# Numerical Simulation and Application of a Channel Heat Sink with Diamond Ribs

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## Abstract

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## 1. Introduction

## 2. Model Description

#### 2.1. Three-Dimensional Model

#### 2.2. Computational Domain and Mathematical Model

- The flow and heat-transfer processes are considered as steady-state;
- The fluid is incompressible and single-phase, and the thermal properties of fluid and solid are constant;
- The thermal resistance between solids is disregarded;
- Heat exchange between the exterior surface and its surroundings is not considered.

_{h}originates from the heat source situated at the base.

#### 2.3. Data Acquisition

_{h}= d.

_{f}is utilized to evaluate the overall performance of the channel radiator. The heat-transfer enhancement factor simultaneously takes into account the heat dissipation capability and fluid flow capacity of the radiator. With this indicator, we can determine which parameter values can yield the greatest heat-transfer performance with the smallest pressure loss. The higher the heat enhancement coefficient, the better the overall performance of the microchannel radiator. The expression of η

_{f}is given by:

_{0}represents the relative Nusselt number. Similarly, f/f

_{0}represents the relative friction factor. If the value of η

_{f}is less than one, it implies that the enhancement of heat-transfer performance is less than the pressure-drop penalty generated in its channel, and vice versa. In all designs of channel radiators, it is desired that heat-transfer enhancement exceeds the pressure-drop penalty. Evidently, this design poses a challenge; thus, it is also acceptable to improve heat-transfer performance with an acceptable pressure-drop penalty.

## 3. Experimental Device

_{w}. Water is introduced into the inlet of the channel heat sink through the operation of the pump, with its temperature maintained at an ideal ambient temperature (299.15 K) via the water bath. The inlet pressure P1 is acquired through a pressure transducer connection, while the outlet pressure P2 is at atmospheric levels. The overall pressure differential ΔP across the heat sink is determined by the subtraction of P2 from P1.

_{Xi}is the uncertainty of the actual measured value. The values of the uncertainty are exhibited in Table 1.

## 4. Result and Discussion

_{t}, apparent friction factor f, and heat-transfer enhancement factor η

_{f}are employed as output data. The range of variables is summarized in Table 2. Before analyzing the channel radiator’s performance, validation is required, including the accuracy of the simulation and grid independence.

^{−3}, and the energy residual is less than 1 × 10

^{−6}. The physical parameters of materials are listed in Table 3.

#### 4.1. Validation

#### 4.1.1. Validation of Grid Independence

^{−1}, water temperature of 299.15 K). The verification indices for mesh independence are the average temperature of the base surface (${T}_{w,ave}$) and the pressure difference between the inlet and outlet of Channel 5 ($\Delta {P}_{5}$).

#### 4.1.2. Validation of Numerical Model

^{−1}to 300 mL·min

^{−1}, and a flow rate gradient of 40 mL·min

^{−1}). The central temperature of the base surface of the flow channel radiator T

_{w}and the overall pressure difference ΔP were taken as the verification indicators. The results shown in Figure 7 reveal that the experimental and simulated values of ΔP are very close across each Reynolds number gradient. However, the simulated values of T

_{w}are consistently lower than the experimental values for each Reynolds number gradient. This discrepancy is attributed to the fact that in the simulation model, the heat source and flow channel radiator are regarded as ideally connected, thus rendering a heat-transfer capability that is somewhat higher than actuality, leading to slightly lower simulated values. The overall maximum error does not exceed 5%. Considering the verification results for both ΔP and T

_{w}, the modeling and simulation of the flow channel radiator in this study are effective and suitable for subsequent simulation research.

#### 4.2. The Characteristics of the Radiator

^{−1}to study the radiator’s characteristics. The simulated cloud diagrams were extracted at the cross-section H/2, and the results are presented in Figure 8a–c. The temperature of the cross-section, the pressure of the channels, and the velocity of the fluid within the channels all exhibit symmetric distribution. Channels closer to the center exhibit higher velocity, carrying away more heat but also manifesting increased pressure. Vortices appear at the four corners of the radiator. As these are distant from the heat-exchange area, they do not impact the cooling performance.

