Entropy Generation in Cu-Water Nanofluid in a Cavity with Chamfer in the Presence of Magnetic Field

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1 I J C T A, 9(38), 2016, pp International Science Pre Entropy Generation in Cu-Water Nanoluid in a Cavity with Chamer in the Preence o Magnetic Field Souad Marzougui*, Ammar Ben Brahim* and Mourad Magherbi*, ** ABSTRACT Thi paper deal with the invetigation o the irreveribility in nanoluid low convection in a cavity with chamer. The nanoluid i ubmitted to a thermal gradient and a magnetic ield applied in dierent direction. The equation o continuity, momentum balance and energy are olved by uing the Comol otware. It wa tudied the eect o the Rayleigh number, and the Hartmann number and the magnetic ield inclination angle, on the entropy generation. Keyword: Entropy generation, nanoluid, Rayleigh number, Hartmann number, magnetic ield, etc. 1. INTRODUCTION Over the lat ew decade many reearch project have dealt with uing nanoluid in the preence o magnetic ield. H. K. Yang and C.P.Yu [1] combined orced and ree convection (MHD) channel low in the entrance region. They ound that an applied tranvere magnetic ield may reduce the entrance length o the velocity coniderably but ha little eect on the temperature development. At high Hartmann number (Ha), the velocity entrance length i inverely proportional to Ha [2]. Mahmoudi et al [2] invetigated the eect o magnetic ield on nanoluid low in a cavity with a linear boundary condition analyzed with Lattice Boltzmann Method, according to the Rayleigh number, the Hartmann number, the volume raction and the direction o the magnetic ield. For variou Rayleigh number and or all magnetic ield direction, heat traner and luid low decline with the increae in Hartmann number. The magnetic ield direction control the eect o nanoparticle in the luid. H. Heidary et al [3] conducted a numerical tudy o the magnetic ield eect on nanoluid orced convection in a channel. A parametric tudy o the eect o nanoluid volume raction and magnetic trength on the enhancement o heat exchange between an iothermal duct and the core low wa carried out. They oberved that the preence o a magnetic ield and the addition o nanoparticle to a pure luid can igniicantly enhance the heat exchange between the wall and the luid. Abolbahari et al [4] ued the Homotopy analyi method (HAM), to tudy the entropy analyi in an unteady magneto hydrodynamic nanoluid regime adjacent to an accelerating tretching permeable urace with water a the bae luid and our dierent type o nanoparticle. HAM i ucceully applied to olve the ytem o ordinary dierential equation. Many other work related to MHD eect on heat traner and entropy generation o nanoluid in mixed, orced and natural convection low are conducted by Meherez et al. [5], Da and Jana [6], Hatami et al. [7], Rahman et al.[8] and Teamah and El-Maghlany [9]. Some recent control method are dicued in [18-25]. * Univerity o Gabe, Chemical and Proce Engineering Department, National School o Engineer Gabè, Applied Thermodynamic Unit, Tuniia ** Univerity o Gabe, Civil Engineering Department, High Intitute o Applied Science and Technology, Gabe, Tuniia

2 292 Souad Marzougui, Ammar Ben Brahim and Mourad Magherbi 2. PROBLEM STATEMENT The chematic o the ytem under conideration i preented in Fig. 1. The temperature Th i uniormly impoed along the vertical wall. Tc along the top wall and the bottom urace a well a the our chamer are aumed to be adiabatic. A magnetic ield with uniorm trength B0 i applied in dierent angle inclinion ã. Alo the encloure i illed with a water baed nanoluid. It i aumed that the nanoparticle are in thermal equilibrium, the nanoluid i Newtonian and incompreible and the low i laminar. 3. MATHEMATICAL FORMULATION Figure 1: Geometric coniguration o the problem Hence and in two-dimenional Carteian coordinate ytem, the dimenionle equation o continuity, momentum and energy are written in teady tate a: U X V Y 0 (1) U U P n U U n U V ( ) Pr Ha V in co U in X Y X X Y n V V P V V ( ) U V ( ) Ra Pr T Pr Ha U in co V co X Y X X Y n n n n n (2) (3) T T n T T U V ( ) X Y X Y Here X, Y dimenionle coordinate; U, V dimenionle velocity; P dimenionle preure.pr, Ha and Ra denoted the Prandtl number, the Hartmann number and the Rayleigh number repectively. g ( T T ) H Pr ; ; ' ' 3 h c Ha B0 H Ra The expreion o denity, thermal expanion, peciic heat coeicient, dynamic vicoity and electrical conductivity o the Nanoluid (Maxwell Model [10])are given a ollow [11]. In the equation below, i the nanoparticle volume raction. (4)

