# Study on Flow and Heat Transfer Characteristics and Anti-Clogging Performance of Tree-Like Branching Microchannels

^{1}

^{2}

^{*}

## Abstract

**:**

_{b}/L

_{0}and fractal dimension D. Nu/Nu

_{s}, f/f

_{s}, and η are increased as m increases from 3 to 5. Furthermore, the tree-like microchannel network exhibits robustness for cooling gas turbine blades. A greater total number of branching levels and a higher Re number are advantageous for enhancing the anti-clogging performance of the tree-like branching microchannel.

## 1. Introduction

## 2. Methodology

#### 2.1. Physical Model

_{0}and L

_{k}are the channel length of the 0th and kth branching levels, d

_{0}and d

_{k}are the channel hydraulic diameters of the 0th and kth branching levels, respectively.

_{k}, except for the initial channel L

_{0}, can be split into the sum of the oblique channel length L

_{b}and the straight channel length L

_{ak}, namely, L

_{k}= L

_{b}+ L

_{ak}.

#### 2.2. Heat Transfer in the Tree-like Branching Networks

_{k}, L

_{b}, and L

_{ak}described before, that is, L

_{k}= L

_{b}+ L

_{ak}, Equation (9) can be simplified into Equation (10)

_{k}can be represented as

_{0}, under the identical heat transfer area, temperature variance and Nu number conditions, the total convection heat transfer is

_{h}/Q

_{hp}

_{1}is

_{h}to Q

_{hp}

_{1}is dependent on both α, β, and L

_{b}/L

_{0}. Thus, Equation (20) was calculated for N = 2, Δ = 2, D = 3, m = 3, 4, 5, and L

_{b}/L

_{0}= 0.1–1, and the results are illustrated in Figure 3a. Also, it was computed for Δ = 3, N = 2, L

_{b}/L

_{0}= 0.2344, m = 3, 4, 5, and D = 1–3.5, and the corresponding results are presented in Figure 3b. It is presented that the tree-like branching microchannels show a superior heat transfer performance than those of the conventional parallel microchannels. Moreover, the larger the ratio of L

_{b}/L

_{0}, the total number of the bifurcating levels m, or the fractal dimension D, the larger the heat transfer performance.

#### 2.3. Flow Resistance in the Tree-like Branching Networks

_{k}denotes the velocity in the channel with the kth branching level and c represents a constant. Based on the law of mass conservation, we can obtain

_{p}

_{2}represents the convection heat transfer area of the parallel net, d denotes the channel diameter, L equals L

_{0}, and n represents the number of channels. Assuming that the flow through parallel channel possesses the identical Nu number as that in the tree-like bifurcating channel, we can obtain hd = h

_{0}d

_{0}. Then, substituting Equations (17) and (28) into Equation (29) yields

_{p}

_{2}follows

_{p}

_{2}of is not only related to α and β, but also to the L

_{b}/L

_{0}.

_{b}/L

_{0}= 0.1–1. The results are shown in Figure 4a. It illustrates that the tree-like bifurcating network requires much lower pumping power than a parallel network. Additionally, as the ratio of L

_{b}/L

_{0}or the total number of branching levels m augments, the required pumping power also increases. Figure 4b exhibits the computed results of Equation (36) for Δ = 3, N = 2, L

_{b}/L

_{0}= 0.239, m = 3, 4, 5 and D = 1–3.5. It is evident that the necessary pumping power in the channel varies depending on the value of D, which represents the length fractal dimension. As the value of D and m increase, the requisite pumping power throughout the flow channel also increases.

## 3. Numerical Approach and Experimental Details

#### 3.1. Numerical Approach

^{2}, while the other walls were set to be adiabatic with non-slip velocity conditions.

^{−5}. The reliability of the computations depends strongly on the choice of the turbulence model, with each model being suitable for different flow patterns. In the case of branching and intersecting flow computations, the selection of an appropriate turbulent flow model is crucial. The SSG turbulence model has been shown to accurately predict the complex features and physics of secondary flow in branching microchannel networks by Shui et al. [29,30], and was therefore chosen for this study. Therefore, the SSG turbulence model was selected in this study. Moreover, a scalable wall function was applied to near-wall modeling.

#### 3.2. Experimental Details

^{2}. The temperature ratio varied between 0.68 and 0.89, depending on the flow and heating conditions.

#### 3.3. Data Reduction

_{in}and d

_{0}are the entrance velocity and hydraulic diameter, respectively.

_{w}and T

_{f}are the wall temperature and the coolant temperature, respectively. The local wall temperature (T

_{w}) used in Equation (39) can be read from the output of the J-type thermocouples. It was assumed that the air temperature rise along the flow duct is linear. The bulk mean temperature of air (T

_{f}) at the l position was calculated by the following equation

_{out}and T

_{in}are the air mean temperature of the outlet and inlet in the branching microchannel, respectively. l is the distance of the air flowing through the channel, and L is the centerline length from the inlet to outlet.

