Mixed convection fluid flow and heat transfer and optimal distribution of discrete heat sources location in a cavity filled with nanofluid

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1 Trans. Phenom. Nano Micro Scales, 5(1): -4, Winter - Spring 17 DOI: 1.758/tpnms ORIGINAL RESEARCH PAPER Mixed convection luid low and heat transer and optimal distribution o discrete heat sources location in a cavity illed with nanoluid A. A Abbasian Arani 1, M. Abbaszadeh *,1, A. Ardeshiri 1 1 Mechanical Engineering Department, University o Kashan, Kashan, I. R.Iran Received 19 August 15; revised 1 October 15; accepted 1 November 15; available online 4 December 16 ABSTRACT: Mixed convection luid low and heat transer o water-al O nanoluid inside a lid-driven square cavity has been examined numerically in order to ind the optimal distribution o discrete heat sources on the wall o a cavity. The eects o dierent heat source length, chardson number and Grasho number on optimal heat source location has been investigated. Moreover, the average Nusselt number on the heat source or two models o nanoluid, constant properties and variable properties, are compared. The obtained results showed that by decreasing the chardson number and increasing the Grasho number, heat transer rate decreases. Also by reducing the chardson number, optimal heat source location move to the top o the wall and with augmentation o chardson number, heat source optimal location move to the middle o the wall. Furthermore, the overall heat transer increases by increasing nanoparticles volume raction. Moreover, it was ound that or two dierent models o nanoluids and in =1, the values o the average Nusselt number are close together. KEYWORDS: Mixed convection; Nanoluids; Heat sources; Optimization Introduction Fluid low and heat transer in a cavity which is driven by buoyancy and shear orce are encountered in a variety o thermal engineering applications [1 ]. Interaction o buoyancy orce due to temperature gradient and orced convection due to shear orces is a complex phenomenon in mixed convection low and heat transer. Numerous researches on this type o problem including the single or double lid-driven cavity low and heat transer involving dierent cavity coigurations, various luids and imposed temperature gradients have been published in the last two decades [-9]. For example, Shari [1] studied laminar mixed convection in a shallow inclined cavity where the top wall is hot and the bottom wall is cool. He demonstrated that the average Nusselt number increases by increasing the cavity inclination angle or orced convection-dominated regime (=.1) while it increases more rapidly or natural convection-dominated regime (=1). Tiwari and Das [11] numerically investigated the mixed convection heat transer and luid low o Cu water nanoluid in a square cavity with top and bottom insulated walls and dierentially-heated moving sidewalls. They ound that when the =1, the average Nusselt number increases substantially with augmentation o the volume raction o the nanoparticles. Muthtamilselvan et al. [1] numerically scrutinized the mixed convection low and heat transer o Cu water nanoluid in a lid-driven rectangular dd *Corresponding Author abbaszadeh.mahmoud@gmail.com Tel.: ; Note. This manuscript was submitted on August 19, 15; approved on October 1, 16; published online December 4, 16. enclosure. The sidewalls o the enclosure were adiabatic while the horizontal walls were ept at constant temperatures and the top wall moved at a constant velocity. Aremanesh and Mahmoodi [1] perormed a numerical study to examine the eects o uncertainties o viscosity models or the Al O water nanoluid on mixed convection in a square cavity with cold let, right, and top walls and hot bottom wall. Their results showed that the average Nusselt number o the hot wall increases by increasing the volume raction o nanoparticles or both viscosity models which are used. Kandaswamy et al. [14] conducted a numerical study on buoyancy-driven convection in a cavity with partially thermally active vertical walls. They obtained that heat transer rate is increased when the heating location is at middle o the hot wall. Mahmoodi [15] investigated mixed convection low o water-al O nanoluid in a rectangular cavity. He observed that the heat transer rate increases due to existence o nanoparticles in the base luid or all range o considered chardson number. Sebdani et al. [16] conducted a numerical simulation to investigate the eect o nanoluid variable properties on mixed convection in a square cavity with moving cold side walls and a constant temperature heater on the bottom wall. Their results showed that the heat transer o the nanoluid could be either enhanced or alleviated with respect to the base luid which is depending on the Reynolds number and Rayleigh number. Amiri et al. [17] provided a numerical simulation o combined thermal and mass transport in a square lid-driven cavity.

