Numerical study of nanofluids natural convection in a rectangular cavity including heated fins

Accepted Manuscript Numerical study of nanofluids natural convection in a rectangular cavity including heated fins

M. Hatami PII: DOI: Reference:

S0167-7322(17)30105-8 doi: 10.1016/j.molliq.2017.02.112 MOLLIQ 7025

To appear in:

Journal of Molecular Liquids

Received date: Revised date: Accepted date:

9 January 2017 11 February 2017 27 February 2017

Please cite this article as: M. Hatami , Numerical study of nanofluids natural convection in a rectangular cavity including heated fins. The address for the corresponding author was captured as affiliation for all authors. Please check if appropriate. Molliq(2017), doi: 10.1016/j.molliq.2017.02.112

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Numerical Study of Nanofluids Natural Convection in a Rectangular Cavity Including Heated Fins M. Hatamia,1 Department of Mechanical Engineering, Esfarayen University of Technology, Esfarayen, North Khorasan, Iran

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1

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Abstract

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In this paper, natural convection heat transfer of nanofluids in a rectangular cavity

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investigated while two heated fins are located in the cavity. The governing equations are solved by finite element method (FEM) using FlexPDE commercial package software. Water is considered as

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the base fluid and two nanoparticles (TiO2 and Al2O3) are added as the second phase or additives to base fluid. The problem is solved for different nanoparticles volume fraction and fin height to study

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the effect of these parameters on the local and average Nusselt numbers. Results show that

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maximum point of local Nusselts numbers (which occurs along the fins locations) for the TiO2 is larger than Al2O3 except when the φ=0.09. Also, TiO2 leads to more average Nusselt number as

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well as larger fins height in most cases.

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1. Introduction

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Keywords: Finned Cavity; Nanofluid; FlexPDE; Nusselt number; FEM

Extended surfaces or fins are devices to increase heat transfer in different industrial applications by increasing the surface in convection heat transfer. In many mechanical engineering applications, chemical engineering, energy, heat recoveries, surface studies, etc. fins have large applications [1], so it motivated the researchers to improve their accuracy by different ways such as 1

a Corresponding author, Tel/Fax:+98-915-718-6374

E-mails: [email protected], [email protected]

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experimentally, numerically or analytically. Not only fins improve the heat transfer mechanism, but also nanofluids can assist the heat transfer in industries which both of these two ways are considered in this literature review. Hatami and Ganji [2] improved the thermal efficiency of circular convective–radiative porous fins by defining different section shapes. Hatami et al. [3]

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studied the thermal performance of longitudinal and compared the results for two different

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materials, Si3N4 and Al. Based on their study, Ghasemi et al. [4] improved the fins efficiency by a

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realistic temperature-dependent thermal conductivity and heat generation. As an application of fins in numerical studies, Hatami and Ganji [5] used fin arrays in a micro-channel heat sink to increase

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the heat rejected from the channel to nanofluids. They used Least Square Method (LSM) as an

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analytical study for their modeling and Ghasemi et al. [6] make a validation of this study by other analytical methods presented by Hatami et al. [7-10] for fins heat transfer modeling.

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Another applicable ways for increasing the heat transfer is using the nanofluids as a working

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fluid for conveying the heat from sources. Nanofluids are made from a base fluid such as water for first phase and solid nanoparticles suspended in it as second phase. Haghshenas Fard et al. [11]

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compared the results of two-phase and single phase fluids in a circular tube, numerically. Also,

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Göktepe et al. [12] compared the single phase and two phase nanofluid modeling at the entrance of a uniformly heated tube and found higher accuracy for two-phase modeling. Mohyud-Din et al. [13]

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in an analytical study, considered the three dimensional heat and mass transfer with magnetic effects for the flow of a nanofluid between two parallel plates in a rotating system. Another three-dimensional flow of nanofluids study under the radiation has been analyzed by Hayat et al. [14] and Khan et al. [15]. They also computed and examined the effects of different parameters on the velocity, temperature, skin friction coefficient and Nusselt number of nanofluid flow. Hatami and Ganji [16] modeled the natural convection heat transfer of a non-Newtonian

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nanofluid flow between the parallel plates and Kefayati [17-20] simulated the natural and mixed convection of nanofluid in enclosures using an efficient numerical method called finite difference based Lattice Boltzmann Method (FDLBM). Also, Hatami et al. [21] analyzed the natural convection heat transfer of nanofluids in a circular-wavy cavity and optimized the cavity geometry