3 Entropy Generation in Cu-Water Nanoluid in a Cavity with Chamer (1 ) (5) n ( ) (1 )( ) ( ) (6) n ( c ) (1 )( c ) ( c ) (7) p n p p n 1 (1 ) 2,5 (8) 3 1 n ; (9) There have been many review o nanoluid thermal conductivity (Lee et al. [12] and Da et al. [13]). The thermal conductivity model o Patel et al. [14] wa ued in thi work. It can be given by: k A A kn k 1 Ck p Pe k A k A The parameter C i et equal to The parameter Pe and A/A are deined a: (10) u d Pe A ; A d d 1 In Eq. (11), d i the molecular ize o water, which i taken 2A, d i the diameter o olid particle and kb i the Boltzmann contant. The variable u, which depend on the temperature, i the Brownian motion velocity o particle and i given by H.E. Patel et al [14]. The appropriate initial and boundary condition o the problem are: u 2 (11) 2k b T d (12) - At dimenionle time equal to zero, T = 0 and U = V = 0 in the whole cavity. - Along the let and the right wall the dimenionle temperature i T = 1. - Along the top wall, the dimenionle temperature i T = 0. T - Along the horizontal inulator wall: 0 y T - Along the vertical inulator wall and the chamer: 0 x - Along the iothermal, the chamer and inulator wall U = V = 0.

4 294 Souad Marzougui, Ammar Ben Brahim and Mourad Magherbi 4. ENTROPY GENERATION According to Wood [15], the dimenionle local entropy generation can be expreed by: 2 kn T T U V U V n 2 Sgen Ha U in V co k X Y X Y Y X In the right ide o Eq. (13), the irt term repreent the heat traner irreveribility, the econd i the vicou eect irreveribility and the third i the magnetic irreveribility. The ditribution irreveribility ratio i given by: 2 (13) n T0 k H T 2 (14) 5. NUMERICAL PROCEDURE AND VALIDATION In the dimenionle orm and taking into account the initial and the boundary condition, the low governing equation were olved uing the inite volume method and COMSOL Multiphyic otware. Reult given by numerical calculation uing licened verion o COMSOL otware were validated with the work o Magherbi et al. [16] in term o ientropic line related to a pure luid (air) and o Ozotop et al. 17 in term o treamline and iotherm related to a nanoluid (Cu-water). 6. RESULTS AND DISCUSSION In thi tudy, the Prandtl number (Pr) i kept contant at Pr = 6.2 with olid volume raction Õ = 4% (water- Cu nanoluid). The numerical reult or the treamline and iothermal contour or variou value o Rayleigh number Ra and Hartmann number Ha. In addition, reult or thermal entropy, vicou entropy, magnetic and total entropy, at variou condition, are preented and dicued Eect o Hartmann and Rayleigh number on treamline and iotherm Fig. 2 repreent the eect o Rayleigh number and Hartmann number on the treamline and iothermal contour or, Õ = 0.04 and Pr = 6.2. a) c=45 Ha=40 Ra=10 5 c=45 Ha=40 Ra=10 6 c=0 Ha=40 Ra=10 6 c=45 ; Ha=10 ; Ra=10 6 (b) c=0 ; Ha=0 ; Ra=10 5 c=0 ; Ha=40 ; Ra=10 5 c=0 ;Ha=40 ;Ra=10 6 c=45 ; Ha=10 ; Ra=10 6 Figure 2: a) Streamline and b) Iotherm

5 Entropy Generation in Cu-Water Nanoluid in a Cavity with Chamer For all Rayleigh and Hartmann number, Fig.2a how a pair o cell in rotation, one in clockwie direction the other in anticlockwie, are ormed inide the cavity. Fig.2 how that, the ymmetry o treamline and iotherm i broken ince magnetic ield inclination angle in not zero. The trength o thee cell increae a the Rayleigh number increae and decreae a the Hartmann number increae and the treamline are elongated in the direction o the magnetic ield. Remark that the bottom cell i more elongated then the upper one. For all value o Rayleigh number, the application o the magnetic ield tend to low down the movement o the luid in the cavity Eect o inclination angle on dierent caue o irreveribility Magnetic irreveribility The value o the magnetic entropy o dierent inclination o the magnetic ield are illutrated in igure 3 A een in Fig.3, at ixed magnetic ield inclination angle (), the irreveribility increae or relatively mall Hartman number and reache a maximum value at critical Hartman number (Hac) then it decreae toward a minimum value at maximum value o Ha. The decreae o magnetic irreveribility, when Ha exceed Hac can be the reult o the igniicant decreae o the velocity in the cavity due to the important (a) (b) (c) Figure 3: Variation o magnetic entropy a a unction o Hartmann number or dierent angle inclination o magnetic ield with = 0.04 a) Ra=10 4 ; b) Ra=10 5 ; c) Ra=10 6

