Keywords: Finite element method; Nanofluid; Inclined magnetic field; Natural convection; Square enclosure; Brownian motion

Transcription

1 Columbia International Publishing American Journal o Heat and Mass Transer doi: /ajhmt Research Article Finite Element Analysis o Unsteady Natural Convective Heat Transer and Fluid Flow o Nanoluids inside a Tilted Square Enclosure in the Presence o Oriented Magnetic Field K. S. Al Kalbani 1, M. S. Alam 1, 2, and M. M. Rahman 1* Received: 7 June 2016; Published online: 30 July 2016 The author(s) Published with open access at Abstract In this paper, the problem o unsteady natural convective heat transer low o nanoluids having various sizes o nanoparticles inside an inclined square enclosure in the presence o oriented magnetic ield is investigated numerically. The Brownian motion o nanoparticles is taken into consideration in the thermal conductivity model construction. Two opposite walls o the enclosure are insulated and the other two walls are kept at dierent temperatures. Galerkin weighted residual inite element technique has been employed to solve the governing nonlinear dimensionless equations. In order to ensure the accuracy o the present numerical code, comparisons with previously published works are perormed and excellent agreement is obtained. The eects o model parameters such as Rayleigh number, Hartmann number, nanoparticles volume raction, inclination angle o magnetic ield, inclination angle o the geometry, diameter and Brownian motion o the nanoparticles on the luid low and heat transer are investigated. The results indicate that an increment in Rayleigh number and nanoparticle volume raction increases the heat transer rate in a signiicant way, whereas, an increment in Hartmann number decreases the overall heat transer rate. It is also observed that the heat transer enhancement strongly depends on the diameter o the nanoparticles as well as the types o the nanoluids. It is observed that the time taken to reach the steady state is controlled by the dierent model parameters and in particular, it is longer or low Rayleigh number and shorter or high Rayleigh number. A comparison between the two studies o with and without Brownian motion shows that when Brownian motion is considered, the solid volume raction has more signiicant eects on the heat transer rate at all Rayleigh numbers considered in the square cavity. Finally, the distribution o average heat transer rate or dierent cavity inclination angle along with various model parameters has been ound as almost parabolic shape. Keywords: Finite element method; Nanoluid; Inclined magnetic ield; Natural convection; Square enclosure; Brownian motion *Corresponding mansurdu@yahoo.com (M. M. Rahman), Phone: , Fax: Department o Mathematics and Statistics, College o Science, Sultan Qaboos University, P. O. Box 36, P.C. 123 Al-Khod, Muscat, Sultanate o Oman. 2 On leave rom the Department o Mathematics, Jagannath University, Dhaka-1100, Bangladesh 186

2 1. Introduction Natural convection heat transer in inclined devices has been the subject o many studies in the past since rarely is the earth s surace aligned with geo-potential lines. The position relative to the vertical direction o the hot active source generating the low plays an essential role in natural convection problems. To properly adopt the convective exchanges to the application requirements, it is necessary to know the iluence o the inclination angle. As an example, electronic devices contained in an airborne cavity change position relative to gravity during take-o, landing and in normal light. To achieve correct thermal regulation o such devices subject to natural convection, it is thus necessary to control the heat exchanges in all positions. The inclination angle relative to the direction o gravity also plays an important role on temperature and low ields especially in electronic system, such as laptop computers and crystal growth process in an inclined cylinder etc. Markham and Rosenberger (1984) showed that improved transport rates in the crystal growth process can be achieved by tilting the cylinder. Aminossadati and Ghasemi (200) numerically investigated the low and temperature ields in an inclined enclosure simulating an inclined electronic device. They showed that placing the enclosure at dierent orientations signiicantly aected the heat transer rate. Ben-Nakhi and Chamkha (2006) argued that tilting the enclosure considerably aect the low and temperature ields as well as the heat transer characteristics o a partitioned enclosure. Jeng et al. (2008) presented an experimental and numerical study o the transient natural convection due to mass transer in inclined enclosures and showed that the streamlines and luid concentration vary with the inclination angle. Recently, Tian et al. (2014) studied numerically the problem o unsteady natural convection in an inclined square enclosure with heat-generating porous medium and their result shows that inclination angle plays an important role in the heat transer characteristics o the walls. When the luid is electrically conducting and exposed to a magnetic ield, the Lorentz orce is also active and it interacts with the buoyancy orce in governing the low and temperature ields. Since the Lorentz orce suppresses the convection currents by reducing the velocities, employment o an external magnetic ield has a wide range o application in dierent context. In some cases, where the heat is transerred by natural convection mechanism, the electrically conducting luid may be in the presence o a magnetic ield with an arbitrary inclination. Some examples o these could include the usion reactors, metal casting, geothermal energy extractions, and crystal growth in luids. Thus, some researchers investigated the eect o magnetic ield orientation within the enclosures or two- or three-dimensional heat transer problems and all o them revealed that the orientation o the magnetic ield changed the low ield and consequently the thermal perormance o the enclosure (see Ozoe and Okada, 1989; Garandet et al., 1992; Venkatachalappa and Subbaraya, 1993; Alchaar et al., 199; Krakova and Nikiorovb, 2002; Pirmohammadi and Ghassemi, 2009). Ece and Buyuk (2006) studied natural convection low under a magnetic ield in an inclined rectangular enclosure heated and cooled on adjacent walls. Their numerical results indicate that the magnetic ield suppresses the convective low and the heat transer rate. They also showed that the orientation and the aspect ratio o the enclosure; the strength and direction o the magnetic ield had signiicant eects on the low and temperature ields. Grosan et al. (2009) considered the inclination angle o magnetic ield on the natural convection within a rectangular enclosure and it was ound that the convection mode depends upon both the strength and the orientation o the magnetic ield. Their results also indicate that the applied magnetic ield in the horizontal direction 187

