# Numerical Computation of Anisotropic Thermal Conductivity in Injection Molded Polymer Heat Sink Filled with Graphite Flakes

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Heat Sink Prototype

^{3}, tensile modulus—14.7 GPa). The heat sink was manufactured by injection molding. The mentioned material was melted and injected into a mold that was designed specifically for this research (see Figure 1). Several variations of heat sinks were produced based on the melt inlet side of the heat sink.

#### 2.2. Measurement Description

#### 2.3. Material Study

- The flakes in the
**core**are arranged in a**parabolic shape**perpendicular to the direction of material flow during fabrication. - Conversely, in the
**shell**they form**dense parallel lines**in the flow direction.

- Good—parallel to the flakes’ orientation,
- Poor—perpendicular to the flakes’ orientation.

**Figure 4.**The internal structure of the test specimen in the study conducted by Grundler et al. [29].

- whole specimen, perpendicular to the flow (in the Figure 4 depicted as (
**a**)—overall through-plane conductivity, - ground specimen (the shells were ground, so that the core itself could be measured), perpendicular to the flow (
**b**)—through-plane conductivity in the core, - ground specimen, parallel to the flow (
**c**)—in-plane conductivity in the core.

- flakes in the core form a parabolic structure,
- flakes in the shell are structured into dense parallel lines.

**Figure 5.**Measured thermal conductivities in the study by Grundler et al. [29].

**d**) using series connected thermal resistances. The procedure is analogous to the resistors in electrical circuits and is well described in [35]. The scheme for the calculation is shown in Figure 6.

#### 2.4. Computational Method

## 3. Numerical Model

## 4. Approaches to Thermal Conductivity

#### 4.1. Isotropy

#### 4.2. Simple Anisotropy

#### 4.3. Fractural Anisotropy

## 5. Results

## 6. Conclusions

**isotropic**—whole geometry with a single conductivity parameter,**anisotropic**—traditional approach with in-plane and through-plane thermal conductivities,**fractural**—thermal conductivity distribution based on the internal structure observed from microscopic images of the fractured heat sink.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

$f$ | Objective function [-] |

$G$ | Irradiation [W/m^{2}] |

$h$ | Heat transfer coefficient [W/m^{2}·K] |

${h}_{c}$ | Contact conductance [W/m^{2}·K] |

$J$ | Radiosity [W/m^{2}] |

$\mathit{K}$ | Thermal conductivity tensor [W/m·K] |

$L$ | Length [m] |

$\mathit{n}$ | Unit normal direction [-] |

${\mathrm{N}}_{\mathrm{T}}$ | Number of observed temperatures (accounted for in the objective function) [-] |

$T$ | Temperature [K, °C] |

$\overline{T}$ | Measured temperature [K, °C] |

$\tilde{T}$ | Simulated temperature [K, °C] |

${T}_{ext}$ | External temperature (ambient, air flow) [K, °C] |

${T}_{s}$ | Prescribed surface temperature [K, °C] |

$\dot{q}$ | Heat flux [W/m^{2}] |

$Q$ | Heat source [W/m^{3}] |

$R$ | Thermal resistance [m^{2}·K/W] |

${w}_{i}$ | Weight in the objective function [-] |

$\epsilon $ | Surface emissivity [-] |

${\lambda}_{1},{\lambda}_{2},{\lambda}_{3}$ | Principal conductivities [W/m·K] |

$\sigma $ | Stefan–Boltzmann constant, 5.68·10^{−8} W/m^{2}·K^{4} |

${\mathit{\lambda}}^{*}$ | Solution vector to optimization problem [W/m·K] |

$\nabla $ | Gradient operator |

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**Figure 1.**Both parts of mold specifically designed for this research (

**left**). Entrance of the melt during the manufacturing process (

**right**). The injection system was removed afterwards.

**Figure 2.**Measurement box inside the thermostatic chamber with attached thermal imaging camera (

**left**), heat sink inside the measurement box (

**middle**), attached heater with the thermocouple on the heat sink’s base (

**right**).

**Figure 3.**Locations of the thermocouples inside the heat sink. The locations in the middle fin will be referred to as Tmx, x being the ordinal number from the base up to the fin, e.g., Tm1—thermocouple in the base under the middle fin. Similarly, the locations of thermocouples in the outer fin will be labelled as Tsx. Dimensions are in mm.

