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A numerical study of magneto-hydrodynamic free convection in a square cavity with
heated elliptic block
M. Jahirul Haque Munshi, M. A. Alim, and A. H. Bhuiyan
Citation: AIP Conference Proceedings 1754, 040022 (2016);
View online: https://doi.org/10.1063/1.4958382
View Table of Contents: http://aip.scitation.org/toc/apc/1754/1
Published by the American Institute of Physics
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A Numerical Studyof Magneto-hydrodynamic Free
Convection in A Square Cavity with Heated Elliptic Block
M. JahirulHaque Munshi1,a), M. A. Alim2, A. H. Bhuiyan2
1
Department of Mathematics, Faculty of Science, Engineering & Technology,
Hamdard University Bangladesh (HUB), Gazaria,Munshigang, Bangladesh
2
Department of Mathematics, Bangladesh University of Engineering and Technology
(BUET), Dhaka- 1000, Bangladesh
a)
Corresponding author: [email protected]
Abstract. The problem of Magneto-hydrodynamic (MHD) field on buoyancy- driven free convection heat transfer in a
square cavity with heated elliptic block at the centre has been investigated in this work. The governing differential
equations are solved by using finite element method (Galerkin weighted residual method). The lower wall is adiabatic.
The left wall is kept at heated Th. The right and upper wall is kept at cold Tc respectively. Also all the wall are assumed
to be no-slip condition. The study is performed for different Rayleigh and Hartmann numbers. A heated elliptic block is
located at the centre of the cavity. The object of this study is to describe the effects of MHD on the field of buoyancydriven and flow in presence of such heated block by visualization of graph. The results are illustrated with the
streamlines, isotherms, velocity and temperature fields as well as local Nusselt number.
INTRODUCTION
Heat transfer using movement of fluids is called convection. Natural convection heat transfer is extensively
importance in science and engineering researcher. Several numerical and experimental methods have been
developed to investigate flow characteristics inside the cavities with and without obstacle, because these types of
geometries have practical engineering and industrial application, cooling of commercial high voltage electrical
power transformers.
A number of studies have been conducted to investigate the flow and heat transfer characteristics in closed
cavities in the past. Basak et al. [1] studied and solved the finite element analysis of natural convection flow in a
isosceles triangular enclosure due to uniform and non-uniform heating at the side walls. Pirmohammadi et al. [2]
shows that the effect of a magnetic field on Buoyancy- Driven convection in differentially heated square cavity.
Rahman et al. [3] offered a numerical model for the simulation of double-diffusive natural convection in a rightangled triangular solar collector. Mahmoodi et al. [4] studied numerically magneto-hydrodynamic free convection
heat transfer in a square enclosure heated from side and cooled from the ceiling. Jani et. al. [5] numerically
investigated magneto hydrodynamic free convection in a square cavity heated from below and cooled from other
walls. Bhuiyan et al. [6] numerically investigated magneto hydrodynamic free convection in a square cavity with
semicircular heated block.
International Conference on Mechanical Engineering
AIP Conf. Proc. 1754, 040022-1–040022-7; doi: 10.1063/1.4958382
Published by AIP Publishing. 978-0-7354-1412-9/$30.00
040022-1
On the basis of the literature review, it appears that no work was reported on the free convection flow in a square
cavity with heated elliptic block with internal heat generation. In the present study, we undertake this task varying
the Rayleigh number Ra ሺͳͲସ ൑ ܴܽ ൑ ͳͲ଺ ሻ and Hartmann number Ha ሺͲ ൑ ‫ ܽܪ‬൑ ͳͷͲሻǤThe obtained numerical
results are presented graphically in terms of streamlines, isotherms, local Nusselt number and average Nusselt
number for different Hartmann numbers and Rayleigh numbers will be done.
PHYSICAL CONFIGURATION
The physical model considered in the present study of free convection in a square cavity with heated elliptic
block is shown in Fig. 1. The height and the width of the cavity are denoted by L. The left wall is kept heated Th and
lower wall adiabatic. The right and upper wall is kept at cold Tc. The magnetic field of strength B0 is applied parallel
to x- axis. The square cavity is filled with an electric conductive fluid are considered Newtonian and incompressible.
