NUMERICAL SIMULATION AND PARAMETRIC STUDY OF LAMINAR MIXED CONVECTION NANOFLUID FLOW IN FLAT TUBES USING TWO PHASE MIXTURE MODEL

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1 Saikhani, H., et al.: Nuerical Siulation and Paraetric Study o THERMAL SCIENCE, Year 2016, Vol. 20, No. 2, pp NUMERICAL SIMULATION AND PARAMETRIC STUDY OF LAMINAR MIXED CONVECTION NANOFLUID FLOW IN FLAT TUBES USING TWO PHASE MIXTURE MODEL by Haed SAFIKHANI a, Abbas ABBASSI a *, and Mohaad KALTEH b a Departent o Mechanical Engineering, Airkabir University o Technology, Tehran, Iran b Departent o Mechanical Engineering, University o Guilan, Rasht, Iran Original scientiic paper DOI: /TSCI S In this article, the lainar ixed convection o Al 2 O 3 -water nanoluid low in a horizontal lat tube has been nuerically siulated. The two-phase ixture odel has been eployed to solve the nanoluid low, and constant heat lux has been considered as the wall boundary condition. The eects o dierent and iportant paraeters such as the Reynolds nuber, Grasho nuber, nanoparticles volue raction, and nanoparticle diaeter on the theral and hydrodynaic perorances o nanoluid low have been analyzed. The results o nuerical siulation were copared with siilar existing data and good agreeent is observed between the. It will be deonstrated that the Nusselt nuber and the riction actor are dierent or each o the upper, lower, let, and right walls o the lat tube. The increase o Reynolds and Grasho nubers, nanoparticles volue raction, and the reduction o nanoparticle diaeter lead to the increase o Nusselt nuber. Siilarly, the increase o Reynolds nuber and nanoparticles volue raction results in the increase o riction actor. Thereore, the best way to increase the aount o heat transer in lat tubes using nanoluids is to increase the Grasho nuber and reduce the nanoparticle diaeter. Key words: ixed convection, nanoluid, lat tube, paraetric study, ixture odel Introduction The use o nanoluids is one o the ost eective echaniss o increasing the aount o heat transer in heat exchangers. The use o lat tubes, in which the luid low has a lower theral resistance, is another way o iproving the rate o heat transer in tubes. The subject o the present paper is cobining o the two entioned ethods or increase o heat transer and paraetric study o theral and hydrodynaic perorances o low ield. The word nanoluid reers to a ixture in which solid particles o nanosize (generally less than 100 n) are added to a base luid and cause the increase o heat transer in that ixture. Dierent echaniss are presented to explain this enhanceent in heat transer. Xuan and Li [1] and Xuan and Roetzel [2] have identiied two reasons o iproved heat transer by nanoluids: the increased theral dispersion due to the chaotic oveent o nanoparticles that accelerates energy exchanges in the luid and the enhanced theral conductivity o nanoluids [3]. Siilarly, Keblinski et al. [4] have studied our possible echaniss that contribute to the increase in nanoluid heat transer: Brownian otion o the nanoparticles, o- * Corresponding author; e-ail: abbasi@aut.ac.ir

