Numerical Analysis of Mixed Convective Heat Transfer from a Square Cylinder Utilizing Nanofluids with Multi-Phase Modelling Approach
Abstract
:1. Introduction
2. Review of Previous Work
3. Mathematical Formulation
3.1. Problem Description and Geometrical Configuration
3.2. Governing Equations
3.2.1. Single-Phase Model (SPM)
- Continuity equation:
- Momentum equation:
- Energy equation:
3.2.2. Nanofluids Modelling
- Effective density:
- Effective viscosity (Brinkman [37]):
- Effective specific heat (Xuan and Roetzel [38]):
- Effective thermal conductivity (Xie et al. [39]):
3.2.3. Multi-Phase Model (MPM)
- Continuity equation:
- Momentum equation:
- Energy equation:
- Volume fraction equation:
3.3. Boundary Conditions
4. Numerical Details
4.1. Grid Sensitivity Analysis and Code Verification
4.2. Numerical Method
5. Results and Discussions
- Reynolds number (Re) = 10, 30, 50, 80, 100, and 150.
- Richardson number (Ri) = 0, 0.5, and 1.
- Volume fraction of nanoparticles (φ) = 0, 1%, 3%, and 5%.
5.1. Fluid Flow Characteristics
5.1.1. Flow Patterns
5.1.2. Time-Averaged Pressure Coefficient
5.1.3. Time-Averaged Drag Coefficients
5.2. Heat Transfer Characteristics
5.2.1. Isotherms
5.2.2. Local and Mean Nusselt Number of the Cylinder
5.2.3. Thermal Performance
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Nomenclature
Notations | |
drag coefficient, | |
lift coefficient, | |
pressure coefficient, | |
specific heat capacity, J/kg-K | |
D | height of the cylinder, m |
E | heat transfer enhancement ratio, |
drag force, N | |
lift force, N | |
g | gravitational acceleration, m/s2 |
Gr | Grashof number, |
K | thermal conductivity of fluid, W/m-K |
length of upstream boundary, m | |
length of downstream boundary, m | |
height of the boundary from the cylinder, m | |
m, n | number of grids in x- and y-direction |
Nu | Nusselt number, |
p | dimensional pressure, N/m2 |
P | non-dimensional pressure, |
Pr | Prandtl number, |
thermal conductivity ratio, | |
Ra | Rayleigh number, Gr Pr |
Re | Reynolds number, |
Ri | Richardson number, |
t | dimensional time, s |
U | non-dimensional x-component of velocity, u/ |
u, v | dimensional x- and y-component of velocity, m/s |
reference velocity, m/s | |
V | non-dimensional y-component of velocity, v/ |
X | dimensionless horizontal distance, x/D |
Y | dimensionless vertical distance, y/D |
x, y | horizontal and vertical coordinate |
Greek symbols | |
Δ | largest grid size |
δ | smallest grid size |
μ | dynamic viscosity, kg/m-s |
ω | vorticity magnitude, 1/s |
φ | nanoparticles volume fraction in nanofluid |
ψ | stream function, m2/s |
ρ | fluid density, kg/m3 |
non-dimensional time, t/D | |
Θ | non-dimensional temperature, |
θ | dimensional temperature, K |
β | thermal expansion coefficient, °C |
Subscripts | |
0 | reference value |
avg | average |
bf | base fluid |
cr | critical |
D | downstream |
dr | drift |
e | effective |
f | fluid |
l | local value of variable |
M | mean value |
m | mixture |
nf | nanofluid |
np | nanoparticles |
nl | nano layer |
p | particle |
s | solid |
U | upstream |
w | wall |
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Property | Water [21] | Al2O3 [44,45] |
---|---|---|
ρ (Kg/m3) | 998.2 | 3790 |
(j/Kg-K) | 4182 | 765 |
K (W/m-K) | 0.6 | 40 |
μ (Kg/m-s) | 10.03 × 10−4 | - |
β (1/K) | 2.1 × 10−4 | 8.5 × 10−6 |
Grid Size | Ri = 0 | Ri = 1 | |||
---|---|---|---|---|---|
(m × n) | |||||
10 | 172 × 102 | 1.896 | 8.3245 | 7.584 | 9.8541 |
