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ABSTRACT
This paper reports on double-diffusive natural convection heat transfer in a porous annulus between concentric horizontal circular and square cylinders. A pressure-based segregated finite volume method is used to solve the problem numerically. The diffusion fluxes are discretized using the MIND fully implicit scheme. Furthermore, a modified pressure correction equation is derived that implicitly accounts for the nonorthogonal diffusion terms, which are usually neglected in the standard SIMPLE algorithm. Results indicate that convection effects increase with an increase in Rayleigh number, Darcy number, porosity, and enclosure aspect ratio. Further, at low Darcy values, porosity has no effect on the flow, temperature, and concentration fields.
Nomenclature
| C | = | main grid point at an element centroid |
| cp | = | specific heat of fluid at constant pressure |
| dCF | = | vector joining the two points C and F |
| dCF | = | magnitude of dCF |
| dp | = | pores diameter |
| Da | = | Darcy number |
| eCF | = | unit vector in the direction of dCF |
| E | = | distance vector in the direction of dCF |
| E | = | magnitude of E |
| F | = | neighbor of element C; also constant in Forchheimer’s extension |
| g | = | gravitational acceleration |
| h | = | local convection heat transfer coefficient |
| = | average convection heat transfer coefficient | |
| j | = | unit vector along the y-axis |
| k | = | thermal conductivity of fluid |
| K | = | permeability of the porous media |
| L | = | width of square duct |
| Le | = | Lewis number |
| n | = | unit vector normal to surface |
| N | = | buoyancy ratio: Ras/Rat |
| NC, NF | = | locations used in the calculation of the nonorthogonal part of the diffusive flux |
| Nu, | = | local and average Nusselt number |
| P, P | = | dimensional and dimensionless pressure |
| Pr | = | Prandtl number |
| R | = | radius of cylinder |
| Ra | = | Rayleigh number |
| R/L | = | enclosure aspect ratio |
| S | = | dimensional solute concentration; also magnitude of S |
| S | = | surface vector |
| Si, So | = | distance along the inner and outer enclosure surfaces |
| Sh, | = | local and average Sherwood number |
| T | = | dimensional temperature |
| T | = | vector equal to S−E |
| u, U | = | dimensional and dimensionless x-velocity component |
| v, V | = | dimensional and dimensionless y-velocity component |
| v, V | = | dimensional and dimensionless velocity vector |
| x, y | = | dimensional coordinates |
| X, Y | = | dimensionless coordinates |
| α | = | thermal diffusivity |
| β | = | thermal expansion coefficient |
| μ | = | dynamic viscosity |
| ρ | = | density |
| θ | = | dimensionless temperature |
| ε | = | porosity |
| ψ | = | stream function |
| σ | = | dimensionless solute concentration |
| Γ | = | diffusion coefficient |
| Subscripts | = | |
| c | = | cold wall or convection heat transfer |
| C | = | refers to main grid point |
| f | = | refers to element face |
| F | = | refers to the F grid point |
| h | = | hot wall |
| i | = | condition at inner surface |
| m | = | mass transfer |
| o | = | condition at outer surface |
| s | = | refers to solute concentration |
| t | = | refers to temperature |
Nomenclature
| C | = | main grid point at an element centroid |
| cp | = | specific heat of fluid at constant pressure |
| dCF | = | vector joining the two points C and F |
| dCF | = | magnitude of dCF |
| dp | = | pores diameter |
| Da | = | Darcy number |
| eCF | = | unit vector in the direction of dCF |
| E | = | distance vector in the direction of dCF |
| E | = | magnitude of E |
| F | = | neighbor of element C; also constant in Forchheimer’s extension |
| g | = | gravitational acceleration |
| h | = | local convection heat transfer coefficient |
| = | average convection heat transfer coefficient | |
| j | = | unit vector along the y-axis |
| k | = | thermal conductivity of fluid |
| K | = | permeability of the porous media |
| L | = | width of square duct |
| Le | = | Lewis number |
| n | = | unit vector normal to surface |
| N | = | buoyancy ratio: Ras/Rat |
| NC, NF | = | locations used in the calculation of the nonorthogonal part of the diffusive flux |
| Nu, | = | local and average Nusselt number |
| P, P | = | dimensional and dimensionless pressure |
| Pr | = | Prandtl number |
| R | = | radius of cylinder |
| Ra | = | Rayleigh number |
| R/L | = | enclosure aspect ratio |
| S | = | dimensional solute concentration; also magnitude of S |
| S | = | surface vector |
| Si, So | = | distance along the inner and outer enclosure surfaces |
| Sh, | = | local and average Sherwood number |
| T | = | dimensional temperature |
| T | = | vector equal to S−E |
| u, U | = | dimensional and dimensionless x-velocity component |
| v, V | = | dimensional and dimensionless y-velocity component |
| v, V | = | dimensional and dimensionless velocity vector |
| x, y | = | dimensional coordinates |
| X, Y | = | dimensionless coordinates |
| α | = | thermal diffusivity |
| β | = | thermal expansion coefficient |
| μ | = | dynamic viscosity |
| ρ | = | density |
| θ | = | dimensionless temperature |
| ε | = | porosity |
| ψ | = | stream function |
| σ | = | dimensionless solute concentration |
| Γ | = | diffusion coefficient |
| Subscripts | = | |
| c | = | cold wall or convection heat transfer |
| C | = | refers to main grid point |
| f | = | refers to element face |
| F | = | refers to the F grid point |
| h | = | hot wall |
| i | = | condition at inner surface |
| m | = | mass transfer |
| o | = | condition at outer surface |
| s | = | refers to solute concentration |
| t | = | refers to temperature |