Thermal Convection in a Heated-Block Duct with Periodic Boundary Conditions by Element-by-Element Treatment
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 (m2) |
A | diffusion matrix in energy equation |
CPU | central processing unit |
Cd | drag coefficient () |
dh | hydraulic diameter (m) (=4A/Pw) |
EBE | element-by-element |
FEM | finite element method |
friction drag per unit depth in x direction | |
pressure drag per unit depth in x direction | |
f friction factor | () |
H | duct height (m) |
H | pressure gradient matrix or divergence matrix |
h | convective coefficient (W/m2-°C) |
K | convection matrix |
L | duct length (m) |
l | block width |
M | mass matrix |
n | number of calculation |
Nu | local Nusselt number (=hl/k) |
) | |
) | |
p* | pressure (kPa) |
p | pressure of the node |
p | ) |
Pr | Prandtl number (=ν/α) |
Pw | 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) |
reference temperature (°C) | |
u | ) |
u∞ | the cross-section mean velocity (m/s) |
u | velocity vector at the node |
v | ) |
x | ) |
y | ) |
Δt | dimensionless time step size |
Subscripts | |
w | block surface |
0 | without rectangular cylinder |
Superscript | |
* | dimensional variables |
Greeks | |
thermal diffusivity (m2/s) | |
thermal performance | |
kinematic viscosity coefficient (m2/s) | |
density (kg/m3) | |
) |
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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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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
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 StyleJue, 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