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ABSTRACT
Thermal radiation is an integral part of the heat transfer process but it is often neglected due to the complexity involved in the analysis of radiative transfer. We use the lattice Boltzmann method as a common computational tool to solve all three modes of heat transfer: conduction, convection, and radiation. This tool is then used to analyze the effect of radiatively participating medium on Rayleigh–Benard convection. We find that increasing the effects of radiation (i) increases the critical Rayleigh number required for the onset of Rayleigh–Benard convection and (ii) affects the temperature and flow patterns of convection rolls significantly changing the net heat transfer between the hot and cold plates. Both these effects are due to the presence of radiation available as an additional mode of heat transfer. Thus, we establish that the unified lattice Boltzmann framework is an effective computational tool for heat transfer and propose to use this method for a large range of problems in science and engineering involving radiative heat transfer.
Nomenclature
| B | = | coefficient of thermal expansion |
| C | = | lattice speed |
| Cp | = | specific heat capacity |
| Cs | = | speed of sound |
| feq | = | equilibrium particle distribution function |
| geq | = | equilibrium energy distribution function |
| gi | = | energy distribution function, EDF |
| Ieq | = | equilibrium intensity distribution function |
| Ieq | = | equilibrium intensity distribution function |
| Ib | = | radiation intensity from a blackbody |
| Ii | = | intensity distribution function, IDF |
| qr | = | radiative heat flux |
| Tw | = | wall temperature |
| e | = | internal energy |
| g | = | gravity |
| G | = | incident radiation |
| H | = | distance between the plates |
| k | = | thermal conductivity |
| Ma | = | Mach number |
| N | = | Stark number |
| Pr | = | Prandtl number |
| p | = | scattering phase function |
| Ra | = | Rayleigh number |
| R | = | universal gas constant |
| T | = | temperature |
| Greek | = | |
| ŝ | = | intensity direction |
| α | = | thermal diffusivity |
| β | = | extinction coefficient |
| δ | = | azimuthal direction |
| γ | = | polar direction |
| κa | = | absorption coefficient |
| 𝒯 | = | optical depth |
| ν | = | kinematic viscosity |
| ω | = | scattering albedo |
| Ω | = | solid angle |
| ωi | = | LB weights for ith direction (for DDF and EDF) |
| ωri | = | LB weights for ith direction (for IDF) |
| ρ | = | fluid density |
| σ | = | Stefan-Boltzmann constant (5.67 × 10−8 W/m2K) |
| τri | = | relaxation time for Ii |
| τα | = | relaxation time for gi |
| τν | = | relaxation time for fi |
| Subscripts | = | |
| α | = | energy related |
| ν | = | flow related |
| i | = | index for discrete direction for DDF and EDF |
| ri | = | index for discrete direction for IDF |
| w | = | wall |
Nomenclature
| B | = | coefficient of thermal expansion |
| C | = | lattice speed |
| Cp | = | specific heat capacity |
| Cs | = | speed of sound |
| feq | = | equilibrium particle distribution function |
| geq | = | equilibrium energy distribution function |
| gi | = | energy distribution function, EDF |
| Ieq | = | equilibrium intensity distribution function |
| Ieq | = | equilibrium intensity distribution function |
| Ib | = | radiation intensity from a blackbody |
| Ii | = | intensity distribution function, IDF |
| qr | = | radiative heat flux |
| Tw | = | wall temperature |
| e | = | internal energy |
| g | = | gravity |
| G | = | incident radiation |
| H | = | distance between the plates |
| k | = | thermal conductivity |
| Ma | = | Mach number |
| N | = | Stark number |
| Pr | = | Prandtl number |
| p | = | scattering phase function |
| Ra | = | Rayleigh number |
| R | = | universal gas constant |
| T | = | temperature |
| Greek | = | |
| ŝ | = | intensity direction |
| α | = | thermal diffusivity |
| β | = | extinction coefficient |
| δ | = | azimuthal direction |
| γ | = | polar direction |
| κa | = | absorption coefficient |
| 𝒯 | = | optical depth |
| ν | = | kinematic viscosity |
| ω | = | scattering albedo |
| Ω | = | solid angle |
| ωi | = | LB weights for ith direction (for DDF and EDF) |
| ωri | = | LB weights for ith direction (for IDF) |
| ρ | = | fluid density |
| σ | = | Stefan-Boltzmann constant (5.67 × 10−8 W/m2K) |
| τri | = | relaxation time for Ii |
| τα | = | relaxation time for gi |
| τν | = | relaxation time for fi |
| Subscripts | = | |
| α | = | energy related |
| ν | = | flow related |
| i | = | index for discrete direction for DDF and EDF |
| ri | = | index for discrete direction for IDF |
| w | = | wall |
Acknowledgments
The authors would like to thank H. N. Dixit, V. K. Ponnulakshmi, A. Pattamatta, Sampath, and B. Nitin Kumar for insightful discussions. Email correspondence with P. Asinari that helped us start radiation in LBM is gratefully acknowledged. Authors also thank P. Ripesi, S. Toppaladoddi, Rohan Vernekar, Bittagopal Mondal, and T. Kruger for their help through Email correspondence.