2 A. A Abbasian Arani et al. Nomenclature u velocity component in the direction o x(m s) C Overall heat transer v velocity component in the direction o y(m s) D Heat source length U Dimensionless velocity component in the direction o X c P Speciic heat at constant pressure(j gk) V Dimensionless velocity component in the direction o Y F Buoyancy orce(kgms - ) Gree Symbols g Acceleration o gravity(m s ) α Thermal diusivity(m s) Gr Grasho Number β Coeicient o thermal expansion(1 K) H The height o cavity(m) μ Viscosity(g ms) K Conduction heat transer coeicient(w mk) ν Kinematic viscosity(m s) Nu avg The average Nusselt number θ Dimensionless temperature Nu y Local Nusselt number ρ Density(g m ) Ma Mach number φ Volume raction o nanoparticles p Pressure(N m ) Subscripts p Dimensionless pressure avg Average p r Prandtl number c Cold Re Reynolds number Opt Optimum Ra Rayleigh number Fluid chardson number h Hot T Temperatuer(K) Nanoluid p Nanoparticle Their results demonstrated that the heat and mass transer rates inside the cavity increases or low values o chardson numbers. The other applications o the mixed convection and also, using nanoluids as an innovative approach to enhance heat transer can be ound in reerences [18-6]. The convective heat transer problem and optimal heat sources locations in enclosures have been studied in many literatures due to wide range o engineering applications in such processes. For instance, the thermal perormance o electronic pacages containing a number o discrete heat sources has been studied extensively. The design problem in electronic pacages is to maintain cooling o chips in an eective way to prevent overheating and hot spots. This is achieved generally by eective cooling by natural convection, mixed convection and in certain cases by other means such as heat pipes, and inally by better design. In the latter case, the objective is to maximize heat transer density so that the maximum temperature speciied or sae operation o a chip is not exceeded. Thus, optimum placement o discrete heaters may be required with respect to usual equidistant placement. Mutuoglu and Bilgen [7] determined the optimum position o a discrete heater by maximizing the conductance and then studied heat transer and volume low rate with the discrete heater at its optimum position in open cavities by using the inite dierence-control volume numerical method. The optimum position o single and multiple chips has been studied in reerences [8-9]. Recently, Dias and Milanez [4] have used genetic algorithms in order to optimize the heat sources in a cavity. In this paper, a numerical study has been done in order to ind the optimal distribution o discrete heat sources on the wall in a lid-driven square cavity illed with nanoluid. The eects o dierent heat source length values, chardson number and Grasho number on optimal heat source location has been investigated. Results o low ield and heat transer simulation in dierent conditions such as heat source movement, variation o heat sources length, chardson number, Grasho number, nanoparticles volume raction have been investigated. Also the average Nusselt number on the heat source or two models o nanoluid, constant properties and variable properties, are compared. The geometry, governing equations and boundary conditions In the present study a steady two-dimensional and laminar nanoluid low inside the cavity is considered. The geometry is shown in Figure 1. Fig. 1. Schematic diagram o the physical system Length and height o the cavity are equal (L=H). The top wall is insulated and moves in constant velocity U rom 1

