PHYSICS OF FLUIDS 19, 054102 s2007d
Thermal convection in a rotating porous layer using a thermal nonequilibrium model M. S. Malashetty,a! Mahantesh Swamy, and Sridhar Kulkarni Department of Mathematics, Gulbarga University, Jnana Ganga, Gulbarga 585 106, India
sReceived 10 August 2006; accepted 5 February 2007; published online 7 May 2007d Linear stability of a rotating fluid-saturated porous layer heated from below and cooled from above is studied when the fluid and solid phases are not in local thermal equilibrium. The extended Darcy model, which includes the time derivative and Coriolis terms, is employed as a momentum equation. A two-field model that represents the fluid and solid phase temperature fields separately is used for energy equation. The onset criterion for both stationary and oscillatory convection is derived analytically. It is found that a small interphase heat transfer coefficient has significant effect on the stability of the system. There is a competition between the processes of rotation and thermal diffusion that causes the convection to set in through the oscillatory mode rather than the stationary one. The rotation inhibits the onset of convection in both stationary and oscillatory mode. In addition, the effect of porosity modified conductivity ratio, Darcy-Prandtl number, and the ratio of diffusivities on the stability of the system is investigated. A weak nonlinear theory based on the truncated representation of Fourier series method is used to find the Nusselt number. The effect of thermal nonequilibrium on heat transfer is brought out. The transient behavior of the Nusselt number is also investigated by solving the finite amplitude equations using Runge-Kutta method. © 2007 American Institute of Physics. fDOI: 10.1063/1.2723155g I. INTRODUCTION
Thermal convection in fluid-saturated porous media is of considerable interest due to its numerous applications in different fields, such as geothermal energy utilization, oil reservoir modeling, building thermal insulation, and nuclear waste disposal, to mention a few. The problem of convective instability of a horizontal fluid-saturated porous layer heated from below has been extensively investigated and the growing volume of work devoted to this area is well documented by Ingham and Pop,1,2 Nield and Bejan,3 and Vafai.4,5 In modeling a fluid-saturated porous medium, most of the investigations performed assumed a state of local thermal equilibrium sLTEd between the fluid and the solid phase at any point in the medium. This is a common practice for most of the studies where the temperature gradient at any location between the two phases is assumed to be negligible. For many practical applications, involving high-speed flows or large temperature differences between the fluid and solid phases, the assumption of local thermal equilibrium is inadequate and it is important to take account of the thermal nonequilibrium effects. Due to applications of porous media theory in drying and freezing of foods, and other mundane materials and applications in everyday technology, such as microwave heating, rapid heat transfer from computer chips via use of porous metal foams and their use in heat pipes, it is believed that local thermal nonequilibrium sLTNEd theory will play a major role in future developments. Recently, attention has been given to the LTNE model in the study of convection heat transfer in porous media. Much ad
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of this work has been reviewed in recent books by Ingham and Pop1,2 and Nield and Bejan.3 Criteria for heat and mass transfer models in metal hydride packed beds have been investigated by Kuznetsov and Vafai,6 and effects of nonequilibrium were suggested to be more significant at high Reynolds number and for high porosity. Kuznetsov7 studied a perturbation solution for a thermal nonequilibrium fluid flow through a three-dimensional sensible storage packed bed. Vafai and Amiri8 gave detailed information about the work on thermal nonequilibrium effects of fluid flow through a porous packed bed. The review of Kuznetsov9 gives detailed information about the most but very latest works on thermal nonequilibrium effects on internal forced convection flows. An excellent review of research on local thermal nonequilibrium phenomena in porous medium convection, primarily free and forced convection boundary layers and free convection within cavities, is given by Rees and Pop.10 Nield and Bejan3 have discussed a two-field model for energy equation as the simplest way in which LTNE may be modeled. Instead of having a single energy equation that describes the common temperature of the saturated porous media, two equations are used for fluid and solid phases separately. In the two-field model, the energy equations are coupled by the terms that account for the heat lost to or gained from the other phase. Rees and co-workers sRees and Pop,11 Rees,12 and Banu and Rees13d in a series of studies have investigated the thermal nonequilibrium sLTNEd effect on free convective flows in a porous medium. Free convection in a square porous cavity using a thermal nonequilibrium model is studied by Baytas and Pop,14 while Baytas15 investigated the thermal nonequilibrium natural convection in a square enclosure filled with a heat generating solid phase
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© 2007 American Institute of Physics
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