**Figure 7.**Internal structure along the height of a fractured fin of the heat sink prototype (

**left**) with highlighted flakes’ orientation (

**right**).

**Figure 8.**Geometry of the numerical model. Dark grey—heat sink, light grey—aluminum body of the heater, orange—active heater domain.

**Figure 11.**The mesh of the numerical model, view on the xy-plane (

**left**) and on the yz-plane (

**right**).

**Figure 12.**Temperature fields of optimized numerical models. Isotropic approach (

**top left**)—${T}_{\mathrm{min}}=43\xb0\mathrm{C}$, ${T}_{\mathrm{max}}=77.3\xb0\mathrm{C}$, anisotropic (

**top right**)—${T}_{\mathrm{min}}=45.6\xb0\mathrm{C}$, ${T}_{\mathrm{max}}=78.6\xb0\mathrm{C}$, fractural (

**below**)—${T}_{\mathrm{min}}=45.3\xb0\mathrm{C}$, ${T}_{\mathrm{max}}=78.5\xb0\mathrm{C}$.

**Figure 13.**Thermal picture from the experiment and its comparison with the simulated temperature field (fractural model). The values by the color scale are in °C.

**Figure 14.**Measured and simulated temperatures of the three thermal conductivity approaches. Averaged temperatures on the top of the fins (

**above**), temperature near the heat source (

**below left**), thermocouples inside the heat sink (

**below right**)—their labels correspond to the Figure 3.

Parameter | Value | Description |
---|---|---|

${\mathit{\lambda}}_{\mathit{A}\mathit{l}}$ | 130 W/m·K | Thermal conductivity of the aluminum domain |

${\mathit{\lambda}}_{\mathit{h}\mathit{e}\mathit{a}\mathit{t}}$ | 400 W/m·K | Thermal conductivity of the heater domain |

$\mathit{P}$ | 15 W | Thermal power of the heater domain |

${\mathit{T}}_{\mathit{e}\mathit{x}\mathit{t}}$ | 20 °C | Surrounding temperature (ambient) |

${\mathit{\epsilon}}_{\mathit{A}\mathit{l}}$ | 0.1 | Surface emissivity of the aluminum domain |

${\mathit{\epsilon}}_{\mathit{H}\mathit{S}}$ | 0.85 | Surface emissivity of the composite heat sink |

${\mathit{h}}_{\mathit{c}}$ | 7700 W/m^{2}·K | Contact conductance at the interface between heat sink and the heater’s aluminum body |

**Table 2.**Results of the inverse task for the presented thermal conductivity approaches. Values of conductivities are in W/m·K.

Model | Isotropic | Anisotropic | Fractural | |||
---|---|---|---|---|---|---|

Parameters | $\lambda $ | 8.0 | ${\lambda}_{1}$ | 12.1 | ${\lambda}_{1}^{\mathrm{shell}}$ | 17.9 |

${\lambda}_{2}$ | 3.5 | ${\lambda}_{3}^{\mathrm{shell}}$ | 2.1 | |||

${\lambda}_{3}^{\mathrm{core}}$ | 4.2 | |||||

Objective value | 4.36 | 1.14 | 1.01 |

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**MDPI and ACS Style**

Brachna, R.; Kominek, J.; Guzej, M.; Kotrbacek, P.; Zachar, M.
Numerical Computation of Anisotropic Thermal Conductivity in Injection Molded Polymer Heat Sink Filled with Graphite Flakes. *Polymers* **2022**, *14*, 3284.
https://doi.org/10.3390/polym14163284

**AMA Style**

Brachna R, Kominek J, Guzej M, Kotrbacek P, Zachar M.
Numerical Computation of Anisotropic Thermal Conductivity in Injection Molded Polymer Heat Sink Filled with Graphite Flakes. *Polymers*. 2022; 14(16):3284.
https://doi.org/10.3390/polym14163284

**Chicago/Turabian Style**

Brachna, Robert, Jan Kominek, Michal Guzej, Petr Kotrbacek, and Martin Zachar.
2022. "Numerical Computation of Anisotropic Thermal Conductivity in Injection Molded Polymer Heat Sink Filled with Graphite Flakes" *Polymers* 14, no. 16: 3284.
https://doi.org/10.3390/polym14163284