Alexandria Engineering Journal (2016) xxx, xxx-xxx

HOSTED BY

Alexandria University Alexandria Engineering Journal

www.elsevier.com/locate/aej www.sciencedirect.com

ORIGINAL ARTICLE

CVFEM for free convective heat transfer of CuO-water nanofluid in a tilted semi annulus

M. Sheikholeslamia*, D.D. Ganjib

a Department of Mechanical Engineering, Babol University of Technology, Babol, Iran b Department of Mechanical Engineering, Sari Branch, Islamic Azad University, Sari, Iran

Received 26 July 2016; revised 4 October 2016; accepted 6 November 2016

KEYWORDS

Half-annulus enclosure;

Nanofluid;

Natural convection;

CVFEM;

Inclined enclosure

Abstract Influence of adding CuO nanoparticles in the base fluid on flow and heat transfer in an inclined half-annulus was studied considering constant heat flux as boundary condition of hot wall. Control Volume based Finite Element Method (CVFEM) is applied in order to simulate procedure. Pressure gradient source terms are eliminated by using vorticity stream function formulation. Influences of CuO volume fraction, inclination angle and Rayleigh number on hydrothermal manners are presented. Results indicate that inclination angle makes changes in flow style. The strength of eddies reaches to its minimum value when the upper wall is hot. Temperature gradient enhances with rise of buoyancy forces while it reduces with augment of inclination angle. © 2016 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

Nanotechnology was proposed as new way to improve heat transfer. 3D flow over plate has been examined by Mustafa et al. [1] considering radiation influence. They proved that temperature ratio has significant influence on thermal boundary layer. Magnetohydrodynamic (MHD) flow in a cavity with oscillating wall was examined by Selimefendigil and Oztop [2]. They illustrated that inclination angle of 90° has maximum performance. Sheikholeslami and Ganji [3] presented various applications of nanofluid in their review paper. Sheremet et al. [4] simulated the MHD unsteady free convective flow

* Corresponding author.

E-mail addresses: mohsen.sheikholeslami@yahoo.com,

m.sheikholeslami1367@gmail.com (M. Sheikholeslami), mirgang@ nit.ac.ir (D.D. Ganji).

Peer review under responsibility of Faculty of Engineering, Alexandria University.

in a wavy cavity. They used finite difference method (FDM) to simulate that paper. Influence of asymmetric heating on the heat transfer improvement in a microchannel was examined by Malvandi et al. [5]. Their results illustrated that Lor-entz forces enhance the Nusselt number about 42%. Influence of axial magnetic field on nanofluid thermal management has been analyzed by Sheikholeslami and Abelman [6]. Influences of space reliant magnetic field on ferrofluid motion were investigated by Sheikholeslami et al. [7]. They concluded that the higher speed of lid wall causes temperature gradient to enhance. Influence of single magnetic source on Fe3O4-water flow style has been reported by Sheikholeslami and Ganji [8]. They concluded that Kelvin force has various behaviors according to buoyancy forces and Lorentz forces reduce the nanofluid motion.

Kandasamy et al. [9] investigated the reaction of nanofluid versus chemical reaction. Sheikholeslami and Rashidi [10] applied single phase model for Fe3O4-water nanofluid. They indicated that Lorentz forces make velocity to reduce. Radia-

http://dx.doi.org/10.1016/j.aej.2016.11.012

1110-0168 © 2016 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Nomenclature

Nuloc local Nusselt number l dynamic viscosity

Nuave average Nusselt number a thermal diffusivity

Pr Prandtl number ( = u/a) / volume fraction

q" heat flux P fluid density

X, Y dimensionless space coordinates

U; V horizontal and vertical velocity components Subscripts

Ra Rayleigh number (= gpq'(rom - rin)4/(kav j) nf nanofluid

! gravitational acceleration vector c cold

k thermal conductivity f base fluid

s solid particles

Greek symbols h hot

ß thermal expansion coefficient

Table 1 The coefficient CuO — Water nanofluid [36].