2 Saikhani, H., et al.: Nuerical Siulation and Paraetric Study o 416 THERMAL SCIENCE, Year 2016, Vol. 20, No. 2, pp lecular-level layering o the base luid/nanoparticle interace, heat transport in the nanoparticles, and nanoparticles clustering. Due to the any advantages o using nanoluids, researchers have conducted nuerous experiental and nuerical studies on nanoluids in recent years [5-7]. Kuar Das et al. [5] experientally investigated the eects o dierent paraeters (e. g., teperature, nanoparticle volue raction, etc.) on the theral conductivity o nanoluids. They ultiately presented a relation or theral conductivity o nanoluids as a unction o teperature, nanoparticle volue raction, etc. In addition to costly experiental studies, the nuerical siulation o nanoluids using CFD techniques is another eective approach in analyzing the perorance o nanoluids [8, 9]. In general, or the nuerical siulation o nanoluids, there are two ethods called the single-phase and two-phase, with the twophase ethod being uch ore exact [10]. The two-phase ethod itsel has dierent categories, including the Eulerian-Eulerian ethod, ixture ethod, etc. Using the three ethods o two-phase Eulerian-Eulerian, two-phase ixture, and single-phase hoogeneous, Loti et al. [11] siulated the Al 2 O 3 -water nanoluid low in circular tubes. By coparing the siulation results with the experiental data, they cae to the conclusion that the two-phase ixture ethod is the ost exact ethod aong the existing approaches. In this article also, the twophase ixture ethod is used or the nuerical siulation o nanoluid low in lat tubes. Another eective ethod or increasing the aount o heat transer in tubes is the use o lat tubes instead o circular ones. In these tubes, the ratio o cross-sectional area to lateral surace area is less than that o circular tubes, and this actor reduces the theral resistance o luid low and thus increases the aount o heat transer. O course, next to increasing the degree o heat transer, which is a useul aspect, the use o lat tubes also causes the pressure drop and riction actor to increase in these tubes, which is a negative point. Thereore, the behaviors o heat transer and riction in tubes should be investigated siultaneously. Based on our knowledge, up to now, no nuerical siulation o two-phase nanoluid low in lat tubes has been perored. Altogether, ew research works have been carried out on lat tubes, which ost o the have concerned the ultiphase lows [12-14] and very ew o the have dealt with nanoluids [15, 16]. Razi et al. [15] considered CuO-oil nanoluid experientally in various lat tubes and inally presented relations or Nusselt nuber and pressure drop o nanoluid low in horizontal lat tubes. Vajjha et al. [16] investigated nanoluid low in a lat tube o an autoobile radiator using the single phase nuerical ethod. They used convection heat transer coeicient or the wall boundary condition and inally presented the correlation or local Nusselt nuber and riction actor o the autoobile lat tube. In this article, the siultaneous use o nanoluids and lat tubes to increase the heat transer in heat exchangers has been nuerically investigated by eans o the two-phase ixture ethod. The eects o dierent and iportant paraeters including the Reynolds nuber, Grasho nuber, nanoparticle volue raction (ϕ ), and nanoparticle diaeter (d p ) on the theral and hydrodynaic perorances o the nanoluid have been evaluated. Matheatical odeling Figure 1. Geoetry cross-section o a lat tube Geoetry Figure 1 shows the geoetry cross-section o the considered proble in this paper. The coputation doain is a straight horizontal lat tube with length o L, width and height o W and H, respectively, at the lat parts.

3 Saikhani, H., et al.: Nuerical Siulation and Paraetric Study o THERMAL SCIENCE, Year 2016, Vol. 20, No. 2, pp Mixture odel In the present study, nuerical siulation o nanoluid low is perored using two- -phase ixture odel. The ixture odel is a two phase nuerical ethod that assues local equilibriu over short spatial length scales. The two phases are treated to be interpenetrating continua, eaning that each phase has its own velocity vector ield, and within a given control volue there is a certain raction o each phase. Instead o utilizing the governing equations o each phase separately, it solves the continuity, oentu and energy equations or the ixture, and the volue raction equation or the secondary phases, as well as algebraic expressions or the relative velocities. The equations or the steady-state conditions and ean low are: continuity equation ( ρ V ) = 0 (1) oentu equation [17] n ( rvv ) = P+ ( V ) + ϕρ k kvdr, kvdr, k ρ,iβg( T Ti ) (2) k = 1 energy equation [17] volue raction [17] + = n ϕkvk( ρkhk P) ( k T) (3) k = 1 ( ϕ r V ) = ( ϕ r V ) (4) P P P P dr,p where V is the ass average velocity and deined: V = n k = 1 ϕρv ρ k k k In eq. (2), V dr,k is the drit velocity or the secondary phase k, i. e. the nanoparticles in the present paper and deined: dr, k k (5) V = V V (6) The slip velocity (relative velocity) is deined as the velocity o nanoparticles relative to the velocity o base luid: Vp = Vp V (7) The relation between drit velocity and relative velocity is: V = V ϕr V (8) n k k dr,p p k k = 1 r The relative velocity and drag unction are calculated using Manninen et al. [18] and Schiller and Nauann [19] relations, respectively:

