Numerical Study of Hydro-Magnetic Nanofluid Mixed Convection in a Square Lid-Driven Cavity Heated From Top and Cooled From Bottom

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1 Tran. Phenom. Nano Micro Scale, (1): 9-4, Winter - Spring 014 DOI: /tpnm ORIGINAL RESEARCH PAPER. Numerical Study o Hydro-Magnetic Nanoluid Mixed Convection in a Square Lid-Driven Cavity Heated From Top and Cooled From Bottom A. Zare Ghadi 1, M. Sadegh Valipour 1,* 1 Mechanical Engineering Department, Univerity o Semnan,Semnan, I.R. Iran ARTICLE INFO. Article hitory Received 1 April 013 Accepted 9 November 013 Keyword Lid-Driven Cavity MHD Flow Mixed Convection Nanoluid, Abtract In the preent reearch mixed convection low through a copper-water nanoluid in a driven cavity in the preence o magnetic ield i invetigated numerically. The cavity i heated rom top and cooled rom bottom while it two vertical wall are inulated. The governing equation including continuity, N-S and energy equation are olved over a taggered grid ytem. The tudy i conducted or Graho number10 3 to 10 5, Hartmann number 0 to 100 and volume raction number 0 to 5% while Reynold number i ixed at 100. Hamilton Croer and Brinkman model have etimated eective thermal conductivity and eective vicoity o nanoluid, repectively. It i oberved that magnetic ield ha uncontructive eect on heat traner proce wherea nanoparticle increae heat traner rate. 1. Introduction.The prediction o luid low and heat traner in a lid-driven cavity have been invetigated through the pat decade, becaue o it igniicance in many engineering application uch a building application, olar collector, nuclear reactor and crytal growth. In the latter cae, luid experience magnetic ield. There are ome tudie about mixed convection in lid-driven cavity under the eect o magnetic ield. To review reearche related to our tudy, we divide them into three categorie: 1-Mixed convection in lid-driven encloure with variou boundary condition. -Magnetic eect on mixed convection in lid- driven encloure 3-Mixed convection in lid-driven encloure uing *Correponding author: addre:mvalipour@emnan.ac.ir nanoluid. 1- There have been numerou tudie in the pat on mixed convection in lid-driven encloure. Oztop and Dagtekin [1] dicued on mixed convection in a twoided lid-driven dierentially heated cavity. It i ound that both Richardon number and direction o moving wall aect the luid low and heat traner in the cavity. Shari [] tudied the mixed convection in a hallow inclined driven cavity or variou Richardon number. He indicated that the average convection in deep lid-driven cavitie heated rom below. They oberved that the heat traner wa rather inenitive to the Richardon number. - There are limited tudie on eect o MHD mixed convection in lid-driven cavitie. Rahman et al. [7] numerically tudied the MHD combined convection low in an obtructed driven cavity under the eect o Joule heating. Their numerical reult indicated that the Hartmann number, Reynold number and Richardon number have trong inluence on the 9

