# Heat Transfer Performance of a Novel Multi-Baffle-Type Heat Sink

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

^{2}K with a pressure drop of 2.96 × 10

^{4}Pa, and the temperature difference between the maximum and the minimum temperature of the heating surface is 51.7 K. The results showed that the coolant for M6 is distributed evenly to each channel at the maximal degree. The phenomena of the maldistribution of temperature is effectively improved. Moreover, the thermal resistance and thermal enhancement factor for the six models is also examined. M6 possesses the lowest total thermal resistance and largest thermal enhancement factor compared to the other five models. Furthermore, an experimental platform is set up to verify the simulation results obtained for M6. The simulated heat-transfer coefficient and pressure drop values agree well with the experimental results.

## 1. Introduction

## 2. Problem Description

#### 2.1. Geometric Configurations and Computational Domain

_{1}is the depth of the counter sink. These configurations are chosen to see the effect of having baffles on the flow and temperature distribution.

#### 2.2. Mathematical Model

_{1}) and the corresponding Re are inserted in Table 2. It should be noticed that this study is limited to the conditions applied for the simulation and experiment including:

- (1)
- The volume force and the effect of surface tension are all neglected.
- (2)
- No radiation and gravity is assumed.
- (3)
- The thermo-physical properties of water are considered as constant and incompressible.
- (4)
- Axial conduction and viscous dissipation are not considered.

- (1)
- The coolant used is water and the multi-baffle-type heat sink (MBHS) is fabricated using aluminum.
- (2)
- A uniform heat flux of q = 58000 W/m
^{2}is applied to the bottom of heat sinks, and other surfaces are considered to be adiabatic. - (3)
- The inlet velocity u
_{1}(Table 2) and is assumed to remain constant. The inlet temperature T_{in}= 293 K. - (4)
- The outflow condition in the software is set at the outlet.

#### 2.3. Parameter Definition

_{m}is the average water velocity and µ is the fluid dynamic viscosity and the D

_{h}is the channel hydraulic diameter.

_{f}can be evaluated as

_{n}is the length of channel and u is the fluid velocity.

_{j}can be evaluated as

#### 2.4. Field Synergy Principle

## 3. Experiment Apparatus and Procedure

_{in}and P

_{out}) are measured using HG-80K intelligent digital pressure gauges that measures pressures ranging from 0 to 1MPa. The relative accuracy is ±2%. The pressure drop of the MBHS ΔP is equal to $\Delta {P}_{in}-\Delta {P}_{out}$ as shown in Figure 4. The pressure drop between the inlet and outlet, ΔP can be expressed as

^{2}, five stainless steel heating bars are placed on an aluminum plate. The heating-surface is tightly contacted with the bottom of the heat sink, and the heat flux is transferred to the heat sink from the heating unit. This is controlled by setting the power to each rod to a value between 0 to 300 W using a rheostat. The heat flux input section is 60 × 80 mm for which the corresponding heat flux of the heat sink can reach 0–60,000 W/m

^{2}.

_{1}) supplied by electrical heating source should be equal to the sum of that (Q

_{2}) taken away by cooling fluid (water) and the heat loss (Q

_{loss}) into environment:

_{p}is isobaric specific heat capacity of water, G is mass flow rate of water, T

_{out}is outlet water temperature, and T

_{in}is inlet water temperature. So the maximum relative error of heat balance is

_{out}− T

_{in}) is over 275 K. That is to say the relative error derived from electric heating power insteading of heat quantity flowing into heat sink is not more than 4.1%, which means the heat loss into the surrounding environment is not more than 4.1%. The heating unit is packaged in a layer of insulation material was used over remainder of the heating unit to ensure that 95.9% of the heat was transferred to the heat sink.

_{in}) and the outlet (T

_{out}). Five thermocouples (T

_{1}, T

_{2}, T

_{3}, T

_{4}, and T

_{5}) are embedded in the bottom of the heating unit in order to measure the temperatures of the heating surface of the heat sink. The temperature at the heating surface (T

_{h}) is considered to be the mean value of the five temperature readings. During the measurement, the peristaltic pump with a fixed mass flow rate of 0.04 kg/s first push the coolant into the channels. The steady state stability is defined as the time of the fluctuation range of date less than 0.1% is below twenty minutes. The fluctuation range of temperature is less than 0.2 K.

_{in}, T

_{out}, T

_{h}, P

_{in}, and P

_{out}become stable. Since the measurement accuracy is approximately ±0.1 K and the minimum temperature difference T

_{D}is measured to be 33.7 K.

_{ave}is 1 by definition, and where means there is an ideal flow distribution. Based on Equation (22), a degree of maldistribution parameter [16] was defined as

_{w}is the worst possible maldistribution calculated using Equation (22) with the assumption that the whole flow goes through the first channel of the PHE. The value of DM changes from 0 to 1, where 1 means a uniform mass flow in every channel (no maldistribution).