3 is most eective in suppressing the convection low or a stronger magnetic ield in comparison with the vertical direction. Sathiyammmoorthy and Chamkha (2010) used dierent thermal boundary conditions to examine the steady laminar two-dimensional natural convection in the presence o inclined magnetic ield in a square enclosure illed with a liquid gallium. They ound that heat transer decreases with an increase o the magnetic ield strength. They also noted that vertically and horizontally applied magnetic ields aect the heat transer rate dierently. However, most o the above studies on the natural convection in enclosures with the magnetic ield eect have considered the electrically conducting luid having low thermal conductivity. This, in turn, limits the enhancement o heat transer in the enclosure, particularly, in the presence o a magnetic ield. An innovative technique to enhance heat transer is by using nano-scale particles in the base luid. Nanotechnology has been widely used in industry since materials with sizes o nanometers possess unique physical and chemical properties. Nano-scale particles added to base luids are called as nanoluid which is irstly utilized by Choi (199) in order to develop advanced heat transer luids with substantially higher conductivities. The most important characteristics o this new type o luid are their higher thermal conductivities in comparison with pure luids. Thereore, one o the most signiicant issues regarding these substances is the accuracy o proposed models or calculation o eective thermal conductivity. As revealed in the recent comprehensive reviews by Das et al. (2006) and Yu et al. (2008), over the past decade there have been tremendous attempts to identiy and model mechanisms o thermal conductivity enhancement o nanoluids, including size and shape o nanoparticles, the hydrodynamic interaction between nanoparticles and base luid, clustering o particles, temperature or Brownian motion, and so on. Jou and Tzeng (2006) reported a numerical study o the heat transer perormance o nanoluids inside 2D rectangular enclosures. Their results indicated that increasing the volume raction o nanoparticles produced a signiicant enhancement o the average rate o heat transer. Santra et al. (2008) conducted a study o heat transer augmentation in a dierentially heated square cavity using copper-water nanoluid. Their results show that the Bruggeman model predicts higher heat transer rates than the Maxwell-Garnett model. Abu-Nada (2009) implemented new models or nanoluids properties and examined the heat transer enhancement under a wide range o temperatures and solid volume ractions. He argued that the heat transer enhancement depends on the nanoluid viscosity and thermal conductivity models, and the range o Rayleigh numbers and solid volume ractions. But or thermal conductivity, the above- mentioned models do not consider neither the main mechanisms or heat transer in nanoluids such as Brownian motion nor the nanoparticles size or temperature dependence. Thereore, numerical simulations need more robust model or thermal conductivity that takes into account temperature dependence and nanoparticle size. In order to consider the movement o nanoparticles, some researchers have included the contribution o a dynamic component related to particle Brownian motions in their model development (see Xuan et al., 2003; Koo and Kleinstreuer, 2004; Koo and Kleinstreuer, 200; Palm et al., 2006; Akbarinia and Behzadmehr, 2007). In a quiescent suspension, nanoparticles move randomly and thereby carry relatively large volumes o surrounding liquid with them. As a result o Brownian motion, the eective thermal conductivity, which is composed o the particles conventional static part and the Brownian motion part, increases to result in a lower temperature gradient or a given heat lux. To 188

4 capture these transport phenomena, a new thermal conductivity model (see Koo and Kleinstreuer, 2004) or nanoluids has been considered, which takes into account the eects o particle size, particle volume raction and temperature dependence as well as properties o base liquid and particle phase by considering surrounding liquid traveling with randomly moving nanoparticles. Ghasemi and Aminossadati (2010) studied the Brownian motion o nanoparticles on natural convective heat transer low in a triangular enclosure illed with nanoluid. Their reported results indicate that when the Brownian motion o nanoparticles is taken into account, the resulting average Nusselt numbers are greater than those o not considering the Brownian eects. Sey and Nikaaein (2012) investigated orced convection heat transer o ethylene-glycol based nanoluids in a microchannel heat sink (MCHS) with aluminum-oxide, zinc-oxide and copper-oxide as nanoparticles. Based on their observations, the eect o Brownian motion was more signiicant or smaller nanoparticles. Wang et al. (2012) studied the heat transer enhancement o copper-water nanoluids considering Brownian motion o nanoparticles in a singular cavity. Their results showed that when Brownian motion is considered, the solid volume raction has more signiicant eects on the heat transer rate at all Richardson numbers considered in a singular cavity. Very recently, Ehteram et al. (2016) studied the eect o various conductivity and viscosity models considering Brownian motion on nanoluids mixed convection low and heat transer. Their results showed that when the Brownian motion o nanoparticles is taken into account, the resulting average Nusselt numbers are greater than those o not considering the Brownian eects. Moreover, the eect o temperature, nanoparticle size, and nanoparticles volume raction on thermal conductivity was studied by Chon et al. (200) and they showed that nanoluid thermal conductivity is also aected by temperature, volume raction o nanoparticles, and nanoparticle size. Thus, such physics cannot be neglected and the dependence o nanoluid thermal conductivity on temperature, volume raction and diameter o nanoparticles, must be taken into account in order to predict the correct role o nanoparticles on heat transer enhancement. In a very illuminating paper on nanoluids Khanaer and Vaai (2011) have shown that it is not clear which analytical model should be used to describe the thermal conductivity o nanoluids as there are many models are available in the literature. Additional theoretical and experimental research studies are required to clariy the mechanisms responsible or heat transer enhancement in nanoluids. These authors have established new correlations or eective thermal conductivity and viscosity are synthesized and developed in this study in terms o pertinent physical parameters based on the reported experimental data. The characteristic eature o nanoluids is thermal conductivity enhancement, a phenomenon observed by Masuda et al. (1993). This phenomenon suggests the possibility o using nanoluids in advanced nuclear systems (Buongiorno and Hu 200). Thereore, in this paper a inite element simulation is perormed in order to investigate the eects o magnetic ield strength and its orientation on the thermal perormance o an inclined square enclosure illed with dierent types o nanoluids having various shapes o nanoparticles, where Brownian motion is taken into consideration. We also propose a more appropriate model or the calculation o nanoluid thermal conductivity and study the eect o this model on heat transer through natural convection in an inclined square enclosure. Besides, this model will be compared to requent used model in literature namely Maxwell-Garnett model or thermal conductivity. To the best knowledge o the authors, no study which considers this problem in an inclined square cavity has yet been reported in the open literature. Our numerical results provide iormation that may be 189

