Accepted Manuscript Numerical analysis of nanofluid transportation in porous media under the influence of external magnetic source
M. Sheikholeslami, D.D. Ganji PII: DOI: Reference:
S0167-7322(17)30049-1 doi: 10.1016/j.molliq.2017.03.050 MOLLIQ 7085
To appear in:
Journal of Molecular Liquids
Received date: Revised date: Accepted date:
4 January 2017 13 February 2017 14 March 2017
Please cite this article as: M. Sheikholeslami, D.D. Ganji , Numerical analysis of nanofluid transportation in porous media under the influence of external magnetic source. The address for the corresponding author was captured as affiliation for all authors. Please check if appropriate. Molliq(2017), doi: 10.1016/j.molliq.2017.03.050
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ACCEPTED MANUSCRIPT Numerical analysis of nanofluid transportation in porous media under the influence of external magnetic source
M. Sheikholeslami a,1, D.D. Ganji b a
Department of Mechanical Engineering, Babol Noshirvani University of Technology, Babol, Iran
Department of Mechanical Engineering, Sari Branch, Islamic Azad University, Sari, Iran
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b
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Abstract
This article is made to examine the impact of external magnetic source on Fe3O4 –
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water ferrofluid convective heat transfer in a porous cavity. The solutions of final equations are obtained by Control volume based finite element method (CVFEM). Graphs are shown for various values of Darcy number Da , Fe3O4 -water volume fraction , Rayleigh Ra
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and Hartmann Ha numbers. Results indicate that augmenting in Hartmann number results in reduce in velocity of nanofluid and augment the thermal boundary layer thickness. Adding
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nanoparticles in the based fluid is more effective for higher values of Hartmann number and
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lower values of Darcy number.
Keywords: Porous media; External magnetic source; Nanofluid; MFD viscosity; Natural convection.
Magnetic
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B
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Nomenclature
induction
K
Permeability
Rotation angle
&
dimensionless vorticity
[Tesla]
& stream function Da
Darcy number
Thermal expansion coefficient [1/K]
1
Corresponding author: Email:
[email protected] (M. Sheikholeslami),
[email protected] (D.D. Ganji)
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ACCEPTED MANUSCRIPT Rayleigh
Ra
Fluid density [kg/m3]
Electrical conductivity
Dynamic viscosity
number Fluid
T
temperature [K] Nusselt number
[Pa.s]
Hartmann
Subscripts
number
Gravitational
nf
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g
acceleration
Vertical and
Nanofluid
f
Base fluid
loc
Local
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horizontal
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vector V ,U
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Ha
PT
Nu
dimensionless velocity [m/s] Y ,X
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Vertical and
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horizontal space
coordinates
dimensionless
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Greek symbols
temperature
1. Introduction New types of fluid are needed to obtain more efficient performance in new days. Nanofluid was proposed as innovative way to enhance heat transfer. Beg et al. [1] examined the bio-nanofluid transport phenomena by means of both single and two phase models. Wavy duct in existence of Brownian forces has been examined by Shehzad et al. [2]. They selected Nelder-Mead method to find the solution. Impact of Lorentz force on boundary layer flow has been analyzed by Beg et al. [3]. Sheikholeslami [4] investigated nanofluid forced 2
ACCEPTED MANUSCRIPT convection in a three dimensional porous cavity. He selected LBM for this problem. Rashidi et al. [5] investigated the nanofluid free convection flow over a plate. Garoosi et al. [6] examined the simulation of nanofluid by means of Buongiorno model. Sheikholeslami and Ganji [7] presented various application of nanofluid in their review article. Influence of nonlinear radiative heat transfer has been examined by Hayat et al. [8]. Influence of Coulomb forces on ferrofluid convection was analyzed by Sheikholeslami and Chamkha [9]. They concluded that augmenting Coulomb force has more
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profit in low Reynolds number. Sheikholeslami and Bhatti [10] presented an active method for heat transfer augmentation by means of electric field. Chamkha and Rashad [11] reported
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the nanoparticle migration on porous cone. Ellahi et al. [12] analyzed free convection of carbon nanotubes over a cone. They considered the Lorentz forces impact in governing
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equations. Raju et al. [13] studied the transient ferrofluid flow over a cone. Makinde et al. [14] presented the radiative heat transfer of nanofluid with variable viscosity. Mezrhab et al.
