Heat Transfer Enhancement of Cu-water Nanofluid in a Porous Square Enclosure Driven by an Incessantly Moving Flat Plate

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ScienceDirect Procedia Engineering 127 (2015) 279 – 286

International Conference on Computational Heat and Mass Transfer-2015

Heat Transfer Enhancement of Cu-water Nanofluid in a Porous Square Enclosure Driven by an Incessantly Moving Flat Plate Nithyadevi Nagarajana, Shamadhanibegum Akbarb,* a,b

UGC-DRS Centre for Fluid Dynamics, Department of Mathematics, Bharathiar University, Coimbatore-46, India.

Abstract The existing work is focused on the numerical modelling of mixed convection of Cu-water nanofluid in a square enclosure filled with non-darcian fluid saturated porous medium. The enclosure object has cooled vertical walls and insulated horizontal walls. Finite volume method has been employed to solve the generalised Darcy-Brinkmann Forchheimer extended momentum and energy equations. The parametric study has been taken out for wide ranges of Richardson number, Darcy number and solid volume fraction. The performance of nanofluid is tested inside an enclosure by using solid volume fraction and compared with respect to base fluid (water). A fair degree of precision can be found between the present and previously published work. The results are presented in the form of streamlines, isotherms, average nusselt number and velocity graphs; it clearly explained the influence of flow governing parameters on heat transfer rate and fluid flow within the enclosure. © Published by Elsevier Ltd. B.V. This is an open access article under the CC BY-NC-ND license © 2015 2015The TheAuthors. Authors. Published by Elsevier (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of ICCHMT – 2015. Peer-review under responsibility of the organizing committee of ICCHMT – 2015 Keywords: Mixed convection; numerical modelling; nanofluid; enclosure; porous medium.

1. Introduction Nanofluids paying great attention as a new production of heat transfer in appliances demands rapid and effective heat transfer due to substances with sizes of nanoparticles acquire exclusive physical and chemical possessions in industrial applications. Choi [1] found an ingenious way to enhance the heat transfer by utilizing nano-particles in

* Corresponding author. Tel.:+91-876-017-5080. E-mail address: [email protected]

1877-7058 © 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license

(http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of ICCHMT – 2015

doi:10.1016/j.proeng.2015.11.369

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Nithyadevi Nagarajan and Shamadhanibegum Akbar / Procedia Engineering 127 (2015) 279 – 286

Nomenclature Cp Da g Gr H k K L Nu Nu P Pr Re Ri T U0 u,v U,V x,y X,Y

Specific heat (J/Kg.K) Darcy number Gravitational acceleration (m/s2) Grashof number Height of the enclosure (m) Thermal conductivity (W/m.k) Permeability of the porous medium (m2) Width of the enclosure (m) Local Nusselt number Average Nusselt number Pressure Prandtl number Reynolds number Richardson number Dimensionless temperature Lid velocity (m/s) Velocity components (m/s) Dimensionless velocity components Cartesian coordinates (m) Dimensionless Cartesian coordinates

Greek symbols α E H P Q T U W \

Thermal diffusivity (m2/s) Coefficient of thermal expansion (1/K) Porosity Dynamic viscosity (Ns/m2) Kinematic viscosity (m2/s) Temperature (K) Density (Kg/m3) Dimensionless time Stream function

Subscripts c f h nf 0 p

Cold wall Basefluid Hot wall Nanofluid Reference state Solid particle

low thermal conductivity base fluids like water, propylene glycol. Basak and Chamkha [2], Farhad Talebi et al [3] and Hasan et al [4] are analyzed the square enclosure using nanofluid in natural and mixed convection heat transfer respectively. The influence of nanofluid in porous enclosure is analysed by Nguyen et al [5] and they established that the accumulation of nanoparticles in the porous medium results superior average Nusselt number, also it diminishes with augmenting solid volume fraction in Darcy flow regime. Investigations in mixed convection through more than one moving wall by means of nanofluids is done by Chamkha and Abu-nada [6] and they found the consequence enhancement of heat transfer is achieved for the occurrence of nanoparticles. From the review of the above literatures shows that the much work has been done, but the present configuration model is have not been investigated by using nanofluids in porous medium. This formation finds practical application in the cooling of an extruded plate in a hot rolling process and used as a base line data in the designing system for this kind of model. 2. Formulation of the Problem y,v Tc O

Moving plate U0, Th

Tc

Tc x,u Tc

Fig.1a. Schematic configuration

281

0 .9

Nithyadevi Nagarajan and Shamadhanibegum Akbar / Procedia Engineering 127 (2015) 279 – 286

-1 -3

0.