values of

Coefficient values CuO — Water

a1 —26.593310846

a2 —0.403818333

a3 —33.3516805

a4 — 1.915825591

a5 6.42185846658E-

a6 48.40336955

a7 —9.787756683

a8 190.245610009

a9 10.9285386565

a10 —0.72009983664

tion and magnetic source terms have been added by Sheikholeslami et al. [11] in governing equations. They revealed that Lorentz forces can decrease the temperature gradient. Awais et al. [12] reported the slip impact on nanofluid flow in existence of magnetic field. Influence of chemical reaction on micropolar fluid over a plate has been presented by Rashad et al. [13]. Homotopy Analysis Method (HAM) was used by Joneidi et al. [14] to investigate the mass transfer over a plate. The influence of atherosclerosis on hemodynamics of stenosis has been forecasted by Nadeem and Ijaz [15]. They showed that the velocity gradient on the wall of titled arteries reduces with augment of Strommers number. Ahmad and Mustafa [16] investigated the rotating nanofluid flow induced by an exponentially stretching. Their results revealed that temperature gradient reduces with augment of angular velocity. Hayat et al. [17] presented the influence of radiation on mass transfer of nanofluid. They showed that temperature gradient reduces with augment of thermal radiation. Hussein et al. [18] investigated the free convection of nanofluid in T-shaped cavity. They proved that temperature gradient reduces with rise of heat source length. Radiation heat transfer over a sensor surface has been studied by Hamzah et al. [19]. They showed 30% enhancement in Nusselt number with use of nanofluid. Several authors reported their results about nano-fluid and natural convection decade [20-35].

The goal of this article was to present a simulation of free convective heat transfer in an inclined half annulus using CVFEM. Simulation is carried out for various inclination angles, CuO volume fraction and Rayleigh number. Also a correlation for Nusselt number is presented.

2. Problem definition

Fig. 1 illustrates the important geometric parameters of current geometry. Also sample mesh is presented. Constant heat flux is applied on inner surface and other conditions are clear in Fig. 1.

Figure 2 (a) Comparison of the temperature on axial midline between the present results and numerical results by Sharif et al. [38]; (b) comparison of average Nusselt number between the present results and numerical results by Khanafer et al. [39] Gr = 104, / = 0.1 and Pr = 6.2(Cu - Water).

3. Governing equation and simulation

3.1. Governing formulation

The flow is laminar and steady and in two dimensional. Boussinesq approximation has been considered for momentum

Table 2 Thermo physical properties of water and nanoparticles [36].

P (kg/m3) Cp (j/kgk) k (W/m k) b x 105(K—1) dp (nm) r (X ■ m)—1

Water 997.1 4179 0.613 21 - 0.05

CuO 6500 540 18 29 10—10 6500

Table 3 Comparison of the average Nusselt number Nuave for different grid resolutions at Ra = 105, k = 0° and ф = 0.04.

Mesh size in radial direction x angular direction 51 x 163 61 x 207 71 x 241 81 x 261 7.1086 7.0498 6.9871 6.9230 91 x 291 101 x 321 111 x 351 6:8868 6:8586 6:8358 121 x 381 6.8171 131 x 411 6.8016

Figure 3 Comparison of the streamlines (left) and isotherms (right) contours between nanofluid (/ = 0.04) (---) and pure fluid (/ = 0) (—) for different values of Ra at k = 0° and Pr = 6.2.

equations. Nanofluid is assumed as homogenous fluid. According to these assumptions the governing equations can be presented as follows:

dv du 0

du du 1 dP inf / d2 u

@У dx Pnfdx Pnf Vdy2

dv dv dP 1 inf fd2v „ . . ,

u IT + v яг + TT — = @T1 + — -(T - T)gß,

dx dy dy Pnf pnf Vdx2

(pCp)nf

ßnf, (pCp)nf and pnf are defined as follows:

(1) (2)

ßnf = ßf(1 - /)+ßsФ

(PCP)nf = (PCP)f(1 - Ф) + (pCp)s/

Pnf = pf(1 - /)+ps Ф

(6) (7)

(knf) and (inf) are obtained according to Koo-Kleinstreuer-Li (KKL) model [20]:

knf = 1 +■

3{ f -1

k \ ,k "Y" + 5 x 10У(Ф, TdP)/pfcPfJ-b-

kf+2) - 6 - 0 ф v ppdp

g(ф, T,dp) = (a1 + a2Ln(dp) +a3Ln(ф) + a4Ln(ф) ln(dp) +a5Ln(dp)2^j Ln(T) + (a6 + a7Ln(dp) + a8Ln(ф) + a9 ln(dp)Ln(/) + awLn(dp)2)Rf = dp/kff - dp/kp, Rf = 4 x 10-8km2/W

