A VERTICAL MAGNETO-CONVECTION IN SQUARE CAVITY CONTAINING A Al 2 O 3 +WATER NANOFLUID: COOLING OF ELECTRONIC COMPOUNDS

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1 Available online at Energy Procedia 18 (2012 ) A VERTICAL MAGNETO-CONVECTION IN SQUARE CAVITY CONTAINING A Al 2 O 3 +WATER NANOFLUID: COOLING OF ELECTRONIC COMPOUNDS S.Kadri a,b, R.Mehdaoui b, M.Elmir b a LPDS Laboratory, Bechar University, P.O.Box.417, Bechar,08000, Algeria b ENARGARID Laboratory, Bechar University, P.O.Box 417, Bechar, Algeria Abstract In this paper, we study by numerical simulation natural convection of Al 2 O 3 -water nanofluid in square cavity. In this cavity, the horizontal walls are isothermal, maintained at cold (T c ) and hot (T h ) temperatures. The vertical walls are adiabatic. A vertical magnetic field is externally imposed. The basic equations describing the flow driven by natural convection consist of mass conservation, momentum and energy. These equations are solved numerically by finite element method. A numerical simulation of the problem was performed using the software Comsol Multiphysics. For the physical parameters of Al 2 O 3 +water nanofluid, we use the Brinkman and Wasp model. Numerical results are presented for wide range of Hartmann number (Ha=0 to 100), with a value of Grashof number Gr=10 4 and volume fraction ϕ=0.1 in terms of streamlines, isotherms and the flux ratio. The results show that the magnetic field has effects on the flow and heat transfer Published by by Elsevier Ltd. Ltd. Selection and/or and/or peer peer-review review under under responsibility of The of TerraGreen [name organizer] Society. Open access under CC BY-NC-ND license. Keywords: Nanofluid ; Free convection ; Magneto-convection ;Heat transfer ; Finite element methodsquare cavity ; cooling of electronic compounds 1. Introduction The study of magnetic field effects has important applications in physics and engineering. The heat and mass transfer problems in the presence of magnetic field effects have attracted great interest of engineers and scientists for decades [1-3]. In this context, among previous studies, it may be noted that Ozoe and Maruo [10], Ozoe and Okada [11], Garandet et al. [12] and Venkatachalappa and Subbaraya [13] have been tried to acquire a basic understanding of heat transfer characteristics in an enclosure in the presence of magnetic field. Natural convection of nanofluid in a magnetic field have been studied in resent years by several authors. Rudraiah et al. [14] and Alchaar et al.[15] have shown a specific interest Published by Elsevier Ltd. Selection and/or peer review under responsibility of The TerraGreen Society. Open access under CC BY-NC-ND license. doi: /j.egypro

2 S.Kadri et al. / Energy Procedia 18 ( 2012 ) to focus on a natural convection within a rectangular enclosure with a magnetic field where one vertical wall is cooled and another one heated while the remaining top and bottom walls are well insulated. Recently, Ece and Buyuk [16] illustrated the natural convection flow under a magnetic field in a inclined rectangular cavity for heated and cooled on the adjacent walls. Mahmud and Fraser [17] have investigated magneto-hydrodynamic natural convection flowand entropy generation in a square cavity. On the contrary Grosan et al. [18] has studied effects of magnetic field and internal heat generation on natural convection flow in rectangular cavity filled with porous medium. In most of the previous studies, isothermal or isoflux thermal boundary conditions were applied to side walls of enclosures. There have been published several recent numerical studies on the modelling of natural convection heat transfer in nanofluids: Sathiyamoorthy and Chamkha [4], Hamad et al. [3], Elmir et al [5], Congedo et al. [6], Aminossadati and Ghasemi [7], Ho et al. [8,9], etc. The present work investigates the effects of vertical magnetic field on natural convection within a cavity filled with nanofluid. The main objective is to determine the influence of Hartmann number on streamlines, isotherms and heat transfer.the paragraphs continue from here and are only separated by headings, subheadings, images and formulae. Nomenclature K Thermal conductivity H Side of the square cavity Q Flux P Pressure P* Dimensionless pressure μ Dynamic viscosity β Coefficient of thermal expansion Thermal diffusivity ρ Density (ρc) Heat capacity σ Electrical conductivity ϕ Volume fraction Pr Prandt number Gr Grashof number Ha * Modified Hartmann number R ρ R β R k Density ratio Ratio of thermal expansion coefficients Ratio of thermal conductivities

