Elsevier

Advanced Powder Technology

Volume 24, Issue 6, November 2013, Pages 980-991
Advanced Powder Technology

Original Research Paper
Effect of a magnetic field on natural convection in an inclined half-annulus enclosure filled with Cu–water nanofluid using CVFEM

https://doi.org/10.1016/j.apt.2013.01.012Get rights and content

Abstract

In this paper, the effect of a magnetic field on natural convection in a half-annulus enclosure with one wall under constant heat flux using control volume based finite element method. The fluid in the enclosure is a water-based nanofluid containing Cu nanoparticles. The effective thermal conductivity and viscosity of nanofluid are calculated using the Maxwell–Garnetts (MG) and Brinkman models, respectively. Numerical simulations were performed for different governing parameters namely the Hartmann number, Rayleigh number and inclination angle of enclosure. The results indicate that Hartmann number and the inclination angle of the enclosure can be control parameters at different Rayleigh number. In presence of magnetic field velocity field retarded and hence convection and Nusselt number decreases.

Highlights

MHD nanofluid flow in an inclined half-annulus is investigated. • CVFEM is used to solve this problem. • Nusselt number has direct relationship with ϕ and Ra. • Nusselt number has reverse relationship with Ha. • Inclination angle of the enclosure can be control parameter.

Introduction

The study of magnetohydrodynamics (MHD) flow and heat transfer for an electrically conducting fluid are usually encountered in electrical power generation, astrophysical flows, solar power technology and space vehicle re-entry. The application of magnetic field to convection processes will play as a control factor in the convection by damping both the flow and temperature oscillations in material manufacturing fields. In several energy conversion processes, strong external magnetic fields are applied to liquid flows. Rudraiah et al. [1] investigated numerically the effect of magnetic field on natural convection in a rectangular enclosure. They found that the magnetic field decreases the rate of heat transfer. Natural convection under the influence of a magnetic field is of great importance in many industrial applications such as crystal growth, metal casting and liquid metal cooling blankets for fusion reactors. Pirmohammadi and Ghassemi [2] considered effect of magnetic field on convection heat transfer inside a tilted square enclosure. Their study showed that heat transfer mechanism and flow characteristics inside the enclosure depend strongly upon both magnetic field and inclination angle. The problem of laminar viscous flow in a semi-porous channel in the presence of transverse magnetic field is studied by Sheikholeslami et al. [3]. They show that optimal homotopy asymptotic method was a powerful approach for solving nonlinear differential equations such as this problem. Also they investigated the effects of some important parameters to evaluate how these parameters effect on this fluid behavior. Magnetohydrodynamic natural convection in a vertical cylindrical cavity with sinusoidal upper wall temperature has been investigated by Kakarantzas et al. [4]. They concluded that the increase of Hartmann number damps the fluid motion and thus heat conduction progressively dominates over convection heat transfer. Effect of static radial magnetic field on natural convection heat transfer in a horizontal cylindrical annulus enclosure filled with nanofluid is investigated numerically using the lattice Boltzmann method by Ashorynejad et al. [5]. They found that the average Nusselt number increases as nanoparticle volume fraction and Rayleigh number increase, while it decreases as Hartmann number increases. Sheikholeslami et al. [6] studied the effects of magnetic field and nanoparticle on the Jeffery–Hamel flow. They showed that increasing Hartmann number leads to backflow reduction. In greater angles or Reynolds numbers, high Hartmann number is needed to reduction of backflow. Also the results show that momentum boundary layer thickness increases as nanoparticle volume fraction increases.

Control Volume based Finite Element Method (CVFEM) is a scheme that uses the advantages of both finite volume and finite element methods for simulation of multi-physics problems in complex geometries [7] and [8]. Soleimani et al. [9] studied natural convection heat transfer in a semi-annulus enclosure filled with nanofluid using the control volume based finite element method. They found that the angle of turn has an important effect on the streamlines, isotherms and maximum or minimum values of local Nusselt number. Sheikholeslami et al. [10] studied magnetohydrodynamic flow in a nanofluid filled inclined enclosure with sinusoidal wall. They reported that for all values of Hartmann number, at Ra = 104 and 105 maximum values of enhancement are obtained at γ = 60° and γ = 0°, respectively.

