Abstract
Darcy-Bénard convection in a square porous enclosure with a localized heating from below and lateral cooling is studied numerically in the present paper. A finite-thickness bottom wall is locally heated, the top wall is kept at a lower temperature than the bottom wall temperature, and the lateral walls are cooled. The finite difference method has been used to solve the dimensionless governing equations. The analysis in the undergoing numerical investigation is performed in the following ranges of the associated dimensionless groups: the heat source length—\({0.2\leq H \leq 0.9}\), the wall thickness—\({0.05\leq D \leq 0.4}\), the thermal conductivity ratio—\({0.8\leq K_{\rm r} \leq 9.8}\), and the Biot number—\({0.1\leq Bi \leq 1.1}\). It is observed that the heat transfer rate could increase with increasing heat source lengths, thermal conductivity ratio, and cooling intensity. There exists a critical wall thickness for a high wall conductivity below which the increasing wall thickness increases the heat transfer rate and above which the increasing wall thickness decreases the heat transfer rate.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Abbreviations
- Bi :
-
Biot number
- d, D :
-
Wall thickness, dimensionless wall thickness
- g :
-
Gravitational acceleration
- h, H :
-
Heater size, dimensionless heater size
- h a :
-
Atmospheric convective heat transfer coefficient
- K :
-
Permeability of the porous medium
- K r :
-
Thermal conductivity ratio
- k p :
-
Effective thermal conductivity of porous medium
- k w :
-
Thermal conductivity of wall
- k :
-
Thermal conductivity
- \({\ell}\) :
-
Width and height of cavity
- \({\overline{Nu}}\) :
-
Average Nusselt number
- Pr :
-
Prandtl number
- Ra :
-
Rayleigh number
- T :
-
Temperature
- u,v :
-
Velocity components in the x- and y-directions
- U,V :
-
Dimensionless velocity components in the X- and Y-directions
- x,y & X, Y :
-
Space coordinates & dimensionless space coordinates
- α m :
-
Thermal diffusivity
- β :
-
Thermal expansion coefficient
- \({\Theta}\) :
-
Dimensionless temperature
- \({\nu}\) :
-
Kinematic viscosity
- \({\psi \& \Psi}\) :
-
Stream function & dimensionless stream function
- c:
-
Cold
- h:
-
Hot
- 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)
Al-Farhany K., Turan A.: Non-Darcy effects on conjugate double-diffusive natural convection in a variable porous layer sandwiched by finite thickness walls. Int. J. Heat Mass Transf. 54, 2868–2879 (2011)
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 (1994)
El-Khatib G., Prasad V.: Effects of stratification on thermal convection in horizontal porous layer with localized heating from below. J. Heat Transf. 109, 683–687 (1987)
Elder J.W.: Steady free convection in a porous medium heated from below. J. Fluid Mech. 47, 29–48 (1967)
Elder J.W.: Transient convection in a porous medium. J. Fluid Mech. 47, 609–623 (1967)
Lai F.C., Kulacki F.A.: Experimental study of free and mixed convection in horizontal porous layers locally heated from below. Int. J. Heat Mass Transf. 34, 525–541 (1991)
Molla M.M., Saha S.C., Khan M.A.I.: Natural convection flow in a porous enclosure with localized heating from below. J. Heat Mass Transf. 6, 1–16 (2012)
Prasad V., Kulacki F.A.: Natural convection in horizontal porous layers with localized heating from below. J. Heat Transf. 109(3), 795–798 (1987)
Robillard L., Wang C.H., Vasseur P.: Multiple steady states in a confined porous medium with localized heating from below. Numer. Heat Transf. Part A 13, 91–110 (1988)
Saeid N.H.: Natural convection in porous cavity with sinusoidal bottom wall temperature variation. Int. Commun. Heat Mass Transf. 32, 454–463 (2005)
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. Thermal 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)
Saleh, H., Hashim, I.: Conjugate natural convection in a porous enclosure with non-uniform heat generation. Transp. Porous Media (2012). doi:10.1007/s11242-012-0023-z
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)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Alhashash, A., Saleh, H. & Hashim, I. Effect of Conduction in Bottom Wall on Bénard Convection in a Porous Enclosure with Localized Heating and Lateral Cooling. Transp Porous Med 96, 305–318 (2013). https://doi.org/10.1007/s11242-012-0089-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11242-012-0089-7