An application of CVFEM for nanofluid heat transfer intensification in a porous sinusoidal cavity considering thermal non-equilibrium model

Accepted Manuscript An application of CVFEM for nanofluid heat transfer intensification in a porous sinusoidal cavity considering thermal non-equilibrium model Zhixiong Li, S.A. Shehzad, M. Sheikholeslami

PII: DOI: Reference:

S0045-7825(18)30258-5 https://doi.org/10.1016/j.cma.2018.05.015 CMA 11916

To appear in:

Comput. Methods Appl. Mech. Engrg.

Received date : 16 February 2018 Revised date : 21 April 2018 Accepted date : 14 May 2018 Please cite this article as: Z. Li, S.A. Shehzad, M. Sheikholeslami, An application of CVFEM for nanofluid heat transfer intensification in a porous sinusoidal cavity considering thermal non-equilibrium model, Comput. Methods Appl. Mech. Engrg. (2018), https://doi.org/10.1016/j.cma.2018.05.015 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Highlights (for review)

   

CVFEM is employed to simulate heat transfer intensification inside a permeable cavity. Nanofluid properties are estimated by means of KKL. Darcy- Boussinesq approximation is used for nanofluid flow. Porosity has opposite relationship with temperature gradient.

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An application of CVFEM for nanofluid heat transfer intensification in a 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

porous sinusoidal cavity considering thermal non-equilibrium model Zhixiong Li a,b, S.A. Shehzad 1,c, M. Sheikholeslami d a

b

School of Engineering, Ocean University of China, Qingdao 266110, China

School of Mechanical, Materials, Mechatronic and Biomedical Engineering, University of Wollongong, Wollongong, NSW 2522, Australia c

Department of Mathematics, COMSATS Institute of Information Technology, Sahiwal 57000, Pakistan

d

Department of Mechanical Engineering, Babol Noshiravni University of Technology, Babol, IRAN

Abstract In this article, Control volume based finite element method (CVFEM) is employed to simulate magnetohydrodynamic (MHD) nanofluid convective heat transfer through a porous cavity by means of thermal non-equilibrium model. Nanofluid properties are estimated by means of Koo–Kleinstreuer–Li (KKL) model and Darcy- Boussinesq approximation is employed for momentum equation. Numerical simulations are provided to find the effects of Rayleigh number  Ra  , solid-nanofluid interface heat transfer parameter  Nhs  , Hartmann number  Ha  and porosity    . Outputs depict that velocity augments with increase of Nhs while it increases with augment of Rayleigh number. Temperature gradient reduces with augment of porosity. Keywords: Thermal non-equilibrium; Porous media; Nanofluid; Heat transfer intensification; MHD; CVFEM. 1

Corresponding Author: Email: [email protected] (S.A. Shehzad), [email protected] , [email protected] (M. Sheikholeslami)