_{vi}is the absolute uncertainty of each independent parameter, and n is the total number of parameters.

_{h}and T

_{m}($\Delta T$) is slightly higher than 33 K. According to Table 3 and Equations (25)–(28), the relative uncertainty of the heat transfer coefficients ($\frac{{U}_{h}}{h}\times 100\%$) is less than 8.3%.

## 4. Experimental Results

#### 4.1. Convective Heat Transfer Coefficient and Nusselt Number

#### 4.2. Pressure Drop and Friction Factor

## 5. Results and Discussion

#### 5.1. Simulation Results

#### 5.1.1. Grid Independency

#### 5.1.2. Simulation procedure

#### 5.2. Model Comparison

^{2}·K and the Nusselt number is 37.5 obtained at the highest evaluated flow rate at Re = 2000. These values are the lowest for M1 with the heat-transfer coefficient of 814.84 W/m

^{2}·K and the corresponding Nusselt number of 16.66.

_{0}and f

_{0}denotes the Nusselt number and friction factor of the plain channel, respectively.

#### 5.3. Model Selection and Limits

## 6. Conclusions

^{2}·K and the temperature difference between the maximum and minimum temperature of the heating surface is 51.7 K with a pressure drop of 29.6 × 10

^{3}Pa at Re = 2000. The simulation results show that M6 has the best heat-transfer performance, flow-evenness performance, and temperature uniformity performance comparing the models investigated in this research. By setting up an experimental platform, the simulation results of M6 are verified. The following conclusions can be drawn:

- (1)
- As compared with the five models M1–M5, the velocity field of M6 is more uniformly distributed without the aid of external electronic devices.
- (2)
- M6 takes the advantage of dispersing heat from the high-temperature zone in a well-proportioned way.
- (3)
- The heat-transfer and flow performance of the heat sink can be effectively improved by employing optimally shaped baffles.
- (4)
- The average synergistic field angle decreases with the increase of the multi-baffle.
- (5)
- The heat-transfer coefficients and pressure drop of the experimental and simulation results are consistent with each other, which verifies the correctness of the numerical method and results.
- (6)
- Among six models, M6 possesses the lowest total thermal resistance.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

A_{c} | cross sectional area, m^{2} | T_{b,avg} | average channel base temperature, K |

DM | degree of maldistribution | $\nabla T$ | temperature gradient |

D_{h} | hydraulic diameter, m | ΔT | average temperature difference, K |

f | friction factor | T_{m} | average fluid temperature, K |

f_{0} | friction factor of a plain channel | T_{in} | inlet temperature, K |

H | total height of cooling plate, mm | T_{out} | outlet temperature, K |

H_{1} | height of the counter sink, mm | T_{h} | average temperature of heating surface, K |

h_{conv} | convective heat transfer coefficient, W/m^{2} K | T_{D} | temperature difference between the maximum and minimum temperature of heating surface, K |

k_{s} | thermal conductivity of Al, W/m K | u_{1} | axial velocity, m/s |

K_{f} | thermal conductivity of fluid, W/m K | u_{m} | mean axial velocity, m/s |

L | length of heat sink, mm | $\nabla V$ | velocity vector |

L_{i} | length of flow channel, mm | W_{i} | width of flow channel, mm |

L_{0i} | length of baffle, mm | x_{i} | rectangular coordinates |

MFR | mass flow ratio coefficient | Greek symbols | |

Nu | Nusselt number | $\mu $ | kinematic viscosity, kg/ms |

Nu_{0} | Nusselt number of a plain channel | $\rho $ | density, kg/m^{3} |

p | wetting perimeter, m | ${\theta}_{m}$ | average field synergy angle, ° |

ΔP | pressure drop of heat sink, Pa | $\beta $ | baffles intersection angle, ° |

P_{out} | outlet pressure, Pa | $\zeta $ | head loss coefficient |

P_{in} | inlet pressure, Pa | Subscripts | |

q | heat flux, W/m^{2} | ave | average |

q_{m} | mass flow rate of liquid, kg/s | i | index in x-direction |

Re | Reynolds number | j | index in y-direction |

R_{total} | total thermal resistance, m^{2}·K/W | n | flow channel number |

s | average absolute deviation | w | heating surface |

T | temperature, K | x,y,z | three coordinates shown in Figure 1, mm |

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**Figure 10.**The temperature difference between the maximum and minimum temperature of the heating-surface for six models.