4 Saikhani, H., et al.: Nuerical Siulation and Paraetric Study o 418 THERMAL SCIENCE, Year 2016, Vol. 20, No. 2, pp V p 2 Pdp P r ( r r ) = a (9) 18 r drag P drag ReP or ReP 1000 = ReP or ReP > 1000 (10) The acceleration a in eq. (9) is: Nanoluid ixture properties a = g ( V ) V (11) The ixture properties or Al 2 O 3 -water nanoluid are calculated based on expressions: density [20] speciic heat capacity [2] dynaic viscosity [21] ρ = ϕρ + (1 ϕ) ρ (12) p ( ρc ) = ϕρ ( C ) + (1 ϕ)( ρc ) (13) p p p p 2 pvd B p ρ = + (14) 72Cd where V B and δ are Brownian velocity o nanoparticles and distance between particles, respectively, which can be calculated ro: The C in eq. (14) is deined: where C 1, C 2, C 3, and C 4 are given: V B 1 18kT B = (15) d πρ d d p p p π 6ϕ d = 3 p (16) ( C1dp + C2) ϕ + ( C3dp + C4) C = (17) µ C = , C = , C = , C = (18) theral conductivity [22] where Pr and Re are deined: d kp = ϕ Pr Re p k k d k (19)

5 Saikhani, H., et al.: Nuerical Siulation and Paraetric Study o THERMAL SCIENCE, Year 2016, Vol. 20, No. 2, pp Re Pr η = (20) rα B 2 3π ρk T = (21) ηλ where λ is the ean ree path o water olecular (λ = 0.17 n), k B the Boltzann constant (k B = J/K), and η has been calculated by the equation: B 5 η = A10 T C, A = , B = 247.8, C = 140 (22) theral expansion coeicient [23] 1 β p 1 β = + β (1 ϕρ ) β ρ p 1 ϕ + 1+ ϕρ p 1 ϕ ρ (23) Boundary conditions For nuerical siulation, the equations o previous sections should be solved subject to the ollowing boundary conditions: tubes inlet luid-wall interace V = V, V = V = 0 (24a), z i, x, y T = T i (24b) ϕ = ϕ i (24c) V = V = V = (25a), x, y, z 0 q = k w T n w (25b) tubes outlet: zero gradient is applied to hydrodynaic variables and constant gradient is applied to teperature [24, 25]: Nuerical ethods V z d dz = 0 (26a) dt P q = cte = (26b) dz A ρuc The nuerical siulation is perored using the inite volue ethod by FORTRAN prograing language. A second order upwind ethod is used or the convec- p

6 Saikhani, H., et al.: Nuerical Siulation and Paraetric Study o 420 THERMAL SCIENCE, Year 2016, Vol. 20, No. 2, pp tive and diusive ters and the SIMPLE algorith is eployed to solve the coupling between the velocity and pressure ields. For grid generation, irst, the syste boundaries are deterined and then by interpolating these boundaries, the inner points are generated. For this purpose, the lower and upper boundaries have been considered as r( L ζ, ηin ) and r U ( x, ηax ), respectively. In this case, a linear one directional interpolation or inding the inner points o the ield is deined: r( x, η) = (1 η)r ( ζ, η ) + ηr ( x, η ) (27) L ax in η ηin η = η η In the generated grid, because o the large hydrodynaic and theral gradients, an exponential unction is used near the walls and also at the tube inlet. This unction is expressed: where, β is the clustering ratio. 1 η in U ax 1 α β + 1 ( η) = 1 + β, α = 1 η 1 + α β 1 (28) (29) Figure 2. Grid independency test: (a) θ distribution at x-direction, (b) θ distribution at y-direction, (c) axial velocity at center line, and (d) local Nusselt nuber

