INTRODUCTION
Natural convection in rectangular enclosures has numerous
engineering applications, among which are electronic packages, solar
collectors, thermal design of buildings, storage systems, cooling of
nuclear reactors. A subdivision of the natural convection problem in a
rectangular cavity is the case where one wall is partially/fully heated
and the opposite wall is partially/fully cooled while the other two
walls are kept adiabatic. This cavity configuration is of special
interest in many engineering applications, such as solar receivers,
solar passive design and cooling of electronic equipment.
Azwadi et al. (2010) studied the natural convection heat transfer
inside an inclined enclosure and observed that the natural convection
increased as the inclination angle increased until it reached the
critical angle after which the natural convection started to decrease.
Munir et al. (2011) investigated the natural convection in an inclined
square cavity and found that the vortex formation, size and flow
characteristics were significantly affected by the magnitude of
inclination angles.
Magnetoconvection in square enclosures has numerous engineering
applications. Some of them are solidification process to weaken the
buoyancy-driven fluctuations, to modify the interface shape and the
rates of solidification in the manufacturing process of semiconductor
crystals. Mehmet and Buyuk (2006) studied the steady laminar natural
convection flow in the presence of a magnetic field in an inclined
enclosure heated from one side and cooled from the adjacent side and
observed that the counterclockwise inclination affected the temperature
field significantly. Kandaswamy et al. (2008) investigated the natural
convection in a square cavity filled with an electrically conducting
fluid with partially active vertical walls, for nine different
combinations of active locations in the presence of external magnetic
field parallel to the gravity and observed that the heat transfer is
maximum for the middle-middle thermally active locations while it is
poor for the top-bottom thermally active locations.
From the literature survey, it is observed that the analysis of
buoyancy driven flow in an inclined square cavity with vertical walls
each consisting of alternating equal sized heated and cooled portions
facing each other in the presence of magnetic field has not received
much attention. Hence, in the present study, an attempt is made to
investigate the influence of buoyancy and external magnetic forces in an
inclined square enclosure filled with a perfectly electrically
conducting fluid. The objective of the present study is to investigate
the effect of magnetic field on natural convection in the enclosure for
a wide range of Grashof number, Hartmann number at different angles of
inclinations.
MATERIALS AND METHODS
In the present study a two dimensional inclined square cavity of
length 1 as shown in Fig. 1a is considered. Initially at time t = 0, the
fluid is assumed to be motionless and at a uniform temperature
[[theta].sub.c]. The heated portion is kept at a temperature
[[theta].sub.h] and the cooled portion at temperature [[theta].sub.c]
along the vertical walls with [[theta].sub.h] > [[theta].sub.c] in an
opposed manner. Other two walls are maintained at adiabatic condition.
The heated portion along the left vertical wall is making an angle
[delta] with the horizontal direction. At [delta] = 90[degrees], the
heated and cooled portions are along the vertical walls. It is further
assumed that all other thermodynamic properties are independent of
temperature and that compressibility and dissipation effects are
negligible. The flow within the enclosure is laminar and gravitational
acceleration acts parallel to the vertical walls. Let u, v denote the
velocity components in the x and y directions respectively. An external
magnetic field is assumed to be applied parallel to gravity and the
induced magnetic field is neglected.
The conservation equations for an unsteady laminar two dimensional
flow of fluid in the presence of a magnetic field under Boussinesq
approximation are as given below in Eq. 1-4:
[partial derivative]u/[partial derivative]x + [partial
derivative]v/partial derivative]y = 0 (1)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)
[partial derivative][theta]/[partial derivative]t + u [partial
derivative][theta]/[partial derivative]x - v [partial
derivative][theta]/[partial derivative]y = K/[rho][C.sub.p] [[[partial
derivative].sup.2][theta]/[partial derivative][x.sup.2] + [[partial
derivative].sup.2][theta]/ [partial derivative][y.sup.2]] (4)
The initial and boundary conditions are considered as in Eq. 5:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)
Introducing the following dimensionless variables as defined below:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (6)
The above Eq. 1-4 get modified as:
[partial derivative]U/[partial derivative]X + [partial
derivative]V/[partial derivative]Y = 0 (7)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)