194 × 124 | 1.8152 | 8.1251 | 7.4879 | 9.8574 | |
260 × 190 | 1.7548 | 7.8951 | 7.3215 | 9.5411 | |
15 * | 222 × 112 | 1.7984 | 7.4614 | 5.6412 | 9.2448 |
244 × 134 * | 1.7843 | 7.4566 | 5.6332 | 9.2258 | |
310 × 200 | 1.7843 | 7.4564 | 5.6332 | 9.2256 | |
20 | 272 × 122 | 1.7784 | 7.5842 | 5.6311 | 9.2258 |
294 × 144 | 1.7842 | 7.4566 | 5.6332 | 9.2258 | |
360 × 210 | 1.7843 | 7.4548 | 5.6422 | 9.2255 | |
25 | 322 × 132 | 1.7821 | 7.4667 | 5.6411 | 9.2287 |
344 × 154 | 1.7833 | 7.4545 | 5.6612 | 9.2255 | |
410 × 220 | 1.7844 | 7.4566 | 5.6332 | 9.2258 |
Re | CD | ||||||||
Ri = 0 | Ri = 0.5 | Ri = 1.0 | |||||||
aa | bb | Error (%) | aa | bb | Error (%) | aa | bb | Error (%) | |
1 | 13.995 | 13.836 | 1.137 | 22.125 | 21.883 | 1.094 | 29.32 | 28.965 | 1.211 |
5 | 4.878 | 4.8 | 1.6 | 8.012 | 7.977 | 0.437 | 10.325 | 10.424 | 0.959 |
10 | 3.322 | 3.318 | 0.121 | 5.443 | 5.436 | 0.129 | 7.029 | 7.012 | 0.242 |
20 | 2.351 | 2.353 | 0.086 | 3.861 | 3.859 | 0.052 | 4.981 | 4.989 | 0.161 |
30 | 1.978 | 1.977 | 0.051 | 3.247 | 3.248 | 0.031 | 4.181 | 4.189 | 0.192 |
40 | 1.764 | 1.765 | 0.057 | 2.896 | 2.895 | 0.035 | 3.733 | 3.742 | 0.242 |
Re | NuM | ||||||||
Ri = 0 | Ri = 0.5 | Ri = 1.0 | |||||||
aa | bb | Error (%) | aa | bb | Error (%) | aa | bb | Error (%) | |
1 | 0.681 | 0.696 | 2.203 | 0.771 | 0.758 | 1.687 | 0.781 | 0.793 | 1.537 |
5 | 1.212 | 1.191 | 1.733 | 1.322 | 1.329 | 0.53 | 1.422 | 1.405 | 1.196 |
10 | 1.544 | 1.551 | 0.454 | 1.733 | 1.744 | 0.635 | 1.833 | 1.852 | 1.037 |
20 | 2.051 | 2.035 | 0.781 | 2.322 | 2.312 | 0.431 | 2.433 | 2.457 | 0.987 |
30 | 2.367 | 2.388 | 0.888 | 2.722 | 2.727 | 0.184 | 2.922 | 2.911 | 0.377 |
40 | 2.612 | 2.596 | 0.613 | 3.05 | 3.08 | 0.984 | 3.254 | 3.288 | 1.054 |
φ | NuM | ||||||||
Re = 10 | Re = 40 | ||||||||
aa | cc | Error (%) | aa | cc | Error (%) | ||||
0 | 3.811 | 3.81 | 0.027 | 7.101 | 7.1 | 0.015 | |||
1 | 3.829 | 3.824 | 0.131 | 7.11 | 7.104 | 0.085 | |||
3 | 3.944 | 3.937 | 0.178 | 7.33 | 7.343 | 0.178 | |||
5 | 4.158 | 4.166 | 0.193 | 7.822 | 7.812 | 0.128 |
Re | φ = 1% | φ = 3% | φ = 5% | ||||||
---|---|---|---|---|---|---|---|---|---|
MPM | SPM | Deviation (%) | MPM | SPM | Deviation (%) | MPM | SPM | Deviation (%) | |
10 | 3.392 | 3.381 | 0.326 | 3.585 | 3.55 | 0.986 | 3.789 | 3.768 | 0.558 |
30 | 5.415 | 5.393 | 0.408 | 5.716 | 5.687 | 0.51 | 6.033 | 5.986 | 0.786 |
50 | 6.84 | 6.818 | 0.323 | 7.193 | 7.127 | 0.927 | 7.565 | 7.496 | 0.921 |
80 | 8.331 | 8.325 | 0.073 | 8.718 | 8.702 | 0.184 | 9.124 | 9.1 | 0.264 |
100 | 9.037 | 9.021 | 0.178 | 9.46 | 9.453 | 0.075 | 9.847 | 9.883 | 0.365 |
150 | 10.821 | 10.773 | 0.446 | 11.386 | 11.277 | 0.967 | 11.991 | 11.877 | 0.96 |
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Rajpoot, R.S.; Dhinakaran, S.; Alam, M.M. Numerical Analysis of Mixed Convective Heat Transfer from a Square Cylinder Utilizing Nanofluids with Multi-Phase Modelling Approach. Energies 2021, 14, 5485. https://doi.org/10.3390/en14175485
Rajpoot RS, Dhinakaran S, Alam MM. Numerical Analysis of Mixed Convective Heat Transfer from a Square Cylinder Utilizing Nanofluids with Multi-Phase Modelling Approach. Energies. 2021; 14(17):5485. https://doi.org/10.3390/en14175485
Chicago/Turabian StyleRajpoot, Rajendra S., Shanmugam. Dhinakaran, and Md. Mahbub Alam. 2021. "Numerical Analysis of Mixed Convective Heat Transfer from a Square Cylinder Utilizing Nanofluids with Multi-Phase Modelling Approach" Energies 14, no. 17: 5485. https://doi.org/10.3390/en14175485