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using response surface methodology (RSM). Domairry and Hatami [22] analyzed the nanofluid

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treatment between the parallel plates by using an analytical method. Ahmadi et al. [23] used the

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nanofluids for cooling flat plate by analytical method and Fakour et al. [24] used the analytical methods as a powerful method for analyzing the micropolar fluid flow as Ghasemi et al. [25-26]

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used also the same analytical method. As mentioned above, fins are devices for increasing the heat

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transfer and are used in different shapes and applications such as in microchannel heat sink [27], longitudinal porous area [28], Circular porous shapes [29], semi-spherical shapes [30] and

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rectangular fins [31]. Ashorynejad and Hoseinpour [32] investigated the effect of nanofluids on the

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entropy generation in a porous cavity based on their previous study [33] using Lattice Boltzmann Method or LBM [34-36] to find the treatment of defined nanofluids in the natural convection heat

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transfer. In the present study, a combination of using fins and nanofluids is modeled to enhance the

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heat transfer in a rectangular cavity. Finite Element Method (FEM) is used by FlexPDE code to investigate the natural convection heat transfer in different situations. Nusselt number is reported

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for each case to can make a comparison between the reported case. 2. Problem description Fig. 1-a displays a schematic diagram of a rectangular cavity with two fins. The cavity is filled by water-based nanofluids containing Al2O3 or TiO2 nanoparticles (Table 1 is presented for the physical properties). The density of the nanofluid is approximated by the Boussinesq model and it is considered that water and nanoparticles are in thermal equilibrium and no slip occurs among 3

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them. The cavity contains a cold surface at the top wall and a two heated fins in the bottom wall. Other walls assumed to be insulated and no heat transfers from them. Fig. 1-b shows the sample generated mesh in starting the solution, due to FlexPDE ability the mesh sizes will be regenerated during the solution if its size is not suitable until to reach the convergence solution. The governing

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equations for steady state, laminar, natural convection inside the cavity form of Navier-Stokes and

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energy equation are taken from previous studies as:

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The continuity equation [21]

  2v  2v   2  2   g   nf T  TC   x y 

T T v   nf x y

  2T  2T   2  2 y   x

(4)

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u

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the energy equation

(3)

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 v v  p  v     nf y  y  x

nf  u

(2)

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  2u  2u   2 2  x y 

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The momentum equations in x and y directions:

 u u  p nf  u  v     nf y  x  x

(1)

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u v  0 x y

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The density of the nanofluid, nf , depends on the solid volume fraction,  and can be expressed as

nf  1    f  s

(5)

To model the effective viscosity of the nanofluid, Pak and Cho correlation from their experiments is used and can be expressed as follows

nf   f 1  39.11  533.9 2 

(6)

Thermal expansion coefficient(  ) can be expressed as [21]

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 nf  1      f    s

(7)

The effective thermal diffusivity is

 nf 

(8)

knf

 C 

p nf

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The effective thermal conductivity of the nanofluid, is approximated using Maxwell-Garnetts

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model as [21]

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  ks  2k f   2  k f  ks   knf  k f     ks  2k f     k f  ks  

 C 

p nf

 1     C p     C p  f

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The heat capacitance of the nanofluid s

(9)

(10)

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It is convenient to solve the governing equations in non-dimensional forms and hence following

x L y Y L uL U f vL V f

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scales are used to get the non-dimensional governing equations:

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pL2 f v 2f T  Tc  Tw  Tc

P

(11)

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X

The non-dimensional governing parameters considered in the present simulation are as follows:

Pr 

(12)

vf

f

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Ra 

g  f L3 Tw  Tc  vf f

Using the above mentioned scale, the non-dimensional forms of continuity, momentum and energy equations:

U V  0 X Y

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(13)

f P vnf   2U  2U  U U V      X Y nf X v f  X 2 Y 2 

f P vnf   2V  2V  Ra 1      f     s V V V       X Y nf Y v f  X 2 Y 2  Pr nf  f

(15)

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   nf 1   2  2  V     X Y  f Pr  X 2 Y 2 

(16)

U

(14)

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U

 n

(17)

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Nuloc  

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The local Nusselt number on the upper wall can be expressed as:

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where n is the direction normal to the upper surface. For solution of the governing equations, for all solid boundaries U=V=0, while different temperature conditions are considered for

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boundaries as Fig.1-a. Ass seen in this figure, for top plate, due to colder conditions, θ=0 and for the bottom heated fins θ=1. The average Nusselt number on the upper wall is evaluated as: L

