# Numerical Optimization of a Microchannel Geometry for Nanofluid Flow and Heat Dissipation Assessment

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## Abstract

**:**

^{®}(Burlington, MA, USA) and its results were compared with those obtained from experimental tests performed in microchannel heat sinks of polydimethylsiloxane (PDMS). Distilled water was used as the working fluid under the laminar fluid flow regime, with a maximum Reynolds number of 293. Three sets of geometries were investigated: rectangular, triangular and circular. The different configurations were characterized based on the flow orientation, type of collector and number of parallel channels. The main results show that the rectangular shaped collector was the one that led to a greater uniformity in the distribution of the heat transfer in the microchannels. Similar results were also obtained for the circular shape. For the triangular geometry, however, a disturbance in the jet impingement was observed, leading to the least uniformity. The increase in the number of channels also enhanced the uniformity of the flow distribution and, consequently, improved the heat transfer performance, which must be considered to optimize new microchannel heat sink designs. The achieved optimized design for a heat sink, with microchannels for nanofluid flow and a higher heat dissipation rate, comprised a rectangular collector with eight microchannels and vertical placement of the inlet and outlet.

## 1. Introduction

^{®}(Burlington, MA, USA) [9] software to optimize the fabrication of micro nanofluidic devices through photopolymerization. The model expressed the decomposition of the material that would occur due to the manufacture process. Teodori et al. [10] implemented a volume of fluid (VOF)-based model in the Open FOAM ComputationalFluid Dynamics (CFD) toolbox to compare the heat transfer in straight and finned microchannels. The presence of the fins improved the heat exchanges; however, the optimization of the geometry of the fins is still required.

^{®}software, analyzed a microchannel heat sink device for use as a cooler in a high-concentration photovoltaic system. The authors tested channels with and without triangular ribs on the inner side of the walls. The microchannels with ribs presented better heat transfer, increasing the Nusselt number1.7 times, on average. On the other hand, the simulation also showed that, for the same configuration, the average friction factor was 3.1 times higher.

_{2}O

_{3}nanoparticles. The optimized header geometry showed better results in both numerical simulations and experimental tests compared to the conventional geometry. The thermal performance was also enhanced with an increasein the flow rate and volume concentration of nanoparticles, the optimized geometry being more sensitive to those changes. Bahiraei et al. [22] analyzed numerically the heat transfer on heat sinks with circular, triangular and drop-shaped pin fins. In addition, the impact of a nanofluid with functionalized graphene nanoplatelets on the heat transfer process was also investigated. The heat sink with circular pin fins showed the highest thermal efficiency, while the one with triangular pin fins showed the lowest. The increase in velocity or particle fraction reduces the temperature on the heating surface and the thermal resistance, whilst improving the temperature distribution’s uniformity.

_{2}O

_{3}nanoprticles dispersed in a water base. The results showed a heat transfer enhancement despite the increase in the friction effect. Ganguly et al. [24] also studied the flow of Al

_{2}O

_{3}nanoparticles in water through a microchannel, considering the combined effects of externally applied pressure gradient and electroosmosis, by following a semi-analytical approach. The nanoparticles lead to a reduction in the total entropy generation in the microchannel.

## 2. Numerical Procedure

^{®}software was laminar flow (spf), which enabled the determination of the pressure fields and velocity profiles in a single-phase flow. Initially, the flow was studied by assuming a constant temperature (isothermal) and steady state conditions. The subsequent studies were meant to simulate the heat transfer process. For that purpose, a conjugated heat transfer module was used. This interface simulates the combined solid and fluid heat transfer processes and uses a classical approach for the determination of heat transfer in solids, as determined by the Fourier law given by Equation (5) of the next section, and for the determination of the heat transfer in fluids, given by the governing Equation (4) (also in the next section). We determined the velocity field, the pressure field and the temperature for both solid and liquid domains.

#### 2.1. Numerical Method

#### 2.2. Geometry, Computational Domain and Mesh

^{®}software simulations.

