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 on magnetic field distribution in electrically conducting fluid under magnetohydrodynamic natural

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 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 at a 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 , 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 - 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 . 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 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 Cpu  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 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 , 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   k T T X Y Re m X Y H C

B * x * By k 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  B y 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. Numerical Procedures Rem X Y The governing equations associated with the

V V P  2V  2V boundary conditions were solved numerically using U  V    Pr( 2  2 )  RaPr (10) X Y Y X Y the control volume-based finite volume method. The Pr2 B* B*  Ha 2 B* ( y  x ) hybrid scheme, which is a combination of the central Re x X Y m difference and upwind schemes, was used to discretize convection terms. A staggered grid system, in which    2  2 velocity components are stored midway between scalar U V   (11) X Y X 2 X 2 storage locations, was also employed. The well-known revised semi-implicit method for pressure linked

* * equations algorithm was adopted in coupling the Bx Bx * U * U U  V  Bx  By velocity field and pressure in the momentum X Y X Y (12) Pr 2B* 2B* equations. The solution of the fully coupled discretized  ( x  x ) 2 2 equations was iteratively obtained using the tri- Rem X Y diagonal matrix algorithm. The values of residuals were checked on the basis of physical variables, such 152 M. Pirmohammadi/ JHMTR 4 (2017) 149-155 as temperature, pressure, and velocity. Convergence In the comparison of results, the magnetic field was considered achieved when the summation of induced by the motion of the electrically conducting residuals was less than 10–4, as was the case for most fluid was disregarded. Our findings showed good of the dependent variables. Simulations were agreement with those of Sarris. The relative errors in conducted with cell numbers ranging from 31×31 to the average Nusselt number and the maximum absolute value of the stream function of Sarris work 61×61 at a Rayleigh number (Ra) of 105. Fig. 2 shows [18] and the present model were 0.8% and 1.8%, the grid distributions of the cavity. respectively. In the current research, three magnetic Reynolds numbers, namely, 10–1, 10–3, and 10–5, were 5. Results and Discussion adopted. The fluid used was liquid sodium with Pr=0.01 and Ra=105. To verify the accuracy of the numerical technique Fig. 4 shows the contours of A, Bx, and By at used to solve the problem considered in this work, a Re =10–5 and Ha=80. As expected, because the simulation of magneto-convective flow was carried m magnetic field was very small, the magnetic potential out; the examined structure was a square enclosure from the left to the right sides of the cavity diffused, with a horizontal temperature gradient and was simulated in the presence of the magnetic field and its quantity remained identical to that introduced reported by Sarris [18]. Fig. 3 plots the streamlines into the boundary conditions. Fluid flow did not affect and isotherms of the solution derived in the current the applied magnetic field. Variables Bx and By were research and the results obtained by Sarris for calculated on the basis of the gradient of A; thus, they Ra=7×105, Ha=100, and Pr=0.7. were constant in the entire cavity. Specifically, Bx was approximately equal to 1, and By was equal to 0. The magnetic field in the cavity was preserved. Fig. 5 illustrates the contours of A, Bx, and By at –3 Rem=10 and Ha=80. Given that the Rem was small, the ratio of the diffusion of the magnetic potential to its advection was large. Consequently, the contours of A were approximately horizontal, and Bx and By in the cavity were constant. The maximum deviation of Bx from 1 was 2.73%, and the maximum deviation of By from 0 was 9.19%. –1 The contours of A, Bx, and By at Rem=10 and Ha=80 are presented in Fig. 6. The advection of the magnetic potential was greater than its diffusion, thereby generating horizontal contours. The magnetic Fig. 2. Grid distributions of the cavity field was unequal to the applied magnetic field and was inconstant in the cavity. The magnetic field in the X direction (Bx) therefore varied from 0.55 to 6.6, and the magnetic field in the Y direction (By) varied from –1.164 to 4.05. The concentration of the Bx contours near the horizontal walls was greater than that near the vertical walls. The magnetic field was stronger near Nu  3.20 the adiabatic walls. Furthermore, the concentration of  max  7.23 the By contours near the vertical walls was higher than (a) Sarris’s work [18] that near the horizontal walls. The advection of the magnetic potential was greater than its diffusion, thereby producing non-horizontal contours. The magnetic field was unequal to the applied magnetic field and was inconstant in the cavity. As a result, the magnetic field in the X direction (Bx) varied from 0.55 to 6.6, and the magnetic field in the Y direction (By) varied from –

1.164 to 4.05. The concentration of the Bx contours

 max  7.10 Nu  3.174 near the horizontal walls was higher than that near the (b) Present study vertical walls. The magnetic field was stronger near

