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Abstract
The numerical solution of a one-dimensional, three-phase Stefan problem with a low Stefan number is presented. Joule heating and thermal radiation are demonstrated to be negligible compared to the high power input. The front tracking method is used along with a second-order Lagrangian interpolation of the temperature profile near the moving surface defined by the location of the phase change. Results are compared with analytical, numerical, and experimental solutions available in literature.
NOMENCLATURE
| a | = | length of the slab (m) |
| A | = | coefficient of the Neumann analytical solution |
| B | = | coefficient of the Neumann analytical solution |
| cs | = | specific heat of solid (J/kg-K) |
| cl | = | specific heat of liquid (J/kg-K) |
| Cm | = | latent heat of melting (J/kg) |
| Cv | = | latent heat of vaporization (J/kg) |
| i | = | integer index of the domain mesh |
| f(τ) | = | dimensionless inbound heat flux in Stefan equation |
| fl(τ) | = | dimensionless inbound heat flux in boundary condition for liquid phase |
| fs(τ) | = | dimensionless inbound heat flux in boundary condition for solid phase |
| F(t) | = | inbound heat flux (W/m2) |
| gl | = | dimensionless heat generation term in liquid |
| gs | = | dimensionless heat generation term in solid |
| j | = | current density (A/m2) |
| k | = | integer index for simulation time-step progression |
| = | ratio of solid and liquid thermal conductivities | |
| kl | = | thermal conductivity of liquid (W/m-K) |
| ks | = | thermal conductivity of solid (W/m-K) |
| Ml | = | melted mass (kg) |
| Mv | = | vaporized mass (kg) |
| p | = | fractional parameter for the s1 tracking |
| qb | = | inbound heat flux (W/m2) |
| qr | = | radiative heat flux (W/m2) |
| Qb | = | inbound energy provided during td (W) |
| r | = | arbitrary characteristic length (m) |
| rd | = | diffusion length (m) |
| s1 | = | dimensionless position of the solid–liquid interface |
| s2 | = | dimensionless position of the liquid–vapor interface |
| S1 | = | dimensional position of the solid–liquid interface (m) |
| S2 | = | dimensional position of the liquid–vapor interface (m) |
| Ste | = | Stefan number |
| t | = | time (s) |
| td | = | characteristic time of the phenomenon, i.e. discharge time (s) |
| Ti | = | initial system temperature (anode and environment) (K) |
| Tl | = | liquid temperature (K) |
| Tm | = | melting temperature (K) |
| Ts | = | solid temperature (K) |
| Tsurf | = | temperature of the surface exposed to vapor phase (K) |
| Tv | = | vaporization temperature (K) |
| uB | = | dimensionless melting temperature |
| ul | = | dimensionless liquid temperature |
| us | = | dimensionless solid temperature |
| uv | = | dimensionless vapor temperature |
| v | = | fractional parameter for the s2 tracking |
| x | = | dimensionless spatial coordinate |
| xl | = | dimensionless spatial coordinate for the liquid domain |
| xs | = | dimensionless spatial coordinate for the solid domain |
| X | = | dimensional spatial coordinate (m) |
| Xl | = | dimensional spatial coordinate for the liquid domain (m) |
| Xs | = | dimensional spatial coordinate for the solid domain (m) |
Greek Symbols
| αs | = | diffusivity of solid (m2/s) |
| αl | = | diffusivity of liquid (m2/s) |
| γl | = | dimensionless multiplier proportional to Ste |
| γlv | = | dimensionless multiplier proportional to λv |
| ε | = | emissivity |
| ηs | = | resistivity of solid (Ω −m) |
| ηl | = | resistivity of liquid (Ω −m) |
| λv | = | ratio of sensible heat of solid and latent heat of vaporization |
| ρ | = | density (kg/m3) |
| τ | = | dimensionless time |
Additional information
Notes on contributors
Guido Parissenti
Guido Parissenti graduated in Space Engineering at Politecnico di Milano in 2008. He was a Visiting Student Research Collaborator at the Electric Propulsion and Plasma Dynamic Laboratory (EPPDyLs) at Princeton University (Princeton, NJ) where he performed his M.Sc. Thesis on the topic of anode erosion due to spots formation on MPD Thrusters. He is presently attending his Ph.D. degree in Energetic at Politecnico di Milano about industrial plasma applications in waste treatment and recovery.
Alfonso Niro
Alfonso Niro graduated in Nuclear Engineering in 1982 at Politecnico di Milano where he also received his Ph.D. in 1987; during 1988, he was a visiting scientist at MIT, (Cambridge, MA), within a NATO-CNR grant. In 1990 he started his academic career as an Assistant Professor and he became an Associate Professor in 1997. Since 2001 he has been a Full Professor of Thermodynamics and Heat Transfer. His major research interests are in the field of Heat Transfer and Engineering Thermodynamics. His studies in heat transfer are concerned with both single-phase flows and two-phase flows, flow-boiling and convective condensation of halo-refrigerants inside microfin tubes, critical heat flux in micro-channels, pool-boiling instabilities at low pressure. To these subjects he has contributed with more than 80 scientific papers.