Moving Lids Direction Effects on MHD Mixed Convection in a Two- Sided Lid-Driven Enclosure Using Nanofluid

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1 Tran. Phenom. Nano Micro Scale, 1(2):93-102, Summer Autumn 2013 DOI: /tpnm ORIGINAL RESEARCH PAPER. Moving Lid Direction Effect on MHD Mixed Convection in a Two- Sided Lid-Driven Encloure Uing Nanofluid Javad Rahmannezhad 1, Mohammad Kalteh 2,* 1 Department of Mechanical Engineering, Univerity of Birjand, Iran 2 Department of Mechanical Engineering, Univerity of Guilan, Iran Abtract Magnetohydrodynamic (MHD) mixed convection flow of Cu water nanofluid inide a two-ided lid-driven quare encloure with adiabatic horizontal wall and differentially heated idewall ha been invetigated numerically. The effect of moving lid direction, variation of Richardon number, Hartmann number, and volume fraction of nanoparticle on flow and temperature field have been tudied. The obtained reult how 4 that for a contant Grahof number ( Gr = 10 ), the rate of heat traer increae with a decreae in the Richardon and Hartmann number. Furthermore, an increae of the volume fraction of nanoparticle may reult in enhancement or deterioration of the heat traer performance depending on the value of the Hartmann and Richardon number and the coiguration of the moving lid. Alo, it i found that in the preence of magnetic field, the nanoparticle have their maximum poitive effect when the top lid move rightward and the bottom one move leftward. Keyword: Magnetic field; Mixed convection; Nanofluid; Numerical imulation; Two ided lid-driven 1. Introduction Mixed convection flow and heat traer in encloure i of great importance in many indutrial heating or cooling application namely olar collector, float gla production and electronic device. In thi type of heat traer, complicated pattern of heat and ma traer occur becaue the flow driven by the movement of one or two face create a forced convection condition while temperature difference acro the encloure make econdary buoyancy driven flow [1]. Beide, in application where a great amount of heat need to be ejected from a very mall urface, the coolant hould * Correponding author Addre: mkalteh@guilan.ac.ir have more effective heat traer characteritic. Heat traer in uch area can be improved by uing the nano fluid, owing to their high thermal conductivity and better tability. A comprehenive review of the nanofluid heat traer characteritic can be found in [2] and [3]. Nanofluid natural convection heat traer ha been addreed in a large number of tudie. Many author like Khanafer et al. [4], Ece and Buyk [5], Aminoadati and Ghaemi [6], and Alloui et al.[7]invetigated natural convection of nanofluid in encloure. The reult of tudie about effect of nanofluid on mixed convection heat traer in lid-driven cavitie have been reported in the literature.. 93

2 Nomenclature Greek ymbol B 0 Magnetic field trength α thermal diffuivity C p Specific heat, J/kg (K) β thermal expanion coefficient g gravitational acceleration (m/ 2 ) ν kinematic vicoity L encloure height (m) ξ olid particle volume fractio Ha Hartmann number µ dynamic vicoity k Thermal conductivity (W/m K) ρ Denity Nu av average Nuelt number σ electrical conductivity Nu l local Nuelt number on the left hot wall ψ tream function Pr Prandtl number Ω Vorticity Ra Rayleigh number τ nondimenioal time Re Reynold number θ nondimenioal temperature Ri Richardon number ω nondimenioal vorticity Gr Grahof number Ψ nondimenioal tream function T dimenional temperature (K) u, v dimenional velocity component (m/) Subcript U 0 lid velocity (m/) c Cold U, V nondimenional velocity component f fluid phae x, y dimenional coordinate (m) h Hot X, Y nondimenional coordinate Nanofluid av olid phae Average Tiwari and Da [8] conducted a numerical tudy on the thermal behavior of Cu-water nanofluid in a twoided lid-driven quare encloure with inulated horizontal wall and differentially heated liding vertical wall. They found that the average Nuelt number increae with increae in the olid volume fraction. Another invetigation wa performed by Muthtamilelvan et al. [9], who have numerically examined the mixed convection in a lid-driven cavity filled with Cu-water nanofluid. Their numerical reult howed that both the encloure apect ratio and olid volume fraction affect the fluid flow and heat traer of the encloure. Nemati et al. [10] tudied the mixed convection flow utilizing a waterbaed nanofluid containing Cu, Cuo or Al 2 O 3 nanoparticle. They concluded that the effect of olid volume fraction grow tronger equentially for Al 2 O 3, Cuo and Cu. The effect of uncertaintie of vicoity model for the Al 2 O 3 -water nanofluid on mixed convection in a quare cavity wa conducted by Arefmaneh and Mahmoodi [11]. Their reult indicated that for both ued vicoity model the average Nuelt number increae with increae in the volume fraction of the nanoparticle. Very recently, Abbaian Arani et al. [12] carried out a numerical imulation to invetigate mixed convection flow of Cu-water nanofluid inide a lid-driven quare cavity with inuoidal heating on idewall. Effect of variation in Richardon number, phae deviation of inuoidal heating, and volume fraction of nanoparticle on flow and temperature field were tudied in their reearch. In many applied cae uch a crytal growth, liquid metal cating blanket for fuion reactor, pump, electronic package and 94