Nuave 

(18)

1 Nuloc   d L 0

Where L is the length of wall. 6

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3. Finite Element Method (FEM) A finite element method is characterized by a variational formulation, a discretization strategy, one or more solution algorithms and post-processing procedures. A typical work out of this method involves (1) dividing the domain of the problem into a collection of subdomains, and (2)

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systematically recombining all sets of element equations into a global system of equations for the

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final calculation. In the finite element method, the solution region is considered as built up of many small, interconnected sub-regions called finite elements. The solution of a general continuum

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problem by the finite element method always follows an orderly step-by-step process [21]. In this study FEM is applied based on FlexPDE commercial code which is fast and accurate code because

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it does not need to discretization the partial differential equations and solve them directly.

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Furthermore, this software has self-improvement on meshes until reach to convergence condition

4. Results and discussions

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defined by user.

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As describe above, the goal of this study is enhancement the heat transfer in a cavity using the

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combination of fins and nanofluids. After presenting the governing equation, they are solved by FEM using the FlexPDE package code and results are presented in this section. Fig. 2 shows the

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temperature profile for different nanoparticles volume fraction of Al2O3 –water nanofluid. As seen, thermal spread from the fins towards the upper cold surface is wider for larger nanoparticles volume fraction (φ=0.09). Fig. 3 demonstrates the stream lines for the cavity and it also appear that in larger volume fraction the vortex cavity is greater than other cases due to larger difference in the density of nanofluid. It must be noticed that this larger cavity is not due to larger nanoparticles because the problem is solved single phase and the solid phase motion is not analyzed seperately, so the difference is just due to larger density and viscosity of nanofluid. Fig. 4 and 5 are depicted 7

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for the velocity in X and Y directions, i.e. U and V, respectively. Because the stream line function is obtained from these velocities, the treatment of the contours is the same of streamlines. To see the effect of the fins heights on the heat transfer and flow conditions in natural conditions, Fig. 6 is depicted. When the fins heights are increased the heated surface will be greater and more heat

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transfers to the nanofluids as depicted in the figure. Also this lead to higher average temperature in

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all the domain as seen. For the streamlines, in larger fins, regions of the generated vortex will be

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four regions while in the small fins heights its just divided into two regions as is observable in the contours. To find the effect of nanoparticles type in this study, TiO2 and Al2O3 nanoparticles are

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added to the water separately and the results are presented in Fig. 7 for φ=0.03 which confirm that

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streamlines have not a large difference due to constant naoparticles volume fraction, while temperature profiles are different and for the TiO2 nanoparticles more higher temperatures are

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observed. To make a better comparison from the discussed topics, Figs. 8-10 are plotted for the

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local and average Nusselt number on the upper plate. Fig. 8 reveals that maximum point of local Nusselts numbers (which occurs along the fins locations) for the TiO2 is larger than Al2O3 except

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when the φ=0.09. This fact is also correct for the average Nusselt number as shown in Fig. 9. In

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lower nanoparticles volume fraction, TiO2 has more average Nusselt number while in larger volume fractions Al2O3 nanoparticle has larger Nusselts due to overcome the heat capacity and thermal

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conductivity effects to density effect on heat transfer mechanism. Finally, the average Nusselt Number versus fin heights are depicted in Fig. 10 for different nanoparticles and volume fractions. As described above, this figure also confirms that increasing the fins heights can enhance to increase the Nusselt number in the natural convection heat transfer. It can be understand from this figure that increasing the fin height (by change in vortexes and fluid flow pattern) can also change

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the treatment of φ in an especial case, so finding the effect of φ on heat transfe depends on the fins height and has not a constant results.

5. Conclusion

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In this paper, FlexPDE FEM numerical code have been successfully applied to find the

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solution of 2D modeling of natural convection heat transfer of nanofluids in a rectangular cavity

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containing two heated fins. Two different kind of nanoparticles are considered, TiO2 and Al2O3 to add to water as base fluid. In this study, effect on nanoparticles type, nanoparticles volume fraction

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and height of the fins on the local and average Nusselt number is investigated and it is found that

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TiO2 (especially in lower and moderate φ) in presence of lengthy fins can improve the heat transfer.

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References

[1] Hatami, M., D. D. Ganji, and M. Gorji-Bandpy. "A review of different heat exchangers designs

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Reviews 37 (2014): 168-181.

