# Thermal Convection in a Heated-Block Duct with Periodic Boundary Conditions by Element-by-Element Treatment

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Formulation

#### 2.1. Conservation Equations

#### 2.2. Projection Method

- 1.
- Step 1

- 2.
- Step 2

- 3.
- Step 3

- 4.
- Step 4

## 3. Results and Discussion

#### 3.1. Mesh Independence Test and Model Validation

#### 3.2. The Arrangement Geometries for Periodic Boundary Conditions

#### 3.3. Time-Averaged Nusselt Number on Heated Blocks

#### 3.4. Streamlined Patterns and Temperature Contours

#### 3.5. Friction Factor Enhancement, Nusselt Number Enhancement and Thermal Performance Coefficient

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

A | duct cross-section area (m^{2}) |

A | diffusion matrix in energy equation |

CPU | central processing unit |

C_{d} | drag coefficient ($=\left({\mathrm{F}}_{\mathrm{f}\mathrm{x}}+{\mathrm{F}}_{\mathrm{p}\mathrm{x}}\right)/\left(0.5\mathsf{\rho}{\mathrm{u}}_{\infty}^{2}\mathrm{l}\right)$) |

d_{h} | hydraulic diameter (m) (=4A/P_{w}) |

EBE | element-by-element |

FEM | finite element method |

${\mathrm{F}}_{\mathrm{f}\mathrm{x}}$ | friction drag per unit depth in x direction |

${\mathrm{F}}_{\mathrm{p}\mathrm{x}}$ | pressure drag per unit depth in x direction |

f friction factor | ($\u25b3\mathrm{p}/(0.5\mathsf{\rho}{\mathrm{u}}_{\infty}^{2})\cdot {\mathrm{d}}_{\mathrm{h}}/\mathrm{L}$) |

H | duct height (m) |

H | pressure gradient matrix or divergence matrix |

h | convective coefficient (W/m^{2}-°C) |

K | convection matrix |

L | duct length (m) |

l | block width |

M | mass matrix |

n | number of calculation |

Nu | local Nusselt number (=hl/k) |

$\overline{\mathrm{Nu}}$ | $\mathrm{time}\text{-}\mathrm{mean}\mathrm{Nusselt}\mathrm{number}(={\displaystyle \int \mathrm{N}\mathrm{u}\mathrm{d}\mathrm{t}/{\displaystyle \int \mathrm{d}\mathrm{t}}}$) |

$<\mathrm{Nu}>$ | $\mathrm{area}\mathrm{average}\mathrm{of}\mathrm{time}\text{-}\mathrm{mean}\mathrm{Nusselt}\mathrm{number}(={\displaystyle \int \overline{\mathrm{N}}\mathrm{u}\mathrm{d}\mathrm{A}/{\displaystyle \int \mathrm{d}\mathrm{A}}}$) |

p* | pressure (kPa) |

p | pressure of the node |

p | $\mathrm{dimensionless}\mathrm{pressure}(={\mathrm{p}}^{*}/(\mathsf{\rho}{\mathrm{u}}_{\infty}^{2})$) |

Pr | Prandtl number (=ν/α) |

P_{w} | wetted perimeter (m) |

PCG | preconditioned conjugate gradient |

Re | Reynolds number (=u_{∞}H/ν) |

S | diffusion matrix in momentum equation |

St | Strouhal number |

t* | time (s) |

t | dimensionless time (t*/(l/u_{∞})) |

T | temperature (°C) |

${\mathrm{T}}_{\infty}$ | reference temperature (°C) |

u | $\mathrm{dimensionless}\mathrm{horizontal}\mathrm{velocity}(={\mathrm{u}}^{*}/{\mathrm{u}}_{\infty}$) |

u_{∞} | the cross-section mean velocity (m/s) |

u | velocity vector at the node |

v | $\mathrm{dimensionless}\mathrm{vertical}\mathrm{speed}(={\mathrm{v}}^{*}/{\mathrm{u}}_{\infty}$) |

x | $\mathrm{dimensionless}\mathrm{horizontal}\mathrm{coordinate}(={\mathrm{x}}^{*}/\mathrm{l}$) |

y | $\mathrm{dimensionless}\mathrm{vertical}\mathrm{coordinate}(={\mathrm{y}}^{*}/\mathrm{l}$) |