^{®}commercial software was used to evaluate the quality of the mesh. The value ranges between 0 and 1 depending on the proximity between the mesh elements. An ideal element has thevalue 1, while the value is 0 for a completely distorted element. A mesh is considered acceptable when this parameter is above 0.1 [28]. Details of the mesh quality and other mesh characteristics are presented in Table 1.

#### 2.3. Fluid and Solid Properties

#### 2.4. Boundary Conditions

#### 2.5. Data Evaluation

_{i}and the mean flow rateq

_{med}, was calculated, enabling the comparison of the flow rate distribution’s uniformity within different geometries. An example can be seen in Figure 5, where a higher flow is responsible for the concentration of the flow in the central microchannels, which are aligned with the inlet, showing an effect of the inertia of the fluid on the FDU coefficient S.

## 3. Results and Discussion

#### 3.1. Comparison between the Numerical and the Experimental Results

#### 3.2. Influence of Heat Sink Geometry in Flow Distribution

^{®}software allowed the determination of the flow rate through each section, and the analysis of the quadratic deviations enabled the quantification of the FDU through the FDU coefficient, S. It can already be stated that the rectangular-shaped support manifold is the one that provides the lowest values of flow distribution uniformity coefficient (S) when compared with the circular and triangular shapes. As such, the rectangular shape is the one causing the greater uniformity in the distribution of the heat transfer in the microchannels, as was clearly outlined in the abstract, and because of this fact the rectangular shape is the only one to be considered in the further analysis undertaken in this work. The orientation of the flowing the rectangular-shaped manifold was imposed by the positioning of the inlet and outlet, as presented in Figure 10. After the numerical simulation, the FDU coefficient was determined for the different geometries. The results are compiled in Figure 11. The coefficient increases exponentially with the increase in the flow rate. The vertical placement of the inlet and outlet allowed the development of a more uniform flow. Regarding the horizontal flow, a small difference was noticed when placing the inlet and outlet on the same or opposite faces. Finally, the greatest flow maldistribution was observed with the injection of the fluid frontally.

#### 3.3. Effect on Heat Transfer

## 4. Concluding Remarks

^{®}. From the different relative positions of the inlet and outlet that were tested, the vertical placement allowed the development of a flow with greater uniformity, while the frontal placement showed the highest flow maldistribution. The results also showed that the collector with a rectangular shape was the one leading to greater uniformity, while the triangular shape led to the least uniformity since the geometry causes a disturbance in the jet impingement. Additionally, this study indicates that increasing the number of channels tends to improve the uniformity of the flow distribution. In this way, the microchannel heat sink that showed the greatest flow uniformity and heat transfer performance was the heat sink with the vertical placement of the inlet and outlet, a rectangular collector, and eight microchannels.

- A rectangular-shaped collector, since this causes less flow maldistribution and instability.
- Vertical placement of the inlet and outlet, since, in the particular case of the inlet, this causes less throttling and the easier breaking and spreading of the impinging jet.
- The maximum number of microchannels possible within the limits of the intended overall reduced dimensions of the heat sinks (eight in the case of the current work).

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

A | Surface area (m^{2}) |

Cp | Specific heat(J/kg·K) |

f | Force(N) |

G | Strain rate tensor (1/s) |

H | Channel height (m) |

h | Enthalpy (J) |

k | Thermal conductivity (W/m·K) |

L | Heat sink length (m) |

N | Number of channels |

p | Pressure (Pa) |

q | Flow rate (mL/min or m^{3}/s) |

Q | Heat (W) |

S | Flow distribution uniformity coefficient |

t | Time (s) |

T | Temperature (K) |

u | Velocity field (m/s) |

V | Velocity (m/s) |

W | Heat sink width (m) |

Z | Heat sink height (m) |

Greek Symbols | |

αp | Thermal expansion coefficient (1/K) |

ϕ | Volume fraction (-) |

υ | Kinematic viscosity (m^{2}/s) |

ρ | Density (kg/m^{3}) |

τ | Viscous stress tensor |

Subscripts | |

c | Channel |

w | Wall |

b | Base |

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**Figure 1.**Schematic representation of the acrylonitrile–butadiene–styrene (ABS) master mold and respective dimensions (heat sink length, L; width, W; height, Z).