7 Saikhani, H., et al.: Nuerical Siulation and Paraetric Study o THERMAL SCIENCE, Year 2016, Vol. 20, No. 2, pp To ake sure that the results are independent o the generated grid, our dierent grids with , , , and eleents along the x-, y-, and z-axes have been generated. Dierent paraeters such as diensionless teperature (θ) along the x and y, diensionless axial velocity along the centerline, and the local Nusselt nuber have been calculated or these our grids and their values have been copared with one another in ig. 2. As this igure shows, all the our generated grids successully pass the grid independency test or the teperature along the x-axis and axial velocity, but the grid ails the grid independency test or the other two paraeters. Thereore, the grid will be used in all the siulations. Figure 3 shows the structure non-unior generated grid or the lat tube o present paper. As shown in this igure, the grids are iner near the tubes entrance and near the wall where the velocity and teperature gradients are high. Validations o the nuerical siulations To attain the conidence about the CFD re- Figure 3. Structure non-unior grid sults, it is necessary to copare the siulation generation or the lat tube results with the available data. Figure 4 copares the Nu and C Re vs. lattening, predicted in the present study with analytical data o Shah and London [26] and nuerical siulations o Vajjha et al. [16]. Figure 4. Coparison o: (a) Nu and (b) C Re o the present nuerical calculation with the Shah and London [26] data and nuerical results o Vajjha et al. [16] at Re = 100 The siulations o Vajjha et al. [16] are perored or speciied convection coeicient wall boundary condition whereas present siulations and Shah and London [26] data are

8 Saikhani, H., et al.: Nuerical Siulation and Paraetric Study o 422 THERMAL SCIENCE, Year 2016, Vol. 20, No. 2, pp related to constant heat lux wall boundary condition. As evident ro ig. 4 the present CFD siulations agree well with the available nuerical and analytical data. Results and discussion The lainar ixed convection o Al 2 O 3 -water nanoluid low in a horizontal lat tube has been siulated by using the two-phase ixture odel, and the results have been presented in this section. The Nusselt nuber is one o the ost iportant theral paraeters o the low ield inside a tube and is deined: where hdh Nu = (30) k h = T In ig. 5, the local Nusselt nuber has been presented or each o the upper, lower, let, and right walls o the lat tube. As this igure shows, the Nusselt nuber values vary or dierent walls o the tube, and the lower wall has the highest Nusselt nuber, ollowed by the upper wall. Also, the values o Nusselt nuber or the let and right walls are equal. The reason or a higher Nusselt nuber at the lower wall is that due to the lainar low regie in this region, gravity plays a signiicant role and causes the increase o the Nusselt nuber at the lower wall o tube. Fro now on, to present the Nusselt nuber under dierent conditions, the Nusselt nuber values are averaged along the perieter and the ean values are presented or various walls. In order to study the eects o iportant low paraeters w q T on the heat transer value in lat tubes, the local heat transer coeicient has been presented in ig. 6 or dierent Reynolds and Grasho nubers, nanoparticle volue ractions (φ), and nanoparticle diaeters (d p ). As is illustrated in this igure, the increase o Reynolds nuber, Grasho nuber, φ, and the reduction o d p causes the heat transer coeicient to increase. Although, it sees that the eect o Grasho nuber and φ on the increase o heat transer coeicient is higher than that o the other two paraeters. Another iportant paraeter or the low ield inside a heat exchanger tubes is the value o pressure drop, which has a direct relationship with the riction actor (C ) and wall shear stress (τ w ) with the expression: C τ b w 2 ui (31) Figure 5. Local Nusselt nuber at top, botto, let and right walls o the lat tube using Al 2 O 3 -water nanoluid, Gr = = (32) 1 ρ 2