3 Trans. Phenom. Nano Micro Scales, 5(1) -4, Winter - Spring 17 let to right. The right wall is at a constant temperature T and heat sources with constant lux q " and D length are on the let insulated wall. The bottom wall is adiabatic. The cavity is illed with water-al O nanoluids. Thermophysical properties o base luid and particles are presented in Table 1. In addition, viscosity and density is considered vary with temperature and volume raction. The nanoluid is considered Newtonian and incompressible. The density changes according to Boussinesq approximation [4]. Steady-state continuity, momentum and energy equations in Cartesian coordinates are as ollows: Table 1 Thermophysical properties o water and Al O nanoparticles at 5 o [44]. Physical Properties Water Al O ρ(g m ) cp(j (gk)) (W (mk)).61 4 β 1 (1 K) 1.85 u v x y (1) The eective dynamic viscosity o the water-al O nanoluid is calculated according to the Brinman model [48] and variable properties Khanaer and Vaai model [4]..5 (9) (1- ) 8.87 ( ) AlO p T p.164p.5.1 p T T d d p p p p p (1) Khanaer and Vaai model [4] is used in a wide range o nanoparticle diameter 1nm d p 11nm, volume raction 1% φ p 9% and temperature T( C) 7. The eective thermal conductivity o the nanoluid is determined using the Maxwell model [41], which is: ( u u v v ) p u v x y x x x y y () ( ) ( ) e s s ( ) ( ) s s (11) u v p u ( u v ) x y x x x y v gt ( Tc ) y T T p T cp ( u v ) x y x x x (4) T y y where the density, heat capacity, thermal expansion coeicient, and thermal diusivity o the nanoluid are as ollow[45]: (1 ) (5) P cp 1cp cp p 1 p () (6) (7) c p (8) p dp nm Alo.46.5 ( T) p p p ( T) T T T (1) Temperature range, diameter and volume raction o nanoparticles in equation 1 in % φ p 1%, T( C) 7, 11nm d p 15nm are valid, respectively. Water dynamic viscosity at dierent temperatures can be expressed as ollows [4]: ( T 14) T (1) To convert the governing equations 1 4 into dimension - less orm, the ollowing dimensionless parameters are employed: u v x, yd,, Si U, V, XYD,,, S i U U H '' ( T / x) q '' Tmax Tc p q max, P, '' qh/ U (14)

4 A. A Abbasian Arani et al. The dimensionless orms o the governing equations are: The local Nusselt number is calculated by: U V X Y V U P 1 1 U V X Y X Re U U ( ) ( ) X X Y Y V V P 1 1 U V X Y Y Re V V ( ) ( ) X X Y Y ( ) Gr Re 1 1 U V X Y Pr Re X X Y Y (15) (16) (17) (18) Nu s h L (1) where the heat transer coeicient is: h T max T y y T c () The thermal conductivity is calculated according to the ollowing expression: T y T / x y () By substituting equations and into equation 1, the Nusselt number can be written as: Nu s Y (4) The Grasho number, the Reynolds number and the Prandtl number are deined as: gh Tmax T c UH Gr,Re, Pr (19) The boundary conditions or the governing equations are: Y, X L : Y H, X L : U V Y max, U U V Y,, max X, Y S : max U V, X X, S Y S D : max U V, ( ) q X X, S D Y H : U V X max, '' () By integrating the local Nusselt number along the hot wall, the average Nusselt number is calculated by: Nu avg 1 s s s Nu s ds s (5) The overall heat transer inside the enclosure is deined as ollows: Q C ( T T ) max C (6) Q ' is the total heat lux in the cavity and is obtained according to equation 7: (7) ' '' Q q D N Which N is the number o heat sources and D is the length o heat source. Numerical simulation Governing equations are solved numerically using the inite volume method and SIMPLER algorithm. At irst a uniorm and appropriate grid is coincided to the solution ield and then around each node, a control volume is