2 Zare. Ghadi & Valipour/ TPNMS (014) 9-4 treamline and iotherm. They alo deduced Joule heating parameter ha little eect on the treamline and iotherm plot. Chamkha [8] reported a numerical invetigation on hydromagnetic combined convection low in a lid-driven cavity with internal heat generation or aborption. The preence o the internal heat generation eect wa ound to decreae the average Nuelt number coniderably or aiding low and to increae it or oppoing low. Sivaankaran et al. [9] numerically tudied the Hydro-magnetic mixed convection in a cavity with inuoidal boundary condition at the idewall. They indicated that the low behavior and heat traner rate inide the cavity are. Nomenclature B0 Magnetic lux denity ( Gr Ha Greek ymbol wb. m ). Thermal diuivity (m -1 ),( = ) 3 gth. 1 Graho number(= ) Coeicient o volume expanion ( ) Hartmann number(= ) Thermal conductivity (W.(mK -1 )) Nu Local Nuelt number Solid volume raction Nu ave Average Nuelt number Dynamic vicoity (N.m - ) p Non dimenional Preure Kinematic vicoity (m ), (= ) Pr Prandtel number(=. ) Dimenionle coordinate Re Reynold number( = ) Denity (kg.m -3 ) T Fluid temperature (K) 1 1 Electrical conductivity ( m ) TC Temperature o cold wall (K) Stream unction TH Temperature o hot wall (K) Subcript u, v Dimenional velocitie (m. -1 ) Free tream U, V Dimenionle velocitie Solid U0 Top lid velocity (m. -1 ) w Wall x, y Dimenional coordinate n Nanoluid X, Y Dimenionle coordinate Fluid trongly aected by the preence o the magnetic ield. Oztop et al. [10] invetigated MHD mixed convection in a lid-driven cavity with corner heater with variou length. They howed that the heater length ha igniicant eect on low regime and heat traner. They alo concluded that an increae in Hartmann number lead to a decreae in low trength and heat traner rate. 3- There are ome tudie on eect o nanoparticle on mixed convetion in a lid-driven cavity. Tiwari and Da [11] made a work on heat traner in a two-lided lid-driven encloure uing nanoluid; the let and the right wall are maintained at dierent contant 30 temperature while the upper and the bottom wall are thermally inulated. They illutrated that nanoparticle embedded in luid are capable o increaing the heat traner capacity o bae luid. They alo indicated the variation o average Nuelt number i nonlinear with olid volume raction. Talebi et al. [1] numerically tudied mixed convection low in lid-driven cavity uing copper-water nanoluid. In their work, the top and bottom horizontal wall are inulated while the vertical wall are maintained at contant but dierent temperature. It i oberved that at the ixed Reynold number, the olid concentration aect on the low pattern and thermal behavior particularly or a higher

3 Zare. Ghadi &.Valipour/ TPNMSS (014) 9-4 Rayleigh number. Abu-Nadcombined convection in a tilted lid-driven cavity illed with a nanoluid. It i ound that igniicant heat traner enhancement can be obtained due to the preence o nanoparticle and that thi i accentuated by inclination o the encloure at moderate and large Richardon number..a dicued above, magnetic ield decreae heat traner rate wherea nanoparticle immered in bae luid increae it. Thereore, in uch engineering application where luid i inevitably ubjected to magnetic ield; perormance o heat and luid low i augmented by adding nanoparticle in bae luid. Thi novel idea ha been carcely tudied in the literature. Hamad et al. [14] reported magnetic ield eect on nanoluid low pat a vertical plate. Later on Hamad [15] analytically invetigated on natural convection o nanoluid low or a heet in the preence o magnetic ield. Recently, Ghaemi et al. [16] analyzed natural convection in a nanoluid-illed cavity in the preence o magnetic ield. Their reult howed that the heat traner rate increae with an increae o the Rayleigh number but it decreae with an increae o the Hartmann number..the main motivation o thi work i to tudy the eect o magnetic ield on mixed convection low in a lid-driven cavity illed with copper-water nanoluid. A mentioned beore two horizontal wall o cavity are at contant but dierent temperature, a top moving wall i hot and bottom wall i cold. Let- and right-hand idewall are thermally inulated. and Chamkha [13] invetigated. Mathematical modeling.1. Governing equation.fig. 1 how a chematic coniguration o the computational domain and coordinate ytem ued in the preent tudy. It i preumed that the low i teady, laminar, incompreible and ingle-phaee model i conidered or copper-water nanoluid. With neglectingg induced magnetic ield and auming electrical ield to be zero the Lorentz orce can be impliied a. Dimenional equation are a ollow. Continuity u v 0 x y x-momentum equation u u 1 u v x y n y-momentum equation n Energyy Equation: u T x Eq. (1)-(4) can be rewritten in the dimenionle orm by deinition o the ollowing parameter: x X, Y H T TC, T T H Pr, T v y C y H, P g TH Gr p x n u u x y v v u v x y 1 p v v n n y x y g n B ( T T ) (1 ) U n n p u U U 0 3 0, T x, V, v U U 0H Re Ha B H n T y (1) () (3) (4) (5) Fig. 1. Schematic o coordinate ytem computational domain and Thereore, dimenionle orm o the governing equation (Eq. 1-4) may be expreed a the ollowing: 31