9 Saikhani, H., et al.: Nuerical Siulation and Paraetric Study o THERMAL SCIENCE, Year 2016, Vol. 20, No. 2, pp Figure 6. Eects o changing: (a) Reynolds nuber, (b) Grasho nuber, (c) and (d) d p on peripherally averaged local heat transer coeicient where τ u w = n w (33) The C is shown in ig. 7 or each wall o the lat tube. Siilar to the Nusselt nuber, C also has the highest value at the lower wall, ollowed by the upper wall. Eects o Reynolds nuber, Grasho nuber, φ, and d p changes on the local wall shear stress averaged along the perieter have been illustrated in ig. 8. According to this igure, the increase o Reynolds nuber and φ leads to the increase o wall stress; while the changes o Grasho nuber and d p have no ipact on wall stress. Figure 7. Local C at top, botto, let, and right walls o the lat tube According to igs. 6 and 8, although the increase o Reynolds nuber and φ causes the increase o the h, but they increase the riction actor as well. While the increase o Grasho nuber and the reduction o d p leads to the increase o h without any increase in the

10 424 Saikhani, H., et al.: Nuerical Siulation and Paraetric Study o THERMAL SCIENCE, Year 2016, Vol. 20, No. 2, pp riction actor thereore, the best way o increasing the heat transer o nanoluid low in lat tubes is to increase the Grasho nuber and decrease the dp. Figure 8. Eects o changing; (a) Reynolds nuber, (b) Grasho nuber, (c) and (d) on peripherally averaged local wall shear stress Figure 9. Eects o changing dp and Grasho nuber on secondary low vectors in the lat tube

11 Saikhani, H., et al.: Nuerical Siulation and Paraetric Study o THERMAL SCIENCE, Year 2016, Vol. 20, No. 2, pp In this article, the eect o changing the low paraeters on the secondary vectors has also been investigated. Secondary vectors play a very iportant role in the theral and hydrodynaic characteristics o low. These vectors are ored in a tube due to the existence o a buoyancy orce and dierent densities o nanoluid in the upper and lower regions o tube. The eects o changes o dp, Grasho nuber, φ, and Reynolds nuber on the quality o secondary vectors have been reviewed in igs. 9 and 10. It is observed that under all conditions, two vortices are created inside the tube. Figures 9 and 10 also indicate that the changes o dp and Reynolds nuber have no eect on secondary vectors. Whereas the changes o Grasho nuber and φ have considerable eects on the quality o secondary vectors, and iproving the entioned paraeters leads to the strengthening o these secondary vectors. Figure 10. Eects o changing φ and Reynolds nuber on secondary low vectors in the lat tube Knowing the details o velocity proiles in the lat tubes, helps better understanding the nature o the low ield. Eects o changing dp, on velocity proiles at x-, y-, and z- directions are shown in ig. 11. As shown the changes o dp, do not lead to change o velocity ield and considering the previous discussions about C and Nusselt nuber it can be concluded that the theral low ield is ore sensitive than the hydrodynaic low ield with respect to changing the paraeters such as dp and Grasho nuber. Since in this article, the low o nanoluid has been siulated by eans o a twophase ethod, the distribution o nanoparticles in the lat tube is known. The distribution o nanoparticles along the vertical direction in the lat tube has been illustrated in ig. 12 or various paraeters. As is shown in this igure, the changes o Reynolds nuber and Grasho nuber have a negligible eect on the distribution o nanoparticles. But the increase o nanoparticle diaeter causes severe non-uniority in the distribution o nanoparticles. Conclusions In this article, the lainar ixed convection o Al2O3-water nanoluid low in a horizontal lat tube was nuerically siulated. The two-phase ixture odel was used to solve the nanoluid low. Constant heat lux was assued as the wall boundary condition. The eects o dierent and iportant paraeters such as Reynolds nuber, Grasho nuber, φ,, and dp on the theral and hydrodynaic perorances o lat tubes containing nanoluids were discussed, and ultiately the ollowing results were obtained:

12 426 Saikhani, H., et al.: Nuerical Siulation and Paraetric Study o THERMAL SCIENCE, Year 2016, Vol. 20, No. 2, pp Figure 11. Eect o changing dp on diensionless velocity proile at: (a) horizontal, (b) vertical, and (c) axial directions Figure 12. Eects o changing: (a) Reynolds nuber, (b) Grasho nuber, and (c) on vertical nanoparticle distributions The values o Nusselt nuber and C are not equal or each wall o the lat tube, and the lat tube s lower surace has higher values o Nusselt nuber and C relative to the other suraces. The increase o Reynolds nuber, Grasho nuber, and φ, and the reduction o dp lead to increase o Nusselt nuber. Siilarly, increase o Reynolds nuber and φ leads to in-