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[15] reported the radiation impact in a cavity. Sheikholeslami and Shehzad [16] examined the influence of thermal radiation on ferrofluid flow in existence of uniform magnetic field. Influence of asymmetric heating on the Nusselt number in a microchannel has been reported
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by Malvandi et al. [17]. Their outputs illustrated that Ha augments the Nu about 42%. Sheikholeslami et al. [18] investigated the impact of magnetic field on transportation of nanofluid in a porous media. Sheikholeslami and Rokni [19] reported the influence of
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variable Lorentz force on nanofluid free convection heat transfer.
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Chen et al. [20] studied the performance of solar collectors by using silver nanoparticle. Sheikholeslami and Ellahi [21] selected LBM to simulate Lorentz forces influence on nanofluid convective heat transfer. They depicted that temperature gradient reduces with augment of magnetic strength. Kefayati [22] studied the Soret and Dufour
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influences on MHD natural convection of power-law fluid. He proved that Nu augments with augment of Dufour parameter. Effect of Marangoni convection on nanofluid flow in
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existence of magnetic field has been investigated by Sheikholeslami and Chamkha [23]. Sheikholeslami et al. [24] reported the impact of inconstant Lorentz force on forced convection. They illustrated that higher lid velocity has more sensible Kelvin forces effect. Several papers have been published in recent decade about nanofluid hydrothermal analysis [25-40]. This paper deals with influence of external magnetic source on nanofluid flow in a porous enclosure. CVFEM is selected to simulate this problem. Roles of Darcy number,
Fe3O4 -water volume fraction, Hartmann and Rayleigh numbers are examined.
2. Problem statement 3
ACCEPTED MANUSCRIPT Boundary conditions are depicted in Fig. 1. The inner elliptic wall has constant temperature considered as hot wall. Outer circular wall is cold wall, the others are adiabatic. Magnetic source has been considered as shown in Fig. 2. H x , H y , H can be calculated as follow:
Hx b y
2
2
2
ax
ax
2
2
ax , 2
(1)
y b , 2
(2)
1
1
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Hy b y
2
(3)
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H H x H y .
3. Simulation method
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3.1. Governing formulation
2D laminar nanofluid flow and free convective heat transfer is taken into account.
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The governing PDEs are:
v u 0, y x
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u u 2u 2u P nf u v 2 2 nf x x x y y
(4)
nf H u nf H x H y v 2 0
v
2 y
2 0
v
2v
nf K
u, 2v P
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nf u v nf 2 2 y y y x x 02 H y nf H x u 02 H x nf H x v nf v T T c nf g nf ,
(5)
(6)
K
2 2T 2T T T 2 v u H v H u k 2 p nf nf 0 x y nf 2 x y y x 2 2 u 2 v u v nf 2 , 2 y y x x
C
(7)
nf , C p nf , nf , k n f and nf are calculated as
nf f (1 ) s ,
(8) 4
ACCEPTED MANUSCRIPT C p
nf
C p (1 ) C p , f
(9)
s
nf f (1 ) s ,
(10)
k 2 (k f k s ) 2k f knf kf s , k s ( k f k s ) 2k f
(12)
3 1 1
1 , 1 2 1 1
nf f
(13)
1 s / f .
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n f is obtained as follows [41]:
nf 0.03502 H 2 3.10 H 27886.4807 2 4263.02 316.0629e 0.01T
f
,V
vL
f
,
H
y
,H x ,H H0
T T c , X ,Y T h T c
So equations change to:
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p
f f / L
2
x , y L
.
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/ f 2U 2U U U V Pr nf 2 X Y X 2 nf / f Y / f P Pr nf / f 2 Ha 2 Pr nf H y U H x H yV U , / X Da nf / f nf f
U
(16)
(17)
2V V V 2V nf / f U Pr 2 Y X X 2 nf / f Y
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V
(15)
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V U 0, Y X
,P
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uL
L
y ,H x ,H
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U
b , a , H
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Dimensionless parameters are defined as:
b , a
(14)
/ f 2 Ha 2 Pr nf H xV H x H y U / nf f P Pr nf / f Ra Pr nf V , Y Da f nf / f
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(18)
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ACCEPTED MANUSCRIPT k nf U Y X k f
C P f C P nf
2 2 2 X 2 Y
C P f nf Ha 2 Ec V H x U C P nf f nf 2 f Ec 2 U 2 V C P X Y nf C P f
Hy
2
(19) 2 2 U V Y X
and dimensionless parameters are Raf g f L3 T / f f , Prf f / f , Ha L 0 H 0 f / f
(20)
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, Ec f f / C P f T L2 , Da K / L2 .
