MHD mixed convection and entropy generation of a nanofluid in a vertical porous channel

Highlights

• Entropy generation and MHD mixed convection flow in a porous channel are studied.

• The governing equations are solved by the control volume method.

• Effects of Da, Ha, Ri and ϕ are considered for both assisting and opposing flows.

• The heat transfer is increased with Ha and ϕ, and is decreased with Da and Ec.

• The various irreversibilities are increased by rising the value of Hartmann number.

Abstract

A numerical study of entropy generation and MHD mixed convection flow of a nanofluid in a vertical porous channel is made. The left plate is thermally insulated, whereas four discrete heat sources dissipating a uniform heat flux are mounted on the right wall which is adiabatic elsewhere. Both assisting and opposing flows are considered. The Darcy–Brinkman–Forchheimer model with the Boussinesq approximation is adopted and the finite volume method is used to solve the governing equations with the appropriate boundary conditions. The influence of the magnetic field strength (Hartmann number), Joule heating effect (Eckert number), buoyancy force intensity (Richardson number), nanoparticles volume fraction, as well as porous medium permeability (Darcy number) on velocity profiles, isotherms, isentropic lines, global Nusselt number and total entropy generation are analyzed. The results showed an enhancement on heat transfer rate by using a porous medium, a nanofluid, a magnetic field without taking into account the Joule heating and when mixed convection is assisted. Globally, entropy generation increases with the parameters cited above.

Introduction

MHD mixed convection flow in porous medium has attracted the interest of many researchers in recent years because it is encountered in many industrial applications such as optimization of solidification processes of metals and alloys, waste nuclear processing, dissemination control of chemical waste and pollutants, and design of MHD power generators. This type of problem also arises in electronic packages and microelectronic devices during their operation. Aldoss [1] studied numerically the problem of mixed convection around a cylinder embedded in a porous medium in the presence of a magnetic field. The results showed an increase of heat transfer rate in forced convection dominant regime with an opposite effect in natural convection dominant regime. The problem of heat source/sink effect on laminar MHD mixed convection over a permeable vertical plate placed in a porous media was investigated by Yih [2]. Chamkha [3] considered the simultaneous heat and mass transfer by mixed convection from a semi-infinite, isothermal, isoconcentration, permeable and vertical plate embedded in a porous medium in the presence of a magnetic field and suction or injection effects. Among the obtained results, he found that both Nusselt and Sherwood numbers decrease with the increase of the magnetic field intensity in free and mixed convection regime, whereas they remained constant for forced convection regime. The purpose of the work of Seddeek et al. [4] was the study of the effects of chemical reaction, radiation and variable viscosity on hydromagnetic mixed convection heat and mass transfer for Hiemenz flow through a porous medium. Barletta et al. [5] presented a numerical study of MHD mixed convection in a porous annulus which surrounds a cable carrying an electric current that creates a radially varying magnetic field. It is shown that an electromagnetic force of sizeable intensity tends to inhibit the flow even for high value of pressure gradient. Coupled heat and mass transfer through a porous medium in a vertical channel with a transverse uniform magnetic field, heat sources and viscous dissipation was examined

by Nath et al. [6]. Makinde [7] treated the problem of hydromagnetic mixed convection stagnation point flow past a vertical plate embedded in a highly porous medium by taking into account thermal radiation and internal heat generation. The local values of skin-friction, Nusselt number and Sherwood number increase with the augmentation of magnetic field intensity and radiation parameters. Sahoo [8] presented a theoretical study of laminar MHD mixed convection stagnation point flow and heat transfer on a heated semi-infinite permeable surface embedded in a porous medium. Among the important findings is that, at high Prandtl number, the presence of porous matrix is ineffective to modify the velocity gradient. Recently, Pekmen and Tezer-Sezgin [9] conducted a numerical study of MHD flow and heat transfer in a lid-driven porous enclosure. The induced magnetic field was taken into consideration and the dual reciprocity boundary element method with Houbolt time integration scheme were used.