13 Saikhani, H., et al.: Nuerical Siulation and Paraetric Study o THERMAL SCIENCE, Year 2016, Vol. 20, No. 2, pp crease o C, and the changes o Grasho nuber and d p have no eect on C. Thereore, the best way o increasing the heat transer o nanoluid low in lat tubes is to increase the Grasho nuber and decrease the d p. The increase o Grasho nuber and φ lead to the strengthening o secondary lows, and the changes o Reynolds nuber and d p in a ixed lainar low have negligible eects on secondary lows. The changes o Reynolds nuber and Grasho nuber does not have a signiicant eect on the distribution o nanoparticles. However, the changes o d p has a considerable ipact on the distribution o nanoparticles, and the increase o d p can lead to a severe nonuniority o nanoparticles in lat tubes. Noenclature a acceleration, [s 2 ] C constant in eq. (14), [ ] C skin riction coeicient, [ ] C p speciic heat, [Jkg 1 K 1 ] D h hydraulic diaeter o tubes, [] d p diaeter o nanoparticles, [] g gravitational acceleration, [s 2 ] H lat tube internal height, [] h local heat transer coeicient, [W 2 K 1 ] 4 2 Gr Grasho nuber ( = g βqd h/ kν), [ ] k theral conductivity, [W 1 K 1 ] k B Boltzann constant (= JK 1 ) L length o tubes, [] P pressure, [Pa] Pr Prandtl nuber ( = α/ ν), [ ] q" heat lux, [W 2 ] Re Reynolds nuber ( = VDh/ ν ), [ ] 2 Ri Richardson nuber ( = Gr/Re ), [ ] T teperature, [K] V velocity, [s 1 ] W width o lat area in tubes, [] Greek sybol α theral diusivity, [ 2 s 1 ] β voluetric expansion coeicient, [K 1 ] δ distance between particles, [] φ nanoparticles volue raction, [%] θ diensionless teperature [(=T T i )/(q"d h /k)], [ ] λ ean ree path o water olecular, [] μ dynaic viscosity, [Ns 2 ] ν kineatic viscosity, [ 2 s 1 ] ρ density, [kg 3 ] τ w wall shear stress, [Pa] Subscripts dr drit luid i inlet conditions k indices ixture P particle p nanoparticle phase w wall Reerences [1] Xuan, Y., Li, Q., Heat Transer Enhanceent o Nanoluids, International Journal o Heat and Fluid Flow, 21 (2000), 1, pp [2] Xuan, Y., Roetzel, W., Conceptions or Heat Transer Correlations o Nanoluids, International Journal o Heat and Mass Transer, 43 (2000), 19, pp [3] Choi, U. S., Enhancing Theral Conductivity o Fluid with Nanoparticles, in: D. A. Singiner, H. P. Wang (Eds.), Developent and Application o Non-Newtonian Flows, FED Vol. 231/MD Vol. 66, The Aerican Society o Mechanical Engineers, New York, N. Y., USA, 1995, pp [4] Keblinski, P., et al., Mechaniss o Heat Flow in Suspensions o Nano-Sized Particles (Nanoluid), International Journal o Heat and Mass Transer, 45 (2002), 4, pp [5] Kuar Das, S., et al., Teperature Dependence o Theral Conductivity Enhanceent or Nanoluids, Journal o Heat Transer, 125 (2003), 4, pp [6] Haghshenas Fard, M., et al., Nuerical and Experiental Investigation o Heat Transer o ZnO/Water Nanoluid in the Concentric Tube and Plate Heat Exchangers, Theral Science, 15 (2011), 1, pp [7] Mahoodi, M., Mixed Convection Inside Nanoluid Filled Rectangular Enclosures with Moving botto Wall, Theral Science, 15 (2011), 3, pp [8] Dinarvand, S., et al., Hootopy Analysis Method or Mixed Convective Boundary Layer Flow o a Nanoluid over a Vertical Circular Cylinder, Theral Science, 19 (2015), 2, pp

