Numerical Study of the Magnetic Field Effect on Ferromagnetic Fluid Flow and Heat Transfer in a Square Porous Cavity
Abstract
:1. Introduction
2. Problem Definition
3. Mathematical Modeling
- at , , , , ,
- at inlet , , ;
- at , , , , ;
- at , , , , ;
- at , , , , ;
- at , , , , ,
- at inlet , , .
- at , , , , .
- at , , , , .
- at , , , , .
4. Numerical Method
- -
- The total time interval, [0, T], is divided into a number of time steps, namely, , with a time-step of length .
- -
- For temperature and concentration, each interval, , was divided into a number of subintervals, i.e., .
- -
- The Courant–Friedrichs–Lewy stability condition (CFL < 1) is used to accomplish the time step-size adaptation.
- -
- The time-step size of the pressure was taken to be larger than the time-step sizes of temperature and concentration.
- -
- Calculating the pressure implicitly by coupling the continuity and momentum equations.
- -
- Compute the velocity explicitly.
- -
- Solve energy and concentration equations implicitly.
- -
- Update porosity, permeability, and density.
- -
- The pressure time-step, is taken as an initial time step for both temperature and concentration equations.
- -
- After that, the conditions and are examined. If one of them is satisfied, the temperature/concentration time-step is divided into smaller steps.
- -
- Therefore, the conditions and are recalculated based on the new time steps and so on, until satisfying the conditions and .
5. Results and Discussion
5.1. Results for Case 1
5.2. Results for Case 2
6. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Nomenclature
A | [m] | half of width of the magnet, |
[A m−1] | constants depend on the type of the ferromagnetic material, | |
b | [m] | half of height of the magnet, |
[Tesla] | residual magnetization, | |
[m m−3] | ferrofluid concentration, | |
[m m−3] | ferrofluid initial concentration, | |
[ | heat capacity, | |
D | [m2 s−1] | diffusion coefficient, |
[N] | external magnetic force, | |
[m s−2] | gravitation acceleration, | |
h | [ | thermal conductivity, |
H | [A m−1] | magnetic field strength, |
K | [m2] | permeability of the porous medium, |
L | [m] | distance between the poles of the magnet, |
Nu | [-] | Nusselt number, |
[K] | temperature, | |
[K] | reference temperature, | |
[Pa] | fluid pressure, | |
[kg s−1] | external mass flow rate, | |
[ | rate of change of particle volume of a source/sink term, | |
[ | heat source term, | |
[ | fluid velocity vector, | |
[m] | Cartesian coordinates, | |
[K−1] | thermal expansion coefficient, | |
solutal expansion coefficient, | ||
[s] | time step for the loop k, | |
[s] | time step for the loop l, | |
[s] | time step for the loop m, | |
[-] | porosity of porous media, | |
[ | density of fluid mixture, | |
[ | density of pure water component, | |
[ | density of ferro-particles component, | |
[ | density of solid phase, | |
[m s−2] | viscosity of water-magnetic-particles mixture, | |
[m s−2] | viscosity of water, | |
[N A−2] | magnetic permeability. |
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Parameter | Description | Value | Units |
---|---|---|---|
Constant | 104–105 | m A−1 | |
Constant | 10−6–10−5 | m A−1 | |
a, b | Half of width/height of the magnet | 0.02 | m |
Br | Residual magnetization | 0:0.2 | Tesla |
Initial concentration | 0 | - | |
Inlet concentration | 1 | - | |
Heat capacity | 800 | J/Kg K | |
D | Diffusion coefficient | 5 | /S |
Thermal conductivity of the solid | 0.718 | W/(/K) | |
Thermal conductivity of the ferrofluid | 0.6 | W/(/K) | |
Gravity acceleration | 9.81 | ||
K | Permeability | 100 | md |
Lp | Distance between poles | 2.4 | m |
Lin | Inlet width | 0.3 | m |
L | Cavity side length | 5 | m |
md | Millidarcy | 9.86923 | m2 |
Initial pressure | 1 | ||
Initial temperature | 300 | K | |
Inlet temperature | 360 | K | |
Inlet velocity | /s | ||
Thermal expansion coefficient | 0.005 | ||
Solute expansion coefficient | 0.001 | ||
Porosity | 0.3 | - | |
Water viscosity | 0.001 | Pa.s | |
Magnetic permeability | 1 | N·A−2 | |
Solid media density | 2500 | kg/ | |
Pure water density | 1000 | kg/ | |
Particles density | 8933 | kg/ |
k | ||
---|---|---|
50 | 5.0317 × 10−11 | 0.0198 |
100 | 6.9070 × 10−11 | 0.0099 |
300 | 3.0481 × 10−12 | 0.0033 |
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El-Amin, M.F.; Khaled, U.; Beroual, A. Numerical Study of the Magnetic Field Effect on Ferromagnetic Fluid Flow and Heat Transfer in a Square Porous Cavity. Energies 2018, 11, 3235. https://doi.org/10.3390/en11113235
El-Amin MF, Khaled U, Beroual A. Numerical Study of the Magnetic Field Effect on Ferromagnetic Fluid Flow and Heat Transfer in a Square Porous Cavity. Energies. 2018; 11(11):3235. https://doi.org/10.3390/en11113235
Chicago/Turabian StyleEl-Amin, Mohamed F., Usama Khaled, and Abderrahmane Beroual. 2018. "Numerical Study of the Magnetic Field Effect on Ferromagnetic Fluid Flow and Heat Transfer in a Square Porous Cavity" Energies 11, no. 11: 3235. https://doi.org/10.3390/en11113235