5 Trans. Phenom. Nano Micro Scales, 5(1) -4, Winter - Spring 17 considered and the governing equations are integrated on each control volume and equations are dismissed and a system o algebraic equations is obtained. For separation convection and diusion terms the hybrid method is used. In this method the central dierence scheme is used or the Peclet numbers with absolute value less than and the upwind scheme is used or the Peclet numbers with absolute value more than. In order to achieve the convergence, the under relaxation coeicient is used. It is.5 or velocity components and.7 or temperature. Convergence criteria or pressure, velocity and temperature is obtained by equation 8, where M and N is the number o grids in the x and y direction and "ζ " represents a variable which is solved. is number o iterations and the maximum error is 1-6. M N 1 i, j i, j i1 j1 6 M N 1 i, j i1 j1 Error 1 (8) Grid independency test Numerical code was tested or grid independence by calculating the average Nusselt number on the heat source surace according to Table. It was ound that a grid size o ensures a grid independent solution in nonuniorm mesh. Table Average Nusselt number or dierent grids. Grid size Average Nusselt number Validation o results To validate the computer program results, Silva et al. [46] and Sivaumar et al. [47] solution geometry with our program are simulated. The results o the overall heat transer rates at various locations inside the cavity and the average Nusselt number in Re=1 are compared with their results in Figure and Tables. Fig.. Overall heat transer in various locations along the wall As can be seen or the same conditions, a good agreement exists. Also the small amounts o the numerical dierence represent the computer code accuracy which is used. Table Comparing the average Nusselt number in mixed convection. chardson Number =.1 =1 =1 Distance rom the heat source to the bottom o the cavity Average Nusselt Number Reerence Results[47] Present Study ɛ = ɛ = ɛ = ɛ = ɛ = ɛ = ɛ = ɛ = ɛ = Results and discussion In this section, the results such as the stream lines, the isotherm lines, the average Nusselt number and the overall heat transer in terms o parameters such as the chardson number, Grasho number, Reynolds number and volume raction are presented and discussed. The various conditions are being studied or dierent chardson number =.1, 1, 4 and 1 and in constant Grasho numbers 1, 1 and 1 4. In this study, three dierent heat source lengths (%D =.5,.1,.) are used. Also dierent volume raction values have been used to study the eect o the nanoparticles volume raction on heat transer. Finally the eect o using nanoluids variable properties and constant properties on average Nusselt number are compared. Finding the optimal location or one heat source In Figure, the overall heat transer in term o the distance between the end o the heat source and bottom o the cavity or the dimensionless length.1, φ=.5 and in dierent chardson numbers and constant Grasho numbers Gr=1-1 is shown. As can be seen in this igure, the part o the curves which have the pea o C, represents heat source optimal location (%S.opt ) which is a unique point or each curves. It should be noted that the overall heat transer has a direct relation with the heat source maximum temperature. Also the optimal location and the overall heat transer depend on the chardson number and the Grasho number. When the heat source is located at the bottom o the wall, the overall heat transer is almost independent rom the chardson number. This is due to wea low strength in the subordinate edge o the cavity that reduces the heat transer rate in this area. But by increasing the distance 4

