Numerical Investigation of Mixed Convection Fluid Flow, Heat Transfer and Entropy Generation in Triangular Enclosure Filled with a Nanofluid

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1 Journal of Applied Fluid Mechanics, Vol. 9, No. 1, pp , Available online at ISSN , EISSN Numerical Investigation of Mixed Convection Fluid Flow, Heat Transfer and Entropy Generation in Triangular Enclosure Filled with a Nanofluid A. Aghaei, G. A. Sheikhzadeh, H. Ehteram and M. Hajiahmadi Department of mechanical engineering, University of Kashan, Kashan, Iran, Corresponding Author AlirezaAghaei21@gmail.com (Received August 17, 2014; accepted September 24, 2014) ABSTRACT In this paper, mixed convection fluid flow, heat transfer and entropy generation inside a triangular enclosure filled with CuO-water nanofluid with variable properties are investigated numerically. The inclined walls of enclosure are maintained at a constant temperature T c. The moving bottom wall is kept at a constant temperature T h, which T h >T c.the numerical simulation of mixed convection is carried out using a computer program (FORTRAN language) based on finite volume method and SIMPLER algorithm. This study is performed over a range of nanoparticle volume fractions( ) Richardson numbers(0.01, 0.1, 1, 10, 100), and angles of inclined walls(15 0, 30 0, 45 0, 60 0, 75 0 ) and a fixed Grashof number (Gr=10 4 ).In all investigated aspect ratios and Richardson numbers, average Nusselt number increased by enhancement of volume fraction till 0.02, but it is approximately constant by adding more nanoparticles till 0.04.It is also observed that in all aspect ratios and Richardson numbers, the variation of total entropy generation by enhancement of volume fraction is similar to Nusselt number. Keywords: Nanofluid; Entropy generation; Triangular enclosure; Mixed convection; Numerical solution; Brownian motion. NOMENCLATURE AR c p Gr g H k l L Nu p P Ra aspect ratios heat capacity Grashof number gravitational acceleration (m.s -2 ) height of the triangular (m) thermal conductivity (Wm -1 K -1 ) length of the bottom wall (m) dimensionless Length Nusselt number pressure (kg m -1 s -2 ) dimensionless pressure Rayleigh number Ri Re Pr S gen, h S 0 gen, T u,v U,V U 0 x,y X,Y Richardson number Reynolds number Prandtl number total entropy generation generated entropy due to heat transfer generated entropy due to friction temperature (K) velocity components (ms -1 ) dimensionless velocity velocity of the bottom wall cartesian coordinates (m) dimensionless Cartesian coordinates α β μ ν θ θ s Avg c f thermal diffusivity(m 2 s -1 ) thermal expansion coefficient(k -1 ) viscosity(kgm -2 s -1 ) kinematic viscosity(m 2 s -1 ) dimensionless temperature angle of the inclined walls average cold fluid ρ Ψ Subscripts h nf s density(kgm -3 ) irreversibility coefficient volume fraction of nanoparticle stream function dimensionless stream function hot nanofluid particle

2 gen generation 1. INTRODUCTION Nanofluids are colloidal metal particles, metal oxide or polymers with base fluid such as water or ethylene glycol. Heat transfer in nanofluids increases because of higher conductivity of nanoparticles as compared with pure fluids. Nanofluids can be used for cooling of supercomputers, advanced engines and machines. Numerical study of nanofluid mixed convection in an enclosure with cold moving top wall, cold lateral walls and bottom wall subjected to a constant heat flux performed by Mansour et al (2010). Based on their findings, slowing the movement of fluid flow and augmentation of the average Nusselt number caused by increasing volume fraction. Mixed convection of Al 2 O 3 -water nanofluid in a right-angled triangular enclosure with insulated horizontal wall, hot inclined wall and cold vertical wall which moves(upward or downward) has been investigated by Ghasemi and Aminossadati (2010). They concluded that in all ranges of examined Richardson numbers, the heat transfer rate is increased by enhancement of the nanoparticle