5 useul or design optimization as well as or thermal perormance enhancement o energy systems such as solar thermal collectors, radiators and advanced cooling o nuclear system. 2. Problem Formulation 2.1 Physical modeling We consider an unsteady, laminar, incompressible two-dimensional natural convection low in the presence o oriented magnetic ield in an inclined square enclosure o length L illed with nanoluids. Dimensional coordinates with the x -axis measuring along the bottom wall and y -axis being normal to it along the let wall is used. The geometry and coordinate systems are schematically shown in Fig. 1. The angle o inclination o the enclosure rom the horizontal axis is denoted by. The cavity is permeated by a uniorm magnetic ield B Bxi Byj o constant magnitude B B B, where i, j are the unit vectors along the coordinate axis. The direction x y o the magnetic ield makes an angle with the positive x -axis. The top and bottom walls are insulated and nanoluids are isothermally heated and cooled by the let and right side walls at uniorm temperatures o T H and TC, respectively. Under all situations, TH TC is maintained. In the present study, we have taken water, kerosene and engine oil (EO) as base luids; Cu, Co, and Fe3O4as nanoparticles. It is also assumed that thermal equilibrium exists between the base luids and nanoparticles, and no-slip occurs between the two media. The thermophysical properties o the nanoluids are listed in Table 1. The thermophysical properties o the nanoluids are considered to be constant except the density variation in the body orce term o the momentum equation, which is estimated by the Boussinesq approximation. The gravitational acceleration acts in the negative y direction. All solid boundaries are assumed to be rigid no-slip walls. Table 1 Thermo-physical properties o the base luid and solid nanoparticles. Physical properties -1-1 c Jkg K p Water (H 2O) Engine oil (EO) Kerosene Cu Co Fe 3O kgm k Wm K 1 1 kg m s K Sm Pr

6 2.2 Mathematical modeling Within the ramework o aore-mentioned assumptions, the governing equations or the present study are expressed in dimensional orm as ollows: u v 0 x y 2 2 u u u 1 p u u u v g T T 2 2 C t x y x x y B v sin cos u sin sin 2 2 v v v 1 p v v u v g T T 2 2 C t x y y x y B u sin cos v cos cos T 2 2 u T v T T T 2 2 t x y x y where the variables and the related quantities are deined in the nomenclature. (1) (2) (3) (4) Fig. 1. Schematic view o the square shape enclosure with boundary conditions. Fig. 2. Mesh generation or a square enclosure. 2.3 Initial and boundary conditions The initial and boundary conditions or the above stated model are as ollows: 191

7 For t 0 : u v 0, T 0, p 0 For t 0 : u v 0, T T H or x 0; 0 y L u v 0, T TC or x L; 0 y L T u v 0, 0 or y 0, L; 0 x L y () 2.4 Thermal and physical properties o nanoluids The eective density, speciic heat, thermal expansion coeicient, viscosity and electrical conductivity o nanoluids that appear in equations (1)-(4) are given by the ollowing ormulas (see Rahman et al., 2014): 1 (6) p cp 1 cp cp (7) p 1 p 1 2. p p p 2 2 p 2 where k. c p The eective thermal conductivity o the nanoluid or spherical nanoparticles is introduced by Maxwell (1873) as ollows: k p 2k 2 k k p k k (11) k 2k k k p p In the above Maxwell model (equation (11)), the Brownian motion o nanoparticles has not considered. But experimentally, it has been proved that the Brownian motion o nanoparticles plays an important role on the heat transer enhancement o nanoluid (see Chon et al., 200). Thereore, we propose an appropriate model or the calculation o nanoluid thermal conductivity which is composed o the particle's convectional static part and a Brownian motion part. This two component thermal conductivity model takes into account the eects o particle size, particle volume raction and temperature dependence as well as types o particle and base luid combinations: k k k (12) static Brownian where, kstatic is the static thermal conductivity based on Maxwell classical correlation which is given in equation (11). But k Brownian, the thermal conductivity component enhanced by the irregular motion o suspended nanoparticles can be proposed as ollows: (8) (9) (10) 192

