# Influence of the Fin Shape on Heat Transport in Phase Change Material Heat Sink with Constant Heat Loads

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## Abstract

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## 1. Introduction

_{m}to 252 kJ/kg; paraffin with melting temperatures up to 108 °C; as well as some inorganic compounds [13,14]. The convenience and practicality of using PCM have found wide application in different industries including electronics, aerospace engineering, solar energy, construction, etc. [15,16,17].

_{cr}= 60 °C, T

_{cr}= 70 °C and T

_{cr}= 80 °C). It has been shown that, in some cases, T

_{cr}increases by more than 20 times in comparison with the case of a cooler without fins, which is achieved by an increase in the fins number and its elongation. In [10], the influence of the number of horizontal ribs and their length were analyzed numerically. New correlations were obtained for the liquid fraction in the region with horizontal edges depending on the Stefan, Rayleigh, and Fourier numbers, as well as the geometric parameters of the heat sink. It has been shown that the elongation of the ribs significantly accelerates the melting process. Heat transfer between PCM and the metal profile depends significantly on the ratio of the PCM volume to the profile volume, as well as the geometric parameters of the solid phase, such as thickness, shape, and number of fins [26,27,28,29,30,31]. An increase in the number of fins accelerates the melting process, while a decrease in fins thickness increases the source temperature [27]. In the study [26], the PCM-based heat sinks with different shapes of the pin fins were examined. Tall fins with rectangular, circular, and triangular sections were considered. The triangular-shape pin fins allow reducing the element temperature compared to other analyzed shapes. Two models of passive cooling were considered in [11]; namely, a cooler with vertical plate fins and a tree shape cooler made of aluminum. It was demonstrated that thermal convection significantly enhances the energy transfer. The use of a branched profile design leads to the formation of many vortex cells with an intensification of heat transfer.

## 2. Basic Equations

_{m}, and the enthalpy has a discontinuity in the transition from one phase to another phase using the following relation:

_{out}, while the outer borders Y = 1, X = 0, and X = 2 are cooled by air convection, other boundaries, including the source, are considered to be thermally insulated from the environment. At the initial time, the material has a solid-state (Ψ = 0, Ω = 0), and the entire system has the initial temperature that is equal to the ambient temperature Θ

_{out}.

## 3. Numerical Technique

_{m}. The problem of melting gallium inside a parallelepiped with two isothermal walls [46] was also considered as a benchmark problem. Figure 3b shows a comparison of the phase change interface location at different times.

## 4. Results

^{7}, and the Stefan number is Ste = 1.3.

_{avg}) reflects the efficiency of energy absorption by PCM (see Figure 5). The red profiles illustrate the results without natural convection influence. The intensification of convective phenomenon begins after an appearance of a small volume of melt, which is noticeably reflected in the average heater temperature after 22 min of heating. The temperature of the heater in the model without taking into account convective heat and mass transfer occurs monotonously, while the difference with natural convection process reaches 14 degrees in the absence of horizontal ribs. In the case of l = H/8, the difference in T

_{avg}approaches 10 degrees at 50 min. The phenomenon of convective heat transfer in this case has a significant effect on the melting process and must be taken into account in simulation when the interface moves away from the highly heat-conducting surface of the profile.

_{avg}slows down. The process of transformation of sensible heat into latent heat begins, in which most of the energy is spent on phase transformation. With the emergence and development of the melt zone, the dependence of the source temperature on the geometric characteristics of the profile appears and increases with time. The greatest effect is seen with an elongation, namely, for l = H/8, the temperature in the heater after 40 min of heating is lower by 2.9 degrees compared to the case for l = H/24. However, increasing the surface area of the heat sink characterizes an intensification of melting process. After the zone of solid PCM disappears completely, T

_{avg}again begins to increase rapidly and monotonously, while the temperature of the heater becomes the same for each case.

_{avg}for the cases l = H/24 and l = H/8 exceeds 6 degrees. After PCM is completely melted, this difference diminished due to reduced melting time at large values of l (Table 2).

