# Constructal Optimizations of Line-to-Line Vascular Channels with Turbulent Convection Heat Transfer

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Geometric Model of LVCs

_{i}, and the vertical distance is H

_{i}. The temperature of fluid is T

_{in}(K) at the entrance and T

_{out}(K) at the outlet.

^{−1}), the density is $\rho $ (kg·m

^{−3}), the kinematic viscosity is $\nu $ (m·s

^{−2}), and the mass flow rate is $\dot{m}$ (kg·s

^{−1}) (${\dot{m}}_{i}{}_{+1}=2{\dot{m}}_{i}$). The heat flow on the each axially uniform channel of circular cross-section surface is fixed per unit length, i.e., the linear heat flux ${q}^{\prime}$ (W·m

^{−1}) of channel is taken as a constant. The flow is assumed to be fully developed turbulence. The local pressure losses at the junctions of LVCs are negligible.

## 3. Constructal Optimizations of LVCs with Minimum EGR

#### 3.1. Constructal Optimizations of the First Order LVCs

#### 3.2. Constructal Optimizations of the Second Order LVCs

#### 3.3. Constructal Optimizations of the Third and Higher Order LVCs

## 4. Effects of the Dimensionless Mass Flow Rate on Constructal Optimizations, Dimensionless Total Entropy Generation Rate and EGN

#### 4.1. Effects of the Dimensionless Mass Flow Rate on Constructal Optimizations

#### 4.2. Effects of the Dimensionless Mass Flow Rate on the Dimensionless Total Entropy Generation Rate

#### 4.3. Effects of the Dimensionless Mass Flow Rate on EGN

## 5. Conclusions

- (1)
- The dimensionless total entropy generation rate of LVCs with any order can be significantly decreased by optimizing the angles of LVCs. From the first to fifth order, the dimensionless total entropy generation rate of LVCs with optimal angles were 10.65%, 24.54%, 43.75%, 66.99% and 93.67% smaller than those with fixed angles ($\alpha =45.0\xb0$), respectively. As the order of LVCs is higher, the dimensionless total entropy generation rate of LVCs decreases significantly more.
- (2)
- Based on the minimum dimensionless total entropy generation rate, as the dimensionless mass flow rate increases, the optimal angles of LVCs with any order remain unchanged first, then the optimal angles of LVCs at the entrance increase, and the other optimal angles of LVCs decrease continuously and finally tend to respective stable values. The optimal angles of LVCs continue to increase from the entrance to the outlet, i.e., the LVCs with a certain order gradually spread out from the root to the crown.
- (3)
- As the dimensionless mass flow rate increases, the dimensionless total entropy generation rate and EGN of LVCs with turbulent convection heat transfer decrease first and then increase sharply. There is optimal dimensionless mass flow rate $M$ can make the dimensionless total entropy generation rate and EGN of LVCs with any order obtain their respective minimums.
- (4)
- The dimensionless total entropy generation rate of LVCs increases gradually as the order of LVC increases for the same dimensionless mass flow rate $M$. When the dimensionless mass flow rate $M$ is less than 2, the EGN of LVCs increases as the order of LVC increases; however, when the dimensionless mass flow rate $M$ is greater than 2, this is simply reversed.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclatures

$\mathrm{A}$ | areas (m^{2}) |

${B}_{0}$ | coefficient |

${B}_{0}^{\prime}$ | coefficient |

D | diameter (m) |

d | distance between adjacent outlets (m) |

H | height (m) |

k | thermal conductivity (W·(m·K)^{−1}) |

L | length (m) |

$M$ | dimensionless mass flow rate |

$\dot{m}$ | mass flow rate (kg·s^{−1}) |

${N}_{\mathrm{s}}$ | dimensionless entropy generation number |

${q}^{\prime}$ | linear heat flux (W·m^{−1}) |

${\dot{S}}_{gen}$ | entropy generation rate (J·(K·s)^{−1}) |

${\tilde{S}}_{gen}$ | dimensionless entropy generation rate |

T | temperature (K) |

V | volume (m^{3}) |

Greek letters | |

$\alpha $ | angel (°) |

$\rho $ | density (kg·m^{−3}) |

$\nu $ | kinematic viscosity (m·s^{−2}) |

Superscripts | |

~ | dimensionless |

${}^{*}$ | transform of physical quantity |

Subscripts | |

i | channel rank |

in | inlet |

n | number of construction orders |

out | outlet |

opt | optimal |

Abbreviation | |

EGN | entropy generation number |

EGR | entropy generation rate |

LVC | line-to-line vascular channel |

Nu | Nusselt number |

Pr | Prandtl number |

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**Figure 3.**The third order LVCs with turbulent convection heat transfer. The numbers 1–3 in the figure express the (1–3)th-level channels.