Y
Tc
Bo
L
g
Th
Tc
L
Adiabatic
X
FIGURE 1. Schematic view of the cavity with boundary conditions considered in the present paper
MATHEMATICAL FORMULATION
The flow is considered steady, laminar, incompressible and two-dimensional. The field equations governing the
heat transfer and fluid flow include the continuity equation, the Navier-Stokes equations and the energy equation,
which can be expressed in non-dimensional form as:
wU wV
wX wY
U
U
0
wU
wV
V
wY
wX
wV
wV
V
wX
wY
U
(1)
wT
wT
V
wX
wY
§ w 2U
wP
w 2U
Pr¨
2
¨
wX
wY 2
© wX
§ w 2V
wP
w 2V
Pr¨¨
2
wY
wY 2
© wX
§ w 2T
w 2T
¨
¨ wX 2 wY 2
©
·
¸
¸
¹
(2)
·
¸¸ ( Ra Pr)T Ha 2 Pr V
¹
·
¸
¸
¹
(3)
(4)
where Ra, Pr and Ha are the Rayleigh, Prandtl and Hartman numbers and are defined as:
040022-2
Ra
gE Th Tc L3 ,
Pr
Dv
v , ‫ ܽܪ‬ൌ ‫ܮ ܤ‬ට ఙ
଴
ఘ௩
(5)
D
ሬԦ that is reduced to
The effect of magnetic field into the momentum equation is introduced through the Lorentz force term ‫ܬ‬Ԧ ൈ ‫ܤ‬
െߪ‫ܤ‬଴ ‫ ݒ‬ଶ as shown by Pirmohammadi et al. [2]
To computation of the rate of heat transfer, Nusselt number along the hot wall of the enclosure is used that is as follows:
ܰ‫ݑ‬௬ ൌ
డఏ
ቚ
డ௑ ௑ୀ଴
ଵ
ܰ௨ ൌ ‫׬‬଴ ܰ‫ݑ‬௬ ܻ݀
(
(6)
The boundary conditions are:
On the left wall of the square cavity: U = 0, V = 0, ߠ ൌ ͳ and right wall of the cavity: U = 0, V = 0, ߠ ൌ Ͳ
On the heated elliptic block:
U = 0, V = 0, ߠ ൌ ͳ
NUMERICAL TECHNIQUE
The nonlinear governing partial differential equations, i.e., mass, momentum and energy equations are transferred
into a system of integral equations by using the Galerkin weighted residual method. The integration involved in each
term of these equations is performed with the aid Gauss quadrature method. These modified nonlinear equations are
transferred into linear algebraic equations with the aid of Newton’s method. Lastly, Triangular factorization method
is applied for solving those linear equations.
RESULT AND DISCUSSION
In this section, results of the numerical study on magneto-hydrodynamic free convection fluid flow and heat transfer
in a square cavity filled with an electric conductive fluid with Pr = 0.733 are presented. The results have been
obtained for the Rayleigh number Ra ሺͳͲସ ൑ ܴܽ ൑ ͳͲ଺ ሻ and Hartmann number Ha ሺͲ ൑ ‫ ܽܪ‬൑
ͳͷͲሻǤPirmohammadi et al. [2] was modified and used for the computations in the study. Streamlines for Ha = 0 is
presented in Fig. 2 to undeterstand the effects of Rayleigh number on flow field and temperature distribution. At Ra
= 10000 and in the absence of the magnetic field two cells are formed with two elliptic- shaped eyes topside of the
elliptic heated block of the cavity shown in Fig. 2(a). For higher Rayleigh number two elliptic- shaped eyes are
formed inside the cavity and also flow strength increases are shown in Fig. 2(b), 2(c) and 2(d). Stream function has
symmetrical value about the vertical central line as the elliptic heated block is symmetrical. Again Streamlines for
while Ra = 105 and Pr = 0.733 are presented in Fig. 3 to undeterstand the effects of Hartmann number on flow field
and temperature distribution. At Ha = 0 are two elliptic- shaped eyes right side of the elliptic heated block of the
cavity shown in Fig. 3(a). For higher Hartmann number three elliptic- shaped eyes are formed inside the cavity and
also flow strength increases are shown in Fig. 3(b), 3(c) and 3(d).
Conduction dominant heat transfer is observed from the isotherms in Fig. 4(a) and Fig 4(b) at Ra = 10000 and Ra
= 100000. With increases in Rayleigh number, isotherms are concentrates near the top wall and isotherms lines are
more bending which means increasing heat transfer through convection. Formation of thermal boundary layers can
be found and increases from the isotherms for Ra = 500000 and Ra = 1000000 at Ha = 0 are shown in Fig. 4(c) and
Fig. 4(d). With increases the Hartmann number, isotherms in Fig. 5(a) and 5(b) are concentrates near the top and
right wall isotherms lines are more bending which means increasing heat transfer through convection. Formation of
thermal boundary layers can be found and increases from the isotherms for Ha = 100 and Ha = 150 at Ra = 0 are
shown in Fig. 5(c) and Fig. 5(d).
The local Nusselt number along the bottom wall for different Rayleigh numbers with Pr = 0.733 and Ha = 0 of
the cavity are shown in Fig. 6(a). Minimum and maximum shape curves are obtained here. At Ra minimum we get
shape curve maximum and Ra maximum so shape curve minimum. Again local Nusselt number along the bottom
wall for different Hartmann numbers with Pr = 0.733 and Ra = 10 5 of the cavity are shown in Fig. 6(b). Variations of
the vertical velocity component along the bottom wall for different Rayleigh number with Pr = 0.733 and Ha = 0 of
the cavity are shown in Fig. 7(a). It can be seen from the figure that the absolute value of maximum and minimum
value of velocity increases with increasing the Rayleigh number. The curves are symmetrical parabolic shaped as the
elliptic heated block is symmetrical. For lower Rayleigh number value of the velocity has larger changed. Variations
of the vertical velocity component along the bottom wall for different Hartmann number with Pr = 0.733 and Ra =
105 of the cavity are shown in Fig. 7(b). From the Figure it can be observed that the curves are symmetrical elliptic
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heated block. The absolute value of maximum and minimum value of velocity decreases with increasing the
Hartmann number.