1

Nomenclature 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

h nfs

interface heat transfer coefficient



fluid density

B

magnetic field



dimensionless temperature

Nhs

solid-matrix/ nanofluid interface



stream function

heat transfer parameter Nu

Nusselt number



electrical conductivity

Ha

Hartmann number

s

modified thermal conductivity ratio

CVFEM

Control volume based finite



dynamic viscosity

element method MHD

Magnetohydrodynamic



thermal expansion coefficient

KKL

Koo–Kleinstreuer–Li



kinetic viscosity

Ra

Rayleigh number

Subscripts

K

permeability

s

solid matrix

X ,Y

horizontal and vertical space

p

particle

nf

nanofluid

f

base fluid

coordinates T

fluid temperature

Greek symbols



porosity of the porous medium

2

1. Introduction 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

Nanofluid flow in a porous media has wide uses, such as drying technologies, and solar power collectors. Different applications of porous medium such as energy storage should be simulated by non-equilibrium model. Wu et al. [1] investigated the natural convection in a porous media using this model. Haq et al. [2] investigated the effect of Lorentz forces on nanoparticles migration through a channel. Sheikholeslami [3] employed two passive techniques (nanofluid and fin) to enhance the heat transfer rate in an energy storage system. Soomro et al. [4] investigated nanofluid forced convection over a stretching plate. Sheikholeslami et al. [5] utilized copper oxide nanoparticles to enhance nanofluid heat transfer in a duct with helical twisted tape. Sheikholeslami and Shehzad [6] reported the role of radiative mode on nanofluid treatment in an enclosure. Basak et al. [7] investigated the convective flow in a porous cavity with various boundary conditions. Yadav et al. [8] depicted nanoparticle migration due to external forces. Khan et al. [9] presented the second law analysis of nanofluid mixed convection under the effect of magnetic field. Selimefendigil and Oztop [10] simulated the effect of inclined angle on conjugate heat transfer. Sheikholeslami and Seyednezhad [11] analyzed the electric field influence on nanofluid hydrothermal behavior in a porous media. Khan et al. [12] investigated the thermal radiation impact on nanoparticles flow and heat transfer over a wedge. Sheremet et al. [13] depicted the convective motion of ferrofluid inside a rotating cavity. Usman et al. [14] demonstrated nanofluid flow in a converging duct. They used cooper and silver as nanoparticles. Sheikholeslami and Shehzad [15] employed non-Darcy model for nanofluid motion in a permeable medium under the impact of variable Lorentz forces. Akbar et al. [16] investigated about nanoparticle flow in a channel due to Lorentz forces. Sheikholeslami and Rokni [17] illustrated the nanofluid flow in presence of electric field in a porous cavity. They considered the thermal radiation impact on energy equation. Hayat et al. [18] investigated 3

thermal radiation and joule heating influences on nanofluid hydrothermal behavior. Various 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

articles have been published about nanofluid flow and heat transfer [19-57]. This article intends to investigate nanofluid behavior in a porous cavity considering thermal non-equilibrium model under the impact of uniform magnetic field. CVFEM is employed to show the roles of Rayleigh number, porosity, the solid-matrix/ nanofluid interface heat transfer parameter and Hartmann number.

2. Definition of the problem The porous enclosure and its boundary condition are depicted in Fig. 1. The outer wall formulation is:



rout  rin  A cos N    0 



(1)

Horizontal magnetic field is employed. Working fluid is prepared by dispersing CuO nanoparticles in to water. The enclosure has porous media. 3. Formulation and simulation 3.1. Governing equation The thermal non-equilibrium model and Boussinesq-Darcy law for flow are employed. Considering above models, the governing formulas are: u v  0 x y

(2)

nf

(3)

K

u 

P 2   nf B 02  u  sin    v  sin   cos      x

4

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

nf K

v 

P  T T c  g  nf  nf y

(4)

 nf B 02  v  cos    u  sin   cos      2

hnfs T nf  k nf   2T nf  2T nf  1  T nf u  v    T nf T s        x y   C p   x 2 y 2     C p  nf nf

ks

 C 

p s

  2T s  2T s  2  2 y  x

 C  ,    p

nf

 C  p

  nf

nf

nf

(5)

 hnfs T nf T s   0   1     C p s

(6)

,  nf and  nf can calculated as:

   C p   (1   )  C p  p

(7)

f

 (1   )   f      p

(8)

 nf  f (1   )   p 

(9)

p  MM  1  nf 3  1, MM  f  1  MM    MM  2  f

(10)

k n f ,  n f can be estimated via Koo–Kleinstreuer–Li (KKL) model [58]:

k nf kf

 kp  1   kf  bT   1 3  5 104 g (d p ,T ,  ) f  c  p d p p ,f  kp  kp   2 1     k k  f   f 

g  d p ,T ,    Ln T







a2 Ln d p   a5 Ln d p   a1  a3Ln    a4 Ln d p  Ln   2

 a7 Ln d p   a6  a8 Ln    a9 ln d p  Ln    a10 Ln d p  R f  d p / k p ,eff  d p / k p , R f  4 108 km 2 /W

5

2





(11)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

nf 

f   f  k Brownian / Pr  2.5 1    k f

(12)

Properties of CuO-water and related coefficients are depicted in Tables 1 and 2. Non-dimensional quantities are:

v 

  ,u  , x y

(13)

s  T s T c  / T h T c  , nf  T nf T c  / T h  T c  ,

 X ,Y    x , y  / L ,

   /  nf

According to above definitions, we have:  2  A6 2 2 2  2 2 2    Ha sin   cos   2  sin   cos      2 2 2  2  X Y A5 X X Y  Y  

(14)

A 3 A 2  nf Ra A 4 A 5 X

  2  2   nf2  nf2 X  Y

 nf   nf    Nhs s  nf    Y X Y X 

  2s  2s      Nhs  s nf  s   0 2 X 2   Y

(15)

(16)

where the constant and dimensionless parameters are:

A1 

  nf  nf  , A3  , A 5  nf , f f   f

A2  Ra 

 C P nf  , A 6  nf f  C P f g K   f L T f  f

, A4 

, Ha 

(17)

k nf , kf

 f K B 02 , f

Nhs  hnfs L 2 / k nf ,  s  k nf /  k s 1    

The inner wall is hot and both walls are stationary.