8 k Brownian pcp, p 2KBTre (13) 2 3d which represents the important contribution o the present work. Thus, the apparent thermal conductivity o the nanoluid (that consists o the Maxwell static part and Brownian part) which has been used in the present mathematical model is as ollows: kp 2k 2 k kp p cp, p 2KBTre k k (14) k 2k k k 2 3d p p 2. Dimensional analysis Dimensional analysis is one o the most important mathematical tools in the study o luid mechanics. To describe several transport mechanisms in nanoluids, it is meaningul to make the conservation equations into non-dimensional orm. The advantages o non-dimensionalization are listed as ollows: (i) non-dimensionalization gives reedom to analyze any system irrespective o their material properties. (ii) one can easily understand the controlling low parameters o the system, (iii) make a generalization o the size and shape o the geometry, and (iv) beore doing experiment one can get insight o the physical problem. These aims can be achieved through the appropriate choice o scales. As a scale o distance, we choose the length o the cavity o the region under consideration measured along the x -axis. Thus, in order to reduce the dimensionless orm o the governing equations (1)-(4) with boundary conditions (), we incorporate the ollowing dimensionless variables: 2 x y ul vl pl T T t C X, Y, U, V, P, =, = 2 2 (1) L L TH TC L Introducing the relation (1) into equations (1)-(4), the governing dimensional equations can be written in the ollowing dimensionless orm: U V 0 (16) X Y 2 2 U U U P U U U V Pr 2 2 X Y X X Y Ra Pr sin Pr Ha V sin cos U sin V V V P V V U V Pr 2 2 X Y Y X Y Ra Pr cos Pr Ha U sin cos V cos U V 2 2 X Y X Y (17) (18) (19) 193

9 where, Pr is the Prandtl number, 0 / Ha B L is the Hartmann number. 3 g TH TC L Ra The dimensionless orms o the boundary conditions are as ollows: For 0 : U V 0, 0, P 0 For 0 : U V 0, 1 or X 0; 0 Y 1 U V 0, 0 or X 1; 0 Y 1 U V 0, 0 or Y 0,1; 0 X 1 Y is the Rayleigh number and 3. Calculation o Average Nusselt Number and Average Shear Rate The most important physical quantities or this model are the local and average Nusselt numbers along the let heated wall o the enclosure. The local Nusselt number is deined as Lqw Nu (21) k T T H C where the heat transer rom the let heated wall q T w k x x0 qw is given by The average Nusselt number at the let heated wall o the enclosure can be calculated rom the ollowing expression: 1 k Nuav dy k (23) X 0 Also, the average shear rate at let heated wall o the enclosure can be calculated as 1 V Sr dy (24) X 0 where, V 2 2 U V X X is the average velocity magnitude o the low. 4. Finite Element Formulation and Computational Procedure The inite element method (FEM) is such a powerul method or solving both ordinary and partial dierential equations that arises in science and engineering problems. The basic idea o this method is dividing the whole domain into smaller elements o inite dimensions called inite elements. This method is such a good numerical method in modern engineering analysis, and it can be applied or solving integral equations including heat transer, luid mechanics, chemical processing, electrical systems, and many other ields. Thus, the governing dimensionless equations (20) (22) 194

10 (16)-(19) along with the initial and boundary conditions (20) have been solved numerically by employing Galerkin weighted residual based inite element technique. The method o weighted residual process as described by Zienkiewicz and Taylor (1991) has been applied to (16)-(19) in order to derive the inite element equations as ollows: U V N da 0 X Y Pr N U V da H da N da A 2 2 U U U P U U 2 2 X Y A X A X Y A 2 2 Ra Pr sin N da Ha Pr NV sin cos U sin da A Pr N U V da H da N da 2 2 V V V P V V 2 2 X Y A Y A X Y A 2 2 Ra Pr cos N da Ha Pr NU sin cos V cos da A A A 2 2 N U V da N da 2 2 X Y (28) A X Y A where A is the element area, N ( 1,2,...,6) are the element shape unctions or interpolation unctions or the velocity components and temperature, and H ( 1,2,3) are the element shape unctions or the pressure. Applying Gauss s divergence theorem to the second order derivative terms o the equations (26)- (28) in order to generate the boundary integral terms associated with the surace tractions and heat lux, we obtain the ollowing equations: U U U P Pr N U N U N U V da H da da X Y X X X Y Y A A A 2 2 Ra Pr sin N da Ha Pr NV sin cos U sin da Pr N SxdS0 (29) S0 (2) (26) (27) 19

11 K. S. Al Kalbani, M. S. Alam, and M. M. Rahman / American Journal o Heat and Mass Transer V V V P Pr N V N V N U V da H da da X Y Y X X Y Y A A A 2 2 Ra Pr cos N da Ha Pr NU sin cos V cos da Pr A A N S yds0 (30) S0 N N N U V da da NqwdS w X Y A X X Y Y (31) A Sw where the surace tractions Sx, S y along the outlow boundary S 0 and velocity components and luid temperature or heat lux q w that lows into or out rom the domain along wall boundary S w. The basic unknowns or the above dierential equations (29)-(31) are the velocity components UV,, the temperature and the pressure P. The six node triangular elements are used in this work or the development o the inite element equations. All six nodes are associated with velocities as well as temperature; only corner nodes are associated with pressure. This means that a lower order polynomial is chosen or pressure and which is satisied through the continuity equation. The velocity component and the temperature distributions, and linear interpolation or the pressure distribution according to their highest derivative orders in the dierential equations (16)-(19) as U X, Y N U (32) V X, Y N V (33) X, Y N (34) P X, Y H P (3) where 1,2,...,6 ; =1,2,3 Now substituting the element velocity component distributions, the temperature distribution, and the pressure distribution rom equations (32)-(3) into equations (2) and (29)-(31), the inite element equations can be written in the ollowing orm, K U K V 0 (36) x y Pr x y x xx yy K U K U U K V U R P K K U 2 2 Ra Pr sin K Ha Pr K V sin cos U sin Q u Pr x y y xx yy K V K U V K V V R P K K V 2 2 Ra Pr cos K Ha Pr K U sin cos V cos Q v (37) (38) 196

