Abstract
The problem of double-diffusive flow in a long rotating porous channel has been analysed numerically. The two opposite vertical walls of the channel are maintained at constant but different temperature and concentration, while both horizontal walls are kept insulated. The generalised model is used to mathematically simulate the momentum equations with employing the Boussinesq approximation for the density variation. Moreover, both the fluid and solid phases are assumed to be at a local thermal equilibrium. The Coriolis effect is considered to be the main effect of rotation, which is induced by means of the combined natural heat and mass transfer within the transverse plane. The governing equations are discretised according to the finite volume method with employing the hybrid differencing scheme to calculate the fluxes across the faces of each control volume. The problem of pressure–velocity coupling is sorted out by relying on PISO algorithm. Computations are performed for a wide range of dimensionless parameters such as Darcy–Rayleigh number (100 ≤ Ra* ≤ 10,000), Darcy number (10−6 ≤ Da ≤ 10−4), the buoyancy ratio (−10 ≤ N ≤ 8), and Ekman number (10−7 ≤ Ek ≤ 10−3), while the values of Prandtl and Schmidt numbers are maintained constant and equal to 1.0. The results reveal that the rotation seems to have a dominant role at high levels of porous medium permeability, where it reduces the strength of the secondary flow, and hence the rates of heat and mass transfer. However, this dominance decreases gradually with lessening the permeability for the same level of rotation, but does not completely vanish.
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Abbreviations
- a :
-
Side length of the channel (m)
- c :
-
Dimensional concentration (kg/m3)
- c F :
-
Dimensionless form-drag constant
- c p :
-
Specific heat of fluid phase (J/kg °K)
- C :
-
Dimensionless concentration [C = (c − c c )/(c h − c c )]
- C 1, C 2, C 3 :
-
Coefficients of Eq. (17)
- D :
-
The mass diffusivity of the solute into the solvent (m2/s)
- D m :
-
The effective mass diffusivity (m2/s)
- Da :
-
Darcy number (Da = K/a 2)
- e :
-
Unit vector
- Ek :
-
Ekman number (Ek = ν f /2ω a 2)
- g :
-
Gravitational acceleration (m2 s−1)
- K :
-
Permeability of the porous medium (m2)
- k f :
-
Thermal conductivity of fluid phase (W m−1 °K−1)
- k m :
-
Mean thermal conductivity (W m−1 °K−1)
- k s :
-
Thermal conductivity of solid phase (W m−1 °K−1)
- Le :
-
Lewis number (Le = Sc/Pr = α e /D m )
- N :
-
Buoyancy ratio (N = β c Δc/β T ΔT)
- Nu :
-
Average Nusselt number
- p :
-
Dimensional pressure (Pa)
- p r :
-
Dimensional reduced pressure (Pa)
- P r :
-
Dimensionless reduced pressure
- Pr :
-
Prandtl number (Pr = ν f /α e )
- Ra :
-
Rayleigh number [Ra = g β T ΔT a 3/(ν α)]
- Ra* :
-
Darcy–Rayleigh number [Ra* = Ra Da = g β T ΔT K a/(ν α)]
- Sc :
-
Schmidt number (Sc = ν f /D m )
- Sh :
-
Average Sherwood number
- T :
-
Dimensional temperature (°K)
- u, v, w :
-
Dimensional velocity components (m/s)
- U, V, W :
-
Dimensionless velocity components
- v:
-
Dimensional velocity vector (m/s)
- x:
-
Dimensional position vector (m)
- x, y, z :
-
Dimensional coordinates (m)
- X, Y, Z :
-
Dimensionless coordinates
- θ :
-
The dimensionless temperature
- α e :
-
The effective thermal diffusivity (m2/s)
- ρ f :
-
Fluid density (kg/m3)
- μ f :
-
Dynamic viscosity (N s/m2)
- ν f :
-
Kinematic viscosity (m2/s)
- ε :
-
Porosity of the porous medium
- β c :
-
Coefficient of concentration expansion (°K−1)
- β T :
-
Coefficient of Thermal expansion (°K−1)
- 0:
-
Reference point
- c :
-
Cold
- e :
-
Effective
- f :
-
Fluid phase
- g :
-
Gravity
- h :
-
Hot
- m :
-
Mean
- s :
-
Solid phase
- ω :
-
Rotation
References
Mojtabi A, Charrier-Mojtabi MC (2005) Double-diffusive convection in porous media. In: Vafai K (ed) Handbook of porous media, 2nd edn. Marcel Dekker, New York, pp 269–320