6 A. A Abbasian Arani et al. between the heat source and the bottom wall, the dierence between the values o C are intensiied in dierent chardson numbers C N=1 Gr=1 = S,opt.6.4 D=.5 D=. C N=1 Gr=1 S Gr=1 =.1 =1 =4 = S Gr=1 Fig..The overall heat transer in terms o the distance between the heat source and the bottom wall As shown in Figure, by increasing the chardson number, orce convection reduces and as a result, the heat transer rate inside the cavity, especially at the upper corners, decreases. Thereore, the heat source optimal location is migrated rom the top wall to the middle part o the let wall. Figure 4 elucidates how the optimum heat source location responds to the Grasho number in addition to chardson number. The changes o the optimum heat source location in low Grasho numbers is more than high Grasho numbers as ar as the Grasho number is the proportion o the buoyancy orce to viscosity orce. In general, when the Grasho number increases, the overall heat transer increases inside the cavity. Thus in low Grasho number the heat source should be transmitted to the middle o the cavity in order to increase heat transer. The exact values o the optimal heat source location and maximum overall heat transer in dierent chardson numbers and in constant Grasho number are represented in Table 4. According to these results, the optimal location or high Grasho number is near the top wall and the lid-driven. But in low Grasho number, the optimum location in dierent chardson numbers changes. The changing is near the middle part o the wall up to lid-driven. =1 =4 = N= D=.5.8 S D=. N= Fig. 4. Distance between the heat source and the bottom wall in terms o chardson number Finding the optimal location or two discrete heat sources In this section, the position o two separate heat sources with equal length and heat lux is optimized. Distance between the irst heat source to the bottom wall is S and the distance between the two heat sources is speciied by S. As illustrated in Figure 5, in =4, at irst the overall heat transer increases by increasing S and then at the end o the wall decreases. Also when the irst heat source approaches to the end o the wall, the second heat source location plays an important role in overall heat transer rate in which by closing to the lid-driven, a severe decrease in overall heat transer taes place. C S=. S=.1 S=. S=. S=.4 S=.5 S=.6 S=.7 N= S 1 Fig. 5. Overall heat transer (C) in terms o S or dierent S 5

7 Trans. Phenom. Nano Micro Scales, 5(1) -4, Winter - Spring 17 The optimal heat sources location S and S in term o chardson number and in Gr=1 and φ=.5 are indicated in Figure 6. As can be seen or low chardson number, two heat sources are placed in the upper hal o the wall and or high chardson numbers, optimal heat sources location are placed in middle part o the wall. Figure 7 shows the maximum overall heat transer or two separate heat sources in dierent chardson numbers and in Gr=1 and φ=.5. It is obvious that the maximum overall heat transer inside the cavity reduces by increasing the chardson number. 1.8 S,opt.6.4 D=. D=.5 N=.5.4 S1,opt.. N= D=.5 D= Fig. 6. Optimal location or two separate heat sources in terms o chardson number in Gr=1 and φ=.5 Table 4 Optimum heat source location (%S.opt ) and maximum overall heat transer ( C max ) or dierent chardson numbers and constant Grasho numbers. =.1 =1 =4 =1 %D =.5 C max =1.67 %S,opt =.9 C max =1.14 %S,opt =.8 C max =.9 %S,opt =.65 C max =.911 %S,opt =.55 Gr=1 %D =.1 C max =1.896 %S,opt =.85 C max =1.75 C max =1.49 %S,opt =.6 C max =1.18 %S,opt =.55 %D =. C max =.68 C max =1.895 %S,opt =.65 C max =1.718 %S,opt =.55 C max =1.671 %S,opt =.45 %D =.5 C max =1.965 %S,opt =.9 C max =1.67 %S,opt =.9 C max =1.1 %S,opt =.85 C max =1.14 %S,opt =.8 Gr=1 %D =.1 C max =.771 %S,opt =.85 C max =1.896 %S,opt =.85 C max =1.5 %S,opt =.8 C max =1.75 %D =. C max =.78 C max =.68 C max =.18 %S,opt =.7 C max =1.895 %S,opt =.65 %D =.5 C max =.91 %S,opt =.95 C max =1.965 %S,opt =.9 C max =1.588 %S,opt =.9 C max =1.67 %S,opt =.9 Gr=1 4 %D =.1 C max =4.157 C max =.771 C max =.16 C max =1.896 %S,opt =.9 %D =. C max =5.76 %S,opt =.8 %S,opt =.85 C max = %S,opt =.85 C max =.5 %S,opt =.85 C max =.68 The values obtained or optimum two heat sources locations