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ABSTRACT
Application of the lattice Boltzmann method (LBM) in solving a combined mode conduction, convection, and radiation heat transfer problem in a porous medium is extended. Consideration is given to a 1-D planar porous medium with a localized volumetric heat generation zone. Three particle distribution functions, one each for the solid temperature, the gas temperature, and the intensity of radiation, are simultaneously used to solve the gas- and the solid-phase energy equations. The volumetric radiation source term appears in the solid-phase energy equation, and it is also computed using the LBM. To check the accuracy of the LBM results, the same problem is also solved using the finite volume method (FVM). Effects of convective coupling, flow enthalpy, solid-phase conductivity, scattering albedo porosity, and emissivity on axial temperature distribution are studied and compared with the FVM results. Effects of flow enthalpy, solid-phase conductivity, and emissivity are also studied on radiative output. LBM results are in excellent agreement with those of the FVM.
Nomenclature
| c | = | speed of light |
| cp | = | specific heat at constant pressure |
| e | = | propagation speed along the lattice direction |
| f | = | particle distribution function for the solid-phase energy equation |
| feq | = | equilibrium particle distribution function for the solid-phase energy equation |
| G | = | incident radiation or irradiation |
| Gi | = | non-dimensional emissive power |
| g | = | particle distribution function for the gas-phase energy equation |
| geq | = | equilibrium particle distribution function for the gas-phase energy equation |
| h | = | heat transfer coefficient |
| I | = | intensity of radiation |
| = | particle distribution function for radiation | |
| = | equilibrium particle distribution function for radiation | |
| K | = | thermal conductivity |
| L | = | length |
| M | = | total number of intensities |
| p | = | scattering phase function |
| q | = | heat flux |
| = | amount of heat generated | |
| S | = | source term for radiation |
| = | unit vector direction of ray travel | |
| t | = | time |
| T | = | temperature |
| u | = | velocity |
| w | = | weight for equilibrium distribution function |
| x | = | space variable |
| α | = | thermal diffusivity |
| β | = | extinction coefficient |
| ϵ | = | emissivity |
| φ | = | porosity |
| γ | = | polar angle |
| η | = | non-dimensional distance |
| θ | = | non-dimensional temperature |
| ρ | = | density |
| σ | = | Stefan–Boltzmann constant, 5.67 × 10−8Wm−2K−4 |
| τ | = | optical thickness |
| = | relaxation time in LBM | |
| ω | = | scattering albedo |
| Ω | = | solid angle |
| ξ | = | non-dimensional time |
| ΨR | = | non-dimensional radiative heat flux |
| Subscripts | = | |
| B | = | boundary |
| E | = | east |
| g | = | gas/medium |
| I | = | index for the lattice direction |
| P | = | cell center |
| r | = | related to radiation |
| s | = | solid/medium |
| S | = | source term |
| v | = | volumetric |
| W | = | west |
| Superscript | = | |
| m | = | index for rays in radiation |
Nomenclature
| c | = | speed of light |
| cp | = | specific heat at constant pressure |
| e | = | propagation speed along the lattice direction |
| f | = | particle distribution function for the solid-phase energy equation |
| feq | = | equilibrium particle distribution function for the solid-phase energy equation |
| G | = | incident radiation or irradiation |
| Gi | = | non-dimensional emissive power |
| g | = | particle distribution function for the gas-phase energy equation |
| geq | = | equilibrium particle distribution function for the gas-phase energy equation |
| h | = | heat transfer coefficient |
| I | = | intensity of radiation |
| = | particle distribution function for radiation | |
| = | equilibrium particle distribution function for radiation | |
| K | = | thermal conductivity |
| L | = | length |
| M | = | total number of intensities |
| p | = | scattering phase function |
| q | = | heat flux |
| = | amount of heat generated | |
| S | = | source term for radiation |
| = | unit vector direction of ray travel | |
| t | = | time |
| T | = | temperature |
| u | = | velocity |
| w | = | weight for equilibrium distribution function |
| x | = | space variable |
| α | = | thermal diffusivity |
| β | = | extinction coefficient |
| ϵ | = | emissivity |
| φ | = | porosity |
| γ | = | polar angle |
| η | = | non-dimensional distance |
| θ | = | non-dimensional temperature |
| ρ | = | density |
| σ | = | Stefan–Boltzmann constant, 5.67 × 10−8Wm−2K−4 |
| τ | = | optical thickness |
| = | relaxation time in LBM | |
| ω | = | scattering albedo |
| Ω | = | solid angle |
| ξ | = | non-dimensional time |
| ΨR | = | non-dimensional radiative heat flux |
| Subscripts | = | |
| B | = | boundary |
| E | = | east |
| g | = | gas/medium |
| I | = | index for the lattice direction |
| P | = | cell center |
| r | = | related to radiation |
| s | = | solid/medium |
| S | = | source term |
| v | = | volumetric |
| W | = | west |
| Superscript | = | |
| m | = | index for rays in radiation |