Δt | dimensionless time step size |

Subscripts | |

w | block surface |

0 | without rectangular cylinder |

Superscript | |

* | dimensional variables |

Greeks | |

$\mathsf{\alpha}$ | thermal diffusivity (m^{2}/s) |

$\mathsf{\eta}$ | thermal performance |

$\mathsf{\nu}$ | kinematic viscosity coefficient (m^{2}/s) |

$\mathsf{\rho}$ | density (kg/m^{3}) |

$\mathsf{\varphi}$ | $\mathrm{dimensionless}\mathrm{temperature}(=(\mathrm{T}-{\mathrm{T}}_{\infty})/({\mathrm{T}}_{\mathrm{W}}-{\mathrm{T}}_{\infty})$) |

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**Figure 1.**The geometries for periodic boundary conditions (

**a**) Case 1: with a rectangular cylinder above every block; (

**b**) Case 2: with a rectangular cylinder above the first block for every two blocks; (

**c**) Case 3: with a rectangular cylinder above the first block for every three blocks.

**Figure 2.**Variation in time-mean Nusselt number on a heated block for Case 1 at Re = (

**a**) 100; (

**b**) 175; (

**c**) 250. Numbers: the point along the heated block surface.

**Figure 3.**Variation in time-mean Nusselt number on heated blocks for Case 2 at Re = (

**a**) 100; (

**b**) 175; (

**c**) 250. Numbers: the point along the heated blocks surfaces.

**Figure 4.**Variation in time-mean Nusselt number on heated blocks for Case 3 at Re = (

**a**) 100; (

**b**) 175; (

**c**) 250. Numbers: the point along the heated blocks surfaces.

**Figure 5.**Streamline patterns for periodic boundary conditions on a heated block for Case 1 at Re = (

**a**) 100; (

**b**) 175; (

**c**) 250. Red: recangular cylinder; Black: heated block.

**Figure 6.**Streamline patterns for periodic boundary conditions on a heated block for Case 2 at Re = (

**a**) 100; (

**b**) 175; (

**c**) 250.

**Figure 7.**Streamline patterns for periodic boundary conditions on heated blocks for Case 3 at Re = (

**a**) 100; (

**b**) 175; (

**c**) 250.

**Figure 8.**Temperature contours for periodic boundary conditions on a heated block for Case 1 at Re = (

**a**) 100; (

**b**) 175; (

**c**) 250.

**Figure 9.**Temperature contours for periodic boundary conditions on heated blocks for Case 2 at Re = (

**a**) 100; (

**b**) 175; (

**c**) 250.

**Figure 10.**Temperature contours for periodic boundary conditions on heated blocks for Case 3 at Re = (

**a**) 100; (

**b**) 175; (

**c**) 250.

**Figure 11.**(

**a**) Variation in friction factor enhancement; (

**b**) Variation in Nusselt number enhancement with Reynolds number for periodic boundary conditions with various arrangements of rectangular cylinders.

**Figure 12.**Variation in thermal performance with Reynolds number for periodic boundary conditions with various arrangements of rectangular cylinders.

Case | Mesh |
---|---|

Without Rectangular cylinder | Mesh 1 (element number: 1722; node number: 1621) Mesh 2 (element number: 3314; node number: 3164) Mesh 3 (element number: 4002; node number: 3836) |

1 | Mesh 1 (element number: 727; node number: 700) Mesh 2 (element number: 1451; node number: 1407) Mesh 3 (element number: 2176; node number: 2114) |

2 | Mesh 1 (element number: 1454; node number: 1400) Mesh 2 (element number: 2902; node number: 2814) Mesh 3 (element number: 4352; node number: 4228) |

3 | Mesh 1 (element number: 2181; node number: 2100) Mesh 2 (element number: 4353; node number: 4221) Mesh 3 (element number: 6528; node number: 6342) |

Murata et al. [25] | Present Paper | |
---|---|---|

Strouhal number | 0.30 | 0.31 |

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**MDPI and ACS Style**

Jue, T.-C.; Wu, H.-W.; Hsueh, Y.-C.; Guo, Z.-W.
Thermal Convection in a Heated-Block Duct with Periodic Boundary Conditions by Element-by-Element Treatment. *Inventions* **2023**, *8*, 97.
https://doi.org/10.3390/inventions8040097

**AMA Style**

Jue T-C, Wu H-W, Hsueh Y-C, Guo Z-W.
Thermal Convection in a Heated-Block Duct with Periodic Boundary Conditions by Element-by-Element Treatment. *Inventions*. 2023; 8(4):97.
https://doi.org/10.3390/inventions8040097

**Chicago/Turabian Style**

Jue, Tswen-Chyuan, Horng-Wen Wu, Ying-Chien Hsueh, and Zhi-Wei Guo.
2023. "Thermal Convection in a Heated-Block Duct with Periodic Boundary Conditions by Element-by-Element Treatment" *Inventions* 8, no. 4: 97.
https://doi.org/10.3390/inventions8040097