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for increasing the diesel exhaust waste heat recovery." Renewable and Sustainable Energy

[2] Hatami, M., and D. D. Ganji. "Thermal performance of circular convective–radiative porous fins

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with different section shapes and materials." Energy Conversion and Management 76 (2013): 185-

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193.

[3] Hatami, M., A. Hasanpour, and D. D. Ganji. "Heat transfer study through porous fins (Si 3 N 4 and AL) with temperature-dependent heat generation." Energy Conversion and Management 74 (2013): 9-16. [4] Ghasemi, Seiyed E., M. Hatami, and D. D. Ganji. "Thermal analysis of convective fin with temperature-dependent thermal conductivity and heat generation." Case Studies in Thermal Engineering 4 (2014): 1-8. 9

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[5] Hatami, M., and D. D. Ganji. "Thermal and flow analysis of microchannel heat sink (MCHS) cooled by Cu–water nanofluid using porous media approach and least square method." Energy Conversion and management 78 (2014): 347-358. [6] Ghasemi, S. E., Valipour, P., Hatami, M., & Ganji, D. D. (2014). Heat transfer study on solid

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method. Journal of Central South University, 21(12), 4592-4598

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and porous convective fins with temperature-dependent heat generation using efficient analytical

[7] M. Hatami, G. Domairry, Transient vertically motion of a soluble particle in a Newtonian fluid

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media, Powder Technology 253 (2014) 481-485.

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[8] M. Hatami, D.D. Ganji, Motion of a spherical particle on a rotating parabola using Lagrangian and high accuracy Multi-step Differential Transformation Method, Powder Technology 258 (2014)

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94-98.

ED

[9] M. Hatami, D.D. Ganji, Motion of a spherical particle in a fluid forced vortex by DQM and DTM, Particuology 16 (2014) 206-212.

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[10] A.S. Dogonchi, M. Hatami, G. Domairry, Motion analysis of a spherical solid particle in plane

CE

Couette Newtonian fluid flow, Powder Technology 274 (2015) 186-192. [11] M. Haghshenas Fard, M. Nasr Esfahany, M.R. Talaie, Numerical study of convective heat

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transfer of nanofluids in a circular tube two-phase model versus single-phase model, International Communications in Heat and Mass Transfer 37 (2010) 91–97. [12] Sinan Göktepe, Kunt Atalık, Hakan Ertürk, Comparison of single and two-phase models for nanofluid convection at the entrance of a uniformly heated tube, International Journal of Thermal Sciences 80 (2014) 83-92.

10

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[13] Syed Tauseef Mohyud-Din, Zulfiqar Ali Zaidi, Umar Khan, Naveed Ahmed, On heat and mass transfer analysis for the flow of a nanofluid between rotating parallel plates, Aerosp. Sci. Technol. 46 (2015) 514-522. [14] Tasawar Hayat, Maria Imtiaz, Ahmed Alsaedi, Marwan A. Kutbi, MHD three-dimensional

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flow of nanofluid with velocity slip and nonlinear thermal radiation, Journal of Magnetism and

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Magnetic Materials 396 (2015) 31–37

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[15] J.A. Khan, M. Mustafa, T. Hayat, A. Alsaedi, Three-dimensional flow of nanofluid over a nonlinearly stretching sheet: An application to solar energy, Int. J. Heat. Mass. Trans. 86 (2015) 158-

US

164.

AN

[16] M. Hatami, D.D. Ganji, Natural convection of sodium alginate (SA) non-Newtonian nanofluid flow between two vertical flat plates by analytical and numerical methods, Case Studies in Thermal

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Engineering 2 (2014) 14-22.

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[17] GH. R. Kefayati, FDLBM simulation of entropy generation in double diffusive natural convection of power-law fluids in an enclosure with Soret and Dufour effects, International Journal

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of Heat and Mass Transfer 89 (2015) 267–290.

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[18] GH. R. Kefayati., FDLBM simulation of mixed convection in a lid-driven cavity filled with non-Newtonian nanofluid in the presence of magnetic field, International Journal of Thermal

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Sciences 95 (2015) 29-46.

[19] GH. R. Kefayati, Mesoscopic simulation of mixed convection on non-Newtonian nanofluids in a two sided lid-driven enclosure, Advanced Powder Technology 26 (2015) 576-588. [20] GH. R. Kefayati, FDLBM simulation of entropy generation due to natural convection in an enclosure filled with non-Newtonian nanofluid, Powder Technology 273 (2015) 176– 190.