-4

8

0 .7

-5

0.3

0 .1

0.5 -2

Fig. 1b. Comparision of present results with Basak and Chamkha [2] for nanofluid () and base fluid (}), Ra=104

Consider unsteady, laminar, incompressible and mixed convective flow in a square enclosure of width L and height H. Presenting an incessantly moving flat plate materializing from a channel at a consistent velocity U0 and at temperature Th, which divides the square enclosure into two equal bisects. The enclosure is filled with Cuwater nanofluid and saturated with porous medium of uniform porosity and permeability, isotropic which generates heat at a uniform rate. The vertical walls of the enclosure are maintained at uniform temperature Tc and the horizontal walls are thermally insulated. The Darcy- Brinkman- Forchheimer extended model is used to model the porous medium in the present study. In porous medium, the Local Thermal Equilibrium (LTE) property is applicable and neglecting velocity square term in momentum equations since the problem is treated with the mixed convection flow in an enclosure filled with a saturated porous medium [7]. All the properties of the fluid are assumed to be constant except the density, which changes linearly with temperature in the buoyancy term of the momentum equation, which is treated by Boussinesq approximation. Further the heat generation, viscous dissipation and the heat transfer by radiation effects are negligible in this investigation. Under the above considerations, the dimensionless governing equations of mass, momentum and energy of the system are followed by momentum and energy of the system are followed by

wU wV  0 wX wY wU 1 wU 1 1 wU  U  V H wW H 2 wX H 2 wY 1 wV

H wW



1

H2

U

wV 1 wV  V wX H 2 wY

wT wT wT U V wW wX wY

(1)  

Pnf P nf F wP 1 U   c U U 2 V 2 (’ 2U )  wX U nf Q f H Re U nfQ f Da Re Da

P nf P nf F wP 1 V   c V U 2  V 2  RiT (3) (’ 2V )  wY U nfQ f H Re U nfQ f Da Re Da

D nf 1 (’ 2T ) D f Pr Re

(4)

wU wV  wY wX w\ w\ and V where U wY wX Where the non-dimensional parameters are defined in the following forms, ’ 2\

X

x , Y L

y , U L

u , V U0

(2)

v , U0

P

p , W Unf U 02

U 0t , T L

(5)

T  Tc with T h ! Tc T h  Tc

Appearing in above equations, Fc=(1.75/—150H2) is Forchheimer coefficient, Da=k/L2 is Darcy number, Re=U0L/Qf is Reynolds number, Gr=gE(Th-Tc)L3/Qf2 is Grashof number, Pr =Qf/Df is Prandtl number and Ri=Gr/Re2 is Richardson number.

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Under consideration, the dimensionless form of initial and boundary conditions can be written as,

For , W

0: U

V

0,

For , W ! 0 : U

V

0,

U

1, V

U

V

T

0,

0, T

1,

wT wX

0,

0 d X d 1, 0 d Y d 1,

0,

T

0, 1

X Y 0,

Y

0, r0.5

Thermo physical properties of Cu-water nanofluids like effective density, thermal diffusivity, heat capacitance and thermal expansion coefficient of nanofluids are given by

U nf

(1  M ) U f  MU p

D nf

knf / ( U c p ) nf

( U c p ) nf ( UE ) nf

(1  M )( U c p ) f  M ( U c p ) p (1  M )( UE ) f  M ( UE ) p

Table 1. Comparison of computed average Nusselt number values of present work at the top surface with fixed Grashof number Gr=100. Re

Present Result

Sharif [8]

Waheed [9]

Khanafer & Chamkha [10]

Iwatsu [11]

