Journal of Heat and Mass Transfer Research 4 (2017) 149-155 Journal of Heat and Mass Transfer Research Journal homepage: http://jhmtr.journals.semnan.ac.ir Effects of variations in magnetic Reynolds number on magnetic field distribution in electrically conducting fluid under magnetohydrodynamic natural convection Mohsen Pirmohammadi Department of Mechanical Engineering, Pardis Branch , Islamic Azad University, Pardis, Iran PAPER INFO A B S T R A C T This study numerically investigated the effects of variations in magnetic Reynolds History: number on magnetic field distribution in an enclosure under natural convection heat Submitted 2016-12-15 transfer. The investigated geometry was a two-dimensional enclosure with a hot left Revised 2017-10-07 wall, a cold right wall, and adiabatic top and bottom walls. The fluid was molten Accepted 2017-10-12 sodium with Pr=0.01. Natural convection heat transfer at a Rayleigh number of 105 and magnetic Reynolds numbers (Rem) of 10–1, 10–3, and 10–5 was considered in the Keywords: analysis. The governing equations adopted were continuum, momentum, energy, and Magnetic Reynolds; magnetic induction equations, which were solved concurrently using the finite volume number; method. For the coupling of velocity and pressure, the revised semi-implicit method Natural convection; for pressure linked equations algorithm was employed. Results showed that under a Magnetic field high magnetic Reynolds number, the non-dimensional magnetic fields in the X and Y directions were approximately constant because the diffusion of the magnetic potential was greater than the advection of such potential. As the magnetic Reynolds number increased, however, the magnetic field in the enclosure reached a magnitude unequal to that of the applied magnetic field, exhibited inconstancy, and increased to a value that deviated from 1. Thus, the Rem=10–1 under a non-dimensional magnetic field increased from 0.09 to 6.6 in the X direction and from –1.164 to 4.05 in Y the direction. © 2017 Published by Semnan University Press. All rights reserved. DOI: 10.22075/jhmtr.2017.1503.1100 strength is a critical factor for crystal formation. Hadid 1. Introduction et al. [2] probed into the effects of a strong vertical magnetic field on convection and segregation in The free convection heat transfer of electrically vertical Bridgeman crystal growth. Bessaih et al. [3] conducting fluids in enclosures has been the subject of numerically examined the effects of the electric a considerable number of theoretical, experimental, conductivity of walls and the direction of a magnetic and numerical investigations because of its importance field on gallium flow. The authors found a in many technological applications, such as use in considerable reduction in convection intensity as the liquid metal blankets for reactors. Another important magnetic field increases. application of the aforementioned process is crystal Ciofalo and Cricchio [4] considered the MHD growth in the industrial production of semiconductors. natural convection of liquid Pb–17Li in a cubical This particular application has been explored in a cavity with a uniform volumetric heat source and an number of works. Oreper and Szekely [1], for induced current that significantly stretches to the walls example, showed that a magnetic field suppresses normal to the applied magnetic field flux. Piazza and natural convection currents and that magnetic field Ciofalo [5, 6] developed a simple wall boundary Corresponding Author : Department of Mechanical Engineering, Pardis Branch , Islamic Azad University, Pardis, Iran Email: [email protected] 150 M. Pirmohammadi/ JHMTR 4 (2017) 149-155 condition to solve electromagnetism in a cavity magnetic Reynolds number on the magnetic field without allocating real grid points to electrically distribution and temperature and velocity profiles. conducting walls. Pirmohammadi et al. [7] numerically studied natural convection flow in the 2. Basic Equations presence of a magnetic field in a tilted enclosure heated from below and filled with liquid gallium. The The schematic of the examined enclosure is authors found that at a given inclination angle, depicted in Fig. 1. The cavity is differentially heated; convection heat transfer decreases when magnetic the left and right walls are isothermal at TH and TC, field intensity increases. Pirmohammadi et al. [8] respectively (TH>TC); and the horizontal walls are studied the effects of a magnetic field on buoyancy- adiabatic. A magnetic field was applied in the X driven convection in a