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[21] Hatami, M., D. Song, D. Jing., Optimization of a circular-wavy cavity filled by nanofluid under the natural convection heat transfer condition, International Journal of Heat and Mass Transfer 98 (2016): 758-767. [22] G. Domairry, M. Hatami, Squeezing Cu–water nanofluid flow analysis between parallel plates

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by DTM-Padé Method, Journal of Molecular Liquids 193 (2014) 37-44.

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[23] A.R. Ahmadi, A. Zahmatkesh, M. Hatami, D.D. Ganji, A comprehensive analysis of the flow

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and heat transfer for a nanofluid over an unsteady stretching flat plate, Powder Technology 258 (2014) 125-133.

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[24] M. Fakour, A. Vahabzadeh, D.D. Ganji, M. Hatami, Analytical study of micropolar fluid flow

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and heat transfer in a channel with permeable walls, Journal of Molecular Liquids 204 (2015) 198204.

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[25] Seiyed E. Ghasemi, M. Hatami, A. Kalani Sarokolaie, D.D. Ganji, Study on blood flow

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containing nanoparticles through porous arteries in presence of magnetic field using analytical methods, Physica E: Low-dimensional Systems and Nanostructures 70 (2015) 146-156.

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[26] S.E. Ghasemi, M. Hatami, GH.R. Mehdizadeh Ahangar, D.D. Ganji, Electrohydrodynamic

72 (2014) 47-52.

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flow analysis in a circular cylindrical conduit using least square method, Journal of Electrostatics

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[27] M. Rahimi-Gorji, O. Pourmehran, M. Hatami, D. D. Ganji, Statistical optimization of microchannel heat sink (MCHS) geometry cooled by different nanofluids using RSM analysis, The European Physical Journal Plus (2015) 130-22. [28] M. Hatami, D.D. Ganji, Thermal behavior of longitudinal convective–radiative porous fins with different section shapes and ceramic materials (SiC and Si3N4), Ceramics International 40 (2014) 6765-6775.

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ACCEPTED MANUSCRIPT ``

[29] M. Hatami, D.D. Ganji, Investigation of refrigeration efficiency for fully wet circular porous fins with variable sections by combined heat and mass transfer analysis, International Journal of Refrigeration 40 (2014) 140-151. [30] M. Hatami, GH.R. Mehdizadeh Ahangar, D.D. Ganji, K. Boubaker, Refrigeration efficiency

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analysis for fully wet semi-spherical porous fins, Energy Conversion and Management 84 (2014)

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533-540.

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[31] Keivan Bamdad, H.R. Ashorynejad, “Inverse analysis of a rectangular fin using the lattice Boltzmann method”, Energy Conversion and Management, 97:290–297 (2015)

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[32] H.R. Ashorynejad, B. Hoseinpour, "Investigation of different nanofluids effect on entropy

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generation on natural convection in a porous cavity", European Journal of Mechanics B/Fluids 62: 86–93 (2017)

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[33] B. Hoseinpour, H. R. Ashorynejad, and K. Javaherdeh. "Entropy Generation of Nanofluid in

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a Porous Cavity by Lattice Boltzmann Method", Journal of Thermophysics and Heat Transfer, (2016) 1-8.

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[34] H.R. Ashorynejad, Kurosh Sedighi, Mousa Farhadi, Ehsan Fattahi, “Simulating

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magnetohydrodynamic natural convection flow in a horizontal cylindrical annulus using the lattice Boltzmann method” Heat Transfer-Asian Research, 41(6) :468-483 (2012)

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[35] H.R. Ashorynejad, K. Javaherdeh, HEA Van den Akker, "The effect of pulsating pressure on the performance of a PEM fuel cell with a wavy cathode surface", International Journal of Hydrogen Energy, 41(32): 14239–14251 (2016) [36] H. R. Ashorynejad, and K. Javaherdeh. “ Investigation of a waveform cathode channel on the performance of a PEM fuel cell by means of a pore-scale multi-component lattice Boltzmann method”, Journal of the Taiwan Institute of Chemical Engineers, 66: 126–136 (2016)

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Table 1. Thermal properties of base fluid (water) and nanoparticles Unit Jkg−1∙K−1 kg⋅m−3 Wm−1⋅K−

Water 4179 997.1 0.613

Al2O3 765 3970 40

TiO2 686.2 4250 8.95

K−1

2.1×10−4

0.85×10−5

0.9×10−5

Ns∙m−2

0.001003

-

-

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Properties Heat capacitance Density Thermal conductivity Thermal expansion coefficient Dynamic viscosity