volume fraction and movement of the vertical wall toward both directions. Sheikhzadeh et al (2012), analyzed mixed convection heat transfer and fluid flow of Al2O3- water nanofluid in a square enclosure using finite volume numerical method. Top wall moves at constant velocity, horizontal walls were isolated and vertical walls were kept at low and high constant temperatures in right and left side, respectively. They found that considering various thermal conductivity and viscosity for fluid leads to different average Nusselt numbers compared with those of constant properties. Furthermore, this difference for low Richardson numbers ( ) is more than high Richardson number (10-100). Heat transfer and fluid flow of mixed convection in a finned horizontal canal for cu-water nanofluid was numerically inspected by Pishkar and Ghasemi (2012). They observed that heat transfer incrases by enhancement of volume fraction and Reynolds number. At a constant volume fraction, this increment is more considerable especially for higher Reynolds numbers. Chamkha and Abu-Nada (2012) examined heat transfer and fluid flow of mixed convection in a square cavity with insulated lateral walls, hot top wall and cold bottom wall in two models. In first case only the top wall moves and in second case, top and bottom horizontal walls are moving in opposite direction. They concluded that the average Nusselt number is increased by adding volume fraction and by decreasing the Richardson number. Another study was performed for investigation of mixed convection heat transfer and fluid flow of cuwater nanofluid in a square cavity by Abbasian et al (2012). Horizontal walls were insulated and lateral walls were subjected to sinusoidal temperature distribution. They found that increasing phase difference at constant Richardson number and volume fraction makes the average Nusselt number reduced. Datectin et al (2007), numerically analyzed the entropy generation of natural convection fluid flow in a г shaped enclosure with insulated Top and bottom walls, cold lateral walls and two hot walls. Based on their observation at Ra <105, because of weak convection heat transfer and consequently low velocity of flow, the main part of total entropy is due to heat transfer. While at Ra >105, by increasing the velocity of flow, the main part of total entropy is devoted to entropy generation caused by friction. Ilis et al (2008) numerically studied the effect of aspect ratio on entropy generation in natural convection heat transfer in rectangular enclosure with insulated top and bottom walls and hot and cold lateral walls. In their study, aspect ratio was defined as the ratio of the height the cavity to its length. They found that at lowest Rayleigh number (Ra=102), the total entropy generation was continuously increased by enhancement of enclosure aspect ratio, while for high Rayleigh number (Ra=107)entropy generation is maximum. Numerical investigation of natural convection fluid flow on entropy generation in an enclosure with insulated horizontal walls and thick thermal conductive hot and cold lateral walls was performed by Varol et al (2008) They concluded that enhancement of thermal conductivity ratio that is defined as solid thermal conductivity (sidewalls) divided by fluid thermal conductivity, the total entropy generation is increased. Bejan number which is defined as the ratio of the entropy generation due to heat transfer to total entropy generation, decreased by decrement of the Rayleigh number. Variation the thickness of the wall near to cold boundary has more considerable enhancement effect than the wall near to hot boundary, especially in high Rayleigh numbers. Famouri and Hooman (2008) numerically studied entropy generation due to natural convection flow inside a rectangular enclosure with cold lateral walls, insulated horizontal walls and a hot plate inside the enclosure. Based on their results, generated entropy is increased by enhancement of the Rayleigh number. Furthermore, by enhancement of the vertical position of the hot plate up to half height of the cavity, the generated entropy increased and then decreased. Mukhopadhyay (2010),performed the numerical study of entropy generation on natural convection heat transfer and fluid flow in a square enclosure, which all walls were insulated except bottom 148

