Influence of EFD viscosity on nanofluid forced convection in a cavity with sinusoidal wall

Accepted Manuscript Influence of EFD viscosity on nanofluid forced convection in a cavity with sinusoidal wall

M. Sheikholeslami, Houman B. Rokni PII: DOI: Reference:

S0167-7322(16)34213-1 doi: 10.1016/j.molliq.2017.02.042 MOLLIQ 6955

To appear in:

Journal of Molecular Liquids

Received date: Revised date: Accepted date:

25 December 2016 10 February 2017 13 February 2017

Please cite this article as: M. Sheikholeslami, Houman B. Rokni , Influence of EFD viscosity on nanofluid forced convection in a cavity with sinusoidal wall. The address for the corresponding author was captured as affiliation for all authors. Please check if appropriate. Molliq(2017), doi: 10.1016/j.molliq.2017.02.042

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ACCEPTED MANUSCRIPT Influence of EFD viscosity on nanofluid forced convection in a cavity with sinusoidal wall M. Sheikholeslami1,a, Houman B. Rokni b a

Department of Mechanical Engineering, Babol Noshirvani University of

Department of Mechanical and Materials Engineering, Tennessee Technological

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b

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Technology, Babol, Iran

University, Cookeville, TN 38505, USA

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Abstract

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Impact of Coulomb force on Fe3O4- Ethylene glycol nanofluid convective heat transfer is examined. The positive electrode is considered as moving wall.

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Control Volume based Finite Element Method is selected to obtain the outputs which are the roles of Reynolds number  Re  , nanofluid volume fraction and supplied

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voltage    . EFD viscosity of nanofluid according to experimental data is taken into account. Results reveal that electric field boosts the convention mode so heat transfer rate augments by augmenting Coulomb force. Isotherms become denser near the lid

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wall with augment of Re and  . Using electric field is more useful for lower

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Reynolds number.

Keywords: EHD; Nanofluid; Forced convection; EFD viscosity; Joule heating; CVFEM. Nomenclature De



diffusion number

1

coefficient of

Corresponding author: Email address: [email protected] (M. Sheikholeslami), [email protected] (Houman B. Rokni)

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ACCEPTED MANUSCRIPT expansion [1/K] Lorentz force number



Volume fraction



electric field



Density [kg/m3]

electric force

Subscripts

NE

electric field number

h

Hot

PrE

electric Prandtl number

f

Base fluid

Re

Reynolds number

v ,u

Vertical and horizontal

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velocity

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

FE

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E ,Ex ,E y

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SE

Greek symbols

Dynamic viscosity [Pa.s]



electric field potential



electric conductivity

Solid particles

c

Cold

nf

Nanofluid

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

s

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1. Introduction

One of the useful active way for heat transfer enhancement is

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Electrohydrodynamic. Liu et al. [1] calculated the thermal conductivity of nanofluid in existence of electric field. Sheikholeslami and Ganji [2] investigated the electric field impact on nanofluid flow in an enclosure. Sheikholeslami [3] studied the electric field effect on nanofluid heat transfer augmentation. Sheikholeslami and Ganji [4] presented different application of nanofluid in their recent article. Sheikholeslami et al. [5] examined CuO-water nanofluid hydrothermal analysis in a complex shaped cavity. Hussien et al. [6] reported heat transfer augmentation of nanofluid in a channel. They illustrated the new uses of nanofluid in micro channel. 2

ACCEPTED MANUSCRIPT Khan et al. [6] investigated nanofluid flow with slip motion influence in existence of inclined magnetic field. Sheikholeslami [7] studied the impact of Lorentz forces on nanofluid thermal radiation. Sheikholeslami [8] simulated nanofluid flow in a porous media in existence of uniform magnetic field. Bhatti and Rashidi [9] reported the impact of thermo-diffusion on Williamson nanofluid over a sheet.