**Figure 4.**The fourth order LVCs with turbulent convection heat transfer. The numbers 1–4 in the figure express the (1–4)th-level channels.

**Figure 6.**Effects of the dimensionless mass flow rate on the dimensionless total entropy generation rate.

${\mathit{\alpha}}_{\mathbf{opt}\mathit{,}0}/\xb0$ | ${\mathit{\alpha}}_{\mathbf{opt}\mathit{,}1}/\xb0$ | ${\mathit{\alpha}}_{\mathbf{opt}\mathit{,}2}/\xb0$ | ${\mathit{\alpha}}_{\mathbf{opt}\mathit{,}3}/\xb0$ | ${\mathit{\alpha}}_{\mathbf{opt}\mathit{,}4}/\xb0$ | ${\mathit{\alpha}}_{\mathbf{opt}\mathit{,}5}/\xb0$ | ||
---|---|---|---|---|---|---|---|

$M=0.1$ | $n=1$ | 78.0 | 21.3 | ||||

$n=2$ | 87.0 | 78.0 | 8.6 | ||||

$n=3$ | 88.0 | 86.0 | 77.8 | 5.6 | |||

$n=4$ | 88.6 | 87.8 | 86.0 | 77.0 | 3.6 | ||

$n=5$ | 88.8 | 88.8 | 88.8 | 87.0 | 77.8 | 1.8 | |

$M=1$ | $n=1$ | 72.6 | 27.2 | ||||

$n=2$ | 81.6 | 69.1 | 17.2 | ||||

$n=3$ | 86.2 | 80.6 | 66.1 | 11.6 | |||

$n=4$ | 88.2 | 86.2 | 79.1 | 61.8 | 9.1 | ||

$n=5$ | 88.8 | 88.8 | 85.2 | 77.1 | 61.6 | 6.6 | |

$M=5$ | $n=1$ | 49.9 | 42.9 | ||||

$n=2$ | 60.0 | 47.2 | 40.6 | ||||

$n=3$ | 60.8 | 54.4 | 48 | 39.8 | |||

$n=4$ | 66.2 | 59.0 | 52.2 | 46.2 | 39.4 | ||

$n=5$ | 72.6 | 65.4 | 59.0 | 52.2 | 46.2 | 37.6 |

$\mathit{M}=0.1$ | $\mathit{M}=1$ | $\mathit{M}=5$ | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

${\mathit{\alpha}}_{\mathit{i}}=45\xb0$ | ${\mathit{\alpha}}_{\mathit{i}}=60\xb0$ | ${\mathit{\alpha}}_{\mathit{i}}=75\xb0$ | ${\mathit{\alpha}}_{\mathit{i}}={\mathit{\alpha}}_{\mathit{o}\mathit{p}\mathit{t},\mathit{i}}$ | ${\mathit{\alpha}}_{\mathit{i}}=45\xb0$ | ${\mathit{\alpha}}_{\mathit{i}}=60\xb0$ | ${\mathit{\alpha}}_{\mathit{i}}=75\xb0$ | ${\mathit{\alpha}}_{\mathit{i}}={\mathit{\alpha}}_{\mathit{o}\mathit{p}\mathit{t},\mathit{i}}$ | ${\mathit{\alpha}}_{\mathit{i}}=45\xb0$ | ${\mathit{\alpha}}_{\mathit{i}}=60\xb0$ | ${\mathit{\alpha}}_{\mathit{i}}=75\xb0$ | ${\mathit{\alpha}}_{\mathit{i}}={\mathit{\alpha}}_{\mathit{o}\mathit{p}\mathit{t},\mathit{i}}$ | |

${\tilde{S}}^{*}{}_{gen,1}$ | 23.918 | 24.973 | 29.448 | 21.105 | 4.059 | 4.301 | 5.539 | 3.701 | 25.373 | 32.158 | 80.325 | 25.215 |