1

1.000257

-

1.00033

100

2.005621

-

2.03116

2.01

1.94

400

3.812670

4.05

4.02462

3.91

3.84

500

4.42041

-

4.52671

-

1000

6.298720

6.55

6.48423

6.33

-

Cheng [12]

-

4.14

6.33

6.73

According to the Brinkmann model [13], the effective dynamic viscosity of the nanofluid is agreed by

Pnf

P f / (1  M )2.5

The thermal conductivity of the nano fluid, is modeled by Maxwell-Garnett Model [14] as

knf

k p  2k f  2M (k f  k p )

kf

k p  2k f  M ( k f  k p )

The average Nusselt number at the heated wall is calculated by integrating the local Nusselt number (Nu) is given by 1

Nu



The local Nusselt number is defined as

Nu



knf kf

³ Nu dX 0

knf wT , where n denotes the normal direction on a plane. k f wy

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Nithyadevi Nagarajan and Shamadhanibegum Akbar / Procedia Engineering 127 (2015) 279 – 286 Re=100

Ri=1

Ra=105

Ri=10

1.07

0.01 -0.06

-0.06

-0.04

-0.04

-0.03 -0.01

-0.06

-0.02 -0.03

-0.03 -0.05

0.

3

0.3

0.7

1.0

-0.36

0.5 1.0

1.0 0.8

0.8 0.4

0.5

0.3

0.4

0.6

0.7

0.7

1.0

0.8

0.5

0.36

0.5

0.6

0.6

0.7

-2.50

-0.02

-0.01

0.4

2.50

-0.04

-0.01

-0.05

-1.07

0.3

0.4 0.1

0.1

Fig. 2. Streamlines and isotherms for different convection regime (Ri) for Da =0. 1.( base fluid by solidlines( nanofluid ).

) and dashed lines (

) for

3. Method of Solution and Code Validation: In the current investigation, the dimensionless governing equations are discretized by using FVM method [15] and the SIMPLE algorithm is used to couple the pressure and velocity using under-relaxation technique with powerlaw scheme. The time step is taken to 10-5 for all computations. Uniform grid size 81u81 is taken to acquire the great accuracy in a numerical solution. An iterative process is utilized to find the velocity fields, stream function and temperature fields. The convergence process is continued till

In 1 (i, j )  In (i, j ) d 105. In 1 (i, j ) The exactness of the present numerical code and procedure was checked with some graphical results of Basak and Chamkha [2] in Figure 1b and the solution of Iwatsu et al [11] in the absence of porous medium. A fair degree of accuracy can be seen in graphs, also between the present computed average Nusselt number values in (Table 1). Table 2. Thermo physical properties of base fluid (water) and Copper U(kg/m3)

Cp(J/Kg.K)

k(W/m.K)

E(1/K)

Base fluid

997.1

4179.0

0.613

2.1u10-4

Copper

8933.0

385.0

401.0

1.67u10-5

4. Results and Discussion: This section describes, the numerical results for the mixed convection heat transfer in a square enclosure filled with fluid saturated porous medium by using copper-water nanofluid are explained. Under the mathematical model’s consideration, the problem is governed by the effects of pertinent parameters such as Richardson number

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Nithyadevi Nagarajan and Shamadhanibegum Akbar / Procedia Engineering 127 (2015) 279 – 286 Da=f

-0.07

Da=0.1

-0.04

Da=0.01

-0.01

-0.06

-0.01

-0.03 -0.06

-0.05 -0.02

-0.02

Da=0.001

-0.04

-0.02

-0.01

-0.05

-0.04

-0.02

-0.02

-0.01

-0.03 -0.01 0.5

0.4

0.5

0.3

0.4

0.5 0.7

0.7

0.7

0.6 0.7

0.8 1.0

1.0

1.0

1.0

0.8

0.8

0.6

0.6

0.5

0.6

0. 4

0.3

0.4

0.2

Fig. 3. Streamlines and isotherms for different Darcy number for Ri =1.0 .( base fluid by solidlines(

0.1 0.2

) and dashed lines (

) for nanofluid ).

