect of d

_{f}becomes important at t = 1800 s.

_{f}= 0.05, the solid phase is located at the middle part of the left surface. The primary position of the unmelted region for d

_{f}= 0.20 is the same to d

_{f}= 0.05, apart from the tiny region at the left corner. For other d

_{f}, however, the solid phase is mainly located at the lower left of the enclosure and the size enlarges with moving the fin up. Besides, for d

_{f}= 0.35 and 0.50, the convection flow characterized by velocity vectors forms over the fin but does not accelerate melting process because of the impedance of the fin. Due to the solid phase is above the fin, however, this flow is not blocked for d

_{f}= 0.05 and 0.20. Compared to d

_{f}= 0.05, the unmelted region is much smaller for d

_{f}= 0.20, as a result of the higher strength of vortexes. On account of the stable thermal stratification, this flow is rather weak or even not formed for d

_{f}= 0.65–0.95. The analyses imply that the dominated heat transfer mode is different for different d

_{f}, which is natural convection for d

_{f}= 0.05 and 0.20 and may transform to heat conduction for d

_{f}= 0.35–0.95.

_{f}= 0.05–0.35. While for other cases, the melting just enters the late stage and the unmelted portion is still located at the lower left corner, just with smaller size than t = 4800 s. Besides, an abnormal phenomenon is found that the unmelted region for d

_{f}= 0.95 is slightly larger than the no fin case. It implies that the fin location affects the melting behavior notably and inserting fins is not always effective. The reason for this is that the thermal stratification forms more easily at the upper part for d

_{f}= 0.95, as a result of which the intensity of natural convection near the interface is slightly lower when comparing to the no fin case.

#### 3.2. Melting Time, t_{m}

_{m}with d

_{f}is shown in Figure 6. As d

_{f}increases, t

_{m}decreases before attaining the minimum at d

_{f}= 0.20 after which it increases. Compared to the no fin case of 190.7 min, melting is enhanced significantly with t

_{m}halved for d

_{f}= 0.20. Besides, an interesting result is found that for d

_{f}= 0.95, a scheme mounting fins closing to the top surface, t

_{m}is longer than the no fin case, which is unexpected and never reported by previous studies to the best knowledge of authors. The results demonstrate that the influence of d

_{f}is notable, and fins should be arranged properly in practical applications.

- For a lower γ (less than 10%), inserting fins is effective to enhance melting; the effect of d
_{f}is insignificant because the conduction dominates the heat transfer. - For d
_{f}= 0.05 and 0.20, the melting rate, quite higher than the no fin case, approximately remains the same with rising γ. It is because the fin mounted at a lower place does not hinder the liquid flow. On account of that the influence range of fins as to heat conduction is limited by the bottom surface for d_{f}= 0.05 in comparison to d_{f}= 0.20, t_{m}(d_{f}= 0.05) > t_{m}(d_{f}= 0.20). - For d
_{f}= 0.35–0.80, as γ increases, the melting rate decreases to a value lower than the no fin case at a time. Depending on d_{f}, the larger the d_{f}is, the lower the time is, i.e., the earlier the transition arises. For instance, the transition time is 80 min for d_{f}= 0.35 and reduces to 55 min for d_{f}= 0.65. This behavior is expected since the mounted fin impedes the liquid flow and the higher it is put, the earlier the impedance takes place. Besides, the thermal stratification region enlarges with moving the fin up, as a result of which t_{m}(d_{f}= 0.35) < t_{m}(d_{f}= 0.50) < t_{m}(d_{f}= 0.65) < t_{m}(d_{f}= 0.80). - For d
_{f}= 0.95, the liquid flow intensity weakens considerably, because of forming thermal stratification at the upper part, even though the flow resistance as to fins is rather limited. Thus, the transition of melting rate ahead of d_{f}= 0.80 is presented and the melting process is longer than the no fin case.

#### 3.3. Surface Averaged Nusselt Number Nu

_{f}, the surface averaged Nusselt number, Nu, is used [39]:

_{w}is the heat transfer area.

_{m}, for different d

_{f}and no fin case is shown in Figure 8. In general, Nu decreases with t/t

_{m}because of the increase of thermal resistance between the hot surface and interface with PCM melting. According to the decline rate of Nu, the melting process can be divided into four regimes for d

_{f}= 0.35–0.80, which, in sequence, are heat conduction regime, strong convection regime, convection blockage regime, and weak convection regime. While for other d

_{f}and no fin case, however, dividing the process into three regimes (without regime 3, convection blockage regime) may be more reasonable. Taking d

_{f}= 0.50 as an example, the heat transfer characteristics of these regimes are as follows:

_{f}= 0.05, 0.20 and 0.95, similar to the no fin case [39]. When moving the fin up, the liquid flow resistance enlarges, and the thermal stratification is formed more readily. Hence, both the transition t/t

_{m}from the Regime 2 to the Regime 3 and from the Regime 3 to the Regime 4 decrease, as shown in Figure 8. Figure 8 also shows that the transition t/t

_{m}from the Regime 1 to the Regime 2 is approximately the same for all d

_{f}, indicating that the effect of d

_{f}is limited at the early stage.