3 wall. In bottom wall there were two heat sources and the remaining portion of the wall was insulated. Based on the result of this investigation, the entropy generation due to heat transfer has main part of the total entropy generation. It is also observed that Heat sources locations near to half of the bottom wall, causes more entropy generation. Numerical study was performed to analyze entropy generation on natural convection flow of nanofluid in a square enclosure by Shahi et al (2011). All of the cavity walls were insulated except the cold bottom wall, there was an obstacle inside the enclosure with constant heat flux on its sides. Their results showed that putting hot obstacle on the bottom wall by distance=0.8 from the left one causes minimum entropy generation, while putting the hot obstacle on the top wall by distance=0.2 from the left wall makes the maximum entropy generation achieved. Recently, Malvandi et al (2014), Malvandi et al (2014), performed an analytical investigation of nanofluid flow on stretching sheet. Ganji and Malvandi (2014)examined nanofluid natural convection in vertical enclosure influenced by magnetic field. Based on their results and the latest studies, can be referred to Malvandi and Ganji (2014), Malvandi and Ganji (2014), Malvandi and Ganji (2014),.They studied the effect of Brownian motion and thermophoresis on their researches. One of the most remarkable points in study of convection heat transfer is disagreement between experimental Abbasian et al (2012), Abbasian et al (2013), and numerical results of nanofluids. Maybe one reason for disagreement is ignoring some phenomenon in fluid flow and heat transfer, such as the effect of Brownian motion on convection heat transfer coefficient. Brownian motion of nanoparticles in nanofluids, actually is their continual and random movement. The molecules of fluid, constantly hit and disperse nanoparticles inside the fluid. This event is considered at present work, too. Through a comprehensive review, it is observed that not many studies on mixed convection and entropy generation in triangular enclosures, has been done recently. Considering the effect of Brownian motion and temperature dependence of viscosity and thermal conductivity, examination of entropy generation and triangular enclosure are innovations of this study. Selected geometry can be used as a model of cooling an electronic board located in triangular space junk. At present study, fluid flow and heat transfer of mixed convection and entropy generation of CuOwater nanofluid in a triangular enclosure with hot and moving bottom wall, at various Richardson numbers, volume fraction of nanoparticles and different aspect ratios are numerically investigated. 2. GOVERNING EQUATIONS AND BOUNDARY CONDITIONS Geometry of the problem and boundary conditions has been shown in Fig 1. Bottom wall of enclosure is hot (at temperature of T h ) and moves towards right direction. Lateral walls are cold (at temperature of T c ). Aspect ratio is defined as AR= (2H L) and θ s =tan -1 AR. Height of the enclosure H has been assumed constant and L is changed in order to achieve different aspect ratios. The values of θ s are equal to The enclosure is filled with CuO-water nanofluid. Thermo-physical properties of water as base fluid and copper-oxide nanoparticles are presented in Table 1 (Hwang et al (2007),Ogut (2009). y x T c θ s Fig. 1. Schematic of the enclosure with boundary conditions. Table 1 Thermophysical properties of base fluid and nanoparticles (Hwang et al (2007), Ogut (2009)) k β (K -1 (Wm -1 K - c p ρ ) ) (jkg -1 K -1 ) (kgm -3 ) water CuO Governing continuity, x- and y- components of momentum and energy equations for the twodimensional steady and laminar nanofluid flow are given in Eqs. (1)-(4), respectively. u v 0 (1) x y u u 1 p u v x y nf x (2) 1 u u [ ( nf ) ( nf )] nf x x y y v v 1 p u v x y nf y 1 v v [ ( nf ) ( nf )] (3) nf x x y y ( ) nf gt ( Tc ) nf L T h T c H U 0 149

