Analytical Approach for a Heat Transfer Process through Nanofluid over an Irregular Porous Radially Moving Sheet by Employing KKL Correlation with Magnetic and Radiation Effects: Applications to Thermal System
Abstract
:1. Introduction
2. Mathematical Modeling
2.1. Constitutive Equation
2.2. Basic Equations
2.3. Non-Dimensional Equations
2.4. Gradients
3. Analytic Solutions Methodology
4. Results and Discussion
5. Conclusions
- The suction and magnetic parameters accelerate the velocity in the first branch solution and decelerate in the second branch solution, whilst declining the temperature distribution in both branch solutions.
- The temperature profile uplifts due to the solid nanoparticle volume fraction in both solutions, while the velocity increases and decreases due to the solid nanoparticle volume fraction in the first and second branch solutions, respectively.
- The heat transfer and the friction factor increase due to the Casson parameter.
- The nanoparticle volume fraction augments the heat transfer and declines the friction factor.
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Nomenclature
Velocity of the shrinking disk (m/s) | |
Variable magnetic field strength (Tesla) | |
Constant magnetic field strength (Tesla) | |
Wall mass transfer velocity (m/s) | |
Wall temperature (K) | |
Arbitrary constants | |
Ambient temperature (K) | |
Fluid temperature (K) | |
Radiative heat flux | |
Constant reference temperature (K) | |
Dimensionless velocity | |
Magnetic parameter | |
Prandtl number | |
Mass transfer factor | |
Heat transfer | |
Skin friction coefficient | |
Reynolds number | |
Denotes the roots of the equation | |
Dimensionless velocity | |
Radiation parameter | |
Relative factor of the power-law index | |
Mean absorption coefficient (1/m) | |
Thermal conductivity (W/(m·K)) | |
Velocity components along the rb- and zb-axes (m/s) | |
Cylindrical coordinates (m) | |
Specific heat at constant pressure (J/Kg·K) | |
and | Arbitrary constants |
Greek Symbols
Kinematic viscosity (m2/s) | |
and | Empirical functions |
Dimensionless temperature | |
Non-Newtonian or Casson fluid parameter | |
Electrical conductivity (Ω−1m−1) | |
Dynamic viscosity (N·s/m2) | |
Pseudo-similarity variable | |
Density (kg/m3) | |
Stefan-Boltzmann constant (W/(m2·K4)) | |
Stream function | |
New similarity variable | |
Solid nanoparticle volume fraction |
Acronyms
KKL | Koo–Kleinstreuer–Li |
MHD | Magnetohydrodynamics |
EOF | Electro-osmotic forces |
Al2O3 | Alumina |
2D, 3D | Two and three-dimensional |
BCs | Boundary conditions |
BLT | Boundary layer thickness |
BLF | Boundary layer flow |
VD | Viscous dissipation |
TIR | Thermal interfacial resistance |
Subscripts
Nanoparticles | |
Nanofluid | |
Regular based-fluid | |
Wall boundary condition | |
Far-field condition |
Superscript
‘ | Derivative with respect to ξ |
Appendix A
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Physical Properties | Water | Al2O3 |
---|---|---|
(Wm−1K−1) | 0.613 | 25 |
(J kg−1 K−1) | 4179 | 765 |
(kg m−3) | 997.1 | 3970 |
0.05 | 1 × 10−10 | |
(nm) | - | 47 |
Pr | 6.2 | - |
Coefficient Values | Water-Al2O3 |
---|---|
52.813 | |
6.115 | |
0.695 | |
4.1 × 10−2 | |
0.176 | |
−2.98.198 | |
−34.532 | |
−3.922 | |
−0.235 | |
−0.999 |
3.5 | 4 | 4.5 | 5 | ||||
---|---|---|---|---|---|---|---|
0.5 | 0.5 | 0.025 | First solution | 2.58605 | 3.40613 | 4.09306 | 4.73029 |
Second solution | 1.17470 | 0.89187 | 0.74219 | 0.64220 | |||
1 | 0.5 | 0.025 | First solution | 4.66415 | 5.63891 | 6.55804 | 7.44685 |
Second solution | 0.97697 | 0.80809 | 0.69483 | 0.61190 | |||
2 | 0.5 | 0.025 | First solution | 6.60110 | 7.81896 | 8.99505 | 10.14620 |
Second solution | 0.92040 | 0.77704 | 0.67544 | 0.59881 | |||
5 | 0.5 | 0.025 | First solution | 8.50938 | 9.98435 | 11.42330 | 12.83980 |
Second solution | 0.89249 | 0.76064 | 0.66483 | 0.59149 | |||
∞ | 0.5 | 0.025 | First solution | 10.4065 | 12.1435 | 13.8436 | 15.5307 |
Second solution | 0.87575 | 0.75048 | 0.65812 | 0.58680 |
3.5 | 4 | 4.5 | 5 | ||||||
---|---|---|---|---|---|---|---|---|---|
0.5 | 0.5 | 0.025 | 5 | 1 | First solution | 2.82016 | 3.52056 | 4.10507 | 4.66170 |
Second solution | 2.61293 | 3.32299 | 3.99296 | 4.68492 | |||||
1 | 0.5 | 0.025 | 5 | 1 | First solution | 3.04693 | 3.63183 | 4.18821 | 4.72967 |
Second solution | 2.66077 | 3.44115 | 4.28216 | 5.18029 | |||||
2 | 0.5 | 0.025 | 5 | 1 | First solution | 3.13405 | 3.69660 | 4.24086 | 4.77445 |
Second solution | 2.70750 | 3.72440 | 4.87577 | 6.24120 | |||||
5 | 0.5 | 0.025 | 5 | 1 | First solution | 3.19054 | 3.74125 | 4.27820 | 4.80674 |
Second solution | 2.85264 | 4.31988 | 6.24757 | 9.17801 | |||||
∞ | 0.5 | 0.025 | 5 | 1 | First solution | 3.23132 | 3.77438 | 4.30633 | 4.83129 |
Second solution | 3.18356 | 5.90969 | 11.6249 | 39.6040 |
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Khan, U.; Zaib, A.; Ishak, A.; Waini, I.; Raizah, Z.; Galal, A.M. Analytical Approach for a Heat Transfer Process through Nanofluid over an Irregular Porous Radially Moving Sheet by Employing KKL Correlation with Magnetic and Radiation Effects: Applications to Thermal System. Micromachines 2022, 13, 1109. https://doi.org/10.3390/mi13071109
Khan U, Zaib A, Ishak A, Waini I, Raizah Z, Galal AM. Analytical Approach for a Heat Transfer Process through Nanofluid over an Irregular Porous Radially Moving Sheet by Employing KKL Correlation with Magnetic and Radiation Effects: Applications to Thermal System. Micromachines. 2022; 13(7):1109. https://doi.org/10.3390/mi13071109
Chicago/Turabian StyleKhan, Umair, Aurang Zaib, Anuar Ishak, Iskandar Waini, Zehba Raizah, and Ahmed M. Galal. 2022. "Analytical Approach for a Heat Transfer Process through Nanofluid over an Irregular Porous Radially Moving Sheet by Employing KKL Correlation with Magnetic and Radiation Effects: Applications to Thermal System" Micromachines 13, no. 7: 1109. https://doi.org/10.3390/mi13071109