#### 4.3. The Impact of Geometric Parameters on the Heat Sink

#### 4.3.1. Effects of Fin Angle α

_{t}at various Reynolds numbers Re and different fin angles. The thermal resistance R

_{t}of the radiator is lower under all fin angles compared to the smooth, straight channels, indicating that the ridged fins within the channels effectively enhance heat transfer. As the fin angle decreases and the Reynolds number increases, thermal resistance gradually diminishes. The primary reasons for this can be explained by the velocity of the water flow and the heat-transfer surface area ratio ε between the new channels and the straight channels. As the fin angle decreases, ε increases, signifying a larger exchange area between the fluid and the solid, thereby improving the heat-transfer capability. On the other hand, when the Re increases and the fin angle decreases, the average fluid velocity within the channel rises. Considering the aforementioned factors, thermal resistance gradually decreases, and the channel radiator exhibits the lowest thermal resistance when α = 90°.

_{0}at various Reynolds numbers and different fin angles α. The Nusselt numbers under all fin angles are greater than those in smooth, straight channels. This is because the fins disturb the fluid’s velocity, exerting a perturbing effect on the fluid regardless of the fin angle. The disturbance is weakest, and the heat-exchange area between the fluid and the solid is minimal when α = 150°, resulting in the lowest Nu/Nu

_{0}. From the perspective of Nusselt numbers, the heat-transfer performance is superior at α = 90° compared to smooth, straight channels, reaching up to 1.82 when Re = 315.32.

_{0}at various Reynolds numbers and different fin angles α. It can be observed that regardless of the fin angle, the fluid flow resistance within the channels is higher than that in smooth, straight channels, and it increases with the Reynolds number. When Re remains constant, the variation in fin angle has a dramatic effect on flow resistance and increases sharply as the fin angle decreases. At α = 90° and Re = 630.64, f/f

_{0}is 13.25, indicating that the flow resistance is 13.25 times that of smooth, straight channels. This signifies a marked deterioration in flow performance, primarily due to an excessive pressure difference in the channel when α = 90°. Therefore, excessively small fin angles should be avoided in the current design. Angles of α = 150° and α = 135° may be prioritized, as they result in a flow resistance no more than three times that of smooth, straight channels.

_{f}at various Reynolds numbers and different fin angles α. η

_{f}decreases with increasing Re across all angles, signifying that the radiator’s overall performance is better at lower flow rates. When Re remains constant, η

_{f}first increases and then decreases with the reduction of the fin angle. At α = 90°, η

_{f}is at its lowest, where the pressure-drop penalty far outweighs the heat-transfer enhancement, resulting in η

_{f}being only 0.74 at Re = 630.64. Therefore, in the current design, angles of α = 150° or α = 135° should preferably be utilized.

#### 4.3.2. Effects of Height Ratio β

_{t}at various Re and different height ratios. From the graph, it can be distinctly observed that the thermal resistance of all channels with different rib heights is lower than that of the smooth, straight channels, and furthermore, it diminishes gradually with the increase in Re. When the Re is constant, R

_{t}first decreases with the increase in β, eventually stabilizing. The reason for this may be that when β = 75%, it already provides a sufficiently large heat-transfer area, and even if β further increases to 100%, the resultant enhancement in heat-exchange performance will not increase substantially.

_{0}at various Re and different height ratios. When β = 75% and 100%, Nu/Nu

_{0}tends to decrease with the increase in Re, whereas when β = 25% and 50%, Nu/Nu

_{0}exhibits an ascending trend with the growth of Re. When Re remains constant, Nu/Nu

_{0}gradually increases with the enhancement of β. The reason for this could be the amplification of the heat-exchange area between the fluid and the solid as β grows, coupled with the fact that a larger β results in an increased ability of the ribs to disturb the fluid. Therefore, compared to the smooth, straight channels, the heat-transfer performance of the radiator is optimal when β = 100%, with Nu/Nu

_{0}consistently exceeding 1.4.