[partial derivative]T /[partial derivative][tau] + U[partial
derivative]T/[partial derivative]X + V[[partial
derivative].sup.T]/[partial derivative]Y = 1/Pr [[[partial
derivative].sup.2]T/[partial derivative][X.sup.2] + [[partial
derivative].sup.2]T/[partial derivative][Y.sup.2] (10)
[FIGURE 1 OMITTED]
The initial and boundary conditions are given in the non
dimensional form as:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)
Introducing the vorticity [zeta] and the stream function [psi], the
equations governing the problem can be written as:
[[nabla].sup.2] [psi] = - [zeta] (12)
U = [partial derivative] [psi]/[partial derivative]Y, V = -
[partial derivative] [psi]/ [partial derivative]X (13)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)
[partial derivative]T/[partial derivative][tau] + U [partial
derivative]T/[partial derivative]X + V[partial derivative]T/[partial
derivative]Y = 1/PR [[nabla].sup.2]T (15)
The above Eq. 12-15 are called as the stream function, velocity,
vorticity and temperature equations, respectively. An approximation to
their solution will be obtained at a finite number of grid points at
discrete time [tau] = n [DELTA][tau] where n is an integer. It is
assumed that all quantities are known at a time n[DELTA][tau]. An
implicit alternating direction technique based on suitable finite
difference approximations of the vorticity and temperature equations is
employed to advance the fields of vorticity and temperature at the
interior grid points across a time step n[DELTA][tau] to the new level
(n+2)[DELTA][tau]. The method of Successive over Relaxation is then
employed in conjunction with the newly computed vorticity to solve the
stream function equation. The new fields of U and V are obtained from
space centered finite-difference approximations of the velocity
equations. This computational cycle is then repeated for each of the
next time steps until a steady state situation is obtained, when the
convergence criteria, [[[phi].sub.n+1] (i, j) -[[phi].sub.n] (i,
j)/[[phi].sub.n+1] (i,j)] [less than or equal to] [epsilon] for
temperature, vorticity and stream function have been satisfied. In the
above expression, n is any time level, [epsilon] is of order [10.sup.-5]
and [phi] represents T, [zeta] and [psi]. The numerical solutions
presented in this study were acquired from a 41 x 41 grid system and
with a time increment of order [10.sup.-3]. Further increase in the
number of grids produced essentially the same results as seen in Fig.
1b.
Prior to the present calculations, as a partial verification of the
computational procedure, the results of average Nusselt number at
different Grashof numbers and Hartmann numbers were estimated using the
FORTRAN code developed and compared with the solutions given by Rudraiah
et al. (1995) as seen in Table 1 in which they have studied the natural
convection in a square enclosure and are found to be in good agreement.
RESULTS
Heat transfer in an inclined square cavity for differentially
thermally active locations, inclination angles, Grashof numbers and
Hartmann numbers is studied numerically in the presence of a magnetic
field. The computations are carried out for Grashof numbers (Gr) in the
range from 500-50000, Hartmann number (Ha) from 1-10, angles of
inclination [delta] from 30-180[degrees] and at Prandtl number (Pr) =
0.733. The mid height vertical velocity profiles at different Grashof
numbers, Hartmann numbers and inclination angles are presented.
DISCUSSION
When Gr = 50000, Ha= 5 and all angles of inclination from
30-180[degrees], it is observed in Fig. 2a, two counter rotating cells
appear in the enclosure. Each thermally active location generates an
identical upward buoyancy force and therefore dual cell flow is obtained
with fluid flowing down the middle of the enclosure. The dual cell
structure prohibits direct convective heat transfer between the active
locations. Each cell behaves like an independent one preventing warm
(cool) fluid from the hot (cold) location mixing with cool (warm) fluid
from the cold (hot) location. The higher values of the streamlines
indicate stronger rotation due to the higher value of Grashof number.
The effect of stronger circulation is also displayed by the isotherms as
seen in Fig. 2b.
It is noted in Fig. 3a, when Gr = 50000, Ha = 10 and at [delta] =
30[degrees], two counter rotating cells appear with a primary cell
rotating in clockwise direction. At [delta] = 60[degrees], the secondary
cell rotating in anti-clockwise direction, grows significantly and
suppresses the primary cell. The fluid flow gains strength with
[[psi].sub.max] = 17 at [delta] = 120[degrees]. By increasing angle of
inclination from 120[degrees] in steps of 30[degrees], the change in the
direction of the flow pattern is noticed. Strength of the flow increases
when Gr increases from 5000-50,000 as seen from the higher values of the
streamlines. It is found that in Fig. 3b, the isotherms are greatly
influenced by the inclination of the enclosure.