**Figure 11.**The heat-transfer performance for six models, (

**a**) the heat-transfer coefficient; (

**b**) Nusselt number.

Model | L/W/H (mm) H _{1}/L_{in}/L_{out} | L_{n} (mm) (n = 1, 2, 3, 4) | W_{n} (mm)(n = 1, 2, 3, 4, 5) | L_{0n} (mm) (n = 1, 2, 3, 4) | α/β (°) |
---|---|---|---|---|---|

M1 | 214/139/5 | 198/191/185 | 17.8/17.8/17.8 | -- | -- |

3/18/18 | |||||

M2 | 214/139/5 | 198/191/185 | 9/13/31.4 | -- | -- |

3/18/18 | |||||

M3 | 214/139/5 | 186/173/160 | 17.8/17.8/17.8 | -- | -- |

3/18/18 | |||||

M4 | 214/139/5 | 186/173/160 | 17.8/17.8/17.8 | 22/28 | 80° 120° |

3/18/18 | |||||

M5 | 214/139/5 | 186/173/168 | 9/13/31.4 | 34 | 120° |

3/18/18 | |||||

M6 | 214/139/5 | 170/166 162/158 | 15.5/13.6 12.6/11.7 | 16/30 42/54 | 115° 30° |

3/18/18 |

u_{1} (m/s) | 0.06 | 0.12 | 0.18 | 0.24 | 0.3 |
---|---|---|---|---|---|

Re | 400 | 800 | 1200 | 1600 | 2000 |

Parameter | Absolute Uncertainty | Relative Uncertainty |
---|---|---|

Slot width (l_{1}) | ±0.01 mm | |

Slot depth (w_{1}) | ±0.01 mm | |

Temperature | ±1 K | |

Pressure | ±2% | |

Liquid flow rate | ±1% | |

Power | ±2% | |

Heat loss of the heat unit | 4.1% | |

Heat transfer coefficient | 8.3% | |

Average velocity | 3.9% | |

Friction coefficient | 9.5% |

**Table 4.**Measured (subscribed with exp) and simulated (subscribed with sim) equivalent heat-transfer coeffcients and pressure drop with different velocities for M6.

V (m/s) | ΔT_{exp} (K) | ΔT_{sim} (K) | ΔT Error | ΔP_{exp} (Pa) | ΔP_{sim} (Pa) | ΔP Error | h_{exp} (W/m^{2}·K) | h_{sim} (W/m^{2}·K) | h Error | Nu_{exp} | Nu_{sim} | Nu Error |
---|---|---|---|---|---|---|---|---|---|---|---|---|

0.06 | 118.2 | 115.4 | 2.4% | 5220 | 5110 | 1.3% | 725.42 | 681.15 | 6.1% | 13.3 | 12.9 | 3.0% |

0.12 | 83.5 | 78.0 | 6.6% | 7700 | 7450 | 3.6% | 1349.62 | 1263.31 | 6.4% | 14.8 | 14.1 | 4.7% |

0.18 | 53.8 | 49.5 | 8.0% | 11300 | 11260 | 0.5% | 1612.04 | 1601.53 | 0.7% | 15.9 | 15.5 | 2.5% |

0.24 | 46.5 | 43.1 | 7.3% | 17900 | 17300 | 3.4% | 1706.87 | 1623.32 | 4.9% | 29.4 | 28.8 | 2.0% |

0.3 | 33.7 | 31.8 | 5.6% | 29650 | 28800 | 2.9% | 1758.59 | 1674.85 | 6.3% | 38.2 | 37.5 | 1.8% |

Grid Number | ΔP (KPa) | Relative Error | ΔT (K) | Relative Error |
---|---|---|---|---|

1496772 | 29.63 | 0.0082% | 372.33 | 0.0072% |

2423946 | 29.63 | 0.0064% | 372.35 | 0.0072% |

3422575 | 29.64 | ------- | 372.37 | ------- |

4496728 | 29.65 | 0.0023% | 372.37 | ------- |

5393368 | 29.66 | 0.0023% | 372.35 | 0.0037% |

**Table 6.**The improvement value in field synergy angle, temperature difference, equivalent heat-transfer coefficient, Nusselt number, pressure drop and friction factor of M6.

Factor | θ° | ΔT (K) | h (W/m^{2}·K) | Nu | ΔP (kPa) | f |
---|---|---|---|---|---|---|

M6 | 85.33 | 51.7 | 1758.59 | 42.6 | 29.6 | 0.82 |

Improvements | min | min | max | max | max | min |

Model | M1 | M2 | M3 | M4 | M5 | M6 |
---|---|---|---|---|---|---|

DM | 0.591 | 0.669 | 0.763 | 0.767 | 0.882 | 0.915 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Cao, X.; Liu, H.-l.; Shao, X.-d.
Heat Transfer Performance of a Novel Multi-Baffle-Type Heat Sink. *Entropy* **2018**, *20*, 979.
https://doi.org/10.3390/e20120979

**AMA Style**

Cao X, Liu H-l, Shao X-d.
Heat Transfer Performance of a Novel Multi-Baffle-Type Heat Sink. *Entropy*. 2018; 20(12):979.
https://doi.org/10.3390/e20120979

**Chicago/Turabian Style**

Cao, Xin, Huan-ling Liu, and Xiao-dong Shao.
2018. "Heat Transfer Performance of a Novel Multi-Baffle-Type Heat Sink" *Entropy* 20, no. 12: 979.
https://doi.org/10.3390/e20120979