Natural convection heat transfer in porous media is an important phenomenon in engineering systems due to its wide applications such as cooling of electronics components, heat exchangers, drying processes, building insulations, and geothermal and oil recovery. Due to high surface area, fluid mixing qualities, high thermal conductivity, and wide industrial applications, natural convection through porous media has gained considerable attention of various researchers in the past few decades. Cheng and Minkowycz [1] studied free convection about a vertical flat plate embedded in a porous medium with application to heat transfer from a dike. They used the boundary layer approximations and found the similarity solution for the problem. Evans and Plumb [2] investigated natural convection about a vertical plate embedded in a medium composed of glass beads with diameters ranging from 0.85 to 1.68 mm. Their experimental data was in good agreement with the theory. Cheng [3] and Hsu [4] investigated the Darcian free convection flow about a semi-infinite vertical plate. They used the higher-order approximation theory and confirmed the results of Evans and Plumb [2]. Kim and Vafai [5] analyzed the natural convection about a vertical plate embedded in a porous medium. They took two cases in their analysis, *viz*., constant wall temperature and constant heat flux. They found the analytic solution for the boundary layer flow using the methods of matching asymptotes. Badruddin et al. [6] investigated free convection and radiation for a vertical wall with varying temperatures embedded in a porous medium. Steady and unsteady free convection in a fluid past an inclined plate and immersed in a porous medium was studied by Chamka et al. [7] and Uddin and Kumar [8]. They used the Brinkmann-Forchheimer model for the flow in porous media. Some more details about the theoretical and experimental studies for the convection in porous media can be found in the work of Neild and Bejan [9].

In industries, heat transfer can be enhanced by modifying the design of the devices, e.g., increasing the surface area by addition of fins, applying magnetic field and electric field. In compact-designed devices, these techniques are hard to apply, so the other option for heat transfer enhancement is to use the fluid with high thermal conductivity. However, common fluids like water, ethylene glycol, and oil have low values of thermal conductivities. On the other hand, the metals and their oxide have high thermal conductivities compared to these fluids. Choi [10] proposed that the uniform dispersion of small concentration of nano-sized metal/metal oxides particles into a fluid enhances the thermal conductivity of the base fluid, and such fluids were termed as nanofluids. This concept attracted various researchers towards nanofluids, and various theoretical and experimental studies have been done to find the thermal properties of nanofluids. An extensive review of thermal properties of nanofluids can be found in the study of Wang and Majumdar [11]. From the literature, it is found that the thermal conductivity of nanofluids depends upon various factors, such as particle material, base fluid material, particle volume fraction, particle size, particle shape, temperature, nanoparticle Brownian motion, nanoparticle base fluid interfacial layer, and particle clustering. It is also found in the study of Wang and Majumdar [12] that the insertion of nanoparticle increases the viscosity of the fluid. There are various theoretical relations predicting the thermal conductivity and viscosity of nanofluids, but these empirical relations do not satisfy the experimental data up to a satisfying range. Chon et al. [13] found an empirical correlation for the thermal conductivity of nanofluids within the particle size range of 11 to 150 nm and temperature range of 21°C to 71°C. They reported that the Brownian motion of nanoparticles constitutes a key mechanism of the thermal conductivity enhancement with increasing temperature and decreasing nanoparticle sizes. However, this empirical formula was valid only for water-Al_{2}O_{3} nanofluid. Very recently, Corcione [14] analyzed the experimental data of thermal conductivity and viscosity of nanofluids, which were obtained by various researchers for different types of nanoparticles dispersed in different base fluids, and found an empirical correlating equation for the prediction of effective thermal conductivity and dynamic viscosity of nanofluids.