4 Zare. Ghadi &.Valipour/ TPNMS (014) 9-4 Continuity U V (6) 0 X Y X-Momentum equation U U U V X Y U U X Y P X Y- Momentum equation 1 Re n (1 ).5 (7) The denity o the nanoluid i etimated a the ollow: n ( 1) (1) The heat capacitance o nanoluid i expreed a the ollow: ( c ) p ) n (1 )( c p ) ( c p (13) The eective thermal conductivity o the nanoluid i approximated by the model given by Patel et al. [19] a the ollowing: V U X V V Y.5 V V (1 ) X Y Gr (1 3 ) Ha V Re 1 P Y n 1 Re n (8) n 1 k k A A ck Pe k A A (14) Where c i contant and mut be determined experimentally (or thi tudy = ), A / A and Pe are deined a: A A d u d, Pe d 1 (15) Energy equation U V X X Y Y k k n ( c ( c p p ) ) n 1 Re Pr. Thermo- phyical propertie o nanoluid (9).The electrical conductivity i a below (Maxwell' model) [17]: ( 1 3 ) n (Where ) (10) The vicoity o nanoluid i approximated by Brinkman [18] a the ollowing: n.5 ( 1) (11) With i the diameter o the olid particle that in thi tudy i preumed to be equal to 100 nm, i molecular ize o liquid that i taken a Å or water. Alo i the Brownian motion velocity o nanoparticle which i deined a u kbt (16) d Where i the Boltzmann contant. Table 1 preent thermo-phyical propertie o water and copper at the reerence temperature. Property Water Copper cp

5 Zare. Ghadi &.Valipour/ TPNMS (014) Boundary Condition.According to Fig. 1 the governing equation (1)- (4) are ubjected to the ollowing boundary condition: Top wall ( 0 X Y 1) U 1, V 0, 1 Bottom wall ( 0 X Y 0) U V 0, 0 Let and right wall : (17) (18) ( 0 Y X 0 & 1) (19) U V 0, 0 x.4. Auxiliary equation When the velocity and temperature ield are calculated, the relation decribed by the ollowing are applied to etimate local and average.the local Nuelt number i evaluated at the wall o the cavity a ollow: k Nu k n Y Y 1 (0) The urace averaged value o Nuelt number on the cavity wall i calculated by the ollowing relation: Nu 0NudX 1 ave (1) 3. Numerical olution 3.1. Solution technique.the governing equation mentioned in previou ection are olved with algorithm SIMPLE (emiimplicit method or preure-linked equation) introduced by Patankar [0]. Thi algorithm wa ucceully implemented or many problem o newtonian luid low and heat traner in both laminar and turbulent ituation. Central dierence cheme i ued to dicretize the diuion term wherea a combination o upwind and central dierence i adopted or dicretizing the convection term. The reulting dicretized equation are olved iteratively by Tri-diagonal matrix algorithm (TDMA) [1]. Te iteration proce i terminated when the ollowing condition i atiied t t1 i, j i, j 6 10 t i, j () Where reer to, U,V and t indicate the iteration tep. 3.. Grid independence tudy.to tudy the grid independence examination, the numerical procedure i carried out or dierent grid reolution. The grid enitivity analyi i perormed or Re=100,Gr=10, = 0.05 and Ha=0.Table indicate the impact o the number o grid point on the average Nuelt number on the top wall o the cavity. According to percentage dierence o dierent grid in lat column the uniorm grid ytem o i ine enough to attain accurate outcome in thi work. Table Eect o grid number on 3 Gr 10, and Ha 0 Number o Grid Nuave Validation Nu ave at 100 Re, Pr 6., Percentage dierence 1.03 % 0.64 % 0.15 % 0.07 %.In order to validate our numerical olution, calculated local Nuelt number in top hot wall and U velocity in centerline o encloure are compared with 33

