Coupled Momentum and Heat Transport in Laminar Axisymmetric Pipe Flow of Ferrofluids in Non-Uniform Magnetic Fields: Theory and Simulation MS Candidate: Carlos F. Cruz-Fierro Advisor: Dr. Goran N. Jovanovic Chemical Engineering Department Oregon State University April 2, 2003 Revision 1 33680.05
Presentation Overview Objectives General concepts Transport in the presence of EM fields Simulation Results Conclusions 2
Objectives Show the necessary additions to conservation equations when a fluid is subject to electromagnetic fields Evaluate and interpret the effect of these fields in the particular case of transport of momentum and heat in the axisymmetric pipe flow of a water-based ferrofluid Extrapolate the simulation to predict the behavior of an hypothetical liquid metal-based ferrofluid under similar flow conditions 3
Transport Phenomena Physical quantity A moved through space and generated from or transformed into other physical quantities by known mechanisms Conservation equation - represents a balance of A in a control volume Constitutive relation - associates quantities of different nature; relates transport or generation with one or more driving forces 4
Maxwell Equations Set of four partial differential equations relating fields ( E , D , B , H ) and their sources. D B 0 q D B E H J t t 5
Ferrofluid Colloidal suspension of nanosize ferromagnetic particles, stabilized by surfactant High magnetic susceptibility magnetic particle Remains fluid at high fields surfactant Carrier fluid: water or layer hydrocarbon Current research in liquid- metal ferrofluids 6
Presentation Overview Objectives General concepts Transport in the presence of EM fields Simulation Results Conclusions 7
Mass Conservation Equation u 0 t No change in the presence of electromagnetic fields 8
Momentum Conservation Equation t u uu P g 0 u D B u u t t T P g 0 EM Electromagnetic momentum D B T Maxwell stress tensor EM 9
Maxwell Stress Tensor E D T I DE D E E d D EM 0 B H I BH B H B H d 0 Together with EM momentum can be transformed into Lorentz force density f E J B q 10
Thermal Energy Conservation Equation ˆ ˆ t U U u q P u : u 0 ˆ ˆ 1 1 U D E B H U u 2 2 t t q S u : u 0 P Electromagnetic energy 1 1 D E B H 2 2 Poynting vector S 11
Poynting Vector S E H Together with EM energy can be transformed into Joule heating w J E EM 12
Ferrofluid Viscosity Viscosity at zero field plus effect of magnetic field T , , B T , T , , B h 0 h h 1 0 2 1 a b c h h tanh 2 1.5 sin 0 h tanh 3 mH M Bd 0 T 6 1 T M 13
Vortex Viscosity u B m particle rotation u 14
Vortex Viscosity Particle rotation causes magnetic u misalignment B m particle rotation u 15
Presentation Overview Objectives General concepts Transport in the presence of EM fields Simulation Results Conclusions 16
Simulation Flow R c min R c max Flow R 17
FiRMA code FiRMa: Flow in Response to a Magnetic field Visual Basic code serves as simulation solver and visualization tool for the output 18
Magnetic Field Simulation -direction symmetry Find vector potential A , then B = A NJ=LJ ... First order triangular finite element ... ... Mesh extends radially 4 beyond pipe to include 3 coil(s) and to allow 2 fields lines to close 1 1 2 3 ... LI LI+1 ... ... NI Potential fixed at boundary nodes A = 0 pipe wall 19
Fluid Flow Simulation SIMPLE Method, LJ+1 solution of momentum LJ and mass conservation ... Upwind scheme ... Rectangular grid, ... staggered cells for 4 momentum equation 3 2 1 Additional cells around j=0 to include boundary i=0 1 2 3 ... ... LI LI+1 conditions pipe 20 wall
Heat Transfer Simulation Same grid as pressure Two types of wall cells: constant temperature and adiabatic Upwind scheme 21
Presentation Overview Objectives General concepts Transport in the presence of EM fields Simulation Results Conclusions 22
Vector Potential - Single Coil — 0.002 T·m — 0 — -0.002 23
B r - Single Coil — 0.073 T — 0 — -0.073 24
B z - Single Coil — 0.410 T — 0 — -0.410 25