**Figure 3.**Details of the mesh in the collector.Highlighted: the inlet region and the transition between the collector and the channels.

**Figure 4.**Example of the geometry used in the heat transfer simulation, wherein both domains of the simulation can be seen.

**Figure 6.**View of the temperature profile of a fluid flow at different flow rates:(

**a**) 1 mL/min; (

**b**) 30 mL/min.

**Figure 7.**Temperature evolution throughout each channel, from the entrance to the exit, with a fluid flow of 1 mL/min.

**Figure 8.**Trajectory of the fluid in the collector and respective distribution to the channels at a fluid flow of 100 mL/min and at a scale of 5.

**Figure 9.**Comparison of the difference in temperature generated in the experimental tests and in the numerical simulation.

**Figure 10.**Different configurations for the microchannel device: (

**a**) horizontal frontal; (

**b**) horizontal parallel; (

**c**) horizontal parallel inverted; (

**d**) vertical.

**Figure 11.**Coefficient of the uniformity distribution of the flow for different heat sink configurations.

**Figure 12.**Flow distribution uniformity (FDU) study for the frontal configuration. (

**a**) Schematic view of the velocity profile for a 100 mL/min flow and at a scale of 2. (

**b**) The standardized flow in each channel of the heat sink.

**Figure 13.**FDU study for the inverted horizontal configuration. (

**a**) Schematic view of the velocity profile for a 100 mL/min flow and at a scale of 2. (

**b**) The standardized flow in each channel of the heat sink.

**Figure 14.**FDU study for the vertical configuration. (

**a**) Schematic view of the velocity profile for a 100 mL/min flow and at a scale of 2. (

**b**) The standardized flow in each channel of the heat sink.

**Figure 16.**FDU study for the triangular shape. (

**a**) Schematic view of the velocity profile for a 5 mL/min flow and at a scale of 20. (

**b**) Schematic view of the speed profile for a 100 mL/min flow and at scale of 2.

**Figure 17.**Standardized flow in each channel of the heat sink with triangular-shaped collectors, for different flow rates.