3 micro-electronic device, the flow i iluenced by magnetic field. Hence, tudy on natural or mixed convection fluid flow and heat traer in the encloure in the preence of magnetic force i important in uch cae. There are a large number of tudie on both MHD mixed and natural convection heat traer in conventional fluid-filled encloure. Groon et al. [13] have numerically tudied the teady MHD free convection in a rectangular cavity filled with a fluid-aturated porou medium with internal heat generation. Rahman et al. [14] performed a numerical invetigation into conjugate effect of joule heating and magnetic force, acting normal to the left vertical wall of an obtructed lid-driven cavity aturated with an electrically conducting fluid. Sivaankaran et al. [15] carried out a numerical tudy on mixed convection in a quare cavity of inuoidal boundary temperature at the idewall in the preence of magnetic field. The reult of a numerical tudy on mixed convection of MHD flow with joule effect in two-dimenional lid-driven cavity with corner heater were reported by Oztop et al. [16]. Revnic et al. [17] conducted a numerical invetigation into the effect of an inclined magnetic field and heat generation on unteady free convection within a quare cavity filled with a fluid-aturated medium. The effect of moving lid direction on MHD-mixed convection in a cavity with linearly heated bottom wall were analyzed by Al- Salem et al. [1]. In pite of a large number of paper about natural and mixed convection of conventional fluid in cavitie with magnetic field effect which i mentioned before, there i a eriou lack of iormation about MHD natural or mixed convection of nanofluid in encloure. Ghaemi et al. [18] examined the natural convection in an encloure filled with water-al 2 O 3 nanofluid, iluenced by a magnetic field. Natural convection in a quare cavity filled with different nanofluid i tudied numerically by Teamah et al. [19]. Both upper and lower urface were inulated, whilt a uniform magnetic field wa applied in a horizontal direction. Nemati et al. [20] applied the Lattice Boltzmann method to invetigate the effect of CuO nanoparticle on natural convection with MHD flow in a quare cavity. To the bet knowledge of the author, the problem of mixed convection of a nanofluid in a two-ided lid-driven encloure with magnetohydrodynamic effect ha not been reported o far. Therefore, the aim and cope of the current tudy i to invetigate the problem of mixed convection of a nanofluid in a two-ided lid-driven encloure in the preence of a magnetic field. The iluence of moving lid direction, variation of Richardon number, Hartmann number, and volume fraction of nanoparticle on flow and temperature field are tudied. The reult are preented in term of treamline, iotherm and average Nuelt number. 2. Problem definition and mathematical formulation A two-dimenional quare encloure i conidered for the preent tudy with the phyical dimenion hown in Fig. 1. In Fig. 1-a, the top wall of the encloure move rightward while it bottom wall move leftward. Thi coiguration will be referred to a RL. Similarly, other three coiguration will be referred to a LR (Fig. 1-b), RR (Fig. 1-c) and LL (Fig. 1-d), repectively. In all coiguration, the top and bottom wall are inulated whilt the left and right wall are kept at two different fixed temperature. The encloure i filled by a upenion of copper nanoparticle in water. Fig. 1. Schematic flow coiguration of the problem: a) RL Coiguration, b) LR Coiguration, c) RR Coiguration and d) LL Coiguration. 95

