Natural convection flow of a power-law non-Newtonian nanofluid in inclined open shallow cavities filled with porous media

Highlights

• The variations of the inclination angle lead to decrease the velocities, local and average Nusselt number.

• An increasing the Rayleigh number leaded to a significant enhancement in the strength of streamlines, stronger convection heat transfer.

• An increase in the power-index n reduces the rate of heat transfer and vertical velocity while the average Bejan number is enhanced.

• The local and average Nusselt numbers are decreasing functions of the cavity aspect ratio.

Abstract

A numerical study of the laminar natural convection flow of non-Newtonian nanofluids in an inclined open shallow cavity filled with a porous medium is presented. The two-phase Buongiorno's model is used to simulate the nanofluid case and the Darcy model is applied to the porous medium. The apparent viscosity and stresses are given using the non-Newtonian power-law forms. The partial differential equations governing the flow are solved using the finite volume method with approximating the boundary conditions at the opening. Streamlines and isotherms are produced and local and average Nusselt number, Bejan and total entropy generation, horizontal and velocity components are calculated for the inclination angle α from 0 to π, the power-law index n from 0.4 to 1, the Rayleigh number Ra from 10⁵ to 10⁹ the Darcy number from 10^− 2 to 10^− 4, the cavity aspect ratio from 0.125 to 1 and the variations of the thermophoresis parameter Nt and the Brownian motion parameter Nb from 0.1 to 1. The results indicate that the increase in the power-index n reduces the rate of heat transfer and vertical velocity while the average Bejan number is enhanced. Also, the local and average Nusselt numbers are decreasing functions of the cavity aspect ratio.

Introduction

The effect of nanofluid on natural convection process in a confined porous medium was seen clearly in different industries. Over the last decade, the analysis of natural convection in enclosures filled with nanofluids has been studied extensively using different geometries, equation models, and numerical techniques and can be used in numerous applications of engineering such as the moisture removing the air contained in fibrous insulations, the migration of moistures in grain storage installations, the fuel storage pools, the heating and cooling buildings, the room ventilating, the underground disposal of nuclear wastes, the contaminant transport through water-saturated soil and so on [1]. Therefore, numerous investigations have been conducted in the past on natural convection in a porous medium. For instance, the books by Nield and Bejan [2], Ingham and Pop [3], Pop and Ingham [4], Vadasz [5] and many of published articles have focused on natural convection of nanofluid in closed containers. Bourantas et al. [6] analyzed numerically the natural convection of a nanofluid in a square cavity filled with a porous matrix. They discussed the effect of the porous medium on the cooling effectiveness of the nanofluid system. Sheremet et al. [7] studied numerically natural convective heat transfer in a differentially heated square porous cavity filled with a nanofluid and using a two-temperature model for the heat transfer and the Tiwari and Das nanofluid model by using a second-order accurate finite difference method. Al-Zamily [8] studied the fluid flow, heat transfer and entropy generation inside a cavity with an internal heat generation. He assumed that the cavity was filled with two layers of nanofluid (TiO₂-water) and one center layer of saturated porous media filled with the same nanofluid, and they found that the average Nusselt number decreases as Darcy number or the porous layer thickness increase. Aly [9] solved numerically the double-diffusive natural convection over circular cylinders in a non-Darcy porous enclosure filled with nanofluids using a simple algorithm in a finite volume method.

When the relation between the shear stress and the shear rate in the fluid is nonlinear, the fluid becomes non-Newtonian. So because of the importance of this kind of the fluids in several industrial applications such as compact heat exchangers, polymer engineering, electronic cooling systems, geophysical systems, and chemical reactor design, it has been received considerable attention from many researchers such as [10], [11], [12], [13].

It is demonstrated that many nanofluids exhibit non-Newtonian; therefore, it is necessary to be considered the effect of the shear-thinning behavior of nanofluids. Kefayati [11], [14], [15], [16], [17], [18] studied many problems of natural convection of non-Newtonian power-law fluids in an inclined porous cavity for different power-law indexes.

Natural convection in open enclosures has an important in engineering applications such as cooling of electronic equipment, cooling, and heating of buildings, solar heaters or vehicles and therefore many researchers were considered the open cavities in their researches. For example, Kefayati et al. [19] used the Lattice Boltzmann method to simulate the natural convection in an open enclosure which subjugated to water based copper nanofluid. The authors showed that the mean Nusselt number decreases as the aspect ratio increases at various Ra numbers and different nanoparticles volume fractions. To conduct the influence of thermocapillary forces on the natural convection and entropy generation in an open two dimensional enclosure, Saleem et al. [20] performed a numerical solution by using Alternate Direct Implicit (ADI) method. They observed that the entropy generation rate increases with the increase in the Marangoni number. Holzbecher [21] analyzed the free convection in open-top ended systems which are filled with a porous medium. In this case, the mixed boundary condition is applied and the onset of convection, total heat and mass transfer and the transition from the first to the second mode were examined by solving governing equations via Lattice Boltzmann method for different Prandtl numbers. Numerical simulation of natural convection in partially C-shape open ended enclosure filled with nanofluid has been performed by Bakier [22]. He found that the existence of nanoparticles increases the rate of heat and mass transfer through the opening boundaries for low Rayleigh number. Sheremet et al. [23] investigated the unsteady natural convection in a differentially heated wavy-walled open cavity filled with a nanofluid. The obtained results showed that an increase in the undulations number leads to a decrease in the average Nusselt number at wavy wall due to the significant heating of the wave troughs. Gangawane et al. [24] made a work on natural convection in a partial heater located open ended square cavity. They observed that there is a linear increasing on heat transfer rate with Prandtl number and they also obtained empirical correlations. Sheremet et al. [25] analyzed numerically the problem of MHD natural convection in a wavy open porous tall cavity filled with a Cu–water nanofluid in the presence of an isothermal corner heater. They found that the heat transfer enhancement is obtained with Rayleigh number and heat transfer reduction with Hartmann number, while magnetic field inclination angle leads to non-monotonic changes of the heat transfer rate. Gangawane [26] presented a numerical work on the effect of angle of applied magnetic field on natural convection in a partially active/heated open ended square cavity for incompressible and Newtonian fluid. Natural convection of an alumina-water nanofluid in a partially open rectangular cavity with a left heat conducting solid wall of finite thickness and conductivity had been studied numerically by Bondareva et al. [27]. The buoyancy induced flows in different geometries shape of open cavities can be seen in many studies as in literatures [28], [29].