6

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

  0.0

on all walls

 nf  s  0.0

on outer wall

nf  s  1.0

on inner wall

(18)

Nu loc , Nu ave along the inner wall are:  k   Nu loc   nf  nf  k f  r

Nu ave 

1 2

(19)

2

 Nu

loc

(20)

dr

0

3.2. CVFEM As depicted in Fig. 1(b), triangular element is employed for CVFEM. In this new method both good features of FEM and FVM are combined together. Advection term is discretized via upwind method. Gauss-Seidel method is used in final step. Recently, a reference book was published about this method [59]. 4. Grid independent test and verification The results must not depend on mesh size. Therefore, different meshes should be utilized in each case. For example, according to table 3, the mesh size

81  241

must be utilized. The

FORTRAN code should be verified with previous published papers. Fig. 2 and Table 4 depict the comparisons have been examined with previous articles ([60-62]). According to these comparisons the present CVFEM code has very nice agreement. 5. Results and discussion The thermal non-equilibrium model is used for porous media to analyze nanofluid natural convection heat transfer in existence of Lorentz forces. CVFEM is utilized to show the impacts of Rayleigh number ( Ra  100 ,500 and 10 3 ), the porosity (   0.3 to 0.9 ), the solid-nanofluid interface heat transfer parameter ( Nhs  10 to 1000 ) and Hartmann number ( 7

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

Ha  0 to 20 ) on nanofluid hydrothermal behavior. Streamlines    , isotherms for solid s 

and nanofluid  nf

 are depicted for various

Ra , Nhs , 

and

Ha

in Figs. 3, 4, 5, 6, 7 and 8. When

conduction heat transfer is dominant, all parameters have no significant influence on nanofluid treatment. Isotherms for the nanofluid  nf  are as same as the solid s  . Higher values of Rayleigh number leads to more complication in shape of isotherms  nf  and thermal plume appears near the vertical centerline. But there is no observable change in s  contours. Temperature fields are stratified in existence of magnetic field.   max  increases with rise of Nhs . But temperature gradient of nanofluid decreases and in turn Nu ave reduces. Convective flow improves with rise of porosity. Temperature gradient near the inner wall reduces with rise of  and Nhs . Fig. 9 shows the impact of

Ra , Ha ,  , Nhs

on rate of heat transfer. According to obtained

data Nu ave can be presented as:

Nu ave  5.04  4.95Ra *  0.07Ha *  0.24  2.04Nhs * 1.13Ra *Ha *  1.38Ra *   1.88Ra *Nhs *  0.69Ha *  1.11Ha *Nhs *  1.36 Nhs * 0.69  Ra



* 2

 0.44  Ha



* 2

 1.33     0.12  Nhs 2



(21)

* 2

where Ra *  0.001Ra, Ha *  0.1Ha, Nhs *  0.001Nhs . The Nhs has reverse relationship with Nusselt number because temperature boundary layer thickness increases with augment of Nhs. Nusselt number is a decreasing function of porosity and Hartmann number but it is an increasing function of buoyancy forces. 6. Conclusions Nanofluid free convection in a porous enclosure under the influence of magnetic field is simulated by means of CVFEM. Non-equilibrium model is utilized for porous media. Roles of porosity, Rayleigh number, the solid-matrix/ nanofluid interface heat transfer 8