**Figure 18.**FDU study for the circular shape. (

**a**) Schematic view of the velocity profile for a 100 mL/min flow and at a scale of 2. (

**b**) Standardized flow in each channel of the heat sink.

**Figure 20.**FDU study for different numbers of channels for a 100 mL/min fluid flow at a scale of 5. (

**a**) Velocity profile for four-channel configuration. (

**b**) Velocity profile for eight-channel configuration.

**Figure 22.**Temperature profiles for the studied configurations for heat transfer at a flow rate of 5 mL/min: (

**a**) Vertical Rectangular with 8 channels; (

**b**) Vertical Rectangular with 6 channels; (

**c**) Horizontal Parallel with 6 channels.

**Figure 23.**Heat transfer rate in each channel of the geometries: (

**a**) Vertical rectangular. (

**b**) Horizontal parallel.

Mesh | A1 | A2 | A3 | A4 |
---|---|---|---|---|

Maximum Size (mm) | 0.609 | 0.425 | 0.264 | 0.23 |

Minimum Size (mm) | 0.115 | 0.046 | 0.0172 | 0.015 |

Maximum Growth | 1.13 | 1.1 | 1.08 | 1.08 |

Curve Factor | 0.5 | 0.4 | 0.3 | 0.25 |

Resolution of Narrow Regions | 0.8 | 0.9 | 0.95 | 0.95 |

Elements Number | 125,426 | 402,365 | 1,342,365 | 1,975,310 |

Average Quality | 0.5569 | 0.605 | 0.6359 | 0.6478 |

Simulation Time | 1 min 26 s | 3 min 26 s | 8 min 54 s | 14 min 12 s |

Mesh | B1 | B2 | B3 | B4 | B5 | B6 |
---|---|---|---|---|---|---|

Maximum Size (mm) | 10.5 | 7 | 5.6 | 3.85 | 2.45 | 1.4 |

Minimum Size (mm) | 1.96 | 1.26 | 0.7 | 0.28 | 0.105 | 0.014 |

Maximum Growth | 1.6 | 1.5 | 1.45 | 1.4 | 1.35 | 1.3 |

Curve Factor | 0.7 | 0.6 | 0.5 | 0.4 | 0.3 | 0.2 |

Resolution of Narrow Regions | 0.4 | 0.5 | 0.6 | 0.7 | 0.85 | 1 |

Elements Number | 126,348 | 151,540 | 167,311 | 188,105 | 239,112 | 366,980 |

Average Quality | 0.661 | 0.691 | 0.701 | 0.713 | 0.725 | 0.746 |

Water | PDMS | |
---|---|---|

Viscosity $\left[\mathrm{Pa}\times \mathrm{s}\right]$ | 0.00101 | - |

Density $\left[\mathrm{kg}/{\mathrm{m}}^{3}\right]$ | 1001.6 | 970.0 |

Specific Heat $\left[\mathrm{J}/\mathrm{kg}\times \mathrm{K}\right]$ | 4182.5 | 1460.0 |

Thermal Conductivity $\left[\mathrm{W}/\left(\mathrm{m}\times \mathrm{K}\right)\right]$ | 0.603 | 0.180 |

Specific Heat Ratio | 1 | - |

Surface | Boundary Conditions | Mathematical Expression |
---|---|---|

Inlet | Flow Rate Inlet | ${q}_{IN}\to t,t\in \left[1,100\right]$, T_{mean} = 20 °C |

Outlet | Pressure Outlet | P_{OUT} = 0 Pa (free atmosphere) and ΔT_{OUT} = 0 |

Fluid–Solid Interface | No-slip Interface | ${K}_{f}\times \nabla {T}_{f}={K}_{PDMS}\times \nabla {T}_{PDMS}$ and T_{f} = T_{PDMS} |

Sided Walls | Adiabatic | $-{K}_{i}\times \left(\frac{\partial {T}_{i}}{\partial x}\right)=-{K}_{i}\left(\frac{\partial {T}_{i}}{\partial y}\right)=0$ |

Interface between the Heater and PDMS | Thermal Coupled | ${K}_{H}\times \nabla {T}_{H}={K}_{PDMS}\times \nabla {\mathrm{T}}_{\mathrm{PDMS}}$ and T_{H} = T_{PDMS} |

Bottom Surface of the PDMS | Mixed Radiation and Convection | $\begin{array}{ll}-{K}_{PDMS}\times \left(\frac{\partial {T}_{PDMS}}{\partial z}\right)& ={q}_{rad,PDMS}\\ \text{}& \to s+{q}_{cov,PDMS}\\ \text{}& \to a\end{array}$ |

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## Share and Cite

**MDPI and ACS Style**

Gonçalves, I.M.; Rocha, C.; Souza, R.R.; Coutinho, G.; Pereira, J.E.; Moita, A.S.; Moreira, A.L.N.; Lima, R.; Miranda, J.M.
Numerical Optimization of a Microchannel Geometry for Nanofluid Flow and Heat Dissipation Assessment. *Appl. Sci.* **2021**, *11*, 2440.
https://doi.org/10.3390/app11052440

**AMA Style**

Gonçalves IM, Rocha C, Souza RR, Coutinho G, Pereira JE, Moita AS, Moreira ALN, Lima R, Miranda JM.
Numerical Optimization of a Microchannel Geometry for Nanofluid Flow and Heat Dissipation Assessment. *Applied Sciences*. 2021; 11(5):2440.
https://doi.org/10.3390/app11052440

**Chicago/Turabian Style**

Gonçalves, Inês M., César Rocha, Reinaldo R. Souza, Gonçalo Coutinho, Jose E. Pereira, Ana S. Moita, António L. N. Moreira, Rui Lima, and João M. Miranda.
2021. "Numerical Optimization of a Microchannel Geometry for Nanofluid Flow and Heat Dissipation Assessment" *Applied Sciences* 11, no. 5: 2440.
https://doi.org/10.3390/app11052440