4 The hape and ize of nanoparticle are uppoed to be uniform. It i uppoed that both the fluid phae and nanoparticle are in thermal equilibrium and there i no lip between them. The flow i uppoed to be teady, laminar and two-dimenional. Thermophyical propertie of water and copper at the reference temperature are preented in Table 1. Table 1 Thermophyical propertie of water and nanoparticle at 25⁰ C. Phyical propertie Water Cu C p (J/kg K) ρ(kg/m 3 ) k(w/m K) β 10-5 (K -1 ) The propertie of nanoparticle and fluid are aumed to be contant except for the denity which varie according to Bouineq approximation. Thu, the non-dimenional governing equation in a twodimenional Carteian coordinate ytem can be expreed a below: 2 ψ X ψ Y 2 = ω (1) ω τ + U ω Y + V ω Y = 1 R e ν 2 ω + 2 ω ν f X 2 Y 2 (ρβ) θ +R i ρ β f X + Ha2 V R e X (2) + UU + VV = KK ρρcc nnnn pp ff + KK ff ρρcc pp nnnn 1 θθ RRRRRRRR ( 2 XX θθ (3) YY 2) The following non-dimenional parameter are ued to obtain the above equation: X = x L, Y = y L, U = u U 0, V = v U 0, (4) τ = tu 0 L, ω = ΩL U 0, ψ = ψ LU 0 θ = T T c T h T c, Re = U 0L ν F, Pr = ν f α f Ri= GGGG RRee 2, Ha=BB 0LL σσ nnnn ρρ nnnn νν nnnn where the denity, thermal expanion coefficient, electrical conductivity, thermal diffuivity and heat capacity are a following, repectively: ρ = (1 ξ) ρ f + ξρ (5) ( ρβ ) = (1 ξ)( ρβ ) f + ξ( ρβ ) (6) = (1 ξ) f + ξ (7) α = k /( ρ c p) (8) (c) ρ p = (1 ξ)(c) ρ p f +ξ(c) ρ p (9) The effective dynamic vicoity of the Cu water nanofluid i calculated according to the Brinkman model [21] a follow: 2.5 µ =µ f (1 ξ ) (10) The effective thermal conductivity of the nanofluid i determined a follow [22]: k k A A = 1+ + ck Pe (11) k k A ka f f f f f Where k i the thermal conductivity of dipered nanoparticle and k f i the thermal conductivity of pure fluid. c i a contant and equal to for a wide range of experimental data [22], and: A d ξ = A d 1 ξ ud, Pe= α f f f (12) where d i diameter of the olid nanoparticle that in thi tudy i aumed to be equal to 100 nm, df i the molecular ize of liquid that i taken a 2Å for water and alo u i the Brownian motion velocity of nanoparticle given by: u 2k T = (13) πµ b 2 fd where k b i the Boltzmann contant. The non-dimenional boundary condition, ued to olve the equation (1)-(3) are: 96