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Bhatti et al. [10] studied the electric double layer effect on two phase flow in

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presence of magnetic field. CVFEM has been applied by Sheikholeslami [11] to

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simulate nanofluid flow in a porous curved cavity. Hayat et al. [12] investigated the effect of radiation on nanofluid mass transfer. Ahmad and Mustafa [13] studied the

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rotating nanofluid flow induced by a stretching sheet. Their outputs revealed that temperature gradient reduces with augment of angular velocity. Raju et al. [14]

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examined the transient ferrofluid flow over a cone with variable viscosity. MHD nanofluid free convective hydrothermal analysis in a tilted wavy cavity was

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presented by Sheremet et al. [15]. They concluded that change of inclined angle

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causes convective heat transfer to augment. Effect of non-uniform Lorentz forces on nanofluid flow style was reported by Sheikholeslami Kandelousi [16]. He proved that

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enhancement in heat transfer decays with rise of Kelvin forces. Radiation and magnetic source terms have been considered by Sheikholeslami et al. [17] in

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governing equations. Effect of various arrangements of electrode has been examined by Kasayapanand et al. [18]. Electric field impact on nanofluid convective heat transfer was analyzed by Sheikholeslami et al. [19]. Impacts of electric field on Fe3O4-H2O free convective behavior was investigated by Sheikholeslami and Chamkha [20]. They concluded that Coulomb forces can change the flow style. Rashidi et al. [21] studied the nanofluid free convective flow over a plate. Garoosi et al. [22] investigated the simulation of nanofluid by means of Buongiorno model.

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ACCEPTED MANUSCRIPT Sheikholeslami and Rokni [23] simulated nanofluid flow in presence of induced magnetic field by means of two phase model. Beg et al. [24] examined the bionanofluid transport phenomena by means of both single and two phase models. Applications of various types of nanofluid were presented by several researchers [2541].

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This article intends to investigate the influence of Coulomb forces on

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nanofluid hydrothermal treatment via new numerical method. EFD viscosity is

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considered. Roles of Reynolds number, supplied voltage and Fe3O4 volume fraction

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are presented in outputs.

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2. Problem definition

Fig.1 illustrates the geometry of this problem and its boundary conditions.

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Impact of electric field on Fe3O4- Ethylene glycol nanofluid is considered. Fig. 2

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shows the contour of q for various values of  and Re . As  increases the distortion of isoelectric density lines become more and one cell appear in right side.

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Influence of  on q is more sensible than Re .

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3. Formulation and simulation 3.1. Governing equations According to Gauss’s law and Maxwell’s relation, electric field can be

defined as [20]: 

q  . E

(1)

4

ACCEPTED MANUSCRIPT 

E  

(2)

. J 

q 0 t







(3)



J  qV  D q   E

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(4)

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  .V  0       V . V  V   q E  nf  2 V  p    t   nf  nf  nf           k nf T  J .E 2 T   V . T  t   C  p nf  C p nf       E  q    . J  t   q  . E

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The governing equations are [20]:

knf kf

,  C p nf ,  and  nf can be obtained as [42]: nf



2 (k f  k s )  k s  2k f ,  ( k f  k s )  k s  2k f

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knf

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(5)

 C  p

nf  A1  A 2     A3     A 4    2

nf

  C p  (1   )   C p   , f

s

(6)

3

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 nf  f (1   )  s 

Properties of Fe3O4 and ethylene glycol are presented in Table1. EFD viscosity is presented according to [42]. Table2 illustrates the coefficient values of this formula. So, the final dimensionless equations in presence of electric field are:

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ACCEPTED MANUSCRIPT   .V  0      V . V  V   1  nf / f  2 V  p  S E q E     t  Re nf / f  nf / f      k nf / k f 1     V .       1  2  S E Ec  J . E      t  Pr Re  C p  /  C p      C p nf /  C p f nf f   E     q  . E   q . J   t 

RI

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(7)

u ,v 

  0  y , x  T T 0 , y ,x  ,  ,  L T Lid tU P q E t  Lid , p  q  ,E  , 2 L q0 E0 U Lid  T  T1 T 0 ,   1  0 ,



, 



(8)

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u ,v   U

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where

By eliminating pressure gradient, vorticity and stream function can be introduced as:     L2 v u , u,  ,  ,   x y f f x y

(9)

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v 

Nu loc and Nu ave along the left wall are calculated as:

(10)

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 k   Nu loc   nf   k f  X L

(11)

1 Nu loc dY L 0

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Nu ave 

3.2. CVFEM Control Volume based Finite Element Method (CVFEM) is a scheme that uses the advantages of both finite volume and finite element methods for simulation of multiphysics problems in complex geometries. Using the advantage of control volume the method is conservative and the finite element characteristic of the method make the 6

ACCEPTED MANUSCRIPT CVFEM to handle complex geometries and physics. Linear interpolation is utilized for approximation of variables in the triangular element which is considered as building block. Gauss-Seidel Method is utilized to solve the algebraic equations.