14 Saikhani, H., et al.: Nuerical Siulation and Paraetric Study o 428 THERMAL SCIENCE, Year 2016, Vol. 20, No. 2, pp [9] Abbasian Arani, A. A., et al., Free Convection o a Nanoluid in a Square Cavity with a Heat Source on the botto Wall and Partially Cooled ro Sides, Theral Science, 18 (2014), Suppl., 2, pp. S283-S300 [10] Kalteh, M., et al., Eulerian Eulerian Two-Phase Nuerical Siulation o Nanoluid Lainar Forced Convection in a Microchannel, International Journal o Heat and Fluid Flow, 32 (2011), 1, pp [11] Loti, R., et al., Nuerical Study o Forced Convective Heat Transer o Nanoluids: Coparison o Dierent Approaches, International Counication in Heat and Mass Transer, 37 (2010), 1, pp [12] Wilson, M. J., et al., Rerigerant Charge, Pressure Drop and Condensation Heat Transer in Flattened Tubes, International Journal o Rerigeration, 26 (2003), 4, pp [13] Quiben, J., et al., Flow Boiling in Horizontal Flattened Tubes: Part II Two-Phase Frictional Pressure Drop Results and Model, International Journal o Heat and Mass Transer, 52 (2009), 15-16, pp [14] Nasr, M., et al., Perorance Evaluation o Flattened Tube in Boiling Heat Transer Enhanceent and its Eect on Pressure Drop, International Counication in Heat and Mass Transer, 37 (2010), 4, pp [15] Razi, P., et al., Pressure Drop and Theral Characteristics o Cuo Base Oil Nanoluid Lainar Flow in Flattened Tubes under Constant Heat Flux, International Counication in Heat and Mass Transer, 38 (2011), 7, pp [16] Vajjha, R. S., et al., Nuerical Study o Fluid Dynaic and Heat Transer Perorance o Al 2 O 3 and Cuo Nanoluids in the Flat Tubes o a Radiator, International Journal o Heat and Fluid Flow, 31 (2010), 4, pp [17] Shariat, M., et al., Nuerical Study o Two Phase Lainar Mixed Convection Nanoluid in Elliptic Ducts, Applied Theral Engineering, 31 (2011), 14-15, pp [18] Manninen, M., et al., On the Mixture Model or Multiphase Flow, VTT Publications, Esppo, Finland, 1996 [19] Schiller, L., Nauann, A., A Drag Coeicient Correlation, Z. Ver Deutsch Ing, 77 (1935), p. 318 [20] Pak, B. C., Cho, Y. I., Hydrodynaic and Heat Transer Study o Dispersed Fluids with Subicron Metallic Oxide Particles, Experiental Heat Transer, 11 (1998), 2, pp [21] Masoui, N., et al., A New Model or Calculating the Eective Viscosity o Nanoluids, Journal o Applied Physics, 42 (2009), 5, pp [22] Chon, C. H., et al., Epirical Correlation Finding the Role o Teperature and Particle Size or Nanoluid (Al 2 O 3 ) Theral Conductivity Enhanceent, Journal o Applied Physics, 87 (2005), 15, pp [23] Khanaer, K., et al., Buoyancy Driven Heat Transer Enhanceent in a Two Diensional Enclosure Utilizing Nanoluids, International Journal o Heat and Mass Transer, 46 (2003), 19, pp [24] Bejan, A., Convective Heat Transer. John Wiley & Sons, New York, N. Y., USA, 2004 [25] Ebrahinia-Bajestan, E., et al., Nuerical Investigation o Eective Paraeters in Convective Heat Transer o Nanoluids Flowing under a Lainar Flow Regie, International Journal o Heat and Mass Transer, 54 (2011), 19-20, pp [26] Shah, R. K., London, A. L., Lainar Flow Forced Convection in Ducts, Acadeic Press, New York, N. Y., USA, 1978 Paper subitted: April 15, 2013 Paper revised: Septeber 14, 2013 Paper accepted: Deceber 3, 2013