3 726 S.Kadri et al. / Energy Procedia 18 ( 2012 ) g Gravitational acceleration B Magnetic field T Temperature T Time u,v Components of velocity fields u*,v* Dimensionless velocity components x, y Cartesian coordinates x*, y* Dimensionless coordinates f Fluid properties nf Nanofluid properties s Solid properties c Cold wall h Hot wall eff Effective * Dimensional properties 2. Mathematical Formulation Consider the steady laminar two-dimensional flow of a nanofluid in a square cavity of length H in the presence of an applied vertical magnetic field (Fig 1). The two vertical walls are adiabatic. The top and bottom horizontal walls are maintained at cold (T c ) and hot (T h ) temperatures respectively. The fluid is a water based nanofluid containing alumina (Al 2 O 3 ) nanoparticles. The nanofluid is assumed incompressible and the flow is laminar. It is assumed that the base fluid and the nanoparticles are in thermal equilibrium and they flow at same velocity. It is assumed that the induced magnetic field is negligible compared to the applied magnetic field parallel to the gravity. The boundary layer and Boussinesq approximations are assumed to be valid. T=T c,u=v =0 B T/ x=0, u= v =0 Al 2 O 3 -water g T/ x=0, u= v =0 H T=T h,u=v =0 Fig. 1. Physical model

4 S.Kadri et al. / Energy Procedia 18 ( 2012 ) Thermophysical properties of water-alumina nanofluid are included in table 1. Table 1. Thermophysical properties of water-alumina nanofluid [19] Parameters Value Pr f 7 R ρ R β R K ϕ 0.1 These assumptions, the conservation equations of mass, momentum and energy in cartesian coordinates are: Continuity: u/ x+ v/ y=0 (1) Momentum: u/ tt+u u/ x+v u/ y =-1/ρ nf P/ x+μ eff /ρ nf ( 2 u/ x u/ y 2 ) -(σ nf B 2 /ρ nf )u (2) u/ tt+ u v/ x+v v/ y = -1/ρ nf P/ y+μ eff /ρ nf ( 2 v/ x v/ y 2 ) - β nf (T-T 0 )g (3) Energy: T/ tt+ u T/ x+v T/ y = α nf ( 2 T/ x T/ y 2 ) (4) Stream function equation as: 2 ψ/ tx ψ/ y 2 = u/ y- v/ x (5) Where the thermal diffusivity of nanofluid is given by: α nf = K nf /(ρc P ) nf (6) The effective thermal conductivity of nanofluids has been determined by the model proposed by Wasp [20]. K nf = K f [(2-2ϕ)K f +(1-ϕ)K s ]/[(2+ϕ)K f +(1-ϕ)K s ] (7) The effective density of a fluid containing suspended particles is given by: ρ nf =(1-ϕ)ρ f + ϕρ s (8) The effective viscosity of a fluid containing a dilute suspension of small rigid spherical particles is given by Brinkman [21] as: μ eff =μ f /(1-ϕ) 2.5 (9) The heat capacitance of the nanofluid can be calculated as: (ρc P ) nf = (1-ϕ)(ρC P ) f + ϕ(ρc P ) s (10) The heat transfer is characterized by the flux ratio. The overage flux ratio at the bottom wall is computed as follows: Q nf /Q f =- K nf /K f T/ y (11)