4 T T 1 T u v [ ( k nf ) x y ( cp) nf x x (4) T ( knf )] y y Entropy generation is evaluated from (5), Bejan, (1995). 2 2 k nf T T s gen 2 T 0 x y 2 2 nf u v [2 2 (5) T0 x y 2 u v ] y x Above equations can be converted to nondimensional form bythe following dimensionless variables (6). x y v l X, Y, V, L, H H U0 H (6) u T Tc p U,, P U 2 0 T fu0 Using the mentioned dimensionless variables, governing equations and entropy generation relations can be written in non-dimensional form (7)-(12). U V 0 X Y U U P U V X Y X 1 U [ ( nf ) nf f Re X X U ( nf )] Y Y V V P U V X Y Y 1 V [ ( nf ) nff Re X X V ( ) nf ( nf )] Ri Y Y nf f (7) (8) (9) U V X Y (10) 1 [ ( knf ) ( knf )] RePr f ( cp) nf X X Y Y Dimensionless stream function is defined as (13). ( XY, ) UdY (13) Considering the geometry of problem,the dimensionless boundary conditions are (14). U 1, V 0, 1 U V 0, 0 On inclined walls On hot side (14) Properties of nanofluid including density, heat capacity, thermal expansion coefficient, diffusivity coefficient, static viscosity (Brinkman (1952)), and static thermal conduction coefficient (Maxwell (1873)), are obtained from the following equations.(15) (22) nf (1 ) f s (15) ( cpnf ) (1 )( cp) f ( cps ) (16) ( ) nf (1 )( ) f ( ) s (17) knf nf (18) cp nf eff Static Brownian (19) k k k eff Static Brownian (20) (1 ) 2.5 (21) Static f ( ks 2 kf) 2 ( kf ks) kstatic kf (22) ( ks2 kf) ( kf ks) Which µ Brownian and k Brownian are (Polidori (2007)): 4 T Brownian 510 f ( T, ) (23) 2 R T k c T 4 Brownian 510 f p, f (, ) 2sRs s s (24) R s and ρ s are radius and density of nanoparticles, respectively, and is Boltzmann constant (= J/k). For CuO-water nanofluid, ξ and λ functions which are experimentally estimated at the range of 300<T(k)<325 are (Aminossadati (2011)) (100 ) for 1% (100 ) for 1% ( T, ) ( ) T( ) for 1% 4% (25) Convective heat transfer coefficient is calculated from this equation (26). q hnf (26) Th Tc Nusselt number is written as (27). hnf H Nu (27) kf Wall heat flux per unit area is defined as (28). Th Tc qknf (28) H Y Y0 By substituting (26) and (28) in (27), relation of the Nusselt number is obtained. knf Nu (29) kf Y Y 0 Average Nusselt number on the hot wall is (30). 1 1 NuAvg Nu dx L (30) 0 150

5 Stream lines Isotherms Total entropy lines Ri=100 Ri=10 Ri=1 Ri=0. 1 Ri=0. 01 Fig. 2. Stream lines, Isotherms and total entropy lines for fluid and nanofluid in =0.02 and θ s = NUMERICAL SIMULATION Governing equations are solved numerically by finite volume method. The SIMPLER algorithm used to solve pressure-velocity coupling. At first a proper grid is generated for the solution domain, then a control volume is considered around each nodal point. Governing equations are integrated over each control volume so we have a system of discretized algebraic equations. The hybrid scheme is used for discretizing the diffusion and convection terms. In this method for Peclet numbers with absolute values<2, central difference method, and for Peclet numbers with absolute values>2,upstream method is used. In order to approach convergence, under relaxation coefficients equal to0.5 for components of velocity and 0.7 for temperature are used. Convergence criteria for pressure, velocity and temperature is obtained from equation(31).where, M and N are number of grid points in the directions of x- and y-, ζ defines the variable that is solved. K is number of iterations and maximum error is M N k1 k i, j i, j i1 j1 6 Error 10 (31) M N k 1 i, j i1 j1 3. VALIDATION OF PROGRAM INFORMATION In order to validate the results, the study of Ghasemi and Aminossadati (2010) and Olveski et al (2009), are considered and simulated by the present code and the corresponding results are compared in Table 3 and 4. It is observed that