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(a)

(b)

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Fig. 1 a) Schematic of the problem and b) Sample FlexPDE mesh generated

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Frame 001  06 Jan 2017  Convection - T

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T 0.95 0.85 0.75 0.65 Frame 001  06 Jan 2017  Convection - T 0.55 0.45 0.35 0.25 0.15 0.05

φ=0.06

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φ=0.09

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T 0.95 0.85 0.75 0.65 0.55 0.45 0.35 0.25 0.15 0.05

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T 0.95 0.85 0.75 0.65 0.55 0.45 0.35 0.25 0.15 0.05

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φ=0.03

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Fig. 2 Comparison of Temperature profile for different nanoparticles volume fraction of Al2O3

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Frame 001  06 Jan 2017  Convection - S

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φ=0.03

S 2.5 2 1.5 1 0.5 0 -0.5 Frame 001  06 Jan 2017  Convection - S -1 -1.5 -2 -2.5

φ=0.06

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S

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φ=0.09

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2.5 2 1.5 1 0.5 0 -0.5 -1 -1.5 -2 -2.5

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S 2.5 2 1.5 1 0.5 0 -0.5 -1 -1.5 -2 -2.5

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Fig. 3 Comparison of Stream lines profile for different nanoparticles volume fraction of Al2O3

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Frame 001  06 Jan 2017  Convection - U

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φ=0.03

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9 7 5 3 1 -1 06 Jan 2017  Convection - U Frame 001 -3 -5 -7 -9 -11 U

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9 7 5 3 1 -1 -3 -5 -7 -9 -11

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φ=0.09

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9 7 5 3 1 -1 -3 -5 -7 -9 -11 U

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φ=0.06

U

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Fig. 4 Comparison of U velocities profile for different nanoparticles volume fraction of Al2O3

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φ=0.03

V Frame 00114  06 Jan 2017  Convection - V

11 8 5 2 -1 -4 -7 -10

φ=0.06

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φ=0.09

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V 14 11 8 5 2 -1 -4 -7 -10

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V 14 11 8 5 2 -1 -4 -7 -10

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Fig. 5 Comparison of V velocities profile for different nanoparticles volume fraction of Al2O3

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T 0.95 0.85 0.75 Frame 0010.65  06 Jan 2017  Convection - S 0.55 0.45 0.35 0.25 0.15 0.05 T 0.95 0.85 0.75 0.65 0.55 Frame 001  06 Jan 2017  Convection - S 0.45 0.35 0.25 0.15 0.05 S

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T 0.95 0.85 0.75 Frame 001  06 Jan 2017  Convection - T 0.65 0.55 0.45 0.35 0.25 0.15 0.05

h=0.1

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2.5 2 1.5 1 0.5 0 -0.5 -1 -1.5 -2 -2.5

S 2.5 2 1.5 1 0.5 0 -0.5 -1 -1.5 -2 -2.5

Fig. 6 Effect of fin height (h) on the temperature and stream line contours (Al2O3, φ=0.03) 20

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2.5 2 1.5 1 0.5 Frame 001 0  06 Jan 2017  Convection - S -0.5 -1 -1.5 -2 -2.5 T 0.95 0.85 0.75 0.65 0.55 0.45 0.35 0.25 0.15 0.05

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T 0.95 0.85 0.75 0.65 0.55 Frame 001  06 Jan 2017  Convection - T 0.45 0.35 0.25 0.15 0.05 S

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2.5 2 1.5 1 0.5 0 -0.5 -1 -1.5 -2 -2.5

Fig. 7 Effect of different nanoparticles on the temperature and stream line contours (h=0.5 φ=0.03)

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(a) (b) Fig. 8 Local Nusselt number for a) Al2O3 and b) TiO2

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Average Nusselt

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Nanoparticle volume fraction (phi)

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Fig. 9 Nanoparticles effect in average nusselt number for h=0.1

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Frame 001  07 Jan 2017  Internally created data set

Frame 001  07 Jan 2017  Internally created data set

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(a) (b) Fig. 10 Average nusselt number versus fins height in different fin heights for a) Al2O3 and b) TiO2

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ACCEPTED MANUSCRIPT `` Highlights:

Natural convection of nanofluids in rectangular cavity is studied.



Two heat fins are considered in the cavity.



Effect of fins height and nanoparticles concentration is investigated.



Nusselt numbers are reported.

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

25