6 296 Souad Marzougui, Ammar Ben Brahim and Mourad Magherbi lowing eect induced by the Lorentz orce at high Hartman number. Wherea, at mall Ha value, although the lowdown eect o magnetic ield exit, it remain inigniicant. In thi cae the increae o the magnetic entropy generation i the conequence o the intrinic eect o Hartman number on the magnetic entropy generation equation. It i important to note that the critical Hartman number or which entropy generation i maximum depend on the inclination angle (). One can ee, at Ra = 105 or example, that the critical Hartman number increae rom 16 to 29 when increae rom zero to 90. The critical Hartman number (Hac) repreent the rontier between, irt the cae where Ha i preponderant via it intrinic eect on the magnetic entropy generation equation and econdly, the cae where the luid velocity become dominant through the extrinic eect o Ha on the magnetic entropy generation through the momentum equation balance. Additionally, the minimum o entropy generation at maximum Ha i a important a i important Thermal and vicouirreveribilitie Fig. 4 and 5 how that, at ixed Ra and (), the thermal and vicou entropie generation decreae when Hartmann number increae. In act, when Ha increae the convection in the cavity diminihe that lead (a) Figure 4: Variation o thermal entropy generation with Hartmann number or dierent angle inclination o magnetic ield a) Ra=10 5 ; b) Ra=10 6 (b) (a) (b) Figure 5: Evolution o vicou entropy with Hartman number or dierent angle inclination o magnetic ield a) Ra=10 5 b) Ra=10 6

7 Entropy Generation in Cu-Water Nanoluid in a Cavity with Chamer to a decreae in the thermal and velocity gradient and conequently to a decreae in the thermal and vicou entropie generation. It important to note that, at ixed magnetic ield inclination angle, the decreae o thermal irreveribility i le important when Ra increae. Thi i due to the enhancement o the convection phenomenon when increaing Ra, which oppoe the lowing eect o the magnetic ield. Similar obervation are conducted or the eect o Ra on the vicou irreveribility. It important to notice that, at ixed Ha, both vicou and heat traner irreveribilitie increae when the magnetic ield inclination angle increae. 7. CONCLUSION Thi paper invetigate the eect o an inclined magnetic ield on the entropy generation on a natural convective heat traner o CuO-Water nanoluid. The main inding are: The ymmetry o treamline and iotherm i broken ince magnetic ield inclination angle in not zero. The treamline are elongated in the direction o the magnetic ield. At ixed magnetic ield inclination angle (), the irreveribility increae or relatively mall Hartman number and reache a maximum value at critical Hartman number (Ha c ) then it decreae toward a minimum value at maximum value o Ha. The critical Hartman number or which entropy generation i maximum depend on the magnetic ield inclination angle. The critical Hartman number (Ha c ) repreent the rontier between, irt the cae where Ha i preponderant via it intrinic and extrinic eect on the magnetic entropy generation equation. The thermal and vicou entropie generation decreae when Hartmann number increae. The decreae o thermal irreveribility i le important when Ra increae.. REFERENCES [1] H.K. Yang and C.P.Yu, Combined orced and ree convection MHD channel low in entrance region, International Journal o Heat and Ma Traner, 17 (6), , [2] A. Mahmoudi, I. Mejri, M.A. Abbai and A. Omri, Lattice Boltzmann imulation o MHD natural convection in a nanoluid-ille cavity with linear temperature ditribution, Powder Technology, 256, , [3] H.Heidary, R. Hoeini, M. Pirmohammadi and M. J. Kermani, Numerical tudy o magnetic ield eect on nanoluid orced convection in a channel, Journal o Magnetim and Magnetic Materrial, 374, 11-17, [4] M.H. Abolbahai, N. Freidoonimehr, F. Nazari and M.M. Rahidi, Entropy analyi or unteady MHD low pat a tretching permeable urace in nanoluid, Powder Technology, 267, , [5] Z. Mehrez, A.E. Gai, A. Belghith and P.L. Quere, MHD eect on heat traner and entropy generation o noluid low in an open cavity, Journal o magnetim and Magnetic Material, 374, , [6] S. Da and R.N.Jana, Entropy generation due to MHD low in a porou channel with Navier lip, Ain Sham Engineering Journal, 5 (2), , [7] M. Hatami, R. Nouri and D.D. Ganji, Forced convection analyi or Al2O3-water nanoluid low over a horizontal plate, Journal o Molecular Liquid, 187, , [8] M. M. Rahman, H.F. Ozotop, R. Saidur, S. Mekhile and K. Al-Salem, Finite element olution o MHD mixed convection in a channel with a ully or partially heated cavity,compter&fluid,79, 53-64, [9] M.A.Teamah and W.M. El-Maghlany, Augmentation o natural convective heat traner in quare cavity by utilizing nanoluid in the preence o magnetic ield and uniorm heat generation/aborption, International Journal o Thermal Science,58, , [10] J.C. Maxwell, A Treatie on Electricity and Magnetim, Oxord Univerity Pre, Second Edition, Cambridge, U.K

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