_{in}and p

_{out}are the pressure at the inlet and outlet of the channel, using differential pressure and adiabatic pressure sensors for measurement, respectively.

_{s}is the averaged Nu number in a smooth microchannel acquired from Stephan et al. [33] as

## 4. Results

#### 4.1. Experimental Results and Numerical Verification

^{0.25}for Re > 2300. It can be detected that the numerical values significantly increase with m across the studied flow conditions. However, the changing trend of f versus m for the experimental data in the laminar and transition flow regions appears irregular, possibly due to measurement errors. In the flow region where 4000 < Re < 11,000, the influence of m on the flow friction factor is minimal. Nonetheless, the flow friction factors at m = 5 and 4 display remarkable growth as the Re number continues to increase, which is owed to the local flow resistances rising significantly on account of the effect of the branch confluence. In general, the values of f in the tree-like bifurcating microchannels exhibit similar trends in variation to the classical correlation curves at Re < 11,000, namely, f decreases as Re increases. However, the results of the study show a quite different and increasing tendency after that.

#### 4.2. Flow and Heat Transfer in Tree-like Bifurcating Microchannels

#### 4.3. Thermal Enhancement Performance

_{s}tends to rise with the increasing of the Re number. The rising slope is steeper for Re < 5000, and becomes gentler for Re = 5000–20,000. The maximum Nu/Nu

_{s}is obtained for m = 5, while the lowest is for m = 3. For m = 3, 4, and 5, the increases in Nu

_{a}/Nu

_{s}values are approximately 0.20, 0.25, and 0.32 at the lowest Re number, and 0.56, 0.79, and 1.06 at the highest Re number, respectively.

_{s}initially decline with the increase of Re number, and then gradually rise. At Re = 10,000, the lowest values of 0.49, 0.67, and 0.80 are obtained for m = 3, 4, and 5, respectively.

#### 4.4. Anti-Clogging Performance

_{max}for the case without blockage is about 355.3 K, 353.3 K, and 355.3 K for m = 3, 4, 5 at Re = 10,000. Nevertheless, the maximum temperature rises to 450 K, 434.9 K, and 401.4 K for the case with BL = 1, and to 426.3 K, 409.7 K, and 399.2 K for the case with BL = 3. When the Re number increases to 20,000, the maximum temperatures for all the cases are decreased. In general, the predicted maximum temperature ascends up when the sub-channel is blocked, especially for the channels at the low-order branching level, which are higher than those in the high-order branching level. However, the phenomenon of maximum temperature changing with the branching level is more obvious in the branching microchannel for m = 3. The influence brought by the location of blockage on the temperature distribution declines with an increase in m and Re number. By combining Figure 19, it is clear that T

_{max}appears at two opposite corners of the region that cannot be covered by the microchannel network. Despite all the blocked scenarios, the local temperature of the covered region by the cooling tree-like branching microchannel network remains lower than the uncovered region due to the compensation effect of the adjacent branched channel.

## 5. Conclusions

_{b}/L

_{0}, total number of the branching levels m, or fractal dimension D are all observed to result in stronger heat transfer capability but also require a pumping power penalty.

_{s}increases, f/f

_{s}decreases initially until Re = 10,000 and then rises, and η ascends when Re < 15,000, and then gradually declines. Furthermore, Nu/Nu

_{s}, f/f

_{s}and η are increased when m increases from 3 to 5.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

c | constant | $\dot{Q}$ | the total heat removed by air |

D | ractal dimension of channel length distribution | Q_{h} | total convective heat transfer of bifurcating channels |

d_{0} | the 0th channel hydraulic diameters | Q_{hp1} | total convective heat transfer of parallel channels with the same heat transfer area as bifurcating channels and diameter of d_{0} |

d_{k} | the kth channel hydraulic diameters | ||

f | the Fanning friction factor | ||

f_{s} | friction factor in the smooth channel | ||

h_{k} | heat transfer coefficient of the kth level channel | Q_{hp2} | convective heat transfer of parallel channels |

L_{b} | the channel length of the oblique channel | S | the total heat transfer area of a tree-like net |

L_{0} | the channel length of the 0th branching level | S_{k} | the total heat transfer area of the kth level channel |

L_{k} | the channel length of the kth branching level | S_{p2} | total heat transfer area of parallel channels |

L | the centerline length from the channel inlet to the outlet | T_{in} | the air mean temperature of the inlet in the branching microchannel |

L_{ak} | the channel length of the straight channel after bifurcation | T_{out} | the air mean temperature of the outlet in the branching microchannel |

m | the total number of branching levels | $\dot{V}$ | channel volume flow rate of air |

Nu_{s} | the averaged Nu number in a smooth microchannel for a fully developed flow | v_{0} | velocity in the initial channel |

v_{k} | velocity in the kth level channel | ||

n | number of parallel channels | β | the ratio of the diameter of the channel at the (k + 1)th branch level versus the diameter of the channel at the kth branch level |