relative difference of the average Nusselt number and total entropy generation is insignificant, so the results of the present work are valid. Table 3 Average Nusselt number, comparison with the results of Ghasemi and Aminossadati (2010) Ri Present study Ghasemi and Aminossadati (2010) Difference percentage GRID STUDY To find the proper independent solution grid, average Nusselt number for CuO-water nanofluid choosed and calculated in a range of grid points and compared in Table 2.It is observed that grid with number of points is suitable. Table 2 Average Nusselt number on hot wall for nanofluid Ri=1,=0.02, θ s =45 Node Nu Ave Table 4 Total entropy generated, comparison with the results of Olveski et al (2009) Olveski Ra Present study et al (2009) Difference percentage

6 Stream lines Isotherms Total entropy lines Ri=100 Ri=10 Ri=1 Ri=0.1 Ri=0.01 ٤ Fig. 3. Stream lines, Isotherms and total entropy for fluid and nanofluid in =0.02 and θ s =45 0. Stream lines Isotherms Total entropy lines. Ri=100 Ri=10 Ri=1 Ri=0.1 Ri=0.01 Fig. 4. Stream lines, Isotherms and total entropy lines for fluid and nanofluid in =0.02 and θ s =

7 Nu Avg 13 θ s = Ri=0.0 1 Ri= Nu Avg 10 θ s =30 Nu Avg θ s = θ s =60 40 θ s =75 Nu Avg Nu Avg Fig. 5. Changes of average Nusselt number versus volume fraction in various Richardson number and aspect ratios. 4. RESULTS AND DISCUSSION Isotherms, streamlines and total entropy generation contours for water and CuO-water nanofluid are shown at =0.02 and Gr=10 4, various Richardson numbers and aspect ratios(different angles of inclined walls, θ s ) in Fig2-4. At a constant Richardson number, by increasing theθ s at Ri=1, the primary eddies which dragged in direction of the right-moving bottom wall at angles of , are tended to center of the enclosure. While, at Richardson numbers of , primary eddies at angles of are dragged to the left, and then intend to the center of enclosure. Actually, in high Richardson numbers because of low velocity of the bottom wall, eddies are not affected to its moving direction and are not tended to the right. In cases that secondary eddies exist, center of secondary eddies are developed and tended to the top of the enclosure by enhancement of θ s. At angles of , by decreasing the Richardson number, flow is affected by strong movement of the bottom wall and the eddy is dragged to the direction of the wall. This causes occupying smaller region of the enclosure and consequently, proper space in top of the enclosure is available for the subsequent eddies. At the top space of the enclosure, fluid is affected by primary eddies and the secondary eddies are made. By increment of the Richardson number at θ s =60 0, secondary eddy is developed. Actually, by enhancement of Richardson number, velocity of the bottom wall reduced and natural convection regime dominates. Density of Isotherms reduced by increasing the θ s, which in low angles near the hot and cold wall the Isotherms were more dense. This is a good reason for tending of eddies to move toward the center of the enclosure, that is due to domination of natural convection rather than forced convection by enhancement of Richardson number. Accumulation of isotherms beside the hot and cold wall in angles of is one of the characteristics of flow at Richardson numbers of , and represents increased heat transfer. In all aspect ratios and Richardson numbers, total entropy contours become more densely packed be side the walls of the enclosure, which indicates more generated entropy in this regions. This behavior is predictable because of existence of the boundary layer beside the walls and high gradients of temperature and velocity in the boundary layer. Furthermore, entropy generation due to friction and heat transfer is more considerable in this region. Increasing the θ s at constant Richardson number makes total entropy generation contours moved to center of the enclosure. This is acceptable considering declination of the enclosure due to the increasing θ s, and tend of