5 U = 0, V = 0 and θ= 1 at left wall U = 0, V = 0 and θ= 0 at right wall U =± 1, V = 0 and θ = 0 Y at bottom wall U =± 1, V = 0 and θ = 0 Y at top wall (14) The local and average Nuelt number can preent the local and average heat traer rate of the encloure. The local Nuelt number ( Nu l ) i calculated along the left heated wall (Eq. (15)) and the average Nuelt number ( Nu av ) i determined by integrating the local Nuelt number along the heated wall (Eq. (16)): Nu Nu k θ X l = kf 1 av 0 l (15) = Nu dy (16) A hown in Fig. 2 and Table 2 there i a good agreement between the preent reult and the exiting reult in the literature. The grid independence tudy i carried out by conidering different meh ize for Ri=1, Ha=35 and ξ=0.06. Fig. 3 preent the average Nuelt number ( Nu av ) for four different cae (RL), (LR), (RR) and (LL) which were introduced in Fig. 1. Baed on the reult, a grid ize of i found to meet the requirement of both the grid independency tudy and the computational time limit. The computation are performed until teadytate condition are reached while a time-tep equal to 10-3 ec i ued. The convergence criterion ued in the time loop to achieve teady-tate condition i k k 1 6 ϕ ϕ < 10, where ϕ tand for either θ or ω. 3. Numerical analyi and validation The non-dimenional governing equation (Eq. (1-3)) ubjected to the boundary condition (Eq. (14)) are olved by the finite difference cheme [23] and the ytem i numerically modeled in FORTRAN 95. The point Gau Seidel iterative method i ued to olve the elliptic part of PDE (Eq.1), and a fourth order Runge-Kutta method i employed to advance the olution of the parabolic part (Eq. 2-3). To validate the numerical code, two different tet cae are employed and the reult are compared with the exiting reult. in the literature. The firt tet cae i MHD mixed convection of air in a lid-driven quare cavity with linearly heated bottom wall. The obtained reult by the preent code are compared with reult of Al-Salem et al. [1] in Fig. 2. The econd tet cae i mixed convection of Al 2 O 3 water nanofluid in a two-ided lid-driven cavity. The preent reult for the econd tet cae are compared with reult of Aminoadati et al. [24] in Table 2. Fig. 2. Comparion between preent reult (right column) and reult of Al-Salem et al. [1] (left column) for two cae: a)the lid i moving in +X direction and b) The lid i moving in X direction, (Re=100, Gr=10 5, Ha=30). 97

6 pattern o that their addition intenifie the trength of vortexe. Fig. 3. Grid independency tudy: average Nuelt number at different grid denitie. 5. Reult and dicuion In thi ection, reult of the numerical tudy of mixed convection flow and heat traer inide the two-ided lid-driven quare encloure uing Cu-water nanofluid that wa hown in Fig. 1 are preented. The reult have been obtained for Richardon number ranging from 0.1 to 10, the Hartmann number ranging from 0 to 55, and the volume fraction of the nanoparticle ranging from 0 to It mut be noted that in the preent tudy, the Richardon number i varying with variation of Reynold number while Grahof number i kept contant at Fig. 4 how treamline and iotherm for four different coiguration at Ri=1 and Ha=35. The reult are preented for pure fluid (dahed line) and nanofluid with ξ=0.06 (olid line). A can be een from thi figure, in all cae two large fully developed eddie are formed inide the encloure due to hear force exerted on the nanofluid next to the wall. In the cae (a) where the top moving lid move rightward and the bottom one move leftward, the effect of moving lead amplifie buoyancy force and oppoe the effect of magnetic force. Due to the motion of the two wall, two major clockwie recirculation cell are produced inide the encloure. The direction of thi rotation i according to the direction of hear force exerted by moving lead. It i worth mentioning that nanoparticle have a coniderable effect on the flow Fig. 4. Streamline (on the left) and iotherm (on the right) for Cu-water nanofluid with ξ=0.06 (olid line) and pure fluid (dahed line) a) RL, b) LR, c) RR and d) LL coiguration (Ri=1, Ha=35). The iotherm in the left column of Fig.4a how that the heat traer ha a good balance and the heat i well ditributed in the whole region of the encloure. It i obviou from thi figure that the temperature gradient i high in the vicinity of the right wall which mean high rate of heat traer in thi region. In the cae (b), each wall move in the oppoite direction of correponding wall of the cae (a). Similar to the previou cae, the treamline near the moving wall include two major recirculation cell, but the direction of rotation i counterclockwie in thi cae. Moreover, there i a weaker clockwie circulating zone in the middle of the encloure which i generated due to the dominance of magnetic force in thi region. Fig. 4c i related to the third tudied coiguration named a RR. In thi coiguration, both top and bottom wall move rightward and hence the generated vortexe are in oppoite direction. Due to the coormity of the top 98

