Abstract
The Darcy Model with the Boussinesq approximation is used to study natural convection in a horizontal annular porous layer filled with a binary fluid, under the influence of a centrifugal force field. Neumann boundary conditions for temperature and concentration are applied on the inner and outer boundary of the enclosure. The governing parameters for the problem are the Rayleigh number, Ra, the Lewis number, Le, the buoyancy ratio, \({\varphi }\) , the radius ratio of the cavity, R, the normalized porosity, \({\varepsilon }\) , and parameter a defining double-diffusive convection (a = 0) or Soret induced convection (a = 1). For convection in a thin annular layer (R → 1), analytical solutions for the stream function, temperature and concentration fields are obtained using a concentric flow approximation and an integral form of the energy equation. The critical Rayleigh number for the onset of supercritical convection is predicted explicitly by the present model. Also, results are obtained from the analytical model for finite amplitude convection for which the flow and heat and mass transfer are presented in terms of the governing parameters of the problem. Numerical solutions of the full governing equations are obtained for a wide range of the governing parameters. A good agreement is observed between the analytical model and the numerical simulations.
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Abbreviations
- D :
-
Mass diffusivity of species, m2/K
- D′:
-
Thermodiffusion coefficient, m2/sK
- j′:
-
Constant mass flux per unit area, kg/ms
- K′:
-
Permeability of the porous medium, m2
- k :
-
Thermal conductivity, W/(mK)
- Le :
-
Lewis number, (α /D)
- N :
-
Mass fraction
- N 0 :
-
Initial mass fraction of the denser component of the mixture
- ΔN :
-
Characteristic mass fraction difference of the reference component
- Nu :
-
Nusselt number, Eq. 25
- q′:
-
Constant heat flux per unit area, W/m2
- r′:
-
Radius, m
- R :
-
Radius ratio, \({r^{\prime}_{\rm o} /{r}^{\prime}_{\rm i}}\)
- Ra :
-
Rayleigh number based on the inner radius, \({\beta_{T}^{\prime}q_{i}^{\prime} K^{\prime}\Omega^{\prime 2}r_{\rm i}^{\prime 4} /k\alpha \nu }\)
- Ra*:
-
Rayleigh number based on the gap between inner and outer radius, Ra(R − 1)2
- \({Ra_{\rm C}^{\rm sub}}\) :
-
Subcritical Rayleigh number, Eq. 22
- \({Ra_{\rm C}^{\sup}}\) :
-
Supercritical Rayleigh number, Eq. 24
- S :
-
Normalized mass fraction, N/ΔN
- Sh :
-
Sherwood number, Eq. 26
- t :
-
Dimensionless time, \({t^{\prime}\alpha /(\sigma r^{\prime 2}_{\rm i})}\)
- T :
-
Dimensionless temperature, \({(T^{\prime}-T^{\prime}_0 )/\Delta T^{\prime}}\)
- \({{T}^{\prime}_0 }\) :
-
Reference temperature
- ΔT :
-
Characteristic temperature difference, \({{q_{\rm i}^{\prime}r_{\rm i}^{\prime}}/k}\)
- u :
-
Dimensionless velocity in r direction
- v :
-
Dimensionless velocity in θ direction
- α :
-
Thermal diffusivity, m2/s
- β N :
-
Concentration expansion coefficient
- \({\beta^{\prime}_{\rm T}}\) :
-
Thermal expansion coefficient
- \({\varepsilon }\) :
-
Normalized porosity of the porous medium, \({\phi /\sigma }\)
- ν :
-
Kinematic viscosity of fluid, m2/s
- \({\varphi}\) :
-
Buoyancy ratio, \({(\beta _N \Delta N/{\beta }^{\prime}_T \Delta {T}^{\prime})}\)
- ρ :
-
Density of fluid, kg/m3
- (ρ C)f :
-
Heat capacity of the fluid, W/K
- (ρ C)p :
-
Heat capacity of the saturated porous medium, W/K
- σ :
-
Heat capacity ratio, (ρ C) p /(ρ C) f
- θ :
-
Angular coordinate
- \({\phi}\) :
-
Porosity of the porous medium
- Ψ:
-
Dimensionless stream function, Ψ′/α
- 0:
-
Reference state at position \({r=1,\varphi =\pi /2}\)
- i:
-
Inner cylinder
- o:
-
Outer cylinder
- ′:
-
Refers to dimensional variable
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Alloui, Z., Vasseur, P. Natural Convection Induced by a Centrifugal Force Field in a Horizontal Annular Porous Layer Saturated with a Binary Fluid. Transp Porous Med 88, 169–185 (2011). https://doi.org/10.1007/s11242-011-9732-y
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DOI: https://doi.org/10.1007/s11242-011-9732-y