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More details exist in reference book [43].

4. Grid study and code verification

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Different grids have been examined for various cases. Table3 illustrates the

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outputs related to the case of   0.05 ,   10 kV ,Re  6000 . It can be found from this

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table that the size of 61 181 should be selected for satisfying grid independency. The FORTRAN code has been verified by comparing the outputs with those of exist

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in [44] and [45]. As shown in Fig. 3, the written code has good capability to simulate

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these problems.

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5. Results and discussion

Fe3O4 – Ethylene glycol nanofluid forced convection in an enclosure is

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simulated in existence of Coulomb forces. The bottom wall is moving lid. Roles of supplied voltage (   0 to 10 kV ), Reynolds number ( Re  3000 to 6000 ) and

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volume fraction of Fe3O4 (   0% to 5% ) are presented as graphs. Effect of  and Re on streamlines and isotherms are shown in Figs. 4, 5

and 6. In low Reynolds number, one clock wise eddy and three small counter clock wise eddies can be seen in streamline. As electric field increases eddies become stronger. By increasing Re , the strength of rotating eddies enhances and the distortion of isotherms becomes more than before. At high Reynolds number, the

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ACCEPTED MANUSCRIPT clock wise eddy converts to two smaller one. Also increasing Reynolds number makes isotherms more complicated. So, temperate gradient enhances with rise of Reynolds number. As electric field is applied, the two clock wise eddies merge together. Isotherms become more disturb by augmenting  . Effect of adding

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electric field is more pronounced in low Reynolds number. Impacts of Reynolds number and supplied voltage on Nu ave are depicted in

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Fig. 7. A correlation for average Nusselt number is presented as:

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Nu ave  5.24  1.66Re*  0.9  0.09Re*   0.185  Re*   0.029   

2

(25)

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2

where Re*  Re 103 and  is voltage supply in Kilovolt. In absence of electric

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field, as Reynolds number augments, convective heat transfer becomes stronger and temperature gradient enhances with augment of Re . Electric field boosts the

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convention mode. So Nu ave augments with augment  . Effect of adding electric

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field becomes weaker in higher Reynolds number.

6. Conclusions

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Nanofluid hydrothermal behavior in existence of external electric field is investigated by means of CVFEM. Results are presented the effects of  ,  and Re . Results depict that the distortion of isotherms becomes more with rise of Reynolds number and electric field. Effect of electric field on forced convection augmentation is more tangible for lower Reynolds number.

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[45] M. K. Moallemi, K. S. Jang, Int. J. Heat Mass Tran. 35 (1992) 1881–1892.

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Fig.1. Geometry of the problem and boundary conditions

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ACCEPTED MANUSCRIPT   5kV

Re=3000

  10kV

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 0.13 0.1

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 0.13 0.12 0.1

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Re=6000

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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

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Re=4500

RI

PT

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

0.95 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

Fig. 2. Electric density distribution injected by the bottom electrode

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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 0.17 0.15 0.13 0.1

ACCEPTED MANUSCRIPT

30

Present work Moallemi and Jang

25

15

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Nu

loc

20

RI

10

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5

NU

0

(a)

0

0.2

0.4

X

0.6

0.8

1

(b)

Fig. 3. (a) Comparison of average Nusselt number between the present results and numerical results by

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Khanafer et al. [44] Gr  104 ,   0.1 and Pr  6.8 Cu Water  ; (b) Comparison of the local Nusselt number over the lid wall between the present results and Moallemi and Jang [45] at Re=500, Ri=0.4,

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and Pr=1.