isotherms and streamlines to the half of the enclosure because of more heat transfer and circulation of the flow in this region. Decreasing Richardson number which means an increment in velocity of the bottom wall, causes the entropy generation contours more densely packed near the bottom wall because of increasing entropy generation due to fluid friction. Variations of the average Nusselt number of the hot wall in various volume fractions of the nanoparticles and Richardson numbers and aspect ratios, is shown In Fig5. In all investigated Richardson numbers and aspect ratios, by increasing the volume fraction to =0.02 average Nusselt number is increased. But with more increment till =0.04,approximately there was no change in average Nusselt number. By decreasing the Richardson number from 100 to 0.01, the maximum increase of the Nusselt number occurred at θ s =75 0 and =0.04(from to 47.05) and the minimum increase was observed atθ s =15 0 and=0 (from 5.15 to 12.96). 153

8 Ri=0.01 θ=15 θ=30 θ=45 θ= Ri= Ri= Ri= Ri= Fig. 6. variations of total entropy generationversus volume fraction in various Richardson number and aspect ratios. 8 Fig 5 indicates that increasing θ s considerably enhances the average Nusselt number. Because the size of the bottom wall becomes smaller and growth of the boundary layer occurs in a smaller length by increasing θ s. Consequently, temperature gradient has less reduction along the wall, so the average Nusselt number is increased. Total entropy variation versus volume fraction of nanoparticle in various aspect ratios and Richardson numbers showed in Fig 6. In all ranges of aspect ratio and Richardson numbers, by increasing the volume fraction of nanoparticles, total entropy generation is increased. Because the part of entropy generation due to heat transfer is more significant than the part of of fluid friction, total entropy variations versus volume fraction is similar to that of the Nusselt number. By decreasing the θ s from 75 0 to 15 0, maximum increase of the total entropy generation is happened in Ri=0.01 and =0.04 (from to 92.07)and minimum increase is occurred in Ri=100 and =0(from 9.31 to 51.48). By decreasing Richardson number from 10 to 0.01, amount of variations in total entropy generation is more considerable especially by increasing the θ s from 45 0 to Maximum amount of total entropy variation due to change of θ s (θ s =45 0 to θ s =75 0 ) is increased from 4.9 in Ri =10 to 11.9 in Ri=0.01. Maximum amount of stream function Ψ max as a criterion of the flow power, in various aspect ratios and Richardson numbers for nanofluid with volume fraction of =0.02is presented in Table5. Maximum and minimum values of Ψ max in all aspect ratios, Richardson numbers and volume fractions, are equal to0.42 and00.04 and occured in Ri=100, =0.02, θ s =15 0 and Ri=0.01, =0.04, θ s= 75 0 respectively. Table 5 Amounts of Ψ max in Richardson number and Aspect ratios for nanofluid in =0.02 Ri θ S =15 0 θ S =45 0 θ S = Values of entropy generation due to fluid friction in various Richardson numbers and aspect ratios for CuO-water nanofluid in =0.02arepresented in Table6. Maximum part of entropy generation due to friction in all aspect ratios and Richardson numbers and volume fractions is equal to0.06% and happened in Ri=0.01, =0 and θ s =15 0. Minimum part of entropy generation due to friction on total entropy is equal to and is accrued in Ri =100, =0.02 and θ s =75 0. This shows that the effect of fluid friction on total entropy generation in investigated geometry is in significan Table 6 Amounts of entropy due to friction in various Richardson Number and aspect ratios for nanofluid in =0.02 Ri θ S =15 0 θ S =45 0 θ S = In Table 7, values of the total entropy generation on various aspect ratios, Richardson numbers and for CuO-water nanofluid with a volume fraction of=0.02 are presented. Maximum total entropy 154

