Lattice Boltzmann simulation of MHD natural convection in a nanofluid filled enclosure with non-uniform heating on both side walls [men Mejrii, Ahmed Mahmoudii, Mohamed Ammar Abbassii, and Ahmed Omrii i UR:Unite de Recherche Materiaux, Energie et Energies Renouvelables (MEER), Faculte des Sciences de Gafsa, B.P.19, Zarroug, Gafsa, 2112, Tunisie [men Mejri :
[email protected]
Abstract
This paper examines the natural convection m a square enclosure filled with a water-Ab03 nanofluid and is subjected to a magnetic field. The side walls of the cavity have spatially varying sinusoidal temperature distributions. The horizontal walls are adiabatic. Lattice Boltzmann method (LBM) is applied to solve the coupled equations of flow and temperature fields. This study has been carried out for the pertinent parameters in the following ranges: Rayleigh number of the base fluid, Ra= [03 to [05, Hartmann number varied from Ha=O to 90, phase deviation (y=O, n14, n12, 3nl4 and n) and the solid volume fraction of the nanoparticles between
9=
0
and 6%. The results
show that the heat transfer rate increases with an increase of the Rayleigh number but it decreases with an increase of the Hartmann number. For y=nl2 and Ra= I 05 the magnetic field augments the effect of nanoparticles. At Ha=O, the greatest effects of nanoparticles are obtained at y = 0 and nl4 for Ra=104 and [05 respectively. Keywords: Lattice Boltzmann Method, Natural
convection, nanofluid, magnetic field, Sinusoidal temperature distribution I.
Introduction
The problem of natural convection in square enclosures has many engineering applications such as: cooling systems of electronic components, building and thermal insulation systems, built-in-storage solar collectors, nuclear reactor systems, food storage industry and geophysical fluid mechanics [1]. Some practical cases such as the crystal growth in fluids, metal casting, fusion reactors and geothermal energy extractions, natural convection is under the influence of a magnetic field [2-3]. The LBM is an applicable method for simulating fluid flow and heat transfer [4-5]. The aim of the present study is to identify the ability of Lattice Boltzmann Method CLBM) for solving nanofluid, magnetic field simultaneously in the presence of a sinusoidal boundary condition. Moreover, the effects of
magnetic field and phase drviations on the heat transfer in the cavity. In fact, it is endeavored to express the best situation for heat transfer and fluid flow with the considered parameters. Hence, the AbOrwater nanofluid on laminar natural convection heat transfer at the presence of a magnetic field in sinusoidal temperature distribution on vertical side walls of the cavity by LBM was investigated. II. Mathematical formulation A. Problem statement
A two-dimensional square cavity is considered for the present study with the physical dimensions as shown in the Fig. 1. The bottom wall of the cavity is maintained at a uniform temperature and the top wall is insulated, the left and right vertical walls are heated linearly. The cavity is filled with water and Alz03 nanoparticles. The nanofluid is Newtonian and incompressible. The flow is considered to be steady, two dimensional and laminar, and the radiation effects are negligible. The thermo physical properties of the base fluid and the nanoparticles are given in Table 1. TABLE. [ - Thermo-physical properties of water and nanoparticles p
Pure water Ah03
(kg 1m3)
cp
(J/kg K)
997.1
4179
3970
765
K (W/mK)
� (K·')
0.613
21xlO·'
40
0.85x10.5
The density variation in the nanofluid is approximated by the standard Boussinesq model. The magnetic field strength Bo is applied at an angle y with respect to the coordinate system. It is assumed that the induced magnetic field produced by the motion of an electrically conducting fluid is negligible compared to the applied magnetic field. Furthermore, it is assumed that the viscous dissipation and Joule heating are neglected. Therefore, governing equations can be written in dimensional form as follows:
978-1-4799-2516-2/14/$31.00 ©2014 IEEE
dU dX
+
dV =O dy
(\)
dU dU dp d2U d2U p(u-+v-)=--+/1(-+-)+Fx III �I dx2 dy dx dx dl dv
dv
dp
d2v
d2v
P (U-+V-)=--+/1 (-+-)+F nf dx nf dx2 dl Y dy dy aT
a2T
aT
=a ( nt ax2 ay
u - +v -
ax
F,
Ha2 Ji,lj
=
-
a2T
+ -)
(3) (4)
ay2
.
(5)
•. ·
(6)
" ,R (u -----;j2 smycosy- vcos" y)+ (PI' )n/g(T - 1',,,)
IIa =
(2)
IIB"� II :::l
0 II > II :::l
� + I .......
I .......
�
�
N
N
C . (ii
C .(ii
-1
-$
�
�
H
Adiabatic wall
u=v=Q
u
x
Figure I . Geometry of the present study with boundary conditions
a
b
c
Figure2.
Variation of the average Nusselt number and dimensionless average Nusselt number as function of solid volume fraction for different Hartmann number for y=n:12, Ra=1 03 (a) Ra=1 O� (b) and Ra=1 05 (c)
a
1,21)
4,_
---r=O
4,_
--r=O
-o-r=nl4 4,3
-o-r=nl4
----A- r=n12
1,15
--'-r=3n14
4,2
---r=nl2 --'-r=3n14
�Y=1t
�r=n
4,1
" Z
'
4,0
" Z
3,.
1,10
3,' 1,05 3,7 3,. 1,00
3,_ 0,00
0,02
0,00
0,06
0,04
0,02
b
0,04
0,06
4,9
1,20
4,8
---- y=O
4,7
-o-r=nl4
4,6
1,15
--- r=nl2 --'-r=3n14
4,5 4,4
" Z
4,3
. " Z
4,2
1,10
4,1
�y=O
4.0
-o-r="'4
3,9
1,05
�y=x12
3,8
--'-y=3"'4
3,7
----"f--
0,04
0,02
y=1t 0.00
0,02
C
',.
0,04
0,06
----y=O
., -
1,08
-o-r=nl4 ---r=nl2 --'-r=3n/4
.,2 1,06
.,0
Z
0,06
1,10
.,.
::::J
0,04
, " Z
7,8 7,.
�y=1t
1,04
7,_ 7,2
1,02
7,0 .,. 0.00
0,02
0,04
0,06
0,02
Figure3. Variation of the average Nusselt number and dimensionless average Nusselt number as function of
solid volume fractionfor different phase deviations for Ha=O, Ra=1 03 (a) Ra=1 O� (b) and Ra=1 05 (c)