Various thermal systems are the subject of irreversibility phenomena which are expressed by entropy generation and are related to heat transfer, mass transfer, viscous dissipation, magnetic field, etc. In order to optimize these irreversibilities, analysis of the second law of thermaodynamic has been the object of many research works. We will quote in what follows some studies done in porous media with the presence of a magnetic field. Analytical expressions have been developed for velocity, temperature and entropy generation number by Tasnim et al. [10] for MHD mixed convection in a vertical porous channel. They found that the maximum of entropy generation occurs at the walls which act as strong concentrators of irreversibility. An analysis of entropy generation in a square porous cavity for laminar magnetohydrodynamic natural convection was done by Mahmud and Fraser [11]. They concluded that entropy generation rate increases with the augmentation of the magnetic field intensity. Komurgoz et al. [12] investigated the effect of magnetic field on entropy generation in an inclined channel partially filled with a porous medium. The maximum value of entropy generation rate corresponds to the case where there is no magnetic field and porosity is zero. Irreversibility effects in unsteady, free convective hydromagnetic flow past a vertical plate embedded in a porous medium with thermal radiation were examined by Butt and Ali [13]. The results showed an increase in the entropy generation number with the rise of the magnetic field and radiation parameters, and with the decrease of porous medium permeability. Rashidi et al. [14] studied entropy generation in MHD and slip flow over a rotating infinite porous disk with variable properties and found that the disk surface is the major source of entropy generation.

An innovative technique for enhancing heat transfer in thermal systems is the suspension of solid nanoparticles in the base fluids to improve their thermal conductivity; these are the nanofluids. Due to the very small size (1–100 nm) and low volume fraction of suspended nanoparticles, nanofluids form very stable colloidal systems which will prevent rapid settling and reduce clogging in the wall of heat transfer devices. The study of nanofluids in the presence of a magnetic field has attracted the interest of many researchers [15], [16], [17], [18], [19], [20] due to its importance in numerous engineering applications such as microelectronics, microfluidics, transportation, biomedical, etc. The addition of a porous medium has also been investigated as Ellahi et al. [21] who examined the effects of magnetohydrodynamic flow and slip boundary conditions in coaxial porous cylinders for non-Newtonian nanofluids. Variations of MHD, porosity and slip parameters on velocity and temperature fields for constant and variable viscosity have been analyzed. In their work, Rosmila et al. [22] used the Lie Symmetry Group Transformation to analyze MHD free convection flow of a nanofluid over a vertical porous stretching sheet. It is found that nanoparticles volume fraction in the presence of thermal stratification has important effects on flow and heat transfer characteristics. Murthy et al. [23] studied the effect magnetic field on flow, heat and nanoparticles mass transfer characteristics in free convection along a vertical plate immersed in a porous medium saturated by a thermally stratified nanofluid under convective boundary condition. The problem of laminar MHD nanofluid flow in a semi-porous channel was investigated analytically by Sheikholeslami et al. [24] using the Least Square and Galerkin methods. It is observed an increase of velocity boundary layer thickness with augmentation of magnetic field intensity and a minimal effect of nanoparticles volume fraction on velocity profiles. An analytical approach with a numerical method have been applied by Hatami et al. [25] to analyze flow and heat transfer characteristics of MHD blood, considered as non-Newtonian and conveying gold nanoparticles, in the porous area of a hollow vessel. They found an increase of nanoparticles concentration near the inner wall by increasing the thermophoresis parameter and a decrease of velocity profile by augmentation of MHD parameter. Servati et al. [26] utilized the Lattice Boltzmann method to study forced convection flow in a channel partially filled with a porous medium saturated by a nanofluid. The average Nusselt number increases substantially with nanoparticles volume fraction, whereas it rises slightly with magnetic field intensity. Entropy generation of Cu-water nanofluid flow over an inclined transparent plate embedded in a porous medium was instigated by Dehsara et al. [27]. The authors considered the effects of solar radiation, viscous dissipation and variable magnetic field.

The main objective of this work is to examine numerically the influence of an external magnetic field on entropy generation and mixed convection flow in a vertical channel filled with a porous medium saturated by a nanofluid. To the best knowledge of the authors no such study has been reported till date in the literature. It can find application in the field of electronic components cooling.