and the maximum overall heat transer in dierent chardson numbers.1, 1, 4 and 1, constant Grasho numbers 1, 1 and 1 4 and or three dierent heat source lengths are presented in Table 4. In all cases, the volume raction is considered.5. In Figure 8, the maximum overall heat transer in terms o the Grasho number is illustrated in constant chardson number. As a result, by increasing the Grasho number, heat transer and the maximum overall heat transer inside the cavity increases. Augmentation intensity is greater or larger Grasho numbers C max D=. N= D= Fig. 7. Maximum overall heat transer or dierent values o chardson number and in Gr=1 and φ=.5 6

8 A. A Abbasian Arani et al. 6 5 x N= = 1 4 C max D=. D= Gr Fig. 8. Maximum overall heat transer in terms o the Grasho number in constant chardson number As shown in Figure 8, by increasing the Grasho number, optimal heat sources location is migrated toward the ending corner o the let wall and the heat sources become close to each other. This subject is shown well in Table 5. Table 5 Optimum heat source location (%S.opt ) and maximum overall heat transer ( C max ) or dierent chardson numbers and constant Grasho numbers. =.1 =1 =4 =1 %D =.5 C max =.11 %S,opt =.65 %S 1,opt =. C max =1.8 %S,opt =.5 %S 1,opt =.4 C max =1.7 %S,opt =.5 %S 1,opt =.45 C max =1.677 %S,opt =. %S 1,opt =.45 Gr=1 %D =.1 C max =.879 %S,opt =.55 %S 1,opt =. C max =.9 %S,opt =. %S 1,opt =.5 C max =.59 %S,opt =. %S 1,opt =.4 C max =. %S,opt =. %S 1,opt =.4 %D =. C max =.848 %S,opt =.4 %S 1,opt =.15 C max =. %S,opt =. %S 1,opt =. C max =.19 %S,opt =.15 %S 1,opt =. C max =.99 %S,opt =.15 %S 1,opt =. %D =.5 C max =.9 %S 1,opt =.1 C max =.1 %S,opt =.6 %S 1,opt =.5 C max =1.886 %S,opt =.45 %S 1,opt =.5 C max =1.8 %S,opt =.5 %S 1,opt =.4 Gr=1 %D =.1 C max =.879 %S,opt =.65 %S 1,opt =.1 C max =.879 %S,opt =.55 %S 1,opt =. C max =.57 %S,opt =.4 %S 1,opt =. C max =.99 %S,opt =. %S 1,opt =.5 %D =. C max =5.1 %S,opt =. 5 %S 1,opt =.1 C max =.848 %S,opt =.4 %S 1,opt =.15 C max =.415 %S,opt =.5 %S 1,opt =.5 C max =. %S,opt =. %S 1,opt =. %D =.5 C max =4.167 %S,opt =. 85 %S 1,opt =.1 C max =.9 %S,opt =. 75 %S 1,opt =.15 C max =.15 %S,opt =. 7 %S 1,opt =.15 C max =.1 %S,opt =. 6 %S 1,opt =.5 Gr=1 4 %D =.1 C max =5.74 %S,opt =. 75 %S 1,opt =.5 C max =.879 %S,opt =.65 %S 1,opt =.1 C max =.948 %S,opt =. 55 %S 1,opt =. C max =.68 %S,opt =. 55 %S 1,opt =. %D =. C max =7.9 %S,opt =. 6 %S 1,opt =. C max =4.916 %S,opt =. 5 %S 1,opt =.1 C max =.689 %S,opt =. 45 %S 1,opt =.1 C max =.5 %S,opt =. 4 %S 1,opt =.15 According to Table 5, the maximum overall heat transer or a heat source with length.5 and Gr=1 is C max =

9 Trans. Phenom. Nano Micro Scales, 5(1) -4, Winter - Spring 17 It is clear that with an increase o 1% heat source length, the maximum overall heat transer only 8% increases and becomes C max = The important aspect o the maximization o global perormance is observed when the total heated area is held constant or in other words ND = cte. Eect o the length and number o heat sources on overall heat transer Figure 9 shows Maximum overall heat transer or ND =cte in dierent chardson numbers and constant Grasho numbers 1 and 1. In general and based on Figure 9, the maximum overall heat transer increases in a regular pattern or dierent chardson numbers while the number o heat sources increases in a ixed total length. This increasing is more or high chardson numbers than low ones. But the striing thing is the dierence in the augmentation intensity o maximum overall heat transer or Grasho 1 and 1 in which by increasing the Grasho number, discrepancy between C max and C max or ND =.1,. becomes less. Figure 1 shows the overall heat transer in terms o the chardson numbers and in dierent volume ractions (.1,.,.5 and.9).. C max C max ND =.1 ND =.1 C.8 max Cmax.6 ND =. ND =. 1.8 C max.4 Cmax Gr= C max 1.8 Cmax ND =.1 Cmax Cmax.8 4 Cmax Cmax RI ND =. Gr=1.4 Cmax Cmax.5 C max C max Fig. 9. Maximum overall heat transer or ND =cte in dierent chardson numbers and constant Grasho numbers 1 and 1.4. N= 1 C max Fig. 1.Overall heat transer Changes in dierent the volume raction o nanoparticles 8