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

parameter and Hartmann number are demonstrated. Results prove that Nu ave has reverse relationship with Ha,  , Nhs . Convective flow reduces with augment of Hartmann number. Acknowledgements: This research was supported by the National Sciences Foundation of China (NSFC) (No. U1610109), Yingcai Project of CUMT (YC2017001), PAPD and UOW Vice-Chancellor’s Postdoctoral Research Fellowship References [1] M. Wu, A.V. Kuznetsov, W.J. Jasper, Modeling of particle trajectories in an electrostatically charged channel, Phys. Fluids 22 (2010) 043301. [2] R. Haq, N.F.M. Noor, Z.H. Khan, Numerical simulation of water based magnetite nanoparticles between two parallel disks, Adv. Powder Tech. 27 (2016) 1568-1575. [3] M. Sheikholeslami, Numerical modeling of nano enhanced PCM solidification in an enclosure with metallic fin, J. Mol. Liq. 259 (2018) 424–438. [4] F.A. Soomro, R. Haq, Z.H. Khan, Q. Zhang, Passive control of nanoparticle due to convective heat transfer of Prandtl fluid model at the stretching surface, Chin. J. Phys. 55 (2017) 1561-1568. [5] M. Sheikholeslami, M. Jafaryar, Z. Li, Nanofluid turbulent convective flow in a circular duct with helical turbulators considering CuO nanoparticles, Int. J. Heat Mass Transf. 124 (2018) 980–989. [6] M. Sheikholeslami, S.A. Shehzad, Thermal radiation of ferrofluid in existence of Lorentz forces considering variable viscosity, Int. J. Heat Mass Transf. 109 (2017) 82–92. [7] T. Basak, S. Roy, T. Paul, I. Pop, Natural convection in a square cavity filled with a porous medium: effects of various thermal boundary conditions, Int. J. Heat Mass Transf. 49 (2006) 1430–1441.

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in presence of variable magnetic field by means of CVFEM, Int. J. Heat Mass Transf. 114 (2017) 1169–1180. [35] T. Hayat, T. Muhammad, S.A. Shehzad, A. Alsaedi, On magnetohydrodynamic flow of nanofluid due to a rotating disk with slip effect: A numerical study, Comput. Methods Appl. Mech. Eng. 315 (2017) 467-477. [36] M. Sheikholeslami, M.B. Gerdroodbary, D.D. Ganji, Numerical investigation of forced convective heat transfer of Fe3O4-water nanofluid in presence of external magnetic source, Comput. Methods Appl. Mech. Eng. 315 (2017) 831–845. [37] M. Sheikholeslami, A. Zeeshan, Analysis of flow and heat transfer in water based nanofluid due to magnetic field in a porous enclosure with constant heat flux using CVFEM, Comput. Methods Appl. Mech. Eng. 320 (2017) 68–81. [38] M. Sheikholeslami, H.B. Rokni, Numerical modeling of nanofluid natural convection in a semi annulus in existence of Lorentz force, Comput. Methods Appl. Mech. Eng. 317 (2017) 419–430. [39] M. Sheikholeslami, H.B. Rokni, Magnetic nanofluid flow and convective heat transfer in a porous cavity considering Brownian motion effects, Phys. Fluid 30 (2018) 012003. [40] M. Sheikholeslami, M. Darzi, M.K. Sadoughi, Heat transfer improvement and pressure drop during condensation of refrigerant-based Nanofluid; An experimental procedure, Int. J. Heat Mass Transf. 122 (2018) 643–650. [41] M. Sheikholeslami, H.B. Rokni, CVFEM for effect of Lorentz forces on nanofluid flow in a porous complex shaped enclosure by means of Non-equilibrium model, J. Mol. Liq. 254 (2018) 446-462.

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melting heat transfer and radiation effects in stagnation point flow of carbon–water nanofluid, Comput. Methods Appl. Mech. Eng. 315 (2017) 1011-1024. [43] P. Moreno-Navarro, A. Ibrahimbegovic, J.L. Pérez-Aparicio, Plasticity coupled with thermo-electric fields: Thermodynamics framework and finite element method computations, Comput. Methods Appl. Mech. Eng. 315 (2017) 50-72. [44] M. Sheikholeslami, A. Ghasemi, Solidification heat transfer of nanofluid in existence of thermal radiation by means of FEM, Int. J. Heat Mass Transf. 123 (2018) 418–431. [45] M. Sheikholeslami, S.A. Shehzad, CVFEM simulation for nanofluid migration in a porous medium using Darcy model, Int. J. Heat Mass Transf. 122 (2018) 1264–1271. [46] C.S.K. Raju, N. Sandeep, A. Malvandi, Free convective heat and mass transfer of MHD non-Newtonian nanofluids over a cone in the presence of non-uniform heat source/sink, J. Mol. Liq. 221 (2016) 108-115. [47] F. Sun, Y. Yao, X. Li, P. Yu, G. Ding, M. Zou. The flow and heat transfer characteristics of superheated steam in offshore wells and analysis of superheated steam performance. Comput. Chem. Eng. 100 (2017) 80–93. [48] F. Sun, Y. Yao, M. Chen, X. Li, L. Zhao, Y. Meng, S. Zheng, T. Zhang, D. Feng, Performance analysis of superheated steam injection for heavy oil recovery and modeling of wellbore heat efficiency, Energy 125 (2017) 795-804. [49] N.A.M. Zin, I. Khan, S. Shafie, A.S. Alshomrani, Analysis of heat transfer for unsteady MHD free convection flow of rotating Jeffrey nanofluid saturated in a porous medium, Results Phys. 7 (2017) 288-309. [50] M. Sheikholeslami, M.K. Sadoughi, Mesoscopic method for MHD nanofluid flow inside a porous cavity considering various shapes of nanoparticles, Int. J. Heat Mass Transf. 113 (2017) 106–114.