Fig. 3. Comparison of isotherms and streamlines in the presentthe adiabatic walls. The concentration of the By work and Sarris’s study at Ra=7×105 and Ha=100 contours near the vertical walls was greater than that near the horizontal walls. M. Pirmohammadi / JHMTR 4 (2017) 149-155 153

A A 0.9375 0.9375 0.875 0.875 0.8125 0.8125 0.75 0.75 0.6875 0.6875 0.625 0.625 0.5625 0.5625 0.5 0.5 0.4375 0.4375 0.375 0.375 0.3125 0.3125 0.25 0.25 0.1875 0.1875 0.125 0.125 0.0625 0.0625

(a) (a)

BX BX 1.00026 1.02734 1.00023 1.0238 1.00019 1.02026 1.00016 1.01672 1.00012 1.01317 1.00009 1.00963 1.00005 1.00609 1.00002 1.00254 0.999982 0.999001 0.999947 0.995458 0.999912 0.991915 0.999876 0.988372 0.999841 0.984829 0.999806 0.981286 0.99977 0.977743

(b) (b)

BY BY 0.00092 0.96045 0.00083 0.09196 0.00074 0.0837 0.00066 0.0748 0.00057 0.0662 0.00049 0.0576 0.0004 0.04053 0.00032 0.03196 0.00023 0.02333 0.00014 0.01482 6.3E-05 0.00624 -2.27E-05 -0.00231 -0.000188 -0.01088 -0.000194 -0.01945 -0.00028 -0.2811

( c) (c) Fig. 4. Contours of variables: (a) contours of magnetic Fig. 5. Contours of variables: (a) contours of magnetic potential A, (b) contours of magnetic field in the X direction potential A, (b) contours of magnetic field in the x direction (Bx), and (c) contours of magnetic field in the Y direction (Bx), and (c) contours of magnetic field in the Y direction –5 –3 (By) at Rem=10 and Ha=80 (By) at Rem=10 and Ha=80

154 M. Pirmohammadi/ JHMTR 4 (2017) 149-155

A 20 0.9375 0.875 0.8125 15 0.75 0.6875 Rem=1e-1 0.625 10 Rem=1e-3 0.5625 Rem=1e-5 0.5 0.4375 0.375 5 0.3125 0.25

0.1875 V 0 0.125 0.0625 -5

-10

-15 (a) -20 0 0.25 0.5 0.75 1 X 5 BX Fig. 7. Velocity profile at Ra=10 , Ha=80, and different 6.59236 magnetic Reynolds numbers 6.12785 5.66334 5.19883 4.73432 4.26981 3.8053 20 3.34079 2.87627 2.41176 15 1.94725 1.48274 Rem=1e-1 1.01823 10 Rem=1e-3 0.8898 Rem=1e-5 0.5567 5

V 0

-5 (b) -10

-15 BY 4.04857 3.67619 -20 3.30381 2.93142 0 0.25 0.5 0.75 1 2.55904 X 2.18666 1.81427 Fig. 8. Temperature velocity profile at Ra=105, Ha=80, and 1.44189 1.06951 different magnetic Reynolds numbers 0.69712 0.3247 -0.04767 -0.42 -0.7924 Fig. 8 illustrates the mid-height temperature -1.1648 profiles at Ra=105, Ha=80, and different magnetic Reynolds numbers. At Rem=0.1, the temperature gradient and the slope of the temperature profile near –3 the hot wall were higher than those at Rem=10 and Re =10–5. (c) m Fig. 6. Contours of variables: (a) contours of magnetic 6. Conclusion potential A, (b) contours of magnetic field in the X direction (Bx), and (c) contours of magnetic field in the Y –1 This study analyzed the steady, laminar, and direction (By) at Rem=10 and Ha=80 magneto-convective flow of a viscous fluid in a cavity. Temperature gradient was applied on the two opposing regular walls of the enclosure, whereas Fig. 7 presents the mid-height velocity profiles at 5 adiabatic conditions were maintained in the other Ra=10 , Ha=80, and different magnetic Reynolds walls. A magnetic field was applied in the X direction. numbers. At Rem=0.1, the maximum velocity near the Magnetic Reynolds numbers of 10–1, 10–3, and 10–5 –3 hot wall was higher than those at Rem=10 and were adopted, and the fluid used was liquid sodium –5 Rem=10 . The magnetic field near the hot wall was with Pr=0.01 and Ra=105. weaker than that near the adiabatic walls (Fig. 6(b)); As the Rem increased, the advection of the thus, the Lorentz force was very low in these regions. magnetic potential increased, and variations in the M. Pirmohammadi / JHMTR 4 (2017) 149-155 155 magnetic field were augmented because fluid flow [6]. I.D. Piazza, M. Ciofalo, ‘‘MHD free convection in a affected magnetic distribution. 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