12 x y xx yy K K U K V K K Q (39) where superposed dot denotes partial dierentiation with respect to and the coeicients in element matrices are in the orm o the integrals over the element area and along the element edges S 0 and S w as, N N N N N N K N N da, K x N da, y K N da, xx X K da, yy Y K da, X X Y Y N N N N K x NN da, y K NN da, x X R N da, y Y R N da, X Y Pr Pr Q u N SxdS 0, Q v N S yds 0, Q w NqdS w. These element matrices are evaluated in closed-orm ready or numerical simulation. Details o the derivation or these element matrices are omitted herein or brevity. The derived inite element equations (36)-(39) are nonlinear. The nonlinear algebraic equations so obtained are modiied by imposition o boundary conditions. To solve the set o the global nonlinear algebraic equations in the orm o matrix, the Newton-Raphson iteration technique has been adapted through partial dierential equation solver with MATLAB interace. The convergence criterion o the numerical m1 m solution along with error estimation has been set to 10, where is the general dependent variable ( UV,, ) and m is the number o iteration. The main advantages o inite element method (FEM) over inite dierence method (FDM) are that it has ability to deal with complex 2D or 3D domains, higher accuracy and rapid convergence. Other beneit o the inite element method is that o the speciic mode to deduce the equations or each element which are then assembled. Thereore, the addition o new elements by reinement o the existing ones is not a major problem. The computational domains with irregular geometries by a collection o inite elements make the method a valuable practical tool or the solution o boundary value problems arising in various ields o engineering. For the other methods, the mesh reinement is a major task and could involve the rewriting o the program. But or all the methods used or the discrete analogue o the initial equation, the obtained system o simultaneous equations must be solved. That is why, the present work emphasizes the use o inite element technique to solve low and heat transer problems. 4.1 Mesh generation In the inite element method, the mesh generation is the technique to subdivide a domain into a set o sub-domains, called inite element, control volume, etc. The discrete locations are deined by the numerical grid, at which the variables are to be calculated. It is basically a discrete representation o the geometric domain on which the problem is to be solved. Meshing the complicated geometry make the inite element method a powerul technique to solve the boundary value problems occurring in a range o engineering applications. Fig. 2 displays mesh coiguration o the present physical domain with triangular inite elements. 197

13 4.2 Grid independency test A grid reinement study has been perormed or Pr , Ra 10, Ha 20, 30, 0.0, 1 and d 10 nm in a square enclosure. Four dierent non-uniorm grid systems with the ollowing number o elements within the resolution ield: 240, 678, 17038, and are examined. The numerical design is carried out or highly precise key in the average Nusselt number Sr or the aoresaid elements to develop an understanding o the Nu and average shear rate av grid ineness as shown in Fig. 3 and Table 2. The scale o Nuav or elements show a very little dierence with the results obtained or the elements Hence the grid size o and elements can be used to get the accurate results. In the presence study, triangular elements have been considered to get the results. Fig. 3. Convergence o the average Nusselt number (let) and average shear rate (right) with grid reinement or Pr , Ra 10, Ha 20, 30, 0.0, 1 and d 10 nm. Table 2 Grid test using the values Pr , d 10 nm. Ra 10, Ha 20, 0 30, 0.04, 0 1, Wall Let Elements (Nodes) 104 (927) 240 (1487) Nu av Sr (372) (9420) (1401) Code validation In order to veriy the accuracy o the present numerical code, we have compared our result or steady state case with Ghasemi et al. (2011). The physical problem studied by Ghasemi et al. (2011) 198

14 was a steady two-dimensional natural convection low in a square enclosure illed with water-al O nanoluid which is under the iluence o a horizontally applied magnetic ield. Using our code the present numerical predictions have been obtained or Rayleigh number between 10 7 and 10. The comparison o the results obtained by our numerical code with those o Ghasemi et al. (2011) with respect to average Nusselt number (at the hot wall) are shown in Table 3 which shows an excellent agreement. This validation boosts the coidence in the numerical outcome o the present study. Table 3 Comparison o the present data (Nu av) with those o Ghasemi et al. (2011) or dierent values o Ra and. = 0 = 0.02 Ra Present study Ghasemi et al. (2011) Present study Ghasemi et al. (2011) Results and Discussion In this section, simulated numerical results are analyzed to investigate the eects o magnetic ield strength and its orientation as well as the inclination angle o the square enclosure illed with various nanoluids having dierent diameters o the nanoparticles (1nm d 100nm ). Calculations are made or various values o volume raction o nanoparticles 0 0.1, Rayleigh number 10 Ra , Hartmann number 0 Ha 60 and magnetic ield orientation In the numerical simulations, we have considered three dierent types o base luids namely water HO, engine oil (EO) and kerosene with three dierent kinds o nanoparticles such as Cu, Co and 2 Fe O. Streamlines and isotherms evolution with dimensionless time 3 4 as well as the average heat lux and the local Nusselt number on the heated let wall are calculated or dierent model parameters. The results are taken or Cu-H 2O nanoluid and then compare the average Nusselt number with dierent nanoluids or dierent volume ractions and dierent nanoparticle diameters. Figure 4 displays the streamline evolutions with dimensionless time or Ra 10. For a shorter time, an oval-shaped circulation is ormed at the center o the cavity and the streamlines intensiy towards the hot wall. As increases, the circulation zone changes to an elliptical pattern and the streamlines intensiy at both the hot wall as well as the cold wall. This indicates a high downward velocity o the low. As increases, the pattern o the streamlines show no signiicant change until it reaches the steady state. 199