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15

[60] B.S. Kim, D.S. Lee, M.Y. Ha, H.S. Yoon, A numerical study of natural convection in a 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

square enclosure with a circular cylinder at different vertical locations, Int. J. Heat Mass Transf. 51 (2008) 1888-1906. [61] 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. [62] N. Rudraiah, R.M. Barron, M. Venkatachalappa, C.K. Subbaraya, Effect of a magnetic field on free convection in a rectangular enclosure, Int. J. Eng. Sci. 33 (1995) 1075–1084.

(a)

16

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

(b) Fig. 1. (a)Geometry and the boundary conditions with (b) A sample triangular element and its corresponding control volume.

Present result

Kim et al. [24]

(a)

17

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

(b) Fig. 2. Comparison of the present solution with previous work (Kim et al. [60]) for different Rayleigh numbers when Ra= 105, Pr=0.7; (b) Comparison of the temperature on axial midline between the present results and numerical results obtained by Khanafer et al. [61] for Gr  10 4 ,   0.1 and Pr  6.2  Cu  Water  .

 nf



18

s

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

1.4 1.2 1 0.8 0.6 0.4 0.2

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

0.95 0.85 0.75 0.65 0.55 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

1.4 1 0.8 0.6 0.4 0.2

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.3 0.2 0.1

Nhs=10

1.2 1 0.8 0.6 0.4 0.2

Nhs=10

Nhs=1000

ɛ=0.3 ɛ=0.9

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

19

Nhs=1000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

1.5 1.4 1.2 1 0.8 0.6 0.4 0.2

0.9 0.8 0.7 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

Fig. 3. Streamlines    , isotherms for the nanofluid  nf  and the solid s  at

20

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.5 0.45 0.35 0.3 0.2 0.15 0.05