[FIGURE 2 OMITTED]
[FIGURE 3 OMITTED]
[FIGURE 4 OMITTED]
[FIGURE 5 OMITTED]
The mid height velocity profiles at Gr = 5000-25000 and 50000, when
Ha = 3 and [delta] = 30[degrees] are presented in Fig. 4a. It is
observed that for increase in Grashof number, the mid height velocity
increases, as significant increase in the flow strength contribute to
higher velocity values. Also as Hartmann number increases, the heat
transfer is suppressed and hence the vertical velocity decreases, as
seen in Fig. 4b. The effect of inclination angles for [delta] =
30-180[degrees], when 50000 and for fixed Ha = 3 on the mid height
velocity profiles are presented in Fig. 4c. It is observed that the
velocity U increases for angles 30-90[degrees] and decreases for angles
from 120-180[degrees]. At [delta] = 180[degrees], Gr = 50000 and Ha = 3,
the vertical velocity decreases appreciably and hence the velocity
profile is flattened. The effect of the angles of inclination on the
average Nusselt number, [bar.[N.sub.u]], at Gr = 5000, 25000, 50000 and
Ha = 5 are presented in Fig. 5a. The average Nusselt number behaves in a
non-linear fashion with angles of inclination. The heat transfer is
found to be maximum at [delta] = 120[degrees] and minimum at [delta] =
180[degrees], irrespective of the Grashof number. It is also observed
that the average Nusselt number [bar.[N.sub.u]] increases considerably
with Grashof number since the circulation becomes stronger. The
influence of angles of inclination on [bar.[N.sub.u]] for Hartmann
numbers Ha = 3, 5, 7, 10 is illustrated in Fig. 5b. It is observed that
the behavior of average heat transfer coefficient is a non-linear
function of inclination angles at all values of Hartmann numbers. Angle
of inclination has remarkable effect on the average Nusselt number
[bar.[N.sub.u]] at lower values of the Hartmann number Ha = 3, Ha = 5
and the effect is less significant when Hartmann number is increased to
Ha = 10. It is also observed that the average Nusselt number
[bar.[N.sub.u]] is a decreasing function of Hartmann number.
CONCLUSION
The present study considers laminar natural convection flow in the
presence of a magnetic field in an inclined square enclosure with
differentially thermally active vertical walls while the horizontal
walls are kept adiabatic. The associated flow characteristics and heat
transfer inside the titled enclosure are found to depend strongly upon
the strength of the magnetic field and the inclination angles. The mid
height vertical velocity increases when Grashof number increases and
decreases when the Hartmann number increases. Angle of inclination has
remarkable effect on the average Nusselt number [bar.[N.sub.u]] at lower
values of the Hartmann number and the effect is less significant at
higher values of Hartmann number. At higher values of the Grashof
number, the heat transfer is maximum at [delta] = 120[degrees] and
minimum at [delta] = 180[degrees]. The average Nusselt number behaves in
a nonlinear fashion with angles of inclination.
REFERENCES
Azwadi, C.S.N., M.Y.M. Fairus and S. Syahrullail, 2010. Virtual
study of natural convection heat transfer in an inclined square cavity.
J. Applied Sci., 10: 331-336. DOI: 10.3923/jas.2010.331.336
http://docsdrive.com/pdfs/ansinet/jas/2010/331 336.pdf
Kandaswamy, P., S. Malliga Sundari and N. Nithyadevi, 2008. Magneto
convection in an enclosure with partially active vertical walls, Int. J.
Heat. Mass Trans., 51: 1946-1954.
DOI:10.1016/J.IJHEATMASSTRANSFER.2007.06.025
Munir, F.A., N.A.C. Sidik and N.I.N. Ibrahim, 2011. Numerical
simulation of natural convection in an inclined square cavity. J.
Applied Sci., 11: 373378. DOI: 10.3923/jas.2011.373.378
Mehmet, C.E. and E. Buyuk, 2006. Natural convection flow under a
magnetic field in an inclined rectangular enclosure heated and cooled on
adjacent walls. Fluid Dynam. Res., 38: 564. DOI:
10.1016/J.FLUIDDYN.2006.04.002
Rudraiah, N., R.M. Barron, M. Venkatachalappa and C.K. Subbaraya,
1995. Effect of a magnetic field on free convection in a rectangular
enclosure. Int. J. Eng. Sci., 33: 1075-1084.
http://cat.inist.fr/?aModele=afficheN&cpsidt=3495002
(1) Gokila Subbarayalu and (2) Selladurai Velappan
(1) Department of Mathematics,
(2) Department of Mechanical Engineering, Coimbatore Institute of
Technology, Coimbatore-641014, Tamilnadu, India
Corresponding Author: Gokila, Department of Mathematics, Coimbatore
Institute of Technology, Coimbatore-641014, Tamilnadu, India Tel:
0422-2574071, 2574072/(0) 9944145334
Table 1: Comparison of average Nusselt numbers
for different Hartmann numbers at Gr = 20000
Gr Ha [[bar.N].sub.u] [[bar.N].sub.u] * u
20000 0 2.5188 2.1340
10 2.2234 2.0233
50 1.0856 1.0219
100 1.0110 1.0019
[[bar.N].sub.u] = Results of Rudraiah et al.
(1995) [[bar.N].sub.u] * = Present study