_{avg}in the case of l = H/24 begins to increase. This phenomenon is primarily associated with a change in the hydrodynamics of the melt affected the addition of transverse fins.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

Bi | Biot number |

c | specific heat, JK^{−1} kg^{−1} |

g | gravity acceleration, ms^{−2} |

H | chamber size, m |

h | specific enthalpy, Jkg^{−1} |

k | heat conductivity, WK^{−1}m^{−1} |

L | cavity length, m |

L_{f} | fusion energy or latent heat of melting, Jkg^{−1} |

Os | Ostrogradsky number |

p | pressure, Pa |

Pr | Prandtl number |

Q | energy production strength per unit of volume, W m^{−3} |

Ra | Rayleigh number |

Ste | Stefan number |

t | time, s |

T | temperature, K |

T_{m} | melting temperature, K |

u, v | velocity projections in Cartesian coordinate along x and y, ms^{−1} |

U, V | non-dimensional velocity projections |

x, y | Cartesian coordinates, m |

X, Y | non-dimensional Cartesian coordinates |

Greek symbols | |

α | heat diffusivity, m^{2} s^{−1} |

β | coefficient of thermal expansion, K^{−1} |

γ | energy transference coefficient, WK^{−1}m^{−2} |

η | smoothing characteristic (or melting temperature range), K |

Θ | non-dimensional temperature |

μ | dynamic viscosity, Pa s |

ν | kinematic viscosity, m^{2} s^{−1} |

ρ | density, kgm^{−3} |

τ | non-dimensional time |

φ | volume fraction of the melt |

ψ | stream function, m^{2} s^{−1} |

Ψ | non-dimensional stream function |

ω | vorticity, s^{−1} |

Ω | non-dimensional vorticity |

Subscripts | |

0 | initial condition or ambient |

1 | cooler |

2 | heated unit |

l | liquid |

m | melting |

s | solid |

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**Figure 1.**Metal heat sink saturated with PCM (phase change material) and heated from the source. (

**a**) 3D shape of the heat sink with the source of volumetric heat generation, (

**b**) considered 2D model with PCM.

**Figure 2.**The grid independence test: the phase front location at t = 25 and t = 35 min (

**a**), as well as a graph of the dependence of the volume fraction of the melt on the dimensionless time (

**b**).

**Figure 4.**Temperature fields and isolines of the stream function at Os = 0.169 for the cases l = H/24, l = H/12, and l = H/8 at the time moments t = 30, t = 50, and t = 60 min.

**Figure 6.**Temperature fields and streamlines at Os = 0.338 for the cases of l = H/24, l = H/12, and l = H/8 at the time moments t = 15, t = 25, and t = 30 min.

**Figure 7.**Temperature fields and streamlines at Os = 0.676 for the cases l = H/24, l = H/12, and l = H/8 at the time moments t = 10, t = 15, and t = 20 min.

Phase Change Material | Lauric Acid [44] | Copper (Radiator) | Silicon (Heat Source) |
---|---|---|---|

T_{m}, °C | 46 | – | – |

L_{f}, kJ/Kg | 187.2 | – | – |

k_{s}/k_{l}, W/(m∙K) | 0.16/0.14 | 401 | 148 |

ρ_{s}/ρ_{l}, kg/m^{3} | 940/885 | 8900 | 2330 |

c_{s}/c_{l}, J/(kg∙K) | 2180/2390 | 385 | 714 |

μ, Pa∙s | 8 × 10^{–3} | – | – |

β, 1/K | 8 × 10^{–4} | – | – |

Ostrogradsky Number | Length of Horizontal Fins | ||
---|---|---|---|

l = H/24 | l = H/12 | l = H/8 | |

0.169 | 55 min | 53 min 9 s | 51 min 30 s |

0.338 | 30 min 9 s | 29 min 20 s | 27 min 45 s |

0.676 | 17 min 23 s | 17 min 7 s | 16 min 4 s |

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**MDPI and ACS Style**

Bondareva, N.S.; Ghalambaz, M.; Sheremet, M.A.
Influence of the Fin Shape on Heat Transport in Phase Change Material Heat Sink with Constant Heat Loads. *Energies* **2021**, *14*, 1389.
https://doi.org/10.3390/en14051389

**AMA Style**

Bondareva NS, Ghalambaz M, Sheremet MA.
Influence of the Fin Shape on Heat Transport in Phase Change Material Heat Sink with Constant Heat Loads. *Energies*. 2021; 14(5):1389.
https://doi.org/10.3390/en14051389

**Chicago/Turabian Style**

Bondareva, Nadezhda S., Mohammad Ghalambaz, and Mikhail A. Sheremet.
2021. "Influence of the Fin Shape on Heat Transport in Phase Change Material Heat Sink with Constant Heat Loads" *Energies* 14, no. 5: 1389.
https://doi.org/10.3390/en14051389