15 Figure shows the temperature distribution inside the cavity against or Ra 10. For 0.0 the low is unsteady and the contour lines are condensing at the bottom o the hot wall which indicates a high temperature gradient due to buoyancy eect. Over some time, the temperature distribution changes in the enclosure while the contour patterns show a marginal variation until it reaches the steady state..1 Eects o Rayleigh number Figure 6 depicts the average Nusselt number on the let heated wall or dierent Rayleigh numbers ,10,10,10, In all our cases the average Ra with dimensionless time Nusselt number decreases initially. As time progress, it starts to increase and reached its maximum 3 value, then to the steady state. It is obvious that in the steady part, when Ra 10 the heat transer is the lowest and as Ra increases the heat transer increases due to the increase in the buoyancy eect. To ind at which time the low reaches the steady state, the average Nusselt number is calculated numerically and displayed graphically in Fig. 7 or dierent Rayleigh number. From this igure we observe that the time taken to reach the solution in steady state is 1., 0.9, 0.8 and 0. or Ra 10,10,10,10 respectively. It means that as Ra increases, the dimensionless time the low takes to reach the steady state decreases. Thus, strong buoyancy helps the low to reach steady state aster. Figure 8 displays the local Nusselt number along the hot wall o the enclosure. The results are taken at 2 (steady state) and are presented or our values o the Rayleigh numbers Ra 10,10,10,and 10 with solid volume raction 0.0, Hartmann number Ha 20 and the nanoparticle diameter d 10 nm. Except in the vicinity o the top let corner o the enclosure, the results show that due to the strengthened buoyant low the local Nusselt number increases as 3 the Rayleigh number increases. For Ra 10 the local Nusselt number shows a horizontal line 4 since the conduction regime dominated. For Ra 10, the local Nusselt number shows a slight increase and then a slight decrease. This is due to the convection- dominated regime. The increase 6 o the local Nusselt number is noticeable or Ra 10 and more signiicant or Ra 10 at which the local Nusselt number increases sharply until it reaches a peak in the vicinity o the lower let corner and then decreases quite signiicantly. 4 6 Figures 9-10 display the eect o three dierent values o Rayleigh number Ra 10, 10, and 10 on streamlines and isotherms respectively or both unsteady case ( 0.0) and the steady case ( 2). In these igures we have considered solid volume raction 0.0, the nanoparticle diameter d 10 nm, Hartmann number Ha 20 and Cu-H 2O nanoluid. The buoyancy driven circulating lows within the enclosure are evident or all values o the Rayleigh number Ra. For 4 Ra 10, a central circulation cell is observed within the enclosure as a dominant characteristic o the low ield. However, as Ra increases, the circulation pattern gets larger and the streamlines 200

16 intensiies in the vicinity o the hot and cold wall which is evident o high velocity gradient and indicates a strengthen in the natural convection. The isotherms also indicate a regime where the contribution o the convective low ield in the heat transer becomes evident. It is obvious that increasing Rayleigh number is associated with the variation o isotherms pattern. As Rayleigh number increases, the contour lines are condensing at the hot and cold walls which indicate higher temperature gradient. Moreover, the thermal boundary layer near the walls becomes thinner, indicating a higher heat transer rate..2 Eects o solid volume raction Figure 11 presents the average Nusselt number on the let heated wall or dierent nanoparticles volume ractions 0, 0.0, 0.02, 0.1 with dimensionless time, , when Rayleigh number is 10, Hartmann number is Ha 20 and nanoparticle diameter is d 10 nm. For small amount o nanoparticles, the average Nusselt number is high at some short time then it dropped to reach a minimum value and then rises up until it shows a constant value as the low reaches the steady state. For large volume raction, 0.1, this luctuation in the average Nusselt number is not signiicant. This is due to the conductivity dominant o the nanoparticles. At the steady state o the low, as the solid volume raction increases, the average Nusselt number on the let heated wall increases. This is due to the increase o the nanoluid thermal conductivity as the taken to raction o the nanoparticles is increased. Figure 12 shows the dimensionless time reach the low in steady state. From this igure we observe that the time taken to reach the steady state o the low is 1.0, 0.8, 0.8 and 0.6 or 0.0,0.02,0.0,0.1 respectively. Thus, addition o nanoparticles to the base luid helps the unsteady problem to reach it steady state. Figure 13 illustrates the local Nusselt number on the let heated wall o the enclosure or our values o solid volume ractions 0,0.02,0.0 and 0.1, when Ra 10, Ha 20 and d 10nm. The results are taken or the steady state, i.e. when 2. Along the heated wall, as the volume raction increases, the heat transer rate also increases. This is due to the high thermal conductivity o the nanoparticles. The pattern o the local Nusselt number lines is similar or all values o. At vicinity o the heated wall 0 Y 0.0 the local Nusselt number increases to its maximum value, then decreases along the other part o the wall. The peaks o the maximum or dierent occur at Y 0.0. Figures 14-1 present the streamlines and isotherms respectively or dierent volume ractions 0,0.02 and 0.0 at 0.0 (unsteady case) and 2 (steady case) when Ra 10, d 10nm, Ha 20. For all values o, the low is rotating clockwise with a circulation zone at the center o the enclosure. At an unsteady case 0.0, as the volume raction increases, the circulation at the center o the cavity becomes more uniorm. At the steady case, as increases, 201