Nield DA, Bejan A (2013) Convection in porous media, 4th edn. Springer, New York
Trevisan OV, Bejan A (1985) Natural convection with combined heat and mass transfer buoyancy effects in a porous medium. Int J Heat Mass Transf 28(8):1597–1611
Goyeau B, Songbe JP, Gobin D (1996) Numerical study of double-diffusive natural convection in a porous cavity using the Darcy–Brinkman formulation. Int J Heat Mass Transf 39(7):1363–1378
Nithiarasu P, Seetharamu KN, Sundararajan T (1996) Double-diffusive natural convection in an enclosure filled with fluid-saturated porous medium: a generalized non-Darcy approach. Numer Heat Transf Part A Appl 30(4):413–426
Karimi-Fard M, Charrier-Mojtabi MC, Vafai K (1997) Non-darcian effects on double-diffusive convection within a porous medium. Numer Heat Transf Part A Appl 31(8):837–852
Al-Farhany K, Turan A (2012) Numerical study of double diffusive natural convective heat and mass transfer in an inclined rectangular cavity filled with porous medium. Int Commun Heat Mass Transfer 39:174–181
Vadasz P (1997) Flow in rotating porous media. In: du Plessis JP (ed) Fluid transport in porous media from the series advances in fluid mechanics 13. Computational Mechanics Publications, Southampton, pp 161–214
Vadasz P (1993) Fluid flow through heterogeneous porous media in a rotating square channel. Transp Porous Media 12:43–54
Havstad MA, Vadasz P (1999) Numerical solution of the three-dimensional fluid flow in a rotating heterogeneous porous channel. Int J Numer Meth Fluids 31:411–429
Vadasz P (1993) Three-dimensional free convection in a long rotating porous box. J Heat Transfer 115:639–644
Vadasz P (1995) Coriolis effect on free convection in a long rotating porous box subject to uniform heat generation. Int J Heat Mass Transf 38(11):2011–2018
Vadasz P (1994) Centrifugally generated free convection in a rotating porous box. Int J Heat Mass Transf 37(16):2399–2404
Vadasz P (1996) Stability of free convection in a rotating porous layer distant from the axis of rotation. Transp Porous Media 23:153–173
Vadasz P (1996) Convection and stability in a rotating porous layer with alternating direction of the centrifugal body force. Int J Heat Mass Transf 39(8):1639–1647
Vadasz P, Govender S (1998) Two-dimensional convection induced by gravity and centrifugal forces in a rotating porous layer far away from the axis of rotation. Int J Rotating Mach 4(2):73–90
Vadasz P (1998) Coriolis effect on gravity-driven convection in a rotating porous layer heated from below. J Fluid Mech 376:351–375
Straughan B (2001) A sharp nonlinear stability threshold in rotating porous convection. Proc R Soc Lond A 457:87–93
Malashetty MS, Swamy M, Kulkarni S (2007) Thermal convection in a rotating porous layer using a thermal non-equilibrium model. Phys Fluids 19(5):054102-1–054102-16
Bhadauria BS, Hashim I, Kumar J, Srivastava A (2013) Cross diffusion convection in a newtonian fluid-saturated rotating porous medium. Transp Porous Med 98:683–697
Beckermann C, Viskanta R, Ramadhyani S (1986) A numerical study of non-darcian natural convection in a vertical enclosure filled with a porous medium. Numer Heat Transf 10(6):557–570
Vadasz P, Govender S (2001) Stability and stationary convection induced by gravity and centrifugal forces in a rotating porous layer distant from the axis of rotation. Int J Eng Sci 39:715–732
Spalding DB (1972) A novel finite-difference formulation for differential expressions involving both first and second derivatives. Int J Numer Methods Eng 4:551–559
Issa RI (1986) Solution of the implicitly discretised fluid flow equations by operator-splitting. J Comput Phys 62:40–65
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Alhusseny, A., Turan, A. A numerical study of double-diffusive flow in a long rotating porous channel. Heat Mass Transfer 51, 497–505 (2015). https://doi.org/10.1007/s00231-014-1426-x
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DOI: https://doi.org/10.1007/s00231-014-1426-x