10 A. A Abbasian Arani et al. Constant heat source length is.1. As shown in this igure, heat transer increases by increasing nanoparticles volume ractions and this increase is independent o the chardson number. Streamlines Eect o the chardson number and Grasho number on stream lines and isotherm In Figures 11 and 1, stream lines and isothermal lines are depicted or constant nanoparticles volume ractions φ=.5 and equal heat source length and location. Isotherm lines = ψ max =.16 θ max = = ψ max =.119 θ max = = ψ max =.186 θ max =.186 Fig. 11. Streamlines and isothermal lines in dierent values o chardson number and in constant Gr= As illustrated in these igures, by reducing the chardson number, vortex center moves toward the upper right corner and the lines agglomeration in this area increases and the small size counterclocwise vortex which is ormed in the starting corner, becomes larger. Also by augmentation the Grasho number, vortex in the middle o the cavity becomes larger and transmits to the center o cavity. This subject represents an increase o luid velocity near the walls. In addition by increasing the Grasho number, small counterclocwise vortices are ormed in the bottom corner o the cavity in which the size o the vortices is larger on the right side o the cavity. Also stream lines density near the suraces rise and thus the mass low rate passing near the surace, increases. In Gr=1 and =1 elements clocwise rotation is wea and the isothermal lines will become vertical. This issue represents the increasing o conduction heat transer in the cavity. It should be noted that in general, by augmentation the chardson number, isothermal lines with the same values approach to each other. Also when the chardson number decreases, the 9

11 Trans. Phenom. Nano Micro Scales, 5(1) -4, Winter - Spring 17 isothermal lines will be near to the horizontal state and the isothermal lines agglomeration besides the heat sources and cold wall increases. Thus the luid which is in more contact to the walls causes enhancement in heat transer and reduces the maximum heat source temperature. Streamlines By increasing the Grasho number due to strengthen o the buoyancy orce, the stream unction increases and the maximum temperature will be reduced. Isotherm lines = ψ max =.165 θ max =.5 = ψ max =.1899 θ max = = ψ max =.1441 θ max =.191 Fig. 1. Streamlines and isothermal lines in dierent values o chardson number and in constant Gr=1 4. Comparison o the average Nusselt number or two models o nanoluids In this section the average Nusselt number compares or two dierent models; variable properties and constant properties. In the irst case the properties o the water-al O varies with temperature, Khanaer and Vaai model [4] according to equations 1, 1 and 1 has been used. In the second case, the nanoluid properties are considered constant, Maxwell model [41] according to equation 11 and Brinman model[48] according to equation 9 have been used. As can be seen in Figure 1, in Maxwell and Brinman models, by increasing the volume raction, the average Nusselt number in a regular basis with constant slope increases but in Khanaer and Vaai model [4], the 4