17 the elliptical eye o the circulation at the center o the enclosure gets smaller. The volume raction plays an insigniicant role on the shape o the isotherms as shown in Fig. 1. Nevertheless, the temperature distribution within the low domain intensiies is varying due the increased thermal conduction ability o the nanoparticles..3 Eects o nanoparticle diameter Figure 16 displays the variation o the average Nusselt number with dimensionless time or dierent nanoparticle diameters d 1 nm,10 nm,0 nm and 100 nm. The results are calculated on the let heated wall o the enclosure or , 0.0, Ha 20, Ra 10 and or Cu-H 2O nanoluid. For a short time the average heat transer rate dropped rom a high value to its minimum and then increases until it reaches to a constant value at steady state. For d 100nm, 0nm, and 10nm the pattern is similar but it shows dierent behavior or d 1nm at which the average heat transer rate increases then shows a slight decrease beore taking a constant value. As the nanoparticle diameter decreases, the average Nusselt number increases. This is because as the nanoparticle diameter decreases, the speciic area increases which enhance the thermal taken to conductivity o the nanoluid to increase. Figure 17 presents the dimensionless time reach the low in steady state or the same parameter values as in Fig. 16. For 1nm particle diameter, it takes 0. which is the shortest time when compared to the larger diameters. Figure 18 illustrates the eect o the nanoparticle diameter on the local Nusselt number on the let heated wall o the enclosure. The Rayleigh number is Ra 10, Hartmann number Ha 20 and the volume raction 0.0 are kept constant. The results are taken or Cu-H 2O nanoluid at the steady case 2. The local Nusselt number becomes higher as the nanoparticle diameter gets smaller. In all cases, the local Nusselt number increases near the bottom let corner o the heated wall and dropped down along the rest o the wall. Figures show the eect o dierent nanoparticle diameters on the streamlines and isotherms respectively or Cu-H 2O nanoluid. The results are calculated or the unsteady state 0.0 as well as or the steady state 2 under the same parameters conditions as in Fig. 18. At an unsteady case, ord 1nm, the streamlines are elongated along the diagonal and condensed at the hot let and cold right walls with an elliptical circulation at the center o the cavity. As the nanoparticle diameter increases, the circulation changes to an oval shape and the streamlines show more condensation at the hot wall than the cold one. For the steady state 2, the streamlines distribution becomes more uniorm and more condense or d 1nm which clearly indicates that as the particle size gets smaller, the mixture becomes more homogeneous. No signiicant dierences in the pattern o the isotherms are noticed with the variation o d as can be seen Fig. 20. A slight deormation in contours can be observed as the nanoparticle diameter increases. Nevertheless, the hot region near the hot wall gets larger with decreasing nanoparticle diameter. 202

18 Fig. 4. Streamlines evolution with dimensionless time Pr , Ha 20, d 10nm, Ra 10. when 0 1, 0 30, 0.0, 203

19 Fig.. Isotherms evolution with dimensionless time ( ), 0.0, Pr , Ha 20, d 10 nm, Ra , 30, 204

20 Fig. 6. Average Nusselt number on the let heated wall or dierent Rayleigh numbers when 0 0 1, 30, 0.0, Pr , Ha 20, d 10nm. Fig. 7. Dimensionless time to reach the solution in steady state or dierent Ra. 20

21 Fig. 8. Local Nusselt number on the heated wall in the steady state or dierent Ra. 4 Ra 10 Ra 10 6 Ra Fig. 9. Streamlines or dierent Rayleigh numbers Ra at 0.0 (unsteady state) and 2 ( steady state ), 0.0, Pr , Ha 20, d 10nm. 206

22 4 Ra 10 Ra 10 6 Ra Fig. 10. Isotherms or dierent Rayleigh numbers Ra at 0.0 (unsteady state) and 2 ( steady state ), 0.0, Pr , Ha 20, d 10nm. Fig. 11. Average Nusselt number on the let heated wall or dierent solid volume ractions when Ra 10, 0.0, Pr , Ha 20, d 10nm. 207

23 Fig. 12. Dimensionless time to reach the solution in the steady state or dierent. Fig. 13. Local Nusselt number on the heated wall at steady state or dierent. 208

24 Fig. 14. Streamlines or dierent volume raction at 0.0 (unsteady state) and 2 ( steady state ), when Ra 10, Pr , 20 Ha, d 10nm. 209

25 Fig. 1. Isotherms or dierent volume raction at 0.0 (unsteady state) and 2 ( steady state ), when Ra 10, Pr , 20 Ha, d 10nm. Fig. 16. Average Nusselt number on the let heated wall or dierent nanoparticle diameter d when Ra Pr Ha 10, , 20,

26 Fig. 17. Dimensionless time to reach the solution in steady state or dierent d. Fig. 18. Local Nusselt number on the heated wall at steady state or dierent d. 211

27 d 1nm d 10nm d 100nm Fig. 19. Streamlines or dierent particle diameter d at 0.0 (unsteady state) and 2 ( steady state ), when Ra 10, Pr , Ha 20,

28 d 1nm d 10nm d 100nm Fig. 20. Isotherms or dierent particle diameter d at 0.0 (unsteady state) and 2 ( steady state ), when Ra 10, Pr , Ha 20, 0.0. Fig. 21. Average Nusselt number on the let heated wall or dierent Hartmann number ( Ha ) when Ra 10, Pr , d 10nm,

29 (a) (b) Fig. 22. (a) Local Nusselt number on the heated wall at the steady state or dierent Hartmann number (let), and (b) dierent magnetic ield inclination angle (right). (a) (b) Fig. 23. (a) Average Nusselt number with τ, and (b) local Nusselt number on the heated wall at steady state or two cases (with Brownian eect) and (without Brownian eect). 214

30 (a) (b) (c) (d) (e) () Fig. 24. Average Nusselt number on the heated wall or dierent geometry inclination angles or Cu-H O nanoluid. 2 21