With the growing demand for efficient cooling systems, particularly in the electronics industry, more effective coolants are required to keep the temperature of electronic components below safe limits. Use of nanofluids is a potential solution to improve heat transfer. Khanafer et al. [11] conducted a numerical investigation on the heat transfer enhancement due to adding nanoparticles in a differentially heated enclosure. They found that the suspended nanoparticles substantially increase the heat transfer rate at any given Grashof number. Sheikholeslami et al. [12] performed a numerical analysis for natural convection heat transfer of Cu–water nanofluid in a cold outer circular enclosure containing a hot inner sinusoidal circular cylinder in presence of horizontal magnetic field. They concluded that in absence of magnetic field, enhancement ratio decreases as Rayleigh number increases while an opposite trend was observed in the presence of magnetic field. Bararnia et al. [13] studied the natural convection in a nanofluid filled portion cavity with a heated built in plate by lattice Boltzmann method. Their results have been obtained for different inclination angles and lengths of the inner plate. Jalaal et al. [14] used homotopy analysis method to solve the problem of acceleration motion of a single spherical particle moving in a continuous fluid phase. They found that homotopy analysis method was very accurate and flexible for solving problem. The unsteady motion of a spherical particle falling in a Newtonian fluid was analyzed by Jalaal et al. [15]. Jalaal and Ganji [16] unsteady motion of a spherical particle rolling down an inclined plane submerged in a Newtonian environment. They present showed the effectiveness of HPM and exhibits a new application of this method for nonlinear problems. Recently novel solutions for acceleration motion of a vertically falling spherical particle were presented [17] and [18]. Jalaal et al. [19] applied He’s homotopy perturbation method to obtain exact analytical solutions for the motion of a spherical particle in a plane couette flow. Heat transfer of a nanofluid flow which is squeezed between parallel plates was investigated analytically by Sheikholeslami and Ganji [20]. They reported that Nusselt number has reverse relationship with the squeeze number when two plates are squeezed. Ashorynejad et al. [21] studied the flow and heat transfer of a nanofluid over a stretching cylinder in the presence of magnetic field has been investigated. They found that choosing copper (for small of magnetic parameter) and alumina (for large values of magnetic parameter) leads to the highest cooling performance for this problem.

The present paper represents the results of a numerical investigation on natural convection nanofluid flow in a half-annulus enclosure with one wall under constant heat flux in presence of magnetic flied using the control volume based finite element method. The numerical investigation is carried out for different governing parameters such as the Hartmann number, Rayleigh number, nanoparticle volume fraction and inclination angle of the enclosure.

Section snippets

Geometry definition and boundary conditions

The physical model along with the important geometric parameters and the mesh of the half-annulus enclosure used in the present CVFEM program are shown in Fig. 1a. The inner wall is under constant heat flux (q), the outer wall is maintained at constant temperatures (Tc) and the two other walls are thermally insulated. It is also assumed that the uniform magnetic field (B=Bxex+Byey) of constant magnitude B=Bx2+By2 is applied, where ex and ey are unit vectors in the Cartesian coordinate

Problem formulation

The flow is two-dimensional, laminar and incompressible. The radiation, viscous dissipation, induced electric current and Joule heating are neglected. The magnetic Reynolds number is assumed to be small so that the induced magnetic field can be neglected compared to the applied magnetic field. The flow is considered to be steady, two dimensional and laminar. Neglecting displacement currents, induced magnetic field, and using the Boussinesq approximation, the governing equations of heat transfer

Grid testing and code validation

To allow grid independent examination, the numerical procedure has been conducted for different grid resolutions. Table 2 demonstrates the influence of number of grid points for the case of at Ra = 105, λ = 0°, Ha = 100, ϕ = 0.06 and Pr = 6.2. The present code is tested for grid independence by calculating the average Nusselt number on the hot wall. In harmony with this, it was found that a grid size of 101 × 331 ensures a grid-independent solution. The convergence criterion for the termination of all

Results and discussion

In this paper, numerical simulations of natural convection nanofluid flow in a half-annulus enclosure with one wall under constant heat flux in presence of magnetic field were performed. Calculations are made for various values of Hartmann number (Ha = 0, 20, 60 and 100), Rayleigh number (Ra = 103, 104 and 105), volume fraction of nanoparticles (ϕ = 0%, 2%, 4% and 6%) and inclination angle (λ = 0°, 45° and 90°) at constant Prandtl number (Pr = 6.2). The fluid in the enclosure is a water-based nanofluid

Conclusions

In this paper, control volume based finite element method is applied to solve the problem of heat and fluid flow of a nanofluid in a half-annulus enclosure with one wall under constant heat flux in presence of magnetic flied. The effects of Hartmann number, Rayleigh number, volume fraction of nanoparticle and inclination angle on the flow and heat transfer characteristics have been investigated. The results indicate that the magnetic field damps the flow and the temperature oscillations by

References (27)

Cited by (230)

View all citing articles on Scopus
View full text