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V
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The thermo-physical properties of Fe3O4 and water are presented in Table1. Pressure gradient source terms discard by vorticity stream function. L2 u v , , , u ,v , . f f y x x y
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(21)
According to Fig. 1, boundary conditions are
1.0
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on inner wall
(22)
0.0
on outer wall
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on other walls
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on all walls
0 y
0.0
Nu loc , Nu ave along cold wall are: (23)
1 0.5
0.5
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Nu ave
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k Nu loc nf , k f r
(24)
Nu loc d ,
0
3.2. Numerical procedure Linear interpolation is utilized for approximation of variables in the triangular element which is considered as building block (Fig. 1(b)). Algebraic equations are solved via Gauss-Seidel method. More details exist in reference book [42].
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ACCEPTED MANUSCRIPT 4. Grid independency and code accuracy Different grids are tested in order to find the mesh independent out puts. For instance, according to table 2, the mesh size of 71 211 should be selected. The written codes have been verified with previous works for magnetic nanofluid convective heat transfer (see Fig. 3
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and Table3) [43-44].
5. Results and discussion
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In this article, the influence of magnetic field on Fe3O4-water nanofluid in a porous enclosure is reported. The governing equations have been solved via Control volume based
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finite element method and the outputs are depicted in several plots for the influence of various parameters on the flow and heat transfer. These parameters are Darcy number ( Da ),
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Rayleigh number ( Ra ), Hartmann number ( Ha ) and volume fraction of Fe3O4 ( ). Pr and
Ec are 6.8 and10-5, respectively.
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Fig. 4 demonstrates the influence of adding Fe3O4 in to water on hydrothermal characteristic. This figure depicts that an increase in nanoparticle volume fraction results in increase in nanofluid velocity. It is also found that the thermal boundary layer thickness of
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water based nanofluid is higher than pure fluid.
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Figs. 5, 6 and 7 illustrate the impact of Darcy, Hartmann and Rayleigh numbers on isotherms and streamlines. In domination of conduction modes, one main clock wise cell appears in half of the enclosure. An augment in magnetic field results in generate secondary
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cell near the vertical center line. As permeability of porous media enhances, convective heat transfer improves and thermal plume appears. An increase in buoyancy force results in
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enhance in strength of main eddy and generates thermal plume near the vertical centerline. As Lorentz forces augments, the position of thermal plume become far from vertical centerline. It is fantastic observation in case of Da=100, Ra=105, max reaches to its maximum value and the main eddy stretch horizontally. Also one powerful thermal plume generates near the 90 . Applying magnetic field for such case, converts the main eddy to three smaller ones. The middle one rotates counter clock wise. Existence of such eddies generates two thermal plume over the hot elliptic wall. Rate of heat transfer is depicted in Fig. 8. The formula of Nu ave corresponding to important parameters is:
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ACCEPTED MANUSCRIPT Nu ave 4.37 0.15Da * 2.77 log Ra 0.03Ha * 0.03Da * log Ra 0.21Da *Ha * 0.39log Ra Ha *
(25)
0.07Da *2 0.51 log Ra 0.61Ha *2 2
where Da* 0.01Da, Ha* 0.1Ha . Increasing in permeability of porous media results in augments in rate of heat transfer. Enhancing Rayleigh number makes the convective heat transfer to augments. So this non dimension parameter has similar effect on Nusselt number
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with that of obtained for Darcy number. As Lorentz force increases Nu ave reduces due to domination of conduction mode. Adding Fe3O4 nanoparticle into base fluid enhances the
100 * Nu ave
0.04
Nu ave
0
/ Nu
ave 0
. In conduction
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improvement. This output is defined as E
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Nusselt number. Table 4 demonstrates the influence of Da, Ha and Ra on heat transfer
mode, influence of adding nanoparticles has more benefit because of more changes in thermal conductivity.
Therefore, heat transfer improvement enhances with enhance of
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NU
Hartmann number but it detracts with rise of Darcy and Rayleigh numbers.
6. Conclusions
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In this article, influence of variable magnetic field on Fe3O4 – water nanofluid heat
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transfer in a porous enclosure has been conducted. The problem is modeled and then solved via CVFEM. Graphs are reported for various values of volume fraction of Fe3O4, Darcy number, Rayleigh and Hartmann numbers. Results depict that velocity of Fe3O4 based
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nanofluid is higher than based fluid. Increasing Lorentz forces results in augment in thermal
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boundary layer thickness. Adding Fe3O4 nanoparticles are more effective in presence of Lorentz forces. Rate of heat transfer improves with rise of Rayleigh and Darcy numbers.