9 generation in all aspect ratios, Richardson numbers and volume fractions, is equal to and is occurred in Ri=.01, =0.04 andθ s =15 0, and also itsminimum value is 9.31 in Ri=100, =0 andθ s =75 0. Table 7 Amounts of total entropy in various Richardson numberand aspect ratios for nanofluid in =0.02 Ri θ S =15 0 θ S =45 0 θ S = The values of Bejan number that represent entropy generated ratio due to heat transfer to total entropy generated presented in table 8. As can be seen, the values of Bejan number in all Richardson numbers, aspect ratios and volume fractions are approximately equal to 1. Based on this result, the main part of total entropy generated is due to heat transfer. Although entropy generated due to friction increased with enhancement of volume fraction, but it is not apparent increment. Table 8 Amounts of Bejan number invarious Richardson number And aspect ratios for nanofluid in =0.02 Ri θ S =15 0 θ S =45 0 θ S = Table 9 indicates values of Nu Ave in various aspect ratios and Richardson numbers for CuO-water nanofluid in =0.02.Maximum average Nusselt number in all aspect ratios, Richardson numbers and volume fractions is equal to and occurred in Ri=0.01, =0.04 and θ s =75 0, also minimum value is 5.15 in Ri =10, =0 and θ s =15 0. Table 9 Amounts of Nu Ave in various Richardson number Andaspect ratios for nanofluid in =0.02 Ri θ S =15 0 θ S =45 0 θ S = CONCLUSION Mixed convection Flow field, heat transfer and entropy generation CuO-water nanofluid with variable properties in triangular enclosure is investigated by finite volume numerical method and using SIMPLER algorithm. Present study is conducted for nanoparticle volume fractions of =0, 0.02, 0.04, Richardson number=0.01, 0.1, 1, 10, 100,angles of inclined walls=15 0, 30 0, 45 0, 60 0, Grashof number was constant at 10 4.The results of the numerical analysis lead to the following conclusions: In all aspect ratio and Richardson numbers, by increment of volume fraction till to =0.02, the average Nusselt number was increased, then it approximately remain constant by adding more nanoparticles till to =0.04.By decreasing Richardson number from 100 to.01, maximum increase of the Nusselt number happened in θ s =75 0, =0.04 (4 times) and minimum increase occurred in θ s =15 0, =0 (2.5 times)in all investigated aspect ratios and Richardson numbers, variations of the total entropy generation versus the volume fraction of nanoparticles was similar to that of Nusselt number. By reducing θ s from 75 0 to 15 0, maximum increase of the total entropy generation was observed in Ri=0.01 and =0.04 (from to 92.07), and minimum increase occurred in Ri=100and =0 (from 9.31 to 51.48). REFERENCES Abbasian Arani, A. A., S. MazroueiSebdani, b. M. Mahmoodi A. Ardeshiri and M. Aliakbari. (2012). Numerical study of mixed convection flow in a lid-driven cavity with sinusoidal heating on sidewallsusing nanofluid. Superlattices and Microstructures 51, Abbasian Arani, A. A. and J. Amani (2012). Experimental study on the effect of TiO2 water nanofluid on heat transfer and pressure drop. Experimental Thermal and Fluid Science 42(1), Abbasian Arani, A. A. and J. Amani (2013). Experimental investigation of diameter effect on heat transfer performance and pressure drop of TiO2 water nanofluid. Experimental Thermal and Fluid Science 44(1), Aminossadati, S. M. and B. Ghasemi (2011). Natural convection of water CuO nanofluid in a cavity with two pairs of heat source sink. International Communications in Heat and Mass Transfer 38(5), Bejan, A. (1995). Entropy Generation Minimization, CRC Press, New York. Brinkman H. C. (1952). the viscosity of concentrated suspensions and solution, The Journal of Chemical Physics 20, Chamkhaa, A. J., Abu-Nada, Eiyad (2012). mixed convection flow in single- and double-lid driven square cavities filled with water Al2O3 nanofluid: Effect of viscosity models. European Journal of Mechanics B/Fluids 36, Dagtekin, E., H. F. Oztop and A. Bahloul (2007). Entropy generation for natural convection in Γ- shaped enclosures. International Communications in Heat and Mass Transfer 155

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