Effect of Inclination Angle on Magneto-Convection ... - IEEE Xplore

IEEE TRANSACTIONS ON MAGNETICS, VOL. 46, NO. 9, SEPTEMBER 2010

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Effect of Inclination Angle on Magneto-Convection Inside a Tilted Enclosure* Mohsen Pirmohammadi1 , Majid Ghassemi2 , and Mohsen Hamedi3 Research Management of R&D Deputy, Mapna Group, Tehran, Iran Mechanical Engineering Department, K.N. Toosi University of Technology, Tehran, Iran School of Mechanical Engineering, University of Tehran, Tehran, Iran Numerically investigation is carried out for magnetohydrodynamics natural convection in an inclined partitioned enclosure. The vertical walls are maintained isothermal at different temperatures and other walls are adiabatic. Two insulated partitions are located on horizontal walls. Non linear governing equations for the fluid flow and heat transfer are solved for inclination angle varying from 0 deg to 90 deg, three different Hartmann numbers (100, 200, and 300), and three nondimensional partition heights (0.166, 0.25, and 0.33). Rayleigh number and Prandtl number are 106 and 0.054, respectively. A finite volume code based on PATANKAR’s SIMPLER method is utilized. It is found as nondimensional partition height (Hp ) and Hartmann number (Ha) increase the mean Nusselt number decreases and this means that the total heat transfer between two isothermal walls is reduced. Also the variation of mean Nusselt number with inclination angle in low Hartmann number is more considerable compare to high Hartmann number. Index Terms—Inclination angle, magnetic field, natural convection, partition.

I. INTRODUCTION

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ATURAL convection in enclosures has received considerable attention in recent years because of numerous applications in engineering. Recently studies of heat transfer and fluid flow characteristics of a partitioned enclosure have come under scrutiny both numerically and experimentally [1]–[5]. Ghassemi et al. [4] studied the effect of two insulated horizontal baffles on the flow field and heat transfer in an enclosure. It was found that placing baffles on the vertical walls generally causes flow and thermal modification occurs due to