(Ri), Darcy number (Da) and solid volume fraction (M) of the nanoparticle on the fluid flow and heat transfer inside the enclosure are investigated. The results are offered in the three different convection regimes namely forced convection (Re=100), mixed convection (Ri=1&10) and pure convection (Ra=105). The Darcy number is taken over a range of 10-3 to f, Da=f means the nonexistence of porous medium. The porosity value H=0.9 is fixed throughout this study. Figure 2 depicts the shear driven flow and heat transfer attributes for the various convection regime, for the fixed Darcy effect Da=0.1 and the enclosure is filled the base fluid (solid line) and copper nanofluid (dashed line) with M=0.1. Here the graphs in the forced convection and natural convection regimes are presented for the comparison intention with the graphs in mixed convection regime. The basic flow structure has a pair of anticlockwise cells at the upper and lower half of the enclosure and the cells are accumulating at the centre due to the driving mechanism of the mid moving plate at Re=100. For the low Richardson number, the flow is homogeneous with the forced convection one. Enhancing buoyancy force results the formation of positive secondary cells at the left top and bottom corners of the enclosure. At the natural convection regime, the strong dual cell equally shares the upper half of the enclosure and the weak cells are placing at the lower half of the enclosure due to the dominating buoyancy force. Increasing Richardson number results the convection vigorous its strength. Effect of Darcy number in the enclosure filled nanofluid is plotted in Fig.3 at the mixed convection regime Ri=1. In the beginning, the absence of Darcy effect (Da=f) pointing that the existence of dual cells with weak circulations and observed the dominating convection mode of heat transfer in enclosure. When the Darcy number starts to decrease, the magnitude of the stream function diminishes and core cells are elongated near the moving plate of the enclosure. This is connected with the reason of reducing permeability of the porous medium, controls the fluid flow through porous medium. Also, the conduction mode prevails for the Darcy number Da=0.001 due to the high flow resistance in porous bed. Evaluating heat transfer rate over the heated wall of the enclosure is most important part of this problem which is applicable in many engineering fields. The average heat transfer rate is calculated over the various solid volume fractions for different convection regime and Darcy number in Fig.4. Enhancing solid volume fraction results the augmenting heat transfer rate for all the convection regime and Darcy number. The high heat transfer

Nithyadevi Nagarajan and Shamadhanibegum Akbar / Procedia Engineering 127 (2015) 279 – 286

26

285

24

24

22

Ri=1, Re=100

22 20

20

Ri=10

18

Da = 0.001, 0.01, 0.1, f

18 Nu Nu16 16 14 5 Ra=10 12 14 10 12 8 6 10 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 M M (a) (b) Fig. 4. Average Nusselt number for various (a) Convection regime (b) Darcy number

0.5

Da=0.1, M=0.1

0.4

Y

0.5

0.3

0.3

0.2

0.2

0.1

0.1 Y -0.0

-0.0 -0.1 -0.2 -0.3 -0.4

Ri=10, M=0.1

0.4

-0.1 Re=100 Ri=1.0 Ri=10 Ra=10

-0.2 -0.3

5

-0.4

-0.5

-1.0 -0.8 -0.6 -0.4 -0.2 -0.0 0.2

U

0.4

0.6

0.8

1.0

Da=f Da=0.1 Da=0.01 Da=0.001

-0.5 -1.0 -0.8 -0.6 -0.4 -0.2 -0.0 0.2 0.4 0.6 0.8 1.0 U

(a) (b) Fig. 5. Horizontal velocities for various (a) Convection regime (b) Darcy number

rate occurs at the forced convection regime due to the dominated shear force and the low buoyancy force produces the less heating rate at the natural convection regime in Fig. 4(a). The maximum heat transfer rate comes about the absence of Darcy effect and minimum one is at Da=0.001 by the nature of low permeability in porous medium in Fig. 4(b). To measure the strength of the flow within the enclosure, the horizontal velocity profiles are designed at the mid-section of the enclosure for various convection regime and Darcy number in Fig. 5. Fig 5(a) is drawn for Da=0.1 and M=0.1, the augmenting Richardson number stimulates the flow and the high flow velocity is obtained for the natural convection regime due to the leading buoyancy force. Figure 5(b) is drawn at the mixed convection regime (Ri=10) for the solid volume fraction M=0.1. It is observed that, the decreasing Da be a sign of depression on the flow by the nature that the permeability of porous bed has the ability to decelerate the moment of fluid particle.