CVFEM for free convective heat transfer of CuO-water nanofluid

Ra = 103

Ra = 104

Ra = 105

= 1.49

(F. ) = 0.00

\ mm / nj

(F ) = 1.587

\ max > nf

(F ) = 0.00

\ mm / nf

(F ) = 8.27

\ max > nf

= -16.68

(Fmax ) = 18.78

\ max / nf

(F. ) = 0.00

\ mm /nf

Figure 4 Isotherms (down) and streamlines (up) contours for different values of Rayleigh number when k = 45° and 90° for CuO-water nanofluid (/ = 0.04).

if . kBrownian if

(1 — /)2:5 kf x Pr

All needed coefficients and properties are illustrated in Tables 1 and 2 [36].

Vorticity and stream function should be defined as follows:

du dv dW OW

x + - = °> =-v, — = u oy ox ox oy

By removing pressure gradient source terms from Eqs. (2) and (3), the final form of equations can be obtained as follows:

/д2т д2а\ f дТ\ да дф да дф

0nf\dX + ~ду2) — bnfg\ßX) = ~дХдУ — ~ду ~дх

д2Т д2Т ду2 дх2

дТ дф дТ дф дх ду ду дх

д2ф С?ф _ 0

ду2 дх2

Dimensionless parameters are as follows: Т-Т,

(«" L/kf)

(Y, X)=(yLX) , (W; Х) = (ф; XL2)

(11) (12)

Ra = 103

Ra = 104

Ra = 105

Pmn ) =-031 (Pmrr ) = 1.66

(Pmn )f =-117

Pma. )f = 117

(Pmn )f =-075

Pmx )f = 7 99

(Pmn )f =-462

Pmx )f = 462

(Pmn )f =-9 92

Pmax ) = 9 92

Figure 5 Isotherms (down) and streamlines (up) contours for different values of Rayleigh number when k = 135° and 180° for CuO-water nanofluid (ф = 0:04).

Using the above formulae the governing equations change to the following:

RafPrf

d© dX

№ +(1

+ Prf,

dX dY + dY dX

dY2 dX2

Prandtl and Rayleigh numbers are introduced as follows: Prf — Df/xf, Raf — g[fL4q"/(kf x/Df), respectively. Nu0c and Nuave over the hot wall should be calculated as follows:

Nuave = -

Nuhc(C)df

'f 1 Г

(18) (19)

d2© dX2

d2 © dY2

dW d©

dY dX "

d2W d2 W +--t +--T = -X

+ d Y2 + dX2

dW d©

(pCp): (pCp)

3.2. Numerical procedure

Linear interpolation is utilized for approximation of variables in the triangular element which is considered as building block (Fig. 1(c)). Gauss-Seidel method is utilized to solve the algebraic equations. More details exist in reference book [37].

Ra = 103

Ra = 104

Ra = 105

2.5 2 1.5

Figure 6 Effects of the nanoparticle volume fraction, Rayleigh number and inclination angle for CuO-water nanofluids on local Nusselt number.

log( Ra)

(a) Я = 180°

(b) log (Ra) = 5

log( Ra)

(c) log (Ra) = 5 ,ф = 0.04, Л = 180°

Figure 7 Effects of the nanoparticle volume fraction, Rayleigh number and inclination angle for CuO-water nanofluids on average Nusselt number.

4. Mesh independency and verification

Various grids have been considered for the case of / = 0.04, Ra= 105and k = 0° as presented in Table 3. This table indicates that the mesh size 71 x 211 can be chosen. Fig. 2(a and b) depicts the validation of current FORTRAN code. These figures prove the accuracy of this code in comparison with [38,39].

5. Results and discussion

In this work, the influence of CuO nanoparticle on the hydrothermal behavior in an inclined half-annulus is studied using CVFEM. Simulation is made for different values of Rayleigh number (Ra — 103,104 and 105), inclination angle

(k — 0°, 45°, 90°, 135° and 180°) and volume fraction of CuO (/ — 0% and 4%).