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ACCEPTED MANUSCRIPT [3] O. A. Beg, A. Y. Bekier, V. R. Prasad, J. Zueco, S. K. Ghosh, International Journal of Thermal Sciences 48(2009) 1596–1606. [4] M. Sheikholeslami, Journal of Molecular Liquids, 2017, 10.1016/j.molliq.2017.02.020. [5] M. M. Rashidi, E. Momoniat, M. Ferdows, and A. Basiriparsa, Mathematical Problems in Engineering (2014) 239082, http://dx.doi.org/10.1155/2014/239082. [6] Faroogh Garoosi, Leila Jahanshaloo, Mohammad Mehdi Rashidi, Arash Badakhsh,
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Mohammed E. Ali, Applied Mathematics and Computation 254 (2015) 183–203.
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[7] M. Sheikholeslami, D.D. Ganji, Journal of the Taiwan Institute of Chemical Engineers 65
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(2016) 43-77
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Heat and Mass Transfer 102 (2016) 723-732
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2016, VOL. 69, NO. 7, 781–793, http://dx.doi.org/10.1080/10407782.2015.1090819 [10] M. Sheikholeslami, M. M. Bhatti, International Journal of Heat and Mass Transfer 109 (2017) 115–122
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[11] A.J. Chamkha and A. M. Rashad, International Journal for Numerical Methods in Heat
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and Fluid Flow, 22 (2012) 1073-1085. [12] R. Ellahi, Mohsan Hassan, A. Zeeshan, IEEE Transactions on Nanotechnology 14.4
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(2015): 726-734.
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(2016)1183-1191
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ACCEPTED MANUSCRIPT [18] M. Sheikholeslami, Z. Ziabakhsh, D.D. Ganji, Colloids and Surfaces A: Physicochemical and Engineering Aspects, 520 (2017) 201–212. [19] M. Sheikholeslami, Houman B. Rokni, Computer Methods in Applied Mechanics and Engineering, 317 (2017) 419–430 [20] Meijie Chen, Yurong He, Jiaqi Zhu, Dongsheng Wen, Applied Energy, 181 (2016) 6574
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[21] M. Sheikholeslami, R. Ellahi, International Journal of Heat and Mass Transfer 89 (2015)
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[22] G.H.R. Kefayati, Energy 107 (2016) 889-916
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799-808
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[24] Mohsen Sheikholeslami, Kuppalapalle Vajravelu, Mohammad Mehdi Rashidi, International Journal of Heat and Mass Transfer 92 (2016) 339–348.
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[25] M. Sheikholeslami, D.D. Ganji, Journal of Molecular Liquids, 229 (2017) 530–540. [26] M. Sheikholeslami, Physics Letters A, 381 (2017) 494–503
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[27] M. Sheikholeslami, K. Vajravelu, Applied Mathematics and Computation, 298 (2017)
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272–282
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237-246.
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419
[30] M. Sheikholeslami, International Journal of Hydrogen Energy, 42 (2017) 821-829 [31] Mohsen Sheikholeslami, Journal of Molecular Liquids 229 (2017) 137–147 [32] Mohsen Sheikholeslami, Eur. Phys. J. Plus (2016) 131: 413, DOI: 10.1140/epjp/i201616413-y [33] Mohsen Sheikholeslami, Davood Domiri Ganji, Energy 75 (2014) 400-410. [34] Noreen Sher Akbar, Zafar Hayat Khan, Journal of Molecular Liquids, 222 (2016) 279286.
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ACCEPTED MANUSCRIPT [35] Madavan R., Sujatha Balaraman, Journal of Molecular Liquids, 230 (2017) 437-444 [36] Asma Khalid, Ilyas Khan, Sharidan Shafie, Journal of Molecular Liquids 221 (2016) 1175-1183. [37] M. Sheikholeslami, The European Physical Journal Plus, (2017) 132: 55 DOI 10.1140/epjp/i2017-11330-3
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[38] M. Sheikholeslami, D.D. Ganji, Journal of Molecular Liquids 229 (2017) 566–573 [39] Mohsen Sheikholeslami, Journal of Molecular Liquids 225 (2017) 903–912
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[40] M. Sheikholeslami, Houman B. Rokni, International Journal of Heat and Mass Transfer
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107 (2017) 288–299.
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Communications in Heat and Mass Transfer 72 (2016) 23–28 [42] Mohsen Sheikholeslami, Davood Domairry Ganji, William Andrew, Elsevier, Print
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Book, (2016), pp. 1-354, ISBN: 9780323431385
[43] K. Khanafer, K. Vafai, M. Lightstone, Int. J. Heat Mass Transfer 46 (2003) 3639–3653.
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CE
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(1995) 1075–1084.
