Asymmetrical Thermal Boundary Condition Influence on the Flow Structure and Heat Transfer Performance of Paramagnetic Fluid-Forced Convection in the Strong Magnetic Field
Abstract
:1. Introduction
2. Materials and Methods
2.1. Mathematical Model
2.1.1. Important Dimensionless Parameters
2.1.2. Dimensionless Conservation Equations
- -
- Stationary flow,
- -
- Three-dimensional flow,
- -
- Laminar flow,
- -
- Incompressible flow,
- -
- No additional mass or energy sources,
- -
- The thermo-magnetic force was treated as the body force,
- -
- Constant values of thermo-physical properties,
- -
- Paramagnetic and electrically non-conductive fluid.
2.1.3. Biot-Savart Law
2.1.4. Energy Budget
2.2. The Studied Case
- At the inlet: ; ,
- At the outlet: ,At the heated wall: ; ,
- At the adiabatic wall: ; ,
- In the center of the magnetic coil: .
2.3. Numerical Approach
3. Results and Discussion
4. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Nomenclature
Latin symbols | |
dimensionless magnetic induction vector ( ) (-) | |
magnetic induction vector (T) | |
Conv | convective term of energy equation (-) |
D | dimensionless diameter (-) |
diameter (m) | |
Diff | diffusive term of energy equation (-) |
infinitely small element of the coil (m) | |
G | dimensionless gravitational acceleration (-) |
gravitational acceleration (m/s2) | |
Grashof number (-) | |
convective heat transfer coefficient (W/(m2K)) | |
electrical current (A) | |
thermal conductivity (W/(mK)) | |
dimensionless pipe length ( (-) | |
pipe length (m) | |
Nu | Nusselt number (-) |
dimensionless pressure () (-) | |
pressure (Pa) | |
Prandtl number (-) | |
q | heat flux magnitude (W/m2) |
r | position vector (m) |
Reynolds number (-) | |
Ri | Richardson number (-) |
T | temperature (K) |
dimensionless flow velocity vector (, ) (-) | |
flow velocity vector (m/s) | |
u | velocity magnitude (m/s) |
y | dimensionless radial distance (y-coordinate) (-) |
dimensionless axial distance (z-coordinate) (-) | |
Greek symbols | |
thermal diffusivity (m2/s) | |
thermal expansion coefficient (1/K) | |
dimensionless thermal expansion coefficient (-) | |
dimensionless temperature ( ) (-) | |
N | dimensionless kinematic viscosity (-) |
magnetic permeability (H/m) | |
fluid kinematic viscosity (m2/s) | |
density (kg/m3) | |
dimensionless density () (-) | |
volumetric magnetic susceptibility (-) | |
dimensionless vorticity magnitude () (-) | |
vorticity magnitude (1/s) | |
Subscripts | |
0 | reference value |
b | bulk |
C | cold (inlet) fluid |
center | center |
coil | coil |
f | fluid |
H | heated (hot) wall |
m | magnetic |
max | maximal |
n | normal |
p | pipe |
v | vacuum |
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Property | Symbol | Unit | Value |
---|---|---|---|
Pipe diameter | dp | m | 0.01 |
Coil diameter | dcoil | m | 0.02 |
Density | ρ0 | kg/m3 | 1.225 |
Kinematic viscosity | ν0 | m2/s | 1.461 × 10−5 |
Specific heat | cp | J/(kg·K) | 1006.43 |
Thermal conductivity | k0 | W/(m·K) | 2.42 × 10−2 |
Thermal expansion coefficient | β0 | K−1 | 3.33 × 10−3 |
Volumetric magnetic susceptibility | χ0 | (-) | 3.77 × 10−7 |
Magnetic permeability of the vacuum | μv | H/m | 4π × 10−7 |
Electric current | i | A | 159,155 |
Magnetic induction in the center of the system | bcenter | T | 10 |
Pr | (Nu–Nu0)/Nu0 (%) | |
---|---|---|
Bottom | Top | |
0.7 | 24.48 | 24.73 |
10 | 14.94 | 14.91 |
100 | 9.82 | 9.82 |
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Pleskacz, L.; Fornalik-Wajs, E.; Gurgul, S. Asymmetrical Thermal Boundary Condition Influence on the Flow Structure and Heat Transfer Performance of Paramagnetic Fluid-Forced Convection in the Strong Magnetic Field. Fluids 2020, 5, 246. https://doi.org/10.3390/fluids5040246
Pleskacz L, Fornalik-Wajs E, Gurgul S. Asymmetrical Thermal Boundary Condition Influence on the Flow Structure and Heat Transfer Performance of Paramagnetic Fluid-Forced Convection in the Strong Magnetic Field. Fluids. 2020; 5(4):246. https://doi.org/10.3390/fluids5040246
Chicago/Turabian StylePleskacz, Lukasz, Elzbieta Fornalik-Wajs, and Sebastian Gurgul. 2020. "Asymmetrical Thermal Boundary Condition Influence on the Flow Structure and Heat Transfer Performance of Paramagnetic Fluid-Forced Convection in the Strong Magnetic Field" Fluids 5, no. 4: 246. https://doi.org/10.3390/fluids5040246