_{0}at diverse Re and different height ratios, revealing that the fluid flow resistance within channels at all height ratios surpasses that of the smooth, straight channels. When β = 75% and 100%, f/f

_{0}progressively escalates with the increase in Re; in contrast, when β = 25% and 50%, f/f

_{0}exhibits a tendency to stabilize with the growth of Re. This phenomenon might be attributed to the comparatively small pressure differential in the channels when β = 25% and 50%. Under the prevailing conditions, f/f

_{0}signifies that any value of β will influence flow performance; however, compared to smooth, straight channels, at β = 25% and 50%, f/f

_{0}does not exceed 2. Hence, in the current design, consideration can be prioritized for β = 25% and 50%.

_{f}at various Re and different height ratios. With the exception of β = 25%, all heat-transfer enhancement factors diminish with the growth in Re. The rate of reduction for the heat-transfer enhancement factor at β = 50% is markedly attenuated compared to β = 75% and 100%, whereas at β = 25%, η

_{f}initially declines and then steadily ascends with the increase in Re, and when Re > 507.5, its curve becomes the highest. In summary, if Re > 507.5, a preference for β = 25% is recommended; if Re < 507.5, a priority for β = 50% and 75% is advised.

#### 4.3.3. Effects of Rib Space s

_{t}at varying rib spacings. It is salient that the thermal resistance of the radiators at all rib spacings is inferior to that of the smooth, straight channels, and all the curves are closely congruent. The reason for this is that the alteration in rib spacing nearly fails to modify the ratio of the heat-transfer area between the new channel and the straight channel, thus resulting in equivalent heat-transfer characteristics.

_{0}at different Re and different rib spacings. Aside from when s = 2, Nu/Nu

_{0}noticeably declines with an increase in Re, reaching its maximum when s = 1.5. Nu/Nu

_{0}at s = 1 and 2.5 are minimal and nearly identical. However, when s = 2, the rate of decrease in Nu/Nu

_{0}lessens when Re exceeds 472.98, and according to the current trend, it can be estimated that Nu/Nu

_{0}will become the greatest when Re surpasses 630.64 at s = 2. In summary, the heat-transfer performance is substantially reduced as Re increases.

_{0}at various Re and different rib spacings. It can be discerned that regardless of the spacing of the ribs, the fluid flow resistance within the channel is higher than that in a smooth, straight channel, and it increases with the enhancement of Re. When Re remains constant, f/f

_{0}initially rises with an increase in s and subsequently diminishes, reaching its minimum when s = 2.5; hence, in the present design, a preference may be given to s = 2.5. Furthermore, the graph reveals that when s = 1, 1.5, or 2, the trends of their curves are consistent and the values differ insignificantly, which could be attributed to the degree of disturbance to the fluid in the channel being almost the same under current conditions.

_{f}at varying Re and different rib spacings. All the curves exhibit a similar trend, with a maximum observed when s = 2.5. Overall, it can be anticipated that when Re exceeds 630.64, the heat-transfer enhancement factor will rapidly diminish following the current trend.

#### 4.4. The Application of Heat Sinks

^{−1}(Re = 525.53) and water temperature of 299.15 K. The measuring apparatus was based on the experimental setup shown in Figure 3, using a data acquisition instrument to capture the temperature of the heat-conducting sheet surface (the contact face between the heat-conducting sheet and the biological reaction chamber, TH), as depicted in Figure 14. The test results indicated that the surface temperature of the heat-conducting sheet nearly reached 368.15 K at its peak during each cycle, with the minimum consistently below 328.15 K; the heating rate reached 7.9 K/s, and the cooling rate achieved 7.8 K/s. Under these temperature conditions, the PCR reaction could be completed. Overall, this channel heat sink is capable of fulfilling the heat dissipation requirements. The symbol "···" shown in Figure 14 indicates that the rest of the curve is the same as the current curve for 45 cycles.