_{Nu}, defined by Equation (12), is introduced to identify the effect of d

_{f}in depth.

_{f}) is the Nu with the fin location, d

_{f}. Nu (no fin) is taken from the no fin case for a reference value. The variation of ε

_{Nu}with t/t

_{m}for different d

_{f}is shown in Figure 9 and the analyses are as follows:

- ε
_{Nu}increases with t/t_{m}in the whole melting process for d_{f}= 0.05 and 0.20. It is expected since mounting fin at a lower position enhances conduction at the beginning and facilitates natural convection at the middle and late stages; besides, the melting in the left corner is reinforced by the lower fin significantly, which is extensively regarded as a factor prolonging melting process seriously [28,46]. A testimony that facilitates natural convection is provided by Figure 10 which shows the variation of the maximum liquid flow velocity, v_{max}, with t/t_{m}. For t/t_{m}> 0.30, the difference between v_{max}(d_{f}) and v_{max}(no fin) enlarges notably with t/t_{m}, precisely corresponding to the sharp increase of ε_{Nu}. - While for d
_{f}= 0.35–0.80, the transition from ε_{Nu}> 0 to ε_{Nu}< 0 arises. That is, the heat transfer is not strengthened all the time. It is because the conduction is improved notably at the early stage or the early and middle stages, depending on d_{f}, but after that natural convection is reduced significantly due to the increase of liquid flow resistance as to fins and the formation of thermal stratification. The impedance of fins is verified by the variation of v_{max}shown in Figure 10. The comparison between Figure 10 and Figure 9 shows that the transition time from v_{max}(d_{f}) > v_{max}(no fin) to v_{max}(d_{f}) < v_{max}(no fin) is approximately equal to the transition time from ε_{Nu}> 0 to ε_{Nu}< 0. - As liquid contours of the no fin case shown (Figure 5), the melting is originally much faster at the upper part in comparison to the lower part. As a result, the improvement of mounting fins near the top surface is not obvious and only limited to the very early stage. Besides, mounting fins near the top surface improves the temperature distribution nearby, facilitating the formation of thermal stratification and weakening the convection flow. Hence, on the one hand, ε
_{Nu}> 0 only arises at t/t_{m}< 0.025 and ε_{Nu}< 0 is presented at the rest of time for d_{f}= 0.95; on the other hand, the influence mechanism is different for d_{f}= 0.35–0.80 and d_{f}= 0.95, even though they share the same trend. As the same to d_{f}= 0.05–0.80, the variation of ε_{Nu}is quite similar to the difference between v_{max}(d_{f}) and v_{max}(no fin).

_{max}can be employed as an indicator to characterize the influence of d

_{f}. The higher the v

_{max}is, the larger the ε

_{Nu}is and the quicker the melting experiences. Thus, a strategy, boosting v

_{max}, is proposed to reinforce melting, which includes promoting liquid flow and/or inhibiting thermal stratification formation. On account of that the liquid flow resistance is rather limited and the formation of thermal stratification is suppressed, the v

_{max}then ε

_{Nu}and melting process is improved significantly for d

_{f}= 0.05 and 0.20. As a result, it is recommended to mount fins at a lower place for vertical enclosures.

## 4. Conclusions

_{f}, on melting behavior were studied numerically. The comparison with the no fin case was conducted. The main conclusions are as follows.

_{f}= 0.05 and 0.20, conduction is improved at the beginning and natural convection is enhanced at the middle and late stages, resulting in halving t

_{m}when comparing to the no fin case; however since the function area of fins is limited by the bottom surface for d

_{f}= 0.05, t

_{m}(d

_{f}= 0.05) > t

_{m}(d

_{f}= 0.20). While for d

_{f}= 0.35–0.80, with an increase of d

_{f}, the effectiveness of inserting fins reduces primarily because of increasing the liquid flow resistance. Besides, t

_{m}(d

_{f}= 0.95) > t

_{m}(no fin) due to the fact that mounting fins in a place next to the top surface accelerates melting at the upper part, facilitating thermal stratification formation, which was never reported in the literature.