Fig. 3 depicts the effect of CuO nanoparticles on hydrothermal characteristics. The nanofluid velocity augments due to enhancement in the solid movements. Temperature gradient augments with rise of volume fraction CuO. Figs. 4 and 5 illustrate the impact of inclination angle, Rayleigh number and CuO volume fraction. Conduction mode is dominated in low Rayleigh number. Two rotating vortexes exist in streamlines. As buoyancy forces augment, temperature gradient enhances and thermal plume appears near the vertical center line. The core of vortexes moves upward. At k — 45°, the counterclockwise vortex is stronger than other one due to more space for circulation. At k — 90°, the two main eddies merged into one counterclockwise eddy. The streamlines and isotherms at k — 135° and 180° are depicted in Fig. 5. As seen, the impact

180.00

135.00

X 90.00

4.00 log(Ra)

0.00500 3.00

4.00 log(Ra)

(a) ф = 0.04

(b) Я= 180°

135.00

180.00

(c) log (Ra) = 5 Figure 8 Contour plots of average Nusselt number.

of Rayleigh number on the size of eddies for k — 135° is opposite to that of k — 45°. By increasing buoyancy forces, the size of secondary eddy decreases and finally at Ra — 105 a small vortex appears on the top of the secondary eddy, which turns in clockwise directions.

This behavior of thermal boundary layer may be due to the stronger flow circulation at higher Rayleigh numbers at these points. The isotherms and streamlines for k — 180°, are almost symmetric to the vertical centerline. However, enhancing the buoyancy forces has no significant influence on the location of two main eddies' core. In fact, when the hot circular wall locates above the cold one, the impact of convection on the velocity and temperature is less pronounced. Similar to the inclination angle of 135°, the temperature gradient becomes thinner near the inner wall except to the two inner corners.

Influences of important parameters on Nuloc and Nuave are depicted in Figs. 6-8. The correlation for this parameter is as follows:

Figure 9 Effects of k and Ra on the heat transfer enhancement due to addition of nanoparticles when Pr — 6:2.

NuaVe = 2.12 - 1.2 log(Ra) + 0.01k - 11.81/

- 4.88 k log(Ra) + 7.94ф log(Ra) - 0.027k/ + 0.38 log (Ra)2 + 2.19k2 + 0.07/2

Response surface methodology (RSM) is utilized to find this correlation. This method presented polynomial formulation according to input data. As volume fraction of CuO augments, Nuloc augments. Nusselt number augments with rise of Ra. Nuloc profiles are symmetric to f — 90° when k — 0° and 180°. At k — 0° and 45° minimum amount of Nu0c is located at f — 90° and 55° respectively. Also Fig. 8 shows that for k — 135° and 180° maximum amount of Nuloc is located at f — 100° and 90° respectively. Nusselt number decreases with enhancement of inclination angle. Fig. 9 shows the influence of k and Ra on the heat transfer enhancement. This parameter can be calculated as follows:

Nu(ф = 0.04) - Nu(basefluid)

Nu(basefluid)

According to this figure, heat transfer enhances with augment of inclination angle. Higher Rayleigh number leads to lower values of E. In low Rayleigh number and k — 180°, the dominant mechanism is conduction and in this way adding nanoparticles has significant impact on thermal conductivity.

6. Conclusions

Nanofluid free convective heat transfer in an inclined half-annulus is studied using CVFEM. Flow style and temperature distribution are presented for different inclination angles, CuO volume fraction and Rayleigh number. Results indicate that temperature gradient enhances with rise of CuO volume fraction and buoyancy forces but it reduces with augment of inclination angle. As inclination angle increases, the maximum value of stream function reduces. Moreover, the impact of CuO volume fraction on temperature gradient is more significant at lower Rayleigh number. The streamlines and isotherms clearly indicate that the inclination angle of enclosure can be utilized as a control parameter.

References

[1] M. Mustafa, A. Mushtaq, T. Hayat, A. Alsaedi, Radiation effects in three-dimensional flow over a bi-directional exponentially stretching sheet, J. Taiwan Inst. Chem. Eng. 47 (2015) 43-49.

[2] F. Selimefendigil, H.F. Oztop, Mixed convection of nanofluid filled cavity with oscillating lid under the influence of an inclined magnetic field, J. Taiwan Inst. Chem. Eng. 63 (2016) 202-215.

[3] M. Sheikholeslami, D.D. Ganji, Nanofluid convective heat transfer using semi analytical and numerical approaches: a review, J. Taiwan Inst. Chem. Eng. 65 (2016) 43-77.

[4] M.A. Sheremet, I. Pop, Natalia C. Rosca, Magnetic field effect on the unsteady natural convection in a wavy-walled cavity filled with a nanofluid: Buongiorno's mathematical model, J. Taiwan Inst. Chem. Eng. 61 (2016) 211-222.