7 wall force, buoyancy force and magnetic force in the middle region, the top recirculation cell penetrated deeper in the encloure. Table 2 Comparion of average Nuelt number from the preent code againt S.M. Aminoadati et al. [22]. Solid volume fraction Same direction Preent tudy S.M. Aminoadati et al. [24] Difference 5.40% 3.97% 2.46% Oppoite direction Preent tudy S.M. Aminoadati et al. [24] Difference 3.76% 2.35% 1.68% The iotherm demontrate that the temperature gradient are higher near the hot wall. Thi implie higher heat traer rate in thi region. In the lat coiguration (Fig. 4d), both top and bottom wall move leftward and hence again the two major generated vortexe are in oppoite direction and alo againt the vortexe in the RR coiguration. Here, the magnetic force augment the bottom wall hear force and hence the underneath recirculation cell penetrate deeper in the encloure. Variation of average Nuelt number with Richardon number at four different coiguration are hown in Fig. 5. Moreover, it can be een that the maximum and minimum average Nuelt number are related to RR and LL coiguration, repectively. Thi i quiet conitent becaue in the RR coiguration the force generated by the two wall are in the ame direction and they oppoe the magnetic force, while, in the cae of LL coiguration, the effect of two moving lid and the magnetic force are all in the ame direction. The Nuelt number of RL and LR coiguration are lower than that of RR coiguration. The reaon i that in the RL and LR cae, the wall move in uch a way that their reultant force are in the oppoite direction. In order to expoe the magnetic field effect on the heat traer rate, variation of the average Nuelt number with the Hartmann number ha been tracked. Fig. 6 how how thi number varie with repect to the direction of moving lid at different value of the Hartmann number. Fig. 5. Variation of average Nuelt number with Richardon number at four different coiguration (ξ=0.06). According to the figure, a Richardon number increae, while the Grahof number i kept contant, the effect of top and bottom wall and forced convection decreae. So the average Nuelt number decreae a a reult of weaker convective flow. Fig. 6. Variation of average Nuelt number with Hartmann number at four different coiguration (ξ=0.06). 99

8 The reult indicate that except for the mall region of the LR coiguration, the heat traer rate decreae with an increae in magnetic field trength. Thi behavior can be decribed according to the combined effect of moving lid force and the magnetic force on each coiguration. Again it i obviou that the maximum and minimum Nuelt number i related to RR and LL coiguration, repectively. Fig. 7 how the variation in the average Nuelt number ratio ( Nu av / Nuav, ξ = ) againt the olid 0 particle volume fraction for different coiguration at different value of the Richardon number. The average Nuelt number ratio at ξ=0 i the reference value and the Hartmann number i aumed to be contant (Ha=35) for thi part of the tudy. A can be een from thi figure, at low Richardon number, the maximum poitive effect of nanoparticle i related to the RL coiguration. It i notable that in ome cae, the addition of nanoparticle deteriorate the heat traer performance in the encloure epecially in higher Richardon number in which natural heat traer i dominant. Variation of the average Nuelt number ratio with the olid particle volume fraction at different Hartmann number and for four different coiguration are depicted in Fig. 8. A illutrated in thi figure, the effect of nanoparticle i more pronounced for lower value of the Hartmann number. For example in the RL coiguration at ξ=0.06 when the Hartmann number varie from 0 to 55, the improvement in the average Nuelt number due to addition of nanoparticle decreae from 5.5% to 1.7%. 4. Concluion -In the exitence of magnetic field, the nanoparticle have their maximum poitive iluenceon the heat traer rate in the RL coiguration. Fig. 7. Variation of the average Nuelt number ratio at hot wall with olid volume fraction at Ha=35 for different coiguration a) Ri=0.1, b) Ri=1, c) Ri=10.. In the preent work the problem of MHD mixed convection flow of Cu water nanofluid inide a twoided lid-driven quare encloure with adiabatic horizontal wall and differentially heated idewall were numerically tudied. The governing equation were olved uing finite difference cheme. According to the obtained reult, it can be concluded that -Depending on the value of Hartmann and Richardon number and the coiguration of the moving lid, both enhanced and deteriorated heat traer were oberved by addition of nanoparticle. -For a contant Grahof number, when the Richardon and Hartmann number decreae, the rate of heat traer increae. 100

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