_{f}= 0.35–0.80. While for d

_{f}= 0.05, 0.20 and 0.95, similar to the no fin case, dividing the process into three regimes is more reasonable.

_{Nu}> 0, in the whole melting process for d

_{f}= 0.05 and 0.20, while for other d

_{f}, the transition from ε

_{Nu}> 0 to ε

_{Nu}< 0 arises and the higher the fin is put, the earlier the transition takes place.

_{max}(d

_{f}) and v

_{max}(no fin) is nearly the same as ε

_{Nu}. To reinforce melting, a strategy is thus proposed that boosts the v

_{max}for the case that natural convection cannot be neglected.

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

A_{mush} | mushy zone constant, kg/(m^{3}·s) |

A_{w} | heat transfer area, m^{2} |

c_{p} | specific heat capacity, J/(kg·K) |

d | distance between the lower surface of fin and the bottom surface of enclosure, m |

d_{f} | fin location, d/H |

g | gravitational acceleration, m/s^{2} |

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

h_{s} | sensible enthalpy, J/kg |

H | enthalpy, J/kg |

H_{c} | height of rectangular enclosure, m |

ΔH | latent enthalpy, J/kg |

L_{c} | length of rectangular enclosure, m |

L_{f} | length of fin, m |

L_{h} | latent heat, J/kg |

Nu | Nusselt number |

p | pressure, Pa |

Q | heat flux, W |

t | time, s |

t_{f} | thickness of fin, m |

t_{m} | melting time, s |

T | temperature, °C |

T_{0} | initial temperature, °C |

T_{w} | surface temperature, °C |

u | velocity, m/s |

v_{max} | maximum liquid flow velocity, m/s |

Greek symbols | |

β | Thermal expansion coefficient, 1/K |

γ | liquid fraction |

λ | thermal conductivity, W/(m·K) |

ε | small number |

ε_{Nu} | Nu enhancement factor |

ρ | density, kg/m^{3} |

Subscripts | |

fin | fin |

l | liquid |

PCM | PCM |

ref | reference |

s | solid |

## Abbreviations

LTES | latent thermal energy storage |

PCM | phase change material |

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**Figure 4.**Validation with the experiment conducted by Kamkari and Shokouhmand [27] for 1 fin case and 3 fins case.

**Figure 5.**Instantaneous liquid fraction contours and velocity vectors for different d

_{f}and no-fin case at (

**a**) t = 300 s, (

**b**) t = 1800 s, (

**c**) t = 4800 s, and (

**d**) t = 8400 s.

Property | Symbol | Value |
---|---|---|

Melting temperature | T_{l} (°C) | 48.2 |

Solidification temperature | T_{s} (°C) | 43.5 |

Latent heat | L_{h} (kJ/kg) | 187.21 |

Specific heat capacity of solid | c_{p,s} (kJ/(kg·K)) | 2.18 |

Specific heat capacity of liquid | c_{p,l} (kJ/(kg·K)) | 2.39 |

Density of solid | ρ_{s} (kg/m^{3}) | 940 |

Density of liquid | ρ_{l} (kg/m^{3}) | 885 |

Thermal conductivity of solid | λ_{s} (W/(m·K)) | 0.16 |

Thermal conductivity of liquid | λ_{l} (W/(m·K)) | 0.14 |

Thermal expansion coefficient | β (1/K) | 0.0008 |

**Table 2.**Thermo-physical properties of aluminum [27].

Property | Symbol | Value |
---|---|---|

Specific heat capacity | c_{p,fin} (kJ/(kg·K)) | 0.871 |

Thermal conductivity | λ_{fin} (W/(m·K)) | 130 |

Density | Ρ_{fin} (kg/m^{3}) | 2179 |

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**MDPI and ACS Style**

Huang, B.; Tian, L.-L.; Yu, Q.-H.; Liu, X.; Shen, Z.-G.
Numerical Analysis of Melting Process in a Rectangular Enclosure with Different Fin Locations. *Energies* **2021**, *14*, 4091.
https://doi.org/10.3390/en14144091

**AMA Style**

Huang B, Tian L-L, Yu Q-H, Liu X, Shen Z-G.
Numerical Analysis of Melting Process in a Rectangular Enclosure with Different Fin Locations. *Energies*. 2021; 14(14):4091.
https://doi.org/10.3390/en14144091

**Chicago/Turabian Style**

Huang, Bin, Lin-Li Tian, Qing-Hua Yu, Xun Liu, and Zu-Guo Shen.
2021. "Numerical Analysis of Melting Process in a Rectangular Enclosure with Different Fin Locations" *Energies* 14, no. 14: 4091.
https://doi.org/10.3390/en14144091