[5] A. Malvandi, M.H. Kaffash, D.D. Ganji, Nanoparticles migration effects on magnetohydrodynamic (MHD) laminar mixed convection of alumina/water nanofluid inside microchannels, J. Taiwan Inst. Chem. Eng. 52 (2015) 40-56.

[6] M. Sheikholeslami, S. Abelman, Two phase simulation of nanofluid flow and heat transfer in an annulus in the presence

of an axial magnetic field, IEEE Trans. Nanotechnol. 14 (3)

(2015) 561-569.

[7] M. Sheikholeslami, K. Vajravelu, Mohammad Mehdi Rashidi, Forced convection heat transfer in a semi annulus under the influence of a variable magnetic field, Int. J. Heat Mass Transf. 92 (2016) 339-348.

[8] M. Sheikholeslami, D.D. Ganji, Ferrohydrodynamic and magnetohydrodynamic effects on ferrofluid flow and convective heat transfer, Energy 75 (2014) 400-410.

[9] R. Kandasamy, Hatem Sabeeh Altaie, Izzatun Nazurah Binti Zaimuddin, Irregular response of nanofluid flow subject to chemical reaction and shape parameter in the presence of variable stream conditions, Alexandria Eng. J. 55 (2016) 24852495.

[10] M. Sheikholeslami, M.M. Rashidi, Effect of space dependent magnetic field on free convection of Fe3O4-water nanofluid, J. Taiwan Inst. Chem. Eng. 56 (2015) 6-15.

[11] M. Sheikholeslami, T. Hayat, A. Alsaedi, MHD free convection of Al2O3-water nanofluid considering thermal radiation: a numerical study, Int. J. Heat Mass Transf. 96 (2016) 513-524.

[12] M. Awais, T. Hayat, Aamir Ali, S. Irum, Velocity, thermal and concentration slip effects on a magneto-hydrodynamic nanofluid flow, Alexandria Eng. J. 55 (2016) 2107-2114.

[13] A.M. Rashad, S. Abbasbandy, Ali J. Chamkha, Mixed convection flow of a micropolar fluid over a continuously moving vertical surface immersed in a thermally and solutally stratified medium with chemical reaction, J. Taiwan Inst. Chem. Eng. 45 (2014) 2163-2169.

[14] A.A. Joneidi, G. Domairry, M. Babaelahi, Analytical treatment of MHD free convective flow and mass transfer over a stretching sheet with chemical reaction, J. Taiwan Inst. Chem. Eng. 41 (2010) 35-43.

[15] S. Nadeem, S. Ijaz, Impulsion of nanoparticles as a drug carrier for the theoretical investigation of stenosed arteries with induced magnetic effects, J. Magn. Magn. Mater. 410 (2016) 230-241.

[16] R. Ahmad, M. Mustafa, Model and comparative study for rotating flow of nanofluids due to convectively heated exponentially stretching sheet, J. Mol. Liq. 220 (2016) 635-641.

[17] T. Hayat, Z. Nisar, H. Yasmin, A. Alsaedi, Peristaltic transport of nanofluid in a compliant wall channel with convective conditions and thermal radiation, J. Mol. Liq. 220 (2016) 448453.

[18] A. Kadhim Hussein, M.A.Y. Bakier, M.B. Ben Hamida, S. Sivasankaran, Magneto-hydrodynamic natural convection in an inclined T-shaped enclosure for different nanofluids and subjected to a uniform heat source, Alexandria Eng. J. 55

(2016) 2157-2169.

[19] N.Sh. bte Amir Hamzah, R. Kandasamy, Radiah Muhammad, Thermal radiation energy on squeezed MHD flow of Cu, Al2O3 and CNTs-nanofluid over a sensor surface, Alexandria Eng. J. 55 (2016) 2405-2421.

[20] A. Malvandi, F. Hedayati, D.D. Ganji, Thermodynamic optimization of fluid flow over an isothermal moving plate, Alexandria Eng. J. 52 (2013) 277-283.

[21] Mohsen Sheikholeslami, Influence of Lorentz forces on nanofluid flow in a porous cylinder considering Darcy model, J. Mol. Liq. (2016), http://dx.doi.org/10.1016/ j.molliq.2016.11.022.