## 5. Conclusions

- The angle of the rhombic fins within the channel has a substantial impact on the flow characteristics. When Re remains constant, the flow resistance at α = 90° is 13.25 times that of a smooth, straight channel, resulting in a pressure-drop penalty that far outweighs the enhancement in heat transfer. Considering both flow and heat-transfer characteristics, the optimal choices are α = 150° or α = 135°.
- With Re held constant, the thermal resistance decreases as β increases. At β = 25% and 50%, f/f
_{0}does not exceed 2. Weighing both flow and heat-transfer characteristics, in the current design, β = 25% and 50% may be considered. If Re > 507.5, the preference should be given to β = 25%, whereas if Re < 507.5, the preference should be given to β = 50% and 75%. - The influence of the design variable s on channel performance indicates that, in the current design, priority can be given to s = 2.5, in which case f/f
_{0}is the lowest and the heat-transfer enhancement factor is the highest. - This channel radiator is employed in PCR devices. Actual experimental results indicate that the radiator can furnish an ample amount of heat-exchange capacity, enabling the TEC to produce a stable cyclical temperature.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 9.**The characteristics of heat sink: (

**a**) The apparent friction factor for each channel; (

**b**) The Nusselt numbers for each channel.

**Figure 10.**The influence of α on thermal hydraulic performance for the channel: (

**a**) Thermal resistance; (

**b**) Nusselt numbers ratio; (

**c**) Apparent friction factor ratio; (

**d**) Heat-transfer enhancement factor.

**Figure 11.**The influence of β on thermal hydraulic performance for the channel: (

**a**) Thermal resistance; (

**b**) Nusselt numbers ratio; (

**c**) Apparent friction factor ratio; (

**d**) Heat-transfer enhancement factor.

**Figure 12.**The influence of s on thermal hydraulic performance for the channel: (

**a**) Thermal resistance; (

**b**) Nusselt numbers ratio; (

**c**) Apparent friction factor ratio; (

**d**) Heat-transfer enhancement factor.

Parameters | Uncertainty |
---|---|

Surface roughness (Channel heat sinks, Cover plate) | ±2 μm |

Thermostatic water tank | ±0.5 K |

Data acquisition instrument | 0.1% |

Power | 1% |

Thermocouple | ±0.1 K |

Pressure transmitter | 0.5% |

Variable | Value |
---|---|

Angle of diamond rib: α | 90, 105, 120, 135, 150 (°) |

The spacing of diamond rib: s | 1, 1.5, 2, 2.5 (mm) |

Ratio of rib height to channel height: β | 25, 50, 75, 100 (%) |

Inlet flow | 60, 70, 80, 90, 100, 110, 120 (mL·min^{−1}) |

Material | Density (kg·m^{−3}) | Specific Heat (J·kg^{−1}·K^{−1}) | Thermal Conductivity (W·m^{−1}·K^{−1}) | Viscosity (kg·m^{−1}·s^{−1}) |
---|---|---|---|---|

Ceramics | 1345 | 710 | 150 | - |

Water | 998.2 | 4182 | 0.6 | 0.001 |

Copper | 8960 | 385 | 401 | - |

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## Share and Cite

**MDPI and ACS Style**

Zhang, D.; Liu, G.; Lai, Y.; Lin, X.; Cai, W.
Numerical Simulation and Application of a Channel Heat Sink with Diamond Ribs. *Water* **2023**, *15*, 3677.
https://doi.org/10.3390/w15203677

**AMA Style**

Zhang D, Liu G, Lai Y, Lin X, Cai W.
Numerical Simulation and Application of a Channel Heat Sink with Diamond Ribs. *Water*. 2023; 15(20):3677.
https://doi.org/10.3390/w15203677

**Chicago/Turabian Style**

Zhang, Dongxu, Guoqiang Liu, Yongkang Lai, Xiaohui Lin, and Weihuang Cai.
2023. "Numerical Simulation and Application of a Channel Heat Sink with Diamond Ribs" *Water* 15, no. 20: 3677.
https://doi.org/10.3390/w15203677