differentially heated square direction. enclosure. The researchers showed that the heat transfer mechanisms and flow characteristics in the The governing equations used in this research are enclosure depend strongly on both the strength of the based on the conservation laws of mass, momentum, magnetic field and the Rayleigh number. They energy, and induction equations. The governing concluded that a magnetic field considerably equations are provided below: decreases the average Nusselt number. An extensive study of mixed convection was u v 0 (1) conducted by Selimefendigil et al. [11] for various x y geometries. Heidary et al. [12] numerically investigated the laminar flow and convective heat 2 2 transfer of Cu-water nanofluid in a channel subjected u u P u u 1 By Bx (2) (u v ) ( ) ( )B x y x x 2 y2 x y y to a uniform magnetic field. The authors revealed that 0 the thickness of the thermal boundary layer increases with the addition of nanoparticles to pure fluid. v v P 2v 2v Applying a magnetic field, however, thins the (u v ) ( ) g (T T ) 2 2 c (3) boundary layer because of the increase in velocity x y y x y B gradient near walls. Seth et al. [13] delved into the 1 y Bx ( )BX combined free and forced convection Couette– 0 x y Hartmann flow of a viscous, incompressible, and electrically conducting fluid. The fluid was placed in a T T 2T 2T rotating channel with arbitrary conducting walls, and Cpu v k( ) (4) 2 2 flow was examined under the presence of the Hall x y x y current. Makinde et al. [14] investigated the steady flow and heat transfer of an electrically conducting 2 2 fluid with variable viscosity and electrical Bx Bx u u 1 Bx Bx (5) u v Bx By ( 2 2 ) conductivity between two parallel plates in the x y x y σμ0 x y presence of a transverse magnetic field. In most of the above-mentioned studies, 2 2 By By v v 1 By By (6) mathematical and numerical methods were intended to u v B B ( ) x y x x y y σμ x 2 y2 determine the influence of any combination of 0 Grashof, Reynolds, Prandtl, and Hartmann numbers on flow and heat transfer. A technique common to these studies was the simplification of the magnetic induction equation on the basis of low Reynolds approximation [15, 16]. The dimensionless magnetic Reynolds number ( Re m 0 ) of flow represents the ratio of advection to diffusion in a magnetic field. At Rem<<1, advection is relatively unimportant, and a magnetic field thus tends to relax toward a purely diffusive state. This relaxation is determined by boundary conditions rather than flow; consequently, fluid motion has no influence on magnetic field distribution. In this work, a simulation of magneto-convective flow was carried out inside a square cavity. The purpose of the current study was to numerically solve full magnetic induction, continuum, momentum, and Fig. 1. Geometry and coordinates of cavity configuration energy equations to investigate the effect of the with magnetic effect M. Pirmohammadi / JHMTR 4 (2017) 149-155 151 More details regarding Equations (5) and (6) can B* B* V V U y V y B* B* be found in [17]. In the equations above, u and v are X Y x X y Y (13) velocity components, p denotes the pressure, T is the Pr 2B* 2B* ( y y ) temperature, represents the density, g is the 2 2 Rem X Y gravitational acceleration, is the viscosity, is the coefficient of thermal expansion, stands for the Bx To calculate the magnetic field instead of solving magnetic field in the X direction, and B represents Equations (12) and (13), we can use the magnetic y the magnetic field in the Y direction. Moreover, k potential vector ( A ) as follows: pertains to the thermal conductivity, Cp is the specific B A (14) heat capacity, is the electrical conductivity, and 0 denotes the magnetic permeability. A z A z A z (15) 3. The dimensionless variables included in the Bx ,By 0,Bz 0 analysis are defined thus: Y X Z x y CpuH CpvH Thus, Equations (12) and (13) are converted into X ,Y , U , V , H H k k Equation (15) in the following manner: 2 2 2 2 A A Pr A A Cp pH T TC (16) , , Re m 0 , U V ( 2 2 ) P 2 X Y Re m X Y k TH TC * B B k x * y The boundary conditions are Bx , By , Pr (7) B0 B0 c p B y at X 0 (17) A Y,Bx 1, 0 3 X c p g(Th T )H , Ra C , Ha B H k 0 By at X 1 (18) A Y, Bx 1, 0 On the basis of these variables, the governing X B x at Y 0 (19) equations are expressed in dimensionless form as A 0,By 0, 0 follows: Y B A 1,B 0, x 0 at Y 1 (20) y Y U V U & V 0 at all walls (21) 0 (8) X Y (X 0, X 1, Y 0, Y 1) (0,Y ) 1, (1,Y ) 0, , (22) 0 0 Y Y U U P 2U 2U Y 0 Y 1 U V Pr( ) X Y X X2 Y2 (9) 2 * * 2 Pr * By Bx Ha By ( ) 4.