6 Zare. Ghadi &.Valipour/ TPNMS (014) 9-4 thoe o Iwatu et al [4]. A it i hown in Fig., the local Nuelt number and u velocity are ound to be in good agreement comparing with the reult o Iwatu et al [4]. Fig.. Comparion o a) the U-proile at the middle o encloure and b) the Nuelt number at the top wall at Re=400 & Gr= RESULTS AND DISCUSSIONS The treamline or variou Graho number and Hartmann number are hown in Fig. 1; The olid and dahed line are correponded to clear luid and nanoluid ( =0.05 ), repectively. A can be een in Fig 3 the eect o lid-driven low i dominant or =10 in the abence o the magnetic ield; The main circulation cover the entire cavity and two minor bubble are ormed near the bottom corner. Becaue the magnetic ield ha the inclination to decelerate the working luid in the cavity, when the Hartmann number i increaed, the main circulation inhibit progreively and econdary bubble i ormed at the bottom o cavity. When the Hartmann number i increaed rom 30 to 100 the primary and econdary circulation in the low are gradually limited to the area at the top and middle o the cavity and the low near the bottom o cavity become almot tagnant. Increae in Graho number to 10 augment the buoyancy eect; but the eect o the orce convection i till oberved. When Graho number i increaed to 10 prominent buoyancy-driven convective characteritic are noticeable and much o the middle and bottom part o working luid i tagnant. In thi cae the augmentation o magnetic ield rom 30 to 100 doe not have ubtantial eect on the low pattern becaue the low i tagnant in the bulk o the cavity. Fig. 3 alo indicate that the trength o circulation increae by increaing olid volume raction or lower Graho number, wherea it remain almot unchanged or higher Graho number. The temperature contour are depicted in Fig. 4 or dierent Gr and Ha number or both clear luid and nanoluid. Where dahed line are ued or nanoluid and olid line are ued or clear luid. For Graho number 10 and in the abence o the magnetic ield temperature ditribution i teep near the bottom o the cavity. In mot o the cavity excluding thi area, however, the temperature gradient are gentle; Thi indicate the remarkable eect o mechanicallydriven circulation. When magnetic ield i applied, the eect o driven lid i decreaed and temperature gradient are gentler at bottom zone o the cavity. A thermally tratiied region in the vertical direction appear in the bulk o the cavity when Hartmann number increae to 100. With increaing Graho number conduction heat traner play more eective role and thereore the temperature ditribution contour demontrate that the luid i vertically tratiied in the tagnant zone at the bottom and middle o the cavity. When Graho number i increaed to10 the eect o the buoyancy predominate the orced convection eect and conduction-dominated regime with horizontal iotherm i ormed in the mot o the cavity. In thi cae, the inteniication o magnetic ield rom 30 to 100 doe not have coniderable eect on the low pattern becaue the low i tagnant in the bulk o the cavity. Fig. 4 alo iner that the addition o nanoparticle i more eective in lower Graho number and weaker magnetic ield. Fig. 5 how the variation o horizontal and vertical velocitie along the mid-plane o the cavity 34