Fig. 8(a) presents the temperature profiles along the bottom wall for different Rayleigh number with Pr = 0.733
and Ha = 0. As seen from the Figure, temperature value is decreased from the increasing of Rayleigh numbers.
Parabolic shape temperature profile is obtained due to symmetric shape of the elliptic shape heated block. For lower
Rayleigh number temperature value has less significant changed but for higher Rayleigh number temperature value
has more significant.Fig. 8(b) present the temperature profiles along the bottom wall for different Hartmann number
with Pr = 0.733 and Ra = 105. As seen from the figure, temperature value is increased from the increasing Hartmann
numbers.
(a)
(b)
(d)
(c)
FIGURE 2. Stream lines for (a) Ra = 10000, (b) Ra = 100000, (c) Ra = 500000, (d) Ra = 1000000 while Ha = 0 and Pr = 0.733
(a)
(b)
(c)
(d)
FIGURE 3 Stream lines for (a) Ha = 0, (b) Ha = 50, (c) Ha = 100, (d) Ha = 150 while Ra = 10 5 and Pr = 0.733.
040022-4
(a)
(b)
(c)
(d)
FIGURE 4. Isotherm for (a) Ra = 10000, (b) Ra = 100000, (c) Ra = 500000, (d) Ra = 1000000 while Ha = 0
and Pr = 0.733.
(a)
(b)
(d)
(c)
FIGURE 5. Isotherm for (a) Ha = 0, (b) Ha = 50, (c) Ha = 100, (d) Ha = 150 while Ra = 10 5 and Pr = 0.733
040022-5
(a)
(b)
FIGURE 6. Variation of local Nusselt number along the bottom wall for different (a) Rayleigh numbers with Pr = 0.733 and
Ha = 0 and (b) Hartmann numbers with Pr = 0.733 and Ra = 10 5
(a)
(b)
FIGURE 7. Variation of velocity profile along the bottom wall at different (a) Rayleigh number with Pr = 0.733 and Ha = 0
(b) Hartmann number with Pr = 0.733 and Ra = 10 5.
(a)
(b)
FIGURE 8. Variation of temperature profiles along the bottom wall at different (a) Rayleigh number with Pr = 0.733 and Ha = 0,
(b) Hartmann number with Pr = 0.733 and Ra = 105
CONCLUSION
A numerical study of magneto-hydrodynamic free convection fluid flow and heat transfer in a square cavity
filled with an electric conductive fluid with heated elliptic block. Finite element method was to solve governing
equations for a heat generation parameters, Rayleigh numbers, Hartmann numbers and Prandtl numbers. Very good
agreements were observed between different Rayleigh numbers, different Hartmann numbers and different Prandtl
numbers. For all cases considered, two or more counter rotating eddies were formed inside the cavity regardless the
Rayleigh, Hartmann and the Prandtl numbers. The obtained results showed that the heat transfer mechanisms,
temperature distribution and the flow characteristics inside the cavity depended strongly upon both the strength of
the magnetic field and the Rayleigh number. From the present investigation the following conclusions may be drawn
as: with increase in the buoyancy force via increase in Rayleigh number, to decrease natural convection, a stronger
magnetic field is needed compared to the lower Rayleigh numbers.
040022-6
ACKNOWLEDGMENTS
The authors would like to express their gratitude to the Department of Mathematics, Bangladesh University of
Engineering and Technology, for providing computer facility during this work.
REFERENCES
[1]
[2]
[3]
T. Basak, S. Roy, S.K. Babu, A.R. Balakrishnan, Int. J. Heat Mass Transfer,51, 4496-4505(2008).
Mohsen Pirmohammadi, Majid Ghassemi, and Ghanbar Ali Sheikhzadeh, IEEE transactions on magnetic, 45(1) (2009).
M.M. Rahman, M.M. Billah, N. A. Rahim, N. Amin, R. Saidur and M.Hasanuzzaman, International Journal of Renewable
Energy Research 1( 50-54) ,(2011).
[4] M. Mahmoodi, Z. Talea’pour,Computational Thermal Science, 3, 219- 226(2011).
[5] S. Jani, M. Mahmoodi, M. Amini, International Journal of Mechanical, Industrial Science and Engineering, 7(4), (2013).
[6] A. H. Bhuiyan, R. Islam, M. A. Alim, International Journal of Engineering Research & Technology (IJERT),3(11). (2014).
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