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ABSTRACT
Combined-mode dual-phase-lag (DPL) heat conduction and radiation heat transfer is analyzed in a concentric cylindrical enclosure filled with a radiatively absorbing, emitting, and scattering medium. The governing energy equation is incorporated with volumetric radiation as a source term, essentially to take the effect of radiative heat flux into account. While the energy equation is solved using the lattice Boltzmann method (LBM), the finite volume method (FVM) is used to calculate the radiative information. To establish the accuracy of the proposed LBM formulation, the governing energy equation is also solved with the finite difference method (FDM). Thermal perturbation is caused by suddenly changing the temperature at the boundaries. Radial temperature distributions during transience as well as steady state (SS) are presented for a wide range of parameters such as lag ratio, extinction coefficient, scattering albedo, conduction–radiation (C-R) parameter, boundary emissivity, and radius ratio. Sample results are benchmarked with those available in the literature, and a good agreement between the present and reported results is found.
Nomenclature
| A | = | - area |
| B | = | - lag ratio |
| b | = | - number of propagation directions |
| C | = | - speed of propagation |
| C1 | = | - coefficient |
| C2 | = | - coefficient |
| C3 | = | - coefficient 2ΓB |
| Cp | = | - dimensionless specific heat at constant pressure |
| cp | = | - specific heat at constant pressure |
| = | - dimensional propagation velocity in a lattice direction | |
| fi | = | - particle distribution function in the lattice direction |
| = | - equilibrium particle distribution function in particle direction | |
| k | = | - thermal conductivity |
| r | = | - space variable |
| T | = | - temperature |
| t | = | - time variable |
| qc | = | - conductive heat flux |
| qr | = | - radiative heat flux |
| qt | = | - total heat flux |
| Vs | = | - speed of thermal wave |
| ΔV | = | - volume of control volumes |
| TE | = | - truncation error |
| N | = | - conduction–radiation parameter |
| G | = | - irradiation |
| I | = | - intensity of radiation |
| D | = | - directional weight |
| Nγ | = | - number of angular divisions in polar space |
| Nδ | = | - number of angular divisions in azimuthal space |
| Sp | = | - source term at the node of the control volume |
| P | = | - present control volume |
| = | - east, west, south, and north neighbor control volumes of P, respectively | |
| = | - direction | |
| = | - outward unit normal vector at face i | |
| = | - unit normal vector at the wall toward the medium | |
| = | - coefficient of discretized RTE in directions m and n at nodal point I | |
| Greek | = | |
| α | = | - thermal diffusivity |
| Γ | = | - nondimensional thermal diffusivity |
| γ | = | - parameter |
| λ | = | - parameter |
| η | = | - scaled nondimensional space variable |
| ζ | = | - scaled nondimensional time |
| θ | = | - scaled nondimensional temperature |
| ρ | = | - density |
| τ | = | - relaxation time in the BGK model |
| = | - dimensional phase lag for heat flow | |
| = | - dimensional phase lag for temperature gradient | |
| τq | = | - dimensionless phase lag for heat flow |
| τT | = | - dimensionless phase lag for temperature gradient |
| Δη | = | - scaled nondimensional lattice size |
| Δζ | = | - scaled nondimensional time step |
| σ | = | - Stefan–Boltzmann constant |
| κa | = | - absorption coefficient |
| β | = | - extinction coefficient |
| ω | = | - scattering albedo |
| ψ | = | - source term |
| ε | = | - emissivity |
| Ω | = | - solid angle |
| γ | = | - polar angle |
| δ | = | - azimuthal angle |
| σs | = | - scattering coefficient |
| Subscripts | = | |
| 1 | = | - inner wall |
| 2 | = | - outer wall |
| c | = | - conductive |
| r | = | - radiative |
| b | = | - blackbody |
| i | = | - direction |
| re | = | - reference state of dimensionless parameter |
| Superscripts | = | |
| * | = | - dimensional quantity |
| (1) | = | - O(ξ) in the Chapman–Enskong expansion |
| (2) | = | - O(ξ2) in the Chapman–Enskong expansion |
| mt | = | - time level for FDM discretization |
Nomenclature
| A | = | - area |
| B | = | - lag ratio |
| b | = | - number of propagation directions |
| C | = | - speed of propagation |
| C1 | = | - coefficient |
| C2 | = | - coefficient |
| C3 | = | - coefficient 2ΓB |
| Cp | = | - dimensionless specific heat at constant pressure |
| cp | = | - specific heat at constant pressure |
| = | - dimensional propagation velocity in a lattice direction | |
| fi | = | - particle distribution function in the lattice direction |
| = | - equilibrium particle distribution function in particle direction | |
| k | = | - thermal conductivity |
| r | = | - space variable |
| T | = | - temperature |
| t | = | - time variable |
| qc | = | - conductive heat flux |
| qr | = | - radiative heat flux |
| qt | = | - total heat flux |
| Vs | = | - speed of thermal wave |
| ΔV | = | - volume of control volumes |
| TE | = | - truncation error |
| N | = | - conduction–radiation parameter |
| G | = | - irradiation |
| I | = | - intensity of radiation |
| D | = | - directional weight |
| Nγ | = | - number of angular divisions in polar space |
| Nδ | = | - number of angular divisions in azimuthal space |
| Sp | = | - source term at the node of the control volume |
| P | = | - present control volume |
| = | - east, west, south, and north neighbor control volumes of P, respectively | |
| = | - direction | |
| = | - outward unit normal vector at face i | |
| = | - unit normal vector at the wall toward the medium | |
| = | - coefficient of discretized RTE in directions m and n at nodal point I | |
| Greek | = | |
| α | = | - thermal diffusivity |
| Γ | = | - nondimensional thermal diffusivity |
| γ | = | - parameter |
| λ | = | - parameter |
| η | = | - scaled nondimensional space variable |
| ζ | = | - scaled nondimensional time |
| θ | = | - scaled nondimensional temperature |
| ρ | = | - density |
| τ | = | - relaxation time in the BGK model |
| = | - dimensional phase lag for heat flow | |
| = | - dimensional phase lag for temperature gradient | |
| τq | = | - dimensionless phase lag for heat flow |
| τT | = | - dimensionless phase lag for temperature gradient |
| Δη | = | - scaled nondimensional lattice size |
| Δζ | = | - scaled nondimensional time step |
| σ | = | - Stefan–Boltzmann constant |
| κa | = | - absorption coefficient |
| β | = | - extinction coefficient |
| ω | = | - scattering albedo |
| ψ | = | - source term |
| ε | = | - emissivity |
| Ω | = | - solid angle |
| γ | = | - polar angle |
| δ | = | - azimuthal angle |
| σs | = | - scattering coefficient |
| Subscripts | = | |
| 1 | = | - inner wall |
| 2 | = | - outer wall |
| c | = | - conductive |
| r | = | - radiative |
| b | = | - blackbody |
| i | = | - direction |
| re | = | - reference state of dimensionless parameter |
| Superscripts | = | |
| * | = | - dimensional quantity |
| (1) | = | - O(ξ) in the Chapman–Enskong expansion |
| (2) | = | - O(ξ2) in the Chapman–Enskong expansion |
| mt | = | - time level for FDM discretization |