12 A. A Abbasian Arani et al. average Nusselt number decreases by increasing the volume raction. It is noteworthy that or both models, the average Nusselt number increases by decreasing the chardson number which increases the heat transer rate inside the cavity. For =1, the average Nusselt numbers or the Khanaer and Vaai [4] model is less than Maxwell [41] and Brinman [48] models and or =.1 this amount is greater. In =1, the values which are obtained rom these three models are almost equal but by increasing the volume raction, the average Nusselt number in Maxwell [41] and Brinman [48] model is less than Khanaer and Vaai [] model. 4 Nuavg Fig. 1.Comparison o the average Nusselt number or constant properties model (Maxwell and Brinman) and variable properties (Khanaer and Vaai model) in dierent chardson numbers and nanoparticles volume ractions. Conclusion =.1 = 1 = 1 N= 1 Khanaer [4] MG [41] & Brinman [48] In the present study, the optimum heat sources locations with constant heat lux on the wall o a lid-driven square cavity containing water-al O nanoluids with variable properties are investigated numerically and the ollowing results are obtained: When the chardson number increases, the natural convection heat transer become stronger, hence the optimal heat source location rom the top wall moves to the midsection. In high Grasho numbers, range o optimum location is in upper part o the wall and near to the lid-driven that or dierent chardson numbers, the optimal location change is negligible but in low Grasho number these changing are intensiied and oriented toward the center o the wall. When the heat source is located on the bottom o the wall the overall heat transer changes are very small or dierent chardson numbers. By increasing nanoparticles volume ractions, the average Nusselt number on heat sources or both constant properties model (Maxwell and Reerences Brinman) and variable properties (Khanaer and Vaai model) increases. The enhancement intensity or Maxwell and Brinman model is constant but or Khanaer and Vaai model, decreases by increasing nanoparticles volume raction. [1] T. Basa, S. Roy, P.K. Sharma, I. Pop: Analysis o mixed convection lows within a square cavity with uniorm and non-uniorm heating o bottom wall, International Journal o Thermal Science 48 (9) [] G. Guo, M.A.R. Shari: Mixed convection in rectangular cavities aspect ratios with moving isothermal sidewalls and constant lux heat source on the bottom wall, International Journal o Thermal Science 4 (4) [] A. Fattahi, M. Alizadeh: Numerical Investigation o Double- Diusive Mixed Convective Flow in a Lid- Driven Enclosure Filled with AlO-Water Nanoluid, Transport Phenomena in Nano and Micro Scales (14) [4] A. Zare Ghadi, M. Sadegh Valipour: Numerical Study o Hydro-Magnetic Nanoluid Mixed Convection in a Square Lid-Driven Cavity Heated From Top and Cooled From Bottom, Transport Phenomena in Nano and Micro Scales (14) 9-4. [5] B. Jaarian, M. Hajipour, R. Khademi: Conjugate Heat Transer o MHD non-darcy Mixed Convection Flow o a Nanoluid over a Vertical Slender Hollow Cylinder Embedded in Porous Media, Transport Phenomena in Nano and Micro Scales 4 (16) 1-1. [6] H. R. Ehteram, A. A. Abbasian Arani, G. A. Sheihzadeh, A. Aghaei, A. R. Malihi: The eect o various conductivity and viscosity models considering Brownian motion on nanoluids mixed convection low and heat transer, Transport Phenomena in Nano and Micro Scales 4 (16) [7] F. Vahidinia, M. Rahmdel: Turbulent Mixed Convection o a Nanoluid in a Horizontal Circular Tube with Non-Uniorm Wall Heat Flux Using a Two-Phase Approach, Transport Phenomena in Nano and Micro Scales (15) [8] A. Aghaei, H. Khorasanizadeh, G. Sheihzadeh, M. Abbaszadeh: Numerical study o magnetic ield on mixed convection and entropy generation o nanoluid in a trapezoidal enclosure, Journal o Magnetism and Magnetic Materials 4 (16) [9] A. Rahmati, A. R. Ronabadi, M. Abbaszadeh: Numerical simulation o mixed convection heat 41

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