7 Zare. Ghadi &.Valipour/ TPNMS (014) 9-4 or variou Gr and Ha number or volume raction o The maximum vertical velocity decreae when the Geraho number increae owing to dominant conduction mode, and it decreae when the Hartmann number increae owing to reitive orce applied by magnetic ield. Fig. 5 (right) depict horizontal velocity i trongly aected by Hartmann number and Graho number. When Graho number i increaed buoyancy-driven low i trengthened and thereore, the bulk o cavity become tagnant and the horizontal velocity i cloe to zero in mot o the cavity. By applying magnetic ield, lid-driven low i retricted to the top portion o the cavity and thereore the horizontal velocity varie in thi area and remain unchanged (cloe to zero) in the bulk o cavity where the low i tagnant..the eect o Hartmann number on horizontal temperature ditribution in mid-plane (right) and Nuelt number (let) on top lid o cavity i illutrated in Fig. 6 or = From the igure it i evident, the Nuelt number at the top lid begin with a high value at the let end and reduce monotonically to a low value toward the right end. Thi how the trong inluence o circulation in the vicinity o the top lid. An increment o Hartmann number decreae Nuelt number, becaue that the circulation i limited cloe to top lid and the middle o cavity experience tagnant low. Temperature ditribution in the middle o the cavity i trongly aected by Hartman number and Graho number. Since the buoyancy eect i negligible in lower Graho number, much o the temperature variation i achieved in mall region near right wall o the cavity. By increaing Hartmann number circulating bubble, move upward and temperature ditribution change linearly throughout the mid-plane. At higher Graho number, the working luid i tationary and the reultant temperature ditribution attain the linear proile due to main conduction mode. In thoe range o Graho number, magnetic ield ha maller eect than that o lower Graho number, a the eect o the magnetic ield i negligible or Graho number o 10..The rate o heat traner augmentation or nanoluid ( =0.05 ) rather than clear luid veru Hartmann number i indicated in Fig. 7. The ormulation i a below: E Nu ( 0.05) Nu ( 0) 100 % Nu ( 0) (3).It i een rom the igure that heat traner enhance with the addition o nanoparticle and the amount o increment, increae with increaing Graho number. A the magnetic ield eect on the low, the rate o heat traner enhancement increae or Graho number o 10 and 10 and become contant in high Hartmann number..fig.8 illutrate the variation o the average Nuelt number on the top lid o cavity veru o Hartmann number at dierent Graho number or = The eect o magnetic ield on the overall heat traner proce i clearly depicted in thi igure. The average Nuelt number decreae gently with increaing Hartmann number or the dominating orced convection mode (Gr =10 ). Sharper increae o the average Nuelt number with enhancing Hartmann number i een in Gr=10 owing to the uppreion orced convective eect and augmentation buoyancy eect..the eect o nanoparticle on average Nuelt number are plotted in Fig. 9 or variou Hartmann number in Gr=10. The increae o the olid concentration increae the eective thermal conductivity and a a reult the heat traner rate. In other word, average Nuelt number enhance with increaing the olid volume raction. The increae o Hartmann number decreae the heat traner rate and hence average Nuelt number. The rate o heat traner uppreion due to magnetic ield eect or variou Graho number at volume raction o 0.05 can be een in Fig. 10. The ormulation i a ollow: Nu( Ha 30,100) Nu( Ha 0) E Ha 100% (4) Nu( Ha 0).It i evident rom the igure that the magnetic ield ha uncontructive impact on heat traner proce. In other word, when Hartmann number increae heat traner uppreion increae. For intance at Gr= 10 3 Heat traner reduction i about 60% or Hartmann number o 30, wherea it i about 70% or Ha=100. The igure alo how increaing in Graho number weaken the impact o magnetic ield. 35

8 Zare. Ghadi & Valipour/ TPNMS (014) 9-4 Gr=1.0E+3 Gr=1.0E+4 Gr=1.0E Ha= Ha= Ha=100 Fig. 3. treamline at variou Hartmann number and graho number (olid line or pure luid and dahed line or nanoluid with volume raction o 0.05) 36

9 Zare. Ghadi &.Valipour/ TPNMSS (014) 9-4 Fig. 4. iotherm at variou Hartmann number and graho number (olid line or puree luid and dahed line or nanoluid with volume raction o 0.05) 37

10 Zare. Ghadi &.Valipour/ TPNMSS (014) 9-4 a b c d e Fig. 5. Variation o u-velocity (let) and v-velocity (right) at the middle o the encloure or volume raction o 0.05 at dierenct graho number. 38

11 Zare. Ghadi & Valipour/ TPNMSS (014) 9-4 a b c d e Fig. 6. Local Nuel number at variou Hartmann number at the hot urace (right) and temperature proile at the middle o the cavity or dierent Graho number at volume raction o

12 Zare. Ghadi &.Valipour/ TPNMSS (014) 9-4 Fig. 7. Ratio o enhancement o heat traner due to addition o nanoparticle. Fig. 9. Variation o the average Nuelt number urace veru volume raction number or Hartmann Graho number o 105 at the hot dierent Fig. 8. Variation o the average Nuelt number at the hot urace with Hartmann number or dierent Graho number at volume raction o 0.05 Fig. 10. Ratio o uppreion o heat traner due to magnetic ield. 40