5 728 S.Kadri et al. / Energy Procedia 18 ( 2012 ) By replacing equation (7) in (11), we obtain: Q nf /Q f =- [(2+(1+ϕ/1-ϕ)R k )/R k +(2+ϕ/1-ϕ)] T/ y (12) Where the ratio of thermal conductivity is defined as follows: R k = K s / K f (13) Using the following non-dimensional variables: (x *, y * ) =(x, y)/h, (u *, v * ) =(u, v)h/α nf, T * = (T-T c )/(T h T c ), P * = PH 2 /ρ f α f 2, t * = α f t /H 2 The governing equations (1)-(4) reduce to dimensionless form as: u * / x * + v * / y * =0 (15) (1-ϕ+ϕR ρ )( u * / tt * +u * u * / x*+v* u*/ y * ) =- P * / x * +Pr/(1-ϕ) 2.5 Δu- (PrHa *2 /(1-ϕ) 2.5 )u * (16) (1-ϕ+ϕR ρ )( v * / tt * +u * v * / x*+v* v*/ y * ) =- P * / y * +Pr/(1-ϕ) 2.5 Δv- (1-ϕ+ϕR ρ R β )GrPr 2 T * (17) T * / tt * + u* T * / x*+v* T*/ y* = [(2+(1+ϕ/1-ϕ)R k )/(R k +(2+ϕ/1-ϕ)](1/(1-ϕ)+ϕR ρ )ΔT * (18) The expressions for dimensionless parameters are given as: Pr = μ f /ρ f α f, Gr = ρ f 2 g β f H 3 ΔT/μ f 2, Ha * = HB σ nf / μ nf,r ρ = ρ s / ρ f, R β = β s /β f Dimensionless boundary conditions used to solve the system of equations (15)-(18) are illustrated in table 2. Table 2. Dimensionless boundary conditions The walls of cavity Boundary conditions Bottom walls u * = v * = 0 and T * = 0 Top walls u * = v * = 0 and T * = 1 Vertical walls u * = v * = 0 and T * / x * = 0 3. Results and discussions Numerical results are obtained by solving the system of steady differential equations (15)-(18), with appropriate boundary and initial conditions, using the Galerkin finite element method. The computational domain consists of bi-quadratic elements which correspond to grid points and a Lagrangequadratic interpolation has been chosen. We fixe the following parameters : Gr=10 4, Pr=7, ϕ=0.1. Ha=0, ψ max =16.72 Ha=20, ψ max =7.001

6 S.Kadri et al. / Energy Procedia 18 ( 2012 ) Ha=50, ψ max =1.19 Ha=60, ψ max = Ha=100, ψ max 0 Fig. 2. Streamlines and Isotherms for different values of the modified Hartmann number Ha * Figure 2 shows the variation of isotherms and streamlines at different values of modified Hartmann number. In absence of magnetic field (Ha * =0), near the horizontal walls, the isotherms are parallel to each other resulting in a conductive regime. In center of the cavity, the isotherms present a heat flux perpendicular to the gravitational acceleration, which favors the convective mode transfer. As the modified Hartmann number is increased, the shape of the isotherms is deformed to become parallel to the horizontal walls. These isotherms are characterized by thermal stratification that fosters conductive transfer mode. Beyond the value of Ha * = 60, the isotherms are characterized by a temperature gradient parallel to the acceleration vector (mechanical condition equilibrium). Transfer mode is purely conductive. The streamlines are characterized by monocellular convective structure and a vortex in the center of the cavity circulating in the anti-clockwise direction. As Ha * is increased, the vortex deform until Ha * <50 and the structure remains single cell. Beyond this value, the structure becomes three-cell or two-cell until complete disappearance of the cell. By increasing the value of Ha * the intensity of the flow becomes weaker still. The flow is characterized by a conductive regime Numerical Mathematical regression 3.0 Qnf/Qf Ha* Fig. 3. Variation of the overage flux ratio according to different values of the modified Hartmann number

7 730 S.Kadri et al. / Energy Procedia 18 ( 2012 ) Figure 3 shows the variation of the Qnf/Qf according to different values of modified Hartmann number. We note that increasing the value of Ha *, the flux ratio decreases, favoring the transfer mode conductive. Beyond the value of Ha * = 60, the flux ratio that characterizes the heat transfer is almost constant and close to the value 1, giving a purely conductive transfer mode. This confirms the results in Figure 2. The numerical values it possible to adopt a mathematical regression equation based on the square methods optimization. A Boltzmann sigmoïdal model is selected: Q nf /Q f =(2.98/(1+0.05exp(0.1Ha * )))+1.17 The test of Student and Fischer were used to determine the significance of parameters of the regression equations. With a confiance interval of 0.99 and a regression coefficient R= Numerical Mathematical regression ψ max Ha * Fig. 4. Variation of maximal stream function according to different values of the modified Hartmann number Figure 4 illustrate the variation of the maximum stream function in absolute value according to different values of Modified Hartmann number. Changes ψ max occurs in a decreasing order with increasing the value of Modified Hartmann number. The values of the maximum stream function show that the intensity of the flow decreases with the increase of Ha *, until a stagnant flow regime. The numerical values of the function of maximum current results suggest a numerical correlation based on the square methods optimization. A Boltzmann sigmoïdal model is selected: ψ max =(36.62/(1+1.16exp(0.06Ha * )))