Ra  100, Ha  0,  0.04

s

 nf



0.9 0.8 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.2 0.1

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

0.04 0.035 0.025 0.02 0.015 0.01 0.005

0.95 0.85 0.8 0.7 0.6 0.5 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

0.04 0.035 0.03 0.025 0.015 0.005

0.95 0.9 0.85 0.8 0.7 0.65 0.6 0.5 0.45 0.4 0.35 0.25 0.15 0.05

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

Nhs=10

0.04 0.03 0.025 0.02 0.015 0.01 0.005

Nhs=10

Nhs=1000

ɛ=0.3 ɛ=0.9

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

21

Nhs=1000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

0.95 0.9 0.8 0.7 0.6 0.5 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

0.04 0.035 0.03 0.025 0.02 0.015 0.01 0.005

Fig. 4. Streamlines    , isotherms for the nanofluid  nf  and the solid s  at

22

0.95 0.9 0.8 0.75 0.7 0.6 0.5 0.4 0.3 0.25 0.2 0.15 0.1 0.05

Ra  100, Ha  20,  0.04

5 4.5 4 3.5 3 2.5 2 1.5 1 0.5

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

7 6 5 4 3 2 1

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.5 0.4 0.3 0.2 0.1 0.05

0.95 0.9 0.85 0.75 0.65 0.55 0.45 0.35 0.25 0.15 0.05

6 5 4 3 2 1

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

Nhs=10

ɛ=0.9

Nhs=1000

ɛ=0.3

Nhs=10

s

 nf

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

23

0.9 0.8 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.2 0.1

0.95 0.9 0.85 0.8 0.75 0.65 0.55 0.5 0.4 0.3 0.2 0.1

7 5 4 3 2 1

Nhs=1000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

Fig. 5. Streamlines    , isotherms for the nanofluid  nf  and the solid s  at

24

Ra  500, Ha  0,  0.04

0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.45 0.35 0.25 0.15 0.05

0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.45 0.35 0.25 0.15 0.05

0.2 0.16 0.1 0.08 0.06 0.04 0.02

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

0.95 0.85 0.75 0.7 0.65 0.6 0.55 0.45 0.35 0.25 0.15 0.05

Nhs=10

ɛ=0.9

Nhs=1000

ɛ=0.3

Nhs=10

s

 nf

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

25

Nhs=1000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.4 0.3 0.2 0.1 0.05

0.21 0.18 0.14 0.12 0.1 0.08 0.06 0.04 0.02

Fig. 6. Streamlines    , isotherms for the nanofluid  nf  and the solid s  at

26

0.95 0.85 0.75 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.25 0.15 0.05

Ra  500, Ha  20,  0.04

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

8 7 6 5 4 3 2 1

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

14 10 8 6 4 2

0.95 0.9 0.8 0.7 0.6 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

10 9 8 7 6 5 4 3 2 1

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

Nhs=10

ɛ=0.9

Nhs=1000

ɛ=0.3

Nhs=10

s

 nf

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

27

12 10 8 6 4 2

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

Nhs=1000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

Fig. 7. Streamlines    , isotherms for the nanofluid  nf  and the solid s  at

28

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

Ra  1000, Ha  0,  0.04

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

0.95 0.85 0.8 0.75 0.65 0.6 0.55 0.45 0.4 0.35 0.25 0.2 0.1 0.05

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.5 0.4 0.3 0.2 0.1

0.4 0.35 0.25 0.2 0.1 0.05

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.45 0.35 0.25 0.15 0.05

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

Nhs=10

ɛ=0.9

Nhs=1000

ɛ=0.3

Nhs=10

s

 nf

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

29

0.4 0.3 0.2 0.15 0.1 0.05

0.95 0.9 0.85 0.8 0.75 0.65 0.55 0.45 0.35 0.25 0.15 0.05

Nhs=1000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

Fig. 8. Streamlines    , isotherms for the nanofluid  nf  and the solid s  at

30

0.95 0.85 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.15 0.05

Ra  1000, Ha  20,  0.04

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

  0.6, Nhs  505,  0.04

Ha  10, Nhs  505,  0.04

Ha  10,   0.6,  0.04

Ra  550, Nhs  505,   0.04

Ra  550,   0.6,  0.04

Ra  550, Ha  10,  0.04

31

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

  0.6, Nhs  505,  0.04

Ha  10, Nhs  505,  0.04

Ha  10,   0.6,  0.04

Ra  550, Nhs  505,   0.04

Ra  550, Ha  10,  0.04

Ra  550,   0.6,  0.04

Fig. 9. Effects of

Ra , Ha ,  , Nhs

32

on average Nusselt number

Table 1: The coefficient values of CuO Water nanofluid 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

Coefficient values

CuO Water

a1

-26.593310846

a2

-0.403818333

a3

-33.3516805

a4

-1.915825591

a5

6.42185846658E-02

a6

48.40336955

a7

-9.787756683

a8

190.245610009

a9

10.9285386565

a10

-0.72009983664

Table 2: Thermo physical properties of water and nanoparticles    m

 ( kg / m 3 )

C p ( j / kgk )

k(W / m.k )

  10 5 ( K 1 )

Water

997.1

4179

0.613

21

0.05

CuO

6500

540

18

29

10-10

Table 3: Comparison of the average Nusselt number Nuave for different grid resolution at

Ra  1000, Ha  20,   0.9, Nhs  1000

and   0.04 .

Mesh size in radial direction  angular direction 61  181

71  211

81  241

4.133954

4.134885

4.135796

33

91  271

4.135601

101  301

4.135807

1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

Table 4: Nuave for various Gr and Ha at Pr=0.733. Gr  2  105

Gr  2  104

Ha

Present

Rudraiah et al. [62]

Present

Rudraiah et al. [62]

0

2.5665

2.5188

5.093205

4.9198

10

2.26626

2.2234

4.9047

4.8053

50

1.09954

1.0856

2.67911

2.8442

100

1.02218

1.011

1.46048

1.4317

34