Magnitude of B - Single Coil — 0.410 T — 0 26
Vector Potential - Double Coil — 0.002 T·m — 0 — -0.002 27
B r - Double Coil — 0.115 T — 0 — -0.115 28
B z - Double Coil — 0.18 T — 0 — -0.18 29
Magnitude of B - Double Coil — 0.18 T — 0 30
Velocity components Water-based ferrofluid, single coil (run W-04s) u r u z — 1 × 10 – 5 m/s — 0.020 m/s — 0 — -1 × 10 – 5 m/s — 0 31
Velocity components Water-based ferrofluid, double coil (run W-04d) u r u z — 3.8 × 10 – 5 m/s — 0.020 m/s — 0 — -3.8 × 10 – 5 m/s — 0 32
Velocity components Mercury-based ferrofluid, single coil (run M-04s) u r u z — 7.5 × 10 – 4 m/s — 0.020 m/s — 0 — -7.5 × 10 – 4 m/s — 0 33
Velocity components Mercury-based ferrofluid, single coil (run M-04s) u r u z — 1.8 × 10 – 3 m/s — 0.019 m/s — 0 — -1.8 × 10 – 3 m/s — 0 34
Apparent Viscosity Water-based ferrofluid single double coil coil — 0.036 Pa·s — 0.032 Pa·s — 0 — 0 35
Apparent Viscosity Mercury-based ferrofluid single double coil coil — 0.037 Pa·s — 0.035 Pa·s — 0 — 0 36
Velocity Profiles Mercury-based ferrofluid, single coil 0.025 2 0.020 1 0.015 Uz [m/s] 2 parabolic 0.010 1 — 0.020 m/s 0.005 0 -0.010 -0.005 0 0.005 0.010 r [m] 37 — 0
Velocity Profiles Mercury-based ferrofluid, single coil 0.020 0.018 1 0.016 0.014 0.012 Uz [m/s] parabolic 2 0.010 0.008 2 1 0.006 — 0.019 m/s 0.004 0.002 0 -0.010 -0.005 0 0.005 0.010 r/R [-] 38 — 0
Average Velocity Water-based ferrofluid 0.020 Single coil without Average velocity [m/s] heating 0.015 Single coil with heating 0.010 Double coil without heating 0.005 Double coil with heating 0 0 2.5E+6 5.0E+6 7.5E+6 1.0E+7 Coil current density [A/m²] 39
Average Velocity Mercury-based ferrofluid 0.015 Single coil without Average velocity [m/s] heating Single coil 0.010 with heating Double coil without 0.005 heating Double coil with heating 0 0 2.5E+6 5.0E+6 7.5E+6 1.0E+7 Coil current density [A/m²] 40
Temperature Profiles Water-based ferrofluid — 100 ° C — 20 No field Single coil Double coil 41
Temperature Profiles Mercury-based ferrofluid — 100 ° C — 20 No field Single coil Double coil 42
Local Heat Transfer Coefficient Water-based ferrofluid 2000 W-00-h W-04s-h coefficient [W/m²·K] W-04d-h 1500 Local heat transfer Heated region 1000 500 0 0 0.025 0.050 0.075 0.100 0.125 0.150 Axial distance from bottom [m] 43
Local Heat Transfer Coefficient Mercury-based ferrofluid 4000 M-00-h 3500 M-04s-h coefficient [W/m²·K] M-04d-h 3000 Local heat transfer Heated region 2500 2000 1500 1000 500 0 0 0.025 0.050 0.075 0.100 0.125 0.150 Axial distance from bottom [m] 44
Average Heat Transfer Coefficient Water-based ferrofluid 800 780 coefficient [W/m²·K] Average heat transfer 760 740 720 700 680 660 640 620 600 0 2.5E+6 5.0E+6 7.5E+6 1.0E+7 Coil current density [A/m²] Single coil with heating Double coil with heating 45
Average Heat Transfer Coefficient Mercury-based ferrofluid 1300 1280 1260 coefficient [W/m²·K] Average heat transfer 1240 1220 1200 1180 1160 1140 1120 1100 0 2.5E+6 5.0E+6 7.5E+6 1.0E+7 Coil current density [A/m²] Single coil with heating Double coil with heating 46
Presentation Overview Objectives General concepts Transport in the presence of EM fields Simulation Results Conclusions 47
Conclusions Very small effect on velocity profile in water-based ferrofluid (vortex viscosity only); much larger effect in mercury-based ferrofluid (Lorentz forces) Increased viscosity caused reduction in average velocity No change in temperature profile of water-based ferrofluid; large changes for mercury-based ferrofluid Peak in local heat transfer coefficient coincides with region of higher radial velocities induced by stronger field gradients 48
Future Work Search for other flow conditions with larger magnetic field effect in momentum and heat transport Examine other geometric configurations Extend simulation to cases without axial symmetry assumption Work into expanding current vision of transport theory 49
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