[22] M. Sheikholeslami, M. Nimafar, D.D. Ganji, M. Pouyandehmehr, CuO-H2O nanofluid hydrothermal analysis in a complex shaped cavity, Int. J. Hydrogen Energy 41 (2016) 17837-17845.

[23] M. Sheikholeslami, D.D. Ganji, M.M. Rashidi, Magnetic field effect on unsteady nanofluid flow and heat transfer using Buongiorno model, J. Magn. Magn. Mater. 416 (2016) 164-173.

[24] M. Sheikholeslami, Ali J. Chamkha, Flow and convective heat transfer of a ferro-nanofluid in a double-sided lid-driven cavity with a wavy wall in the presence of a variable magnetic field,

Num. Heat Transf., Part A 69 (2016) 1186-1200, http://dx.doi. org/10.1080/10407782.2015.1125709.

[25] M. Sheikholeslami, Influence of Coulomb forces on Fe3O4-H2O nanofluid thermal improvement, Int. J. Hydrogen Energy (2016), http://dx.doi.org/10.1016/j.ijhydene.2016.09.185.

[26] M. Sheikholeslami, M.M. Rashidi, T. Hayat, D.D. Ganji, Free convection of magnetic nanofluid considering MFD viscosity effect, J. Mol. Liq. 218 (2016) 393-399.

[27] M. Sheikholeslami, Ali J. Chamkha, Electrohydrodynamic free convection heat transfer of a nanofluid in a semi-annulus enclosure with a sinusoidal wall, Num. Heat Transf., Part A 69 (2016) 781-793, http://dx.doi.org/10.1080/ 10407782.2015.1090819.

[28] Mohsen Sheikholeslami, Ali J. Chamkha, Influence of Lorentz forces on nanofluid forced convection considering Marangoni convection, J. Mol. Liq. (2016), http://dx.doi.org/10.1016/ j.molliq.2016.11.001.

[29] M. Sheikholeslami Kandelousi, Effect of spatially variable magnetic field on ferrofluid flow and heat transfer considering constant heat flux boundary condition, European Phys. J. Plus (2014) 129-248.

[30] B. Mahanthesh, B.J. Gireesha, Rama Subba Reddy Gorla, Heat and mass transfer effects on the mixed convective flow of chemically reacting nanofluid past a moving/stationary vertical plate, Alexandria Eng. J. 55 (2016) 569-581.

[31] M. Sheikholeslami, S.A. Shehzad, Magnetohydrodynamic nanofluid convection in a porous enclosure considering heat

flux boundary condition, Int. J. Heat Mass Transf. (2016), http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.10.107.

[32] M.A. Teamah, A.I. Shehata, Magnetohydrodynamic double diffusive natural convection in trapezoidal cavities, Alexandria Eng. J. 55 (2016) 1037-1046.

[33] M. Sheikholeslami, R. Ellahi, Three dimensional mesoscopic simulation of magnetic field effect on natural convection of nanofluid, Int. J. Heat Mass Transf. 89 (2015) 799-808.

[34] M. Sheikholeslami, H.R. Ashorynejad, P. Rana, Lattice Boltzmann simulation of nanofluid heat transfer enhancement and entropy generation, J. Mol. Liq. 214 (2016) 86-95.

[35] M.A. Teamah, M.M. Sorour, W.M. El-Maghlany, A. Afifi, Numerical simulation of double diffusive laminar mixed convection in shallow inclined cavities with moving lid, Alexandria Eng. J. 52 (2013) 227-239.

[36] M. Sheikholeslami Kandelousi, KKL correlation for simulation of nanofluid flow and heat transfer in a permeable channel, Phys. Lett. A 378 (45) (2014) 3331-3339.

[37] M. Sheikholeslami, D.D. Ganji, Hydrothermal Analysis in Engineering Using Control Volume Finite Element Method, Academic Press, Print Book, 2015, ISBN: 9780128029503.

[38] M.A.R. Sharif, T.R. Mohammad, Natural convection in cavities with constant flux heating at the bottom wall and isothermal cooling from the sidewalls, Int. J. Therm. Sci. 44 (2005) 865-878.

[39] K. Khanafer, K. Vafai, M. Lightstone, Buoyancy-driven heat transfer enhancement in a two-dimensional enclosure utilizing nanofluids, Int. J. Heat Mass Transf. 46 (2003) 3639-3653.