8 S.Kadri et al. / Energy Procedia 18 ( 2012 ) The test of Student and Fischer were used to determine the significance of parameters of the regression equations. With a confiance interval of 0.99 and a regression coefficient R= Conclusions This work is devoted to the study of magneto-convection in square cavity containing alumina-water nanofluid. The governing equations are developed and solved numerically by finite element method with bi-quadratic rectangular elements. The results obtained in the form of streamlines, isotherms, maximum values of stream function and the flux ratio enable us to draw the following main conclusions: 1. By increasing the intensity of the magnetic field, the conductive transfer mode becomes dominant 2. The heat exchange in nanofluids is influenced by the value of Hartmann number and direction of applied magnetic field. References [1] C.Y. Wen, R.T. Tsai, K.P. Leong, Natural Convection of Magnetic Fluid in a Rectangular Hele- Shaw Cell of Different Aspect Ratios, Physics Procedia 9 (2010) [2] A. Mahdy, A.J. Chamkha,Yousef Baba, Double-diffusive convection with variable viscosity from a vertical truncated cone in porous media in the presence of magnetic field and radiation effects, Computers and Mathematics with Applications 59 (2010) [3] M.A.A. Hamada, I. Pop, A.I. Md Ismail, Magnetic field effects on free convection flow of a nanofluid past a vertical semi-infinite flat plate, Nonlinear Analysis: Real World Applications (2010), doi: /j.nonrwa [4] M. Sathiyamoorthy, A. Chamkha, Effect of magnetic field on natural convection flow in a liquid gallium filled square cavity for linearly heated side wall(s), International Journal of Thermal Sciences 49 (2010) [5] M. Elmir, A. Mojtabi, R. Mehdaoui, A. Slimani, Numerical Simulation of Magneto-Convection in an enclosure of Al 2 O 3 -water Nanofluid, ICCM3E, 2009, pp [6] P.M. Congedo, S. Collura, P.M. Congedo, Modeling and analysis of natural convection heat transfer in nanofluids, in: Proc. ASME Summer Heat Transfer Conf. 2009, vol. 3, 2009, pp [7] B. Ghasemi, S.M. Aminossadati, Natural convection heat transfer in an inclined enclosure filled with a water CuO nanofluid, Numer. Heat Transfer, Part A 55 (2009) [8] C.J. Ho, M.W. Chen, Z.W. Li, Numerical simulation of natural convection of nanofluid in a square enclosure: effects due to uncertainties of viscosity and thermal conductivity, Int. J. Heat Mass Transfer 51 (2008) [9] C.J. Ho, M.W. Chen, Z.W. Li, Effect of natural convection heat transfer of nanofluid in an enclosure due to uncertainties of viscosity and thermal conductivity, in: Proc. ASME/JSME Thermal Engng. Summer Heat Transfer Conf. -HT, vol. 1, 2007, pp [10] H. Ozoe, E. Maruo, Magnetic and gravitational natural convection of melted silicon e twodimensional numerical computations for the rate of heat transfer. JSME 30 (1987) [11] H. Ozoe, K. Okada, The effect of the direction of the external magnetic field on the threedimensional natural convection flow in a cubical enclosure. International Journal of Heat and Mass Transfer 32 (1989) [12] J.P. Garandet, T. Albousseiere, M. Moreau, Buoyancy driven convection in a rectangular enclosure with a transverse magnetic field. International Journal of Heat and Mass Transfer 35 (1992)

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