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ABSTRACT
A mathematical model which can describe flows of a number of immiscible fluids at high temperatures, where the radiative heat transfer cannot be neglected, is presented. It combines an interface-capturing multiphase model and the P-1 radiation model chosen for its simplicity. A finite volume method is utilized to discretize the governing equations and the solution methodology is based on the SIMPLE algorithm.
The model implementation is verified on a number of simple problems. The numerical experiments show a good agreement with analytical solutions or results which could be found in literature. A cooling of a gas–liquid system inside a rotating tank is also simulated. The results show that a coupled modeling of the motion of a number of fluids and all fundamental modes of heat transfer are important. Neglecting the convective transport and resulting redistribution of phases, or neglecting the radiative heat transfer, could result in significant modeling errors.
Nomenclature
| a | = | absorption coefficient |
| aφ, bφ | = | coefficients of discretized equations |
| Aφ, bφ | = | coefficient matrix and source vector |
| cv | = | specific heat |
| Cg | = | linear-anisotropic phase function coefficient |
| C, D, EV, ES | = | coefficients in generic transport equation |
| d | = | distance vector |
| fb | = | body force |
| ij | = | Cartesian bese vector |
| I | = | unit tensor |
| G | = | incident radiation |
| k | = | thermal conductivity |
| = | mass flux | |
| n | = | refractive index |
| nph | = | number of phases |
| p | = | pressure |
| qh | = | heat flux vector |
| qG | = | radiative energy flux |
| s | = | surface vector |
| S | = | surface |
| t | = | time |
| T | = | temperature |
| T | = | Cauchy stress tensor |
| v | = | velocity vector |
| vS | = | surface velocity vector |
| vj | = | Cartesian velocity component |
| V | = | volume |
| = | volume flux | |
| xj | = | Cartesian coordinates |
| αi | = | phase volume fraction |
| βp | = | underrelaxation factor |
| γϕ | = | blending factor |
| ε | = | emissivity |
| μ | = | viscosity |
| ρ | = | density |
| σ | = | Stefan–Boltzmann constant |
| σs | = | scattering coefficient |
| ϕ | = | generic variable |
| Subscripts | = | |
| B | = | boundary |
| f | = | cell-face centers |
| i | = | phase index |
| in | = | initial |
| P, N | = | cell centers |
| w | = | wall |
| ϕ | = | solution variable |
| Superscript | = | |
| m | = | time-step counter |
| T | = | transpose |
| ′ | = | correction |
Nomenclature
| a | = | absorption coefficient |
| aφ, bφ | = | coefficients of discretized equations |
| Aφ, bφ | = | coefficient matrix and source vector |
| cv | = | specific heat |
| Cg | = | linear-anisotropic phase function coefficient |
| C, D, EV, ES | = | coefficients in generic transport equation |
| d | = | distance vector |
| fb | = | body force |
| ij | = | Cartesian bese vector |
| I | = | unit tensor |
| G | = | incident radiation |
| k | = | thermal conductivity |
| = | mass flux | |
| n | = | refractive index |
| nph | = | number of phases |
| p | = | pressure |
| qh | = | heat flux vector |
| qG | = | radiative energy flux |
| s | = | surface vector |
| S | = | surface |
| t | = | time |
| T | = | temperature |
| T | = | Cauchy stress tensor |
| v | = | velocity vector |
| vS | = | surface velocity vector |
| vj | = | Cartesian velocity component |
| V | = | volume |
| = | volume flux | |
| xj | = | Cartesian coordinates |
| αi | = | phase volume fraction |
| βp | = | underrelaxation factor |
| γϕ | = | blending factor |
| ε | = | emissivity |
| μ | = | viscosity |
| ρ | = | density |
| σ | = | Stefan–Boltzmann constant |
| σs | = | scattering coefficient |
| ϕ | = | generic variable |
| Subscripts | = | |
| B | = | boundary |
| f | = | cell-face centers |
| i | = | phase index |
| in | = | initial |
| P, N | = | cell centers |
| w | = | wall |
| ϕ | = | solution variable |
| Superscript | = | |
| m | = | time-step counter |
| T | = | transpose |
| ′ | = | correction |
Acknowledgment
We would like to thank Dr Gopalendu Pal for some useful suggestions.