13 Zare. Ghadi & Valipour/ TPNMS (014) Concluion Traner mechanim. Obtained reult releaed ome igniicant point a ollow: The preent work tudied MHD combined convection in a nanoluidilled quare lid-driven cavity. The governing equation were developed by the SIMLPLE algorithm, which i baed on inite volume method. The impact o parameter uch a Graho number, Hartmann number, olid volume raction wa invetigated on the luid low and heat 1-When Graho number increae, the eect o driven lid i decreaed and buoyancy eect i more dominant at high Graho number. So the main circulation i limited near the top moving wall. -Hartmann number decreae the velocity o luid in the cavity and a a reult reduce heat traner rate and the impact o Hartmann number i more dominant at low Graho number. Figure indicate that the low i more tagnant at high Hartmann number. Figure alo illutrate iotherm become vertically tratiied when Hartmann number increae. 3- Nanoparticle ait the mechanim to have a better heat traner. In other word, increaing in volume raction i led to heat traner ampliication, becaue nanoparticle increae thermal conductivity and augment low intenity. Reerence [1] B. Gebhart, L. Pera: The nature o vertical natural convection low reulting rom the combined buoyancy eect o thermal and ma diuion, Int. J. Heat and Ma Traner. 14 (1971) [] A. Bejan: Ma and heat traner by natural convection in a cavity, Int. J. Heat Fluid Flow. 6(3) (1985) [3] C. Beghein, F. Haghighat, F. Allard: Numerical tudy o double-diuive natural convection in a quare cavity, Int. J. Heat Ma Tran. 35 (199) [4] A.J. Chamkha, H. Al-Naer: Hydro magnetic double-diuive convection in a rectangular encloure with oppoing temperature and concentration gradient, Int. J. Heat Ma Tran. 45 (00) [5] Q.H. Deng, J. Zhou, C.Mei, Y.M. Shen: Fluid, heat and contaminant tranport tructure o laminar double-diuive mixed convection in a 41 two-dimenional ventilated encloure, Int. J. Heat Ma Tran. 47(4) (004) [6] A. M. Al-Amiri, Kh. M. Khanaer, I. Pop: Numerical imulation o combined thermal and ma tranport in a quare lid-driven cavity, Int. J. Therm. Sci. 46(7) (007) [7] B. B. Beya, T. Lili: Ocillatory double-diuive mixed convection in a two-dimenional ventilated encloure, Int. J. Heat Ma Tran. 50(3-4) (007) [8] M. A. Teamah, W. M. El-Maghlany: Numerical imulation o double-diuive mixed convective low in rectangular encloure with inulated moving lid, Int. J. Therm. Sci. 49(9) (010) [9] F. Talebi, A. H. Mahmoudi, M. Shahi: Numerical tudy o mixed convection low in a quare lid-driven cavity utilizing nanoluid, Int. Commun. Heat Ma Tran. 37(1) (010) [10] H. Nemati, M. Farhadi, K. Sedighi, E. Fattahi, A.A.R. Darzi: Lattice Boltzmann imulation o nanoluid in lid-driven cavity, Int. Commun. Heat Ma Tran. 37(10) (010) [11] A. J. Chamkha, E. Abu-Nada: Mixed convection low in ingle- and double-lid driven quare cavitie illed with water-al O 3 nanoluid: Eect o vicoity model, Europ. J. Mech. B/Fluid. Available online 19 March 01. [1] F. Talebi, A.H. Mahmoudi, M. Shahi, Numerical tudy o mixed convection low in a quare lid-driven cavity utilizing nanoluid, International Communication in Heat and Ma Traner 37 (010) [13] E. Abu-Nada, A.J. Chamkha, Mixed convection low in a lid-driven inclined quare encloure illed with a nanoluid, European Journal o Mechanic - B/Fluid (9) (010) [14] M.A.A. Hamad, I. Pop, A.I. Md Imail, Magnetic ield eect on ree convection low o a nanoluid pat a vertical emi-ininite lat plate, Nonlinear Analyi: Real World Application 1 (011) [15] M.A.A. Hamad, Analytical olution o natural convection low o a nanoluid over a linearly tretching heet in the preence o magnetic ield, International Communication in Heat and Ma Traner 38 (011) [16] B. Ghaemi, S.M. Aminoadati, A. Raii, Magnetic ield eect on natural convection in a

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