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ACCEPTED MANUSCRIPT   0kV

  10kV

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Isotherm

0.99 0.98 0.97 0.96 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

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Streamline

0.0055 0.005 0.0045 0.004 0.0035 0.003 0.0025 0.002 0.0015 0.001 0.0005 -0.0005 -0.001 -0.0015 -0.002 -0.0025 -0.003 -0.0035 -0.004 -0.0045 -0.005 -0.0055 -0.006 -0.0065

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Fig. 4. Effect of supplied voltage on streamlines and isotherm when Re  3000,   0.05 .

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0.006 0.004 0.002 0.0007 -0.002 -0.004 -0.008 -0.012 -0.016 -0.02 -0.024 -0.028 -0.032 -0.036 -0.04 -0.044 -0.046 -0.048 -0.05 -0.052

0.99 0.98 0.97 0.96 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

ACCEPTED MANUSCRIPT

  0kV

  10kV

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0.99 0.98 0.97 0.96 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

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Isotherm

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Streamline

0.004 0.0035 0.003 0.0025 0.002 0.0015 0.001 0.0005 -0.0005 -0.001 -0.0015 -0.002 -0.0025 -0.003 -0.0035 -0.004 -0.0045 -0.005 -0.0055 -0.006 -0.0065 -0.007

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Fig. 5. Effect of supplied voltage on streamlines and isotherm when Re  4500,   0.05 .

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0.004 0.002 0.0011 0.0008 0.0003 -0.0008 -0.0014 -0.002 -0.004 -0.006 -0.008 -0.01 -0.012 -0.014 -0.016 -0.018 -0.02 -0.022 -0.024 -0.026 -0.028 -0.03 -0.032

0.99 0.98 0.97 0.96 0.95 0.9 0.85 0.8 0.75 0.7 0.655 0.653 0.652 0.64 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.2

ACCEPTED MANUSCRIPT

  0kV

  10kV

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0.99 0.97 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 0.016 0.015 0.013

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Isotherm

NU

Streamline

0.003 0.0025 0.002 0.0015 0.001 0.0005 -0.0001 -0.0005 -0.001 -0.0015 -0.002 -0.0025 -0.003 -0.0035 -0.004 -0.0045 -0.005 -0.0055 -0.006 -0.0065 -0.007 -0.0075 -0.008 -0.0085

AC

CE

Fig. 6. Effect of supplied voltage on streamlines and isotherm when Re  6000,   0.05 .

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0.003 0.002 0.001 0.0005 -0.001 -0.002 -0.003 -0.005 -0.007 -0.009 -0.011 -0.013 -0.015 -0.017 -0.018 -0.019 -0.02 -0.021

0.99 0.97 0.95 0.9 0.85 0.8 0.75 0.74 0.73 0.72 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

AC

CE

PT E

D

MA

NU

SC

RI

PT

ACCEPTED MANUSCRIPT

Fig. 7. Effects of Reynolds number and supplied voltage on average Nusselt number.

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ACCEPTED MANUSCRIPT Table1. Thermo physical properties of Ethylene glycol and nanoparticles C p ( j / kgk )

k(W / m.k )

Ethylene glycol

1110

2400

0.26

Fe3O4

5200

670

6

PT

( kg / m3 )

RI

Table2. The coefficient values of Eq. (6)  0

A1 A2 A3 A4

1.0603E+001 -2.698E-003 2.9082E-006 -1.1876E-008

  0.05

9.5331 -3.4119E-003 5.5228E-006 -4.1344E-008

MA

NU

SC

Coefficient values

Table3. Comparison of the average Nusselt number Nuave along hot wall for different

PT E

D

grid resolution at Re  6000 ,   0.05 ,   10kV and Pr  149.54 . 41 121

51 151

61 181

71  211

81  241

2.54 1254

2.54 6178

2.55 8273

2.56 7491

2.56 8651

2.56 9012

AC

CE

31  91

Highlights    

Coulomb force effect on Fe3O4- Ethylene glycol flow is studied. CVFEM is applied to find the result. Electric field dependent viscosity is considered. Nu enhances with rise of φ and voltage supply.

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