Abstract
Effects of a conductive wall on natural convection in a square porous enclosure having internal heating at a rate proportional to a power of temperature difference is studied numerically in this article. The horizontal heating is considered, where the vertical walls heated isothermally at different temperatures while the horizontal walls are kept adiabatic. The Darcy model is used in the mathematical formulation for the porous layer and finite difference method is applied to solve the dimensionless governing equations. The governing parameters considered are the Rayleigh number (0 ≤ Ra ≤ 1000), the internal heating and the local exponent parameters (0 ≤ γ ≤ 5), (1 ≤ λ ≤ 3), the wall to porous thermal conductivity ratio (0.44 ≤ Kr ≤ 9.9) and the ratio of wall thickness to its width (0.02 ≤ D ≤ 0.5). The results are presented to show the effect of these parameters on the fluid flow and heat transfer characteristics. It is found a strong internal heating can generate significant maximum fluid temperature more than the conductive solid wall. Increasing value thermal conductivity ratio and/or decreasing the thickness of solid wall can increase the maximum fluid temperature. It is also found that at very low Rayleigh number, the heat transfer across the porous enclosure remain stable for any values of the thermal conductivity ratio.
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Abbreviations
- d, D :
-
Wall thickness, dimensionless wall thickness
- g :
-
Gravitational acceleration
- G :
-
Measure of the internal heat generation
- K :
-
Permeability of the porous medium
- Kr :
-
Thermal conductivity ratio
- k p :
-
Effective thermal conductivity of porous medium
- k w :
-
Thermal conductivity of wall
- ℓ :
-
Width and height of enclosure
- \({\overline{Nu}}\) :
-
Average Nusselt number
- Ra :
-
Rayleigh number
- T :
-
Temperature
- u, v :
-
Velocity components in the x- and y-directions
- x, y and X, Y :
-
Space coordinates and dimensionless space coordinates
- α :
-
Effective thermal diffusivity
- β :
-
Thermal expansion coefficient
- γ :
-
Internal heating parameter
- λ :
-
Local heating exponent
- ψ, Ψ:
-
Stream function, dimensionless stream function
- Θ:
-
Dimensionless temperature
- ν :
-
Kinematic viscosity
- c:
-
Cold
- h:
-
Hot
- max:
-
Maximum
- p:
-
Porous
- w:
-
Wall
References
Al-Amiri A., Khanafer K., Pop I.: Steady-state conjugate natural convection in a fluid-saturated porous cavity. Int. J. Heat Mass Transf. 51, 4260–4275 (2008)
Baytas A.C., Liaqat A., Grosan T., Pop I.: Conjugate natural convection in a square porous cavity. Heat Mass Transf. 37, 467–473 (2001)
Chang W.J., Lin H.C.: Natural convection in a finite wall rectangular cavity filled with an anisotropic porous medium. Int. J. Heat Mass Transf. 37, 303–312 (1994a)
Chang W.J., Lin H.C.: Wall heat conduction effect on natural convection in an enclosure filled with a non-darcian porous medium. Numer. Heat Transf. A 25, 671–684 (1994b)
de Lemos M.J.S.: Turbulence in Porous Media: Modeling and Applications. Elsevier, Oxford (2006)
Haajizadeh M., Ozguc A.F., Tien C.L.: Natural convection in a vertical porous enclosure with internal heat generation. Int. J. Heat Mass Transf. 27, 1893–1902 (1984)
Ingham D.B, Pop I.: Transport Phenomena in Porous Media. Elsevier Science, Oxford (1998)
Ingham D.B., Bejan A., Mamut E., Pop I.: Emerging Technologies and Techniques in Porous Media. Kluwer, Dordrecht (2004)
Ingham D.B., Pop I.: Transport Phenomena in Porous Media III. Elsevier Science, Oxford (2005)
Joshi M.V., Gaitonde U.N., Mitra S.K.: Analytical study of natural convection in a cavity with volumetric heat generation. J. Heat Transf. 128, 176–182 (2006)
Manole D.M., Lage J.L.: Numerical benchmark results for natural convection in a porous medium cavity, in: Heat and mass transfer in porous media. ASME HTD 216, 55–60 (1992)
Mealey L.R., Merkin J.H.: Steady finite Rayleigh number convective flows in a porous medium with internal heat generation. Int. J. Therm. Sci. 48, 1068–1080 (2009)
Nield D.A., Bejan A.: Convection in Porous Media, 3rd edn. Springer, New York (2006)
Pop I., Ingham D.B.: Convective Heat Transfer: Mathematical and Computational Modelling of Viscous Fluids and Porous Media. Pergamon, Oxford (2001)
Rao Y.F., Wang B.X.: Natural convection in vertical porous enclosures with internal heat generation. Int. J. Heat Mass Transf. 34, 247–252 (1991)
Saeid N.H.: Conjugate natural convection in a vertical porous layer sandwiched by finite thickness walls. Int. Commun. Heat Mass Transf. 34, 210–216 (2007a)
Saeid N.H.: Conjugate natural convection in a porous enclosure: effect of conduction in one of the vertical walls. Int. J. Therm. Sci. 46, 531–539 (2007b)
Saeid N.H.: Conjugate natural convection in a porous enclosure sandwiched by finite walls under thermal nonequilibrium conditions. J. Porous Media 11, 259–275 (2008)
Saleh H., Saeid N.H., Hashim I., Mustafa Z.: Effect of conduction in bottom wall on Darcy-Bénard convection in a porous enclosure. Transp. Porous Media 88, 357–368 (2011)
Vadasz P.: Emerging Topics in Heat and Mass Transfer in Porous Media. Springer, New York (2008)
Vafai K.: Handbook of Porous Media, 2nd edn. Taylor and Francis, New York (2005)
Varol Y., Oztop H.F., Koca A.: Entropy generation due to conjugate natural convection in enclosures bounded by vertical solid walls with different thicknesses. Int. Commun. Heat Mass Transf. 35, 648–656 (2008)
Varol Y., Oztop H.F., Mobedi M., Pop I.: Visualization of heat flow using Bejan’s heatline due to natural convection of water near 4°C in thick walled porous cavity. Int. J. Heat Mass Transf. 53, 1691–1698 (2010)
Walker K.L., Homsy G.M.: Convection in porous cavity. J. Fluid Mech. 87, 449–474 (1978)
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Saleh, H., Hashim, I. Conjugate Natural Convection in a Porous Enclosure with Non-Uniform Heat Generation. Transp Porous Med 94, 759–774 (2012). https://doi.org/10.1007/s11242-012-0023-z
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DOI: https://doi.org/10.1007/s11242-012-0023-z