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ABSTRACT
This work examines the volumetric effect of convection within a packed bed in the presence of collimated irradiation. Using a modified P-1 approximation incorporating a local thermal nonequilibrium (LTNE) model, the energy transportation through convection and thermal conduction, and collimated and diffuse radiative transfer are investigated. The impact of pertinent parameters such as porosity φ, pore diameter dp, and optical thickness τ on the volumetric effect are analyzed. In addition, the mechanisms of how the volumetric effect impacts LTNE and radiative heat loss are revealed. The effect of the volumetric heat transfer coefficient hv, the fluid flow velocity u, and the ratio of solid to fluid thermal conductivities ζ versus the volumetric effect are systematically analyzed and displayed through a number of contour maps to assess the efficiency η. Our analysis shows that enhancing the volumetric effect and extending the thickness of the porous medium improves the efficiency η.
Nomenclature
| cp | = | specific heat of fluid at constant pressure (J kg−1 K−1) |
| F | = | inertial coefficient |
| dp | = | pore diameter (m) |
| G | = | incident radiation |
| hsf | = | fluid-to-solid heat transfer coefficient (W m−2 K) |
| K | = | permeability (m2) |
| L | = | thickness of a absorber (m) |
| Nu | = | Nusselt number |
| P | = | pressure (Pa) |
| Pr | = | Prandtl number |
| q0 | = | initial heat flux (W m−2) |
| q | = | heat flux |
| = | unit vector in the direction of fluid flow | |
| T | = | temperature (K) |
| u | = | velocity (m s−1) |
| V | = | velocity vector (m s−1) |
| αsf | = | specific surface area of the porous medium (m−1) |
| ε | = | emissivity |
| φ | = | porosity |
| λ | = | thermal conductivity (W m−1 K−1) |
| μ | = | dynamic viscosity (kg m−1 s−1) |
| β | = | extinction coefficient (m−1) |
| σ | = | Stefan−Boltzmann constant |
| σs | = | scattering coefficient |
| θ | = | dimensionless temperature |
| ζ | = | ratio of solid to fluid thermal conductivities |
| ρ | = | density (kg m−3) |
| τ | = | optical thickness |
| ω | = | single scattering albedo |
| Ψ | = | dimensionless heat flux |
| Subscripts | = | |
| a | = | average |
| c | = | collimated |
| d | = | diffuse |
| e | = | effective/environment |
| f | = | fluid phase |
| l | = | heat loss |
| r | = | radiative |
| s | = | solid phase |
| t | = | total |
| v | = | void |
| w | = | wall |
Nomenclature
| cp | = | specific heat of fluid at constant pressure (J kg−1 K−1) |
| F | = | inertial coefficient |
| dp | = | pore diameter (m) |
| G | = | incident radiation |
| hsf | = | fluid-to-solid heat transfer coefficient (W m−2 K) |
| K | = | permeability (m2) |
| L | = | thickness of a absorber (m) |
| Nu | = | Nusselt number |
| P | = | pressure (Pa) |
| Pr | = | Prandtl number |
| q0 | = | initial heat flux (W m−2) |
| q | = | heat flux |
| = | unit vector in the direction of fluid flow | |
| T | = | temperature (K) |
| u | = | velocity (m s−1) |
| V | = | velocity vector (m s−1) |
| αsf | = | specific surface area of the porous medium (m−1) |
| ε | = | emissivity |
| φ | = | porosity |
| λ | = | thermal conductivity (W m−1 K−1) |
| μ | = | dynamic viscosity (kg m−1 s−1) |
| β | = | extinction coefficient (m−1) |
| σ | = | Stefan−Boltzmann constant |
| σs | = | scattering coefficient |
| θ | = | dimensionless temperature |
| ζ | = | ratio of solid to fluid thermal conductivities |
| ρ | = | density (kg m−3) |
| τ | = | optical thickness |
| ω | = | single scattering albedo |
| Ψ | = | dimensionless heat flux |
| Subscripts | = | |
| a | = | average |
| c | = | collimated |
| d | = | diffuse |
| e | = | effective/environment |
| f | = | fluid phase |
| l | = | heat loss |
| r | = | radiative |
| s | = | solid phase |
| t | = | total |
| v | = | void |
| w | = | wall |