4. Conclusion: Systematic numerical simulations of mixed convection flow induced by a continuously moving hot plate through the mid-section of the fluid saturated porous enclosure by using copper-water nanofluid are performed to

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Nithyadevi Nagarajan and Shamadhanibegum Akbar / Procedia Engineering 127 (2015) 279 – 286

characterize the flow and heat transfer behaviours are reported in this study. Fluid velocity and heat transfer characteristics are mainly pretentious by the flow controlling parameters Ri, Da. The considerable revelation of conduction is established by the presence of porous medium. The average Nusselt number enhances for the augmenting Richardson number and the lessen Darcy number.

Acknowledgment: The author A.SHAMADHANI BEGUM is gratefully acknowledges for the financial support from the UGC, New Delhi for BSR-JRF Fellowship (F.25-1/2013-14(BSR)/7-27/2007(BSR) Dated 30.05.2014).

References: [1] S. Choi, Enhancing Thermal Conductivity of Fluids With Nanoparticles, ASME 66 (1995) 99–105. [2] T. Basak, A.J. Chamkha, Heatline analysis on natural convection for nanofluids confined within square cavities with various thermal boundary conditions, International Journal of Heat and Mass Transfer, 55 (2012) 5526–5543. [3] F. Talebi, A.H. Mahmoudi, M. Shahi, Numerical study of mixed convection flows in a square lid-driven cavity utilizing nanofluid, International Communications in Heat and Mass Transfer, 37 (2010) 79 –90. [4] M. N. Hasan, K. Samiuzzaman, H. Haytul Haque, S. Saha, Md. Quamrul Islam, Mixed convection heat transfer Inside a Square Cavity Filled with Cu-water Nanofluid, Procedia Engineering, 105 (2015) 438-445. [5] M.T. Nguyen, A.M. Aly, and Sang-Wook Lee, Natural convection in a non-darcy porous cavity filled with cu–water nanofluid using the characteristic-based split procedure in finite-element method, Numerical Heat Transfer, Part A , 67 (2015) 224–247. [6] A.J. Chamkha, E. Abu-Nada, Mixed convection flow in single-and double-lid driven square cavities filled with water–Al2O3 nanofluid: Effect of viscosity models, European Journal of Mechanics B/Fluids,36 (2012) 82–96. [7] T. Basak, S. Roy, S.K. Singh, I. Pop, Analysis of mixed convection in a lid-driven porous square cavity with linearly heated side walls, International Journal of Thermal Sciences, 53(2010)1819-1840. [8] M.A.R. Sharif, Laminar mixed convection in shallow inclined driven cavities with hot moving lid on top and cooled from bottom, Appl. Therm. Eng., 27 (2007) 1036-1042. [9] M.A. Waheed, Mixed convective heat transfer in rectangular enclosures driven by a continuously moving horizontal plate, Int. J. Heat Mass Transfer, 52 (2009) 5055-5063. [10] K. Khanafer, A.J. Chamkha, Mixed convection flow in a lid-driven enclosure filled with a fluid-saturated porous medium, Int. J. Heat Mass Transfer, 42 (1999) 2465–2481. [11] R. Iwatsu, J.M. Hyun, K. Kuwahara, Mixed convection in a driven cavity with a stable vertical temperature gradient, Int. J. Heat Mass Transfer, 36 (1993) 1601-1608. [12] T.S. Cheng, Characteristics of mixed convection heat transfer in a lid-driven square cavity with various Richardson and Prandtl numbers, Int. J. Therm. Sci. 50 (2011) 197-205. [13] H.C. Brinkman, The Viscosity of Concentrated Suspensions and Solution, J. Chem. Phys., 20 (1952 ) 571– 581. [14] J.C. Maxwell-Garnett, Colours in Metal Glasses and in Metallic Films, Philos.Trans. R. Soc. A, 203 (1904) 385–420. [15] S. V. Patankar, Numerical Heat Transfer and Fluid Flow, Hemisphere , Washington, D.C, 2004. .