31 Table 4 Average Nusselt number on the heated wall or dierent nanoluids and dierent volume 0 0 ractions when 1, 30, Ha 20, d 10 nm, Ra 10. Nanoluids I II III IV ( ) ( ) ( ) Water Cu Water Co Water- Fe 3O Engine oil- Cu Engine oil- Co Engine oil Fe 3O Kerosene - Cu Kerosene- Co Kerosene Fe 3O

32 Table Average Nusselt number on the heated wall or dierent nanoluids and dierent 0 0 nanoparticle diameters d when 1, 30, Ha 20, 0.0, Ra 10. d Nanoluids I II ( ) III ( ) IV ( ) 100nm 0nm 10nm 1nm Water Cu Water Co Water-Fe 3O Engine oil- Cu Engine oil- Co Engine oil Fe 3O Kerosene - Cu Kerosene- Co Kerosene Fe 3O

33 Table 6 Average Nusselt number on the heated wall or dierent volume ractions and dierent Rayleigh number Ra or two cases (with Brownian eect) and (without Brownian eect). Ra Nu av ( ) ( ) ( ) ( ) ( ) I II III IV V VI 3 10 With Brownian eect Without Brownian eect 4 10 With Brownian eect Without Brownian eect 10 With Brownian eect Without Brownian eect 6 10 With Brownian eect Without Brownian eect Eects o magnetic ield and its orientation Figure 21 depicts the eect o the magnetic ield intensity on the average Nusselt number which is calculated on the let heated wall o the enclosure or , taking Hartmann number Ha 0, 20, 40, and 60, Ra 10, d 10nm and 0.0. As Hartmann number ( Ha ) increases, the average Nusselt number in steady state decreases. This is because the magnetic ield suppresses the convective lows as the intensity o the magnetic ield increases, which in turn slow down the heat transer rate. Figure 22 displays the eect o the magnetic ield intensity and its orientation on the local Nusselt number on the let heated wall. The results are calculated when the solution becomes steady 2, or the same parameters as in Fig. 21. Figure 22(a) shows the signiicant eect o Hartmann number on the local Nusselt number. In the case when no magnetic ield aects the low, the heat lux is higher on the wall and as its intensity increased, the heat lux decreases. The magnetic ield inclination angle iluences the heat lux as shown in igure 22(b). The local Nusselt number rises in the vicinity o the heated wall and then declines away rom the wall. The eect o the magnetic ield inclination angle on the top hal o the heated wall is minimal. 218

34 . Average heat transer rate or dierent nanoluids So ar we have discussed the results or water-cu nanoluid. Here, we considered various base luids and nanoparticles to see how the results depend on them. Table 4 illustrates the average Nusselt number on the heated wall o the enclosure or the steady state 2 or three dierent types o base luids namely water (H 2O), engine oil (EG) and Kerosene with three dierent kinds o nanoparticles Cu, Co and Fe 3O 4. Rayleigh number is considered to be Ra 10, Hartmann number Ha 20 and the nanoparticle diameter d 10nm. For all nine types o nanoluids, as the volume raction increases, the average Nusselt number increases. Kerosene-based nanoluids show the highest heat transer when compared to the water-based and engine oil-based nanoluids. Although, Cu nanoparticle has higher thermal conductivity than Co and Fe 3O 4, kerosene-co and kerosene-fe 3O 4 show higher heat transer rate. This is due to the Brownian eect on the thermal conductivity o the nanoluids. Engine oil-based nanoluids show the lowest heat transer due to the high dynamic viscosity and low thermal conductivity o the base luid. The eect o dierent nanoparticle diameters on the heat transer rates is listed in Table. As the nanoparticle diameter increases, the average heat transer rate decreases. Kerosene based nanoluids show a signiicant enhancement in heat transer rate as the nanoparticle diameter decreases. For 1 nm nanoparticles diameter, it reaches %, 169.8% and % enhancement or Cu, Co and Fe 3O 4 when compared with 100 nm nanoparticle diameter. For waterbased nanoluids, the heat transer rate intensiied by around 100% or 1nm nanoparticle diameter compared with 100 nm particle diameter. Engine oil-based nanoluids show around 24 % increased heat transer or 1 nm particle diameter compared with 100 nm particle diameter. This is due to the high dynamic viscosity o engine oil which suppresses the Brownian motion o the nanoparticles..6 Eect o Brownian motion All the results obtained are by taking into account the Brownian motion eect on the eective thermal conductivity o the nanoluid as shown in equation (14). In order to examine the impact o the Brownian motion on heat transer rates, we have calculated the average Nusselt number and local Nusselt number on the heated wall or two cases (with Brownian motion and without Brownian motion) considering Rayleigh number Ra 10, Hartmann number Ha 20 and the nanoparticle diameter d 10nm or water-cu nanoluid. The results are illustrated in Fig. 23. It is obvious that the Brownian motion plays a signiicant role in enhancement o the heat transer rate. The contribution o Brownian motion o nanoparticles to the heat transer is due to the movement o the nanoparticles which can transer heat and micro-convection o the luid around individual nanoparticles. Table 6 presents a comparison study or the increase in the average Nusselt number or the nanoluid with respect to pure water at various Rayleigh numbers 10 3 Ra 10 6 and solid volume ractions The iluence o on the average Nusselt number Nu av is more signiicant at low Rayleigh numbers or both models with and without Brownian motion. For 3 example, at Ra 10, or the nanoluid with 0.0, Nu av increased by % when Brownian motion is taken into account and by 1.22% when neglected it. 219