N | number of branches into which a single channel bifurcates | ||

P | pumping power of the tree-like channels | α | the ratio of the length of the channel at the (k + 1)th branch level versus the length of the channel at the kth branch level |

P_{p2} | pumping power of parallel channels | ||

Q | flow rate | $\u2206$ | fractal dimension of the hydraulic diameter distribution |

Q_{p2} | flow rate of parallel channels | ||

$\u2206{p}_{f}$ | The pressure drop of the tree-like channels | $\lambda $ | the mass-averaged thermal conductivity of air |

$\u2206{p}_{p2}$ | The pressure drop of the parallel channels | $\u2206T$ | temperature difference |

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**Figure 3.**Plot of Q

_{h}/Q

_{hp}

_{1}versus (

**a**) L

_{b}/L

_{0}at N = 2, Δ = 2, D = 3, m = 3, 4, 5, and (

**b**) D at Δ = 3, N = 2, L

_{b}/L

_{0}= 0.2344, m = 3, 4, 5.

**Figure 4.**Plot of P/P

_{p}

_{2}versus (

**a**) L

_{b}/L

_{0}at Δ = 3, N = 2, D = 2, m = 3, 4, 5, and (

**b**) D at N = 2, Δ = 3, L

_{b}/L

_{0}= 0.239, m = 3, 4, 5.

**Figure 8.**The layout of thermocouples measuring points (MP) for (

**a**) m = 3, (

**b**) m = 4, and (

**c**) m = 5.

**Figure 10.**Comparison of experimental results, numerical data, and correlations for the average Nu number of tree-like branching microchannels with m = 3, 4, 5.

**Figure 11.**Comparison of experimental results, numerical data, and correlations for the fiction factors of tree-like branching microchannels with m = 3, 4, 5.

**Figure 12.**Comparison between the theoretical values and experimental data of (

**a**) heat flow and (

**b**) pressure drop.

**Figure 13.**Velocity contour distribution over mid-plane (z = 0.5H) for (

**a**) m = 3, (

**b**) m = 4, and (

**c**) m = 5 at Re = 10,000.

**Figure 14.**Temperature contour distribution over mid-plane (z = 0.5H) for (

**a**) m = 3, (

**b**) m = 4, and (

**c**) m = 5 at Re = 10,000.

**Figure 18.**Velocity vector distribution over mid-plane (z = 0.5H) at (

**a**) BL = 2, (

**b**) BL = 3, and (

**c**) BL = 5 for m = 5.

**Figure 19.**Temperature distribution over mid-plane (z = 0.5H) at (

**a**) BL = 2, (

**b**) BL = 3, and (

**c**) BL = 5 for m = 5.

**Figure 20.**The variations in the maximum temperature and pressure drop of the tree-like branching microchannels with and without blockage at (

**a**) Re = 10,000 and (

**b**) Re = 20,000.

k | m = 3 | m = 4 | m = 5 | ||||||
---|---|---|---|---|---|---|---|---|---|

l_{k}/mm | d_{k}/mm | W_{k}/mm | l_{k}/mm | d_{k}/mm | W_{k}/mm | l_{k}/mm | d_{k}/mm | W_{k}/mm | |

0 | 50.67 | 3 | 3 | 44.56 | 3 | 3 | 40.64 | 3 | 3 |

1 | 43 | 2.38 | 1.97 | 35.37 | 2.38 | 1.97 | 32.25 | 2.38 | 1.97 |

2 | 34.44 | 1.89 | 1.38 | 28.07 | 1.89 | 1.38 | 25.6 | 1.89 | 1.38 |

3 | 27.85 | 1.5 | 1 | 22.28 | 1.5 | 1 | 20.31 | 1.5 | 1 |

4 | - | - | - | 17.68 | 1.19 | 0.74 | 16.12 | 1.19 | 0.74 |

5 | - | - | - | - | - | - | 12.8 | 0.945 | 0.56 |

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**MDPI and ACS Style**

Shui, L.; Hu, Z.; Song, H.; Zhai, Z.; Wang, J.
Study on Flow and Heat Transfer Characteristics and Anti-Clogging Performance of Tree-Like Branching Microchannels. *Energies* **2023**, *16*, 5531.
https://doi.org/10.3390/en16145531

**AMA Style**

Shui L, Hu Z, Song H, Zhai Z, Wang J.
Study on Flow and Heat Transfer Characteristics and Anti-Clogging Performance of Tree-Like Branching Microchannels. *Energies*. 2023; 16(14):5531.
https://doi.org/10.3390/en16145531

**Chicago/Turabian Style**

Shui, Linqi, Zhongkai Hu, Hang Song, Zhi Zhai, and Jiatao Wang.
2023. "Study on Flow and Heat Transfer Characteristics and Anti-Clogging Performance of Tree-Like Branching Microchannels" *Energies* 16, no. 14: 5531.
https://doi.org/10.3390/en16145531