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[44] Rudraiah N, Barron RM, Venkatachalappa M, Subbaraya CK, Int. J. Engrg. Sci. 33
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ACCEPTED MANUSCRIPT
(b)
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(a)
(c)
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Fig. 1. (a)Geometry and the boundary conditions with (b) the mesh of Geometry considered in this
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work; (c) A sample triangular element and its corresponding control volume.
12
ACCEPTED MANUSCRIPT 9 8 7 6 5 4 3 2 1 0.5 -0.5 -1 -2 -3 -4 -5 -6 -7 -8 -9
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19 16 13 11 10 9 8 7 6 5 4 3 2
(a) H x , y
(b) Hx x , y
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-0.6 -1 -2 -3 -4 -5 -6 -7 -8 -11 -12 -15 -16 -19
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(c) Hy x , y
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Fig. 2. Contours of the (a) magnetic field strength H ; (b) magnetic field intensity component in x direction Hx ; (c) magnetic field intensity component in y direction Hy .
13
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ACCEPTED MANUSCRIPT
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Fig. 3. Comparison of temperature profile between the present results and numerical results
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by Khanafer et al. [43] Gr 104 , 0.1 and Pr 6.8 Cu Water .
14
ACCEPTED MANUSCRIPT
-10
15
-25
25
-25
-15
15 5
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-5
0.55 0.5
0.45
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0.45
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0.7
0 .3 5
0.35
NU
0.25
0.2
0.15
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Fig. 4. Impact of nanofluid volume fraction on streamlines (up) and isotherms (bottom) contours (nanofluid ( 0.04 )(––) and pure fluid( 0 ) (- - -)) when
AC
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Ra 10 5 ,Da 100 ,Ha 0
15
Ha=20
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Da=100
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Ha=0
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Ha=20
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Da=0.01
PT
Ha=0
ACCEPTED MANUSCRIPT
Fig. 5. Influence of Da,Ha on streamlines (left) and isotherms (right) contours when
0.04 ,Ra 10 3
16
Ha=20
AC
Da=100
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Ha=0
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Ha=20
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Da=0.01
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Ha=0
ACCEPTED MANUSCRIPT
Fig. 6. Influence of Da,Ha on streamlines (left) and isotherms (right) contours when
0.04 ,Ra 10 4
17
Ha=20
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Da=100
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Ha=0
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Ha=20
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Da=0.01
PT
Ha=0
ACCEPTED MANUSCRIPT
Fig. 7. Influence of Da,Ha on streamlines (left) and isotherms (right) contours when
0.04 ,Ra 10 5
18
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Ra 10 5
Da 100
AC
CE
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NU
Ha 10
Ha 10
Ra 10 5
19
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Da 100
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NU
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Fig. 8. Effects of Da,Ha and Ra on average Nusselt number
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ACCEPTED MANUSCRIPT Table1. Thermo physical properties of water and nanoparticles ( kg / m3 )
C p ( j / kgk )
k(W / m.k )
d p ( nm )
Pure water
997.1
4179
0.613
-
0.05
Fe3O4
5200
670
6
47
25000
PT
m
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Table2. Comparison of Nu ave along curved wall for different grid resolution
51 151
61 181
71 211
81 241
91 271
101 301
2.025558
2.026173
2.026883
NU
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at Ra 105 , Da 100, 0.04 , Ha 20, Ec 105 and Pr 6.8 .
2.027834
2.027903
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2.027783
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Table3. Average Nusselt number versus at different Grashof number under various strengths
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of the magnetic field at P r= 0.733.
Gr 2 104
CE
Ha
Present
2.5665
10
Rudraiah et
Present
al. [44]
Rudraiah et al. [44]
2.5188
5.093205
4.9198
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
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0
Gr 2 105
21
1
ACCEPTED MANUSCRIPT
Table4. Effects of Da,Ha and Ra on heat transfer enhancement Da
Ha
E
103
0.01
0
11.09238
103
0.01
20
11.42401
104
0.01
0
4.20269
104
0.01
20
7.652818
105
0.01
0
2.613391
105
0.01
20
3.245394
103
100
0
103
100
20
104
100
104
100
105
100
105
100
NU
SC
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PT
Ra
5.874633 7.615688 3.787017
20
6.464888
0
3.757416
20
1.560289
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CE
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D
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0
22
ACCEPTED MANUSCRIPT Highlights
Nanoparticles transportation in a porous semi annulus is examined Variable magnetic field effect on hydrothermal behavior is considered. CVFEM is applied to solve the governing equations
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Nu increase with increase of Ra and Da.
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