With the advances in thermal properties and viscosity of nanofluids, various researchers studied the convective flow numerically as well as experimentally. Ho et al. [15] studied the natural convection of nanofluid having a particle concentration within the range of 0% to 4% in a square enclosure and analyzed the effects caused by uncertainties of viscosity and thermal conductivity. This study was limited to Al_{2}O_{3}-water nanofluid only. A detailed study of the natural convection of water-based nanofluids in an inclined enclosure has been done by Elif [16]. In this study, he investigated heat transfer enhancement using five different types of nanoparticles dispersed in water. To model the problem, he used a renovated Maxwell model containing the effect of interfacial layers in the enhanced thermal conductivity of nanofluids, given by Yu and Choi [17]. Abu-Nada and Oztop [18] investigated the effects of inclination angle on natural convection in enclosures filled with Cu-water nanofluid. All these authors reported that the heat transfer rate increases with the increase in nanoparticle concentration in the base fluid. However, in these studies, the effect of temperature and Brownian motion was not considered in the formulation of the problem. Abu-Nada [19] investigated the natural convection heat transfer in horizontal cylindrical annulus filled with Al_{2}O_{3}-water nanofluid taking the effect of variable viscosity and thermal conductivity. In the study, the effective thermal conductivity was calculated by the model of Chon et al. [13], and to formulate the dynamic viscosity of the Al_{2}O_{3}-water nanofluid, the author used the experimental data and found the empirical correlation for the dynamic viscosity as a function of temperature and particle concentration. Ho et al. [20] investigated the natural convection heat transfer of alumina-water nanofluid in vertical square enclosure experimentally. They reported that on higher Rayleigh numbers, the heat transfer rate increases on the dispersion of very small quantity of nanoparticles in water, but a larger quantity of nanoparticles in water decreases the heat transfer rates. The natural convection of nanofluids past vertical plate under different conditions has been studied by Hamad and Pope [21] and Rana and Bhargava [22]. They reported that the Nusselt number as well as the skin friction coefficient both increase with the increase in nanoparticle concentration in the base fluid. Zoubida et al. [23] investigated the effects of thermophoresis and Brownian motion significant in nanofluid heat transfer enhancement and found an enhancement in heat transfer at any volume fraction of nanoparticles. They also reported that the enhancement is more pronounced at low volume fraction of nanoparticles and that the heat transfer decreases by increasing the nanoparticle volume fraction.

The dispersion of nano-sized particles in the traditional fluid increased the thermal conductivity of the fluid, and the presence of porous media enhances the effective thermal conductivity of the base fluid. Thus, the use of nanofluids in porous media would be very much helpful in heat transfer enhancement. So far, very few studies have been done for the natural convection of nanofluids in porous media. Nield and Kuznetsov [24] studied the Cheng-Minkowycz problem for natural convection boundary layer flow in a porous medium saturated by a nanofluid. In the modeling of the problem, they used nanofluids by incorporating the effects of Brownian motion and thermophoresis. For the porous medium, the Darcy model was taken. Aziz et al. [25] found the numerical solution for the free convection boundary layer flow past a horizontal flat plate embedded in porous medium filled by nanofluid containing gyrotactic microorganisms. Recently, Rana et al. [26] found the numerical solution for steady-mixed convection boundary layer flow of a nanofluid along an inclined plate embedded in a porous medium. In the studies of natural convection of nanofluids in porous media, the authors did the parametric study only. However, they did not account any effect of parameters influencing the thermal conductivity and dynamic viscosity, such as particle concentration, particle size, temperature, nature of base fluid, and the nature of nanoparticle, which satisfy the experimental data for the thermal conductivity and dynamic viscosity of the nanofluids. In the best knowledge of the authors of this article, no such study has been done with regard to the natural convection of nanofluids in porous media. It is known that heat transfer in a fluid depends upon the temperature difference in fluid and heated surface and the thermophysical properties of the fluid. Heat transfer also depends upon the fluid flow rate, which depends upon the viscosity of the fluid.

As seen from the literature, most of the experimental studies on the thermal properties of nanofluids proved that the thermal conductivity of nanofluid depends upon the nanoparticle material, base fluid material, particle volume concentration, particle size, temperature, and nanoparticle Brownian motion. In previous works related to the flow of nanofluid in porous media, the authors used the variable thermophysical properties of the nanofluids, but it did not satisfy the experimental data for a wide range of reasons. Also, they did not consider the heat transfer through the two phases, i.e., nanofluid and porous media.

Therefore, the scope of the current research is to implement the appropriate models for the nanofluid properties, which consist the velocity-slip effects of nanoparticles with respect to the base fluid and the heat transfer flow in the two phases, i.e., through porous medium and nanofluid to be taken into account, and to analyze the effect of nanofluids on heat transfer enhancement in the natural convection in porous media.