${\tilde{S}}^{*}{}_{gen,2}$ | 38.356 | 40.048 | 47.223 | 27.710 | 6.468 | 6.843 | 8.747 | 5.193 | 36.880 | 46.706 | 116.439 | 36.407 |

${\tilde{S}}^{*}{}_{gen,3}$ | 58.785 | 61.377 | 72.374 | 33.715 | 9.788 | 10.330 | 13.003 | 6.809 | 45.311 | 57.264 | 142.027 | 44.202 |

${\tilde{S}}^{*}{}_{gen,4}$ | 87.645 | 91.510 | 107.905 | 39.582 | 14.417 | 15.175 | 18.810 | 8.633 | 51.473 | 64.840 | 159.503 | 49.689 |

${\tilde{S}}^{*}{}_{gen,5}$ | 128.261 | 133.917 | 157.910 | 45.052 | 20.888 | 21.939 | 26.845 | 10.785 | 56.317 | 70.612 | 171.657 | 53.726 |

$\mathit{M}=0.1$ | $\mathit{M}=1$ | $\mathit{M}=5$ | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

${\mathit{\alpha}}_{\mathit{i}}=45\xb0$ | ${\mathit{\alpha}}_{\mathit{i}}=60\xb0$ | ${\mathit{\alpha}}_{\mathit{i}}=75\xb0$ | ${\mathit{\alpha}}_{\mathit{i}}={\mathit{\alpha}}_{\mathit{o}\mathit{p}\mathit{t},\mathit{i}}$ | ${\mathit{\alpha}}_{\mathit{i}}=45\xb0$ | ${\mathit{\alpha}}_{\mathit{i}}=60\xb0$ | ${\mathit{\alpha}}_{\mathit{i}}=75\xb0$ | ${\mathit{\alpha}}_{\mathit{i}}={\mathit{\alpha}}_{\mathit{o}\mathit{p}\mathit{t},\mathit{i}}$ | ${\mathit{\alpha}}_{\mathit{i}}=45\xb0$ | ${\mathit{\alpha}}_{\mathit{i}}=60\xb0$ | ${\mathit{\alpha}}_{\mathit{i}}=75\xb0$ | ${\mathit{\alpha}}_{\mathit{i}}={\mathit{\alpha}}_{\mathit{o}\mathit{p}\mathit{t},\mathit{i}}$ | |

${N}_{s}{}_{,1}$ | 10.357 | 10.063 | 9.017 | 4.807 | 1.758 | 1.733 | 1.696 | 1.585 | 11.528 | 13.553 | 25.672 | 10.987 |

${N}_{s}{}_{,2}$ | 11.960 | 11.621 | 10.412 | 5.656 | 2.017 | 1.986 | 1.928 | 1.711 | 11.499 | 12.958 | 24.594 | 10.937 |

${N}_{s}{}_{,3}$ | 14.230 | 13.826 | 12.388 | 5.904 | 2.369 | 2.327 | 2.226 | 1.871 | 10.968 | 12.910 | 24.311 | 10.089 |

${N}_{s}{}_{,4}$ | 17.253 | 16.764 | 15.020 | 6.303 | 2.838 | 2.780 | 2.618 | 2.061 | 10.133 | 11.878 | 22.202 | 8.217 |

${N}_{s}{}_{,5}$ | 21.209 | 20.608 | 18.464 | 6.579 | 3.454 | 3.376 | 3.139 | 2.217 | 9.313 | 10.866 | 20.071 | 6.528 |

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**MDPI and ACS Style**

Lin, D.; Xie, Z.; Nan, G.; Jiang, P.; Ge, Y.
Constructal Optimizations of Line-to-Line Vascular Channels with Turbulent Convection Heat Transfer. *Entropy* **2022**, *24*, 999.
https://doi.org/10.3390/e24070999

**AMA Style**

Lin D, Xie Z, Nan G, Jiang P, Ge Y.
Constructal Optimizations of Line-to-Line Vascular Channels with Turbulent Convection Heat Transfer. *Entropy*. 2022; 24(7):999.
https://doi.org/10.3390/e24070999

**Chicago/Turabian Style**

Lin, Daoguang, Zhihui Xie, Gang Nan, Pan Jiang, and Yanlin Ge.
2022. "Constructal Optimizations of Line-to-Line Vascular Channels with Turbulent Convection Heat Transfer" *Entropy* 24, no. 7: 999.
https://doi.org/10.3390/e24070999