Inclination angle plays an important role on natural convection. Bilgen and Oztop [30] performed a study of natural convection in partially open inclined square cavities using finite volume technique for two-dimensional solution. They found that inclination angle is the most important parameter on volumetric flow rate and heat transfer. Polat and Bilgen [31], [32] carried out two-dimensional studies on conjugate heat transfer in inclined open shallow cavities by including thickness of the wall and they made the similar work for thin walled open cavities. In both cases, they indicated that open cavity problem is a highly sensitive problem for numerical calculations.

The free convection heat transfer and a fluid flow of Cu-water nanofluid within a porous tilted right-angle triangular enclosure were discussed by Sun and Pop [33]. They found that the maximum value of the average Nusselt number achieves at highest Rayleigh number when the tilted angle of the cavity is 150°.

Ahmed et al. [34] investigated the numerical natural convection heat transfer in a porous media filled enclosure with a corner heater including radiation and inclination angle effects. Their results show that both heat transfer, fluid flow and heat transfer are an increasing function of radiation parameter and a decreasing function of Darcy number. The problem of natural convection in an inclined L-shaped enclosure filled with Cu/water nanofluid that operates under differentially heated walls in the presence of an inclined magnetic field is presented by Elshehabey et al. [35]. Li and Tong [36] made a presentation on the natural convective heat transfer in the inclined rectangular cavities with low width to height ratios. They found that increasing the width-to-height ratio and cavity inclination contributes to accelerate the natural convection flow and enhance the convective heat transfer in the cavity. Zhang and Che [37] studied the effects of the inclination angle in a cavity with four square heat sources by using Lattice Boltzmann simulation technique. They observed that both the inclination angle and location of the heat sources are main effects on heat transfer.

Motlagh et al. [38] analyzed the natural convection heat transfer in an inclined square enclosure filled with a porous medium saturated by nanofluid using Buongiorno's mathematical model. Their numerical results demonstrated that there is a mass boundary layer adjacent to enclosure walls and its thickness decreases by reducing porosity or increasing porous Rayleigh number and at porous Rayleigh numbers (Ra = 10², Ra = 10³), heat transfer enhancement continuously increases with increasing inclination angle of the enclosure. Aly [39] introduced double-diffusive natural convection in an enclosure by an incompressible smoothed particle hydrodynamics (ISPH) method. He had been studied two different cases of an enclosure. In the first case, the non-Darcy model for natural convection and heat and mass transfer in an enclosure saturated with anisotropic porous media had been investigated numerically by a stabilized ISPH method. The second case including sloshing rod inside an enclosure filled with free fluid had been studied numerically by a stabilized ISPH method. Rashad et al. [40] studied the effect of magnetic field and internal heat generation on the free convection flow in a rectangular cavity. They found that the average Nusselt number decreases as the Hartmann number or the solid volume fraction increases, while the opposite behavior occurs with the increase in magnetic field inclination angle. Hussein and Mustafa [41] investigated numerically the natural convection in a fully open parallelogrammic cavity filled with Cu–water nanofluid and heated locally from its bottom wall. They found that the inclination angle affects on the isotherms contours shape and maximum stream function for both pure and nanofluids decrease with the increase of the inclination angle. Double-diffusive natural convection and entropy generation of Bingham fluid in an inclined cavity was studied by Kefayati [42] by using finite difference Lattice Boltzmann Method (FDLBM). He found that the rise of the inclined angle from θ = 0^o to 40^o causes heat and mass transfer to augment while the enhancement from θ = 40^o to 80^o declines the heat and mass transfer. But the heat and mass transfer augments as the inclination angle increases to θ = 120^o although the values at θ = 120^o are less than θ = 0^o. Sheremet et al. [43] studied numerically the natural convection of alumina-water nanofluid inside a square cavity with time-sinusoidal temperature. The references [44], [45], [46], [47], [48], [49], [50] consider good related papers for the present study. They concluded that the cavity inclination angle and periodic thermal boundary conditions can be very good control parameters for heat and fluid flow inside the cavity.

Many of the previous references did not address the case of non-Newtonian nanofluid and others whom considered this case neglected the case of an open enclosure filled with porous medium. Therefore the significance of this study appears in investigation the fluid flow and heat transfer characteristics of non-Newtonian nanofluids inside an open enclosure filled with a porous medium using the Buongiorno's nanofluid model. Also, one of the reasons of our study this subject is the importance of this problem in many practical applications as stated in [1]. In the enclosure, it is assumed that the side facing the opening is imposed by a constant heat flux while the other sides perpendicular to the heated wall are adiabatic. The obtained results are validated with previous numerical investigations and the effects of the main parameters (Rayleigh number, power-law index, Darcy number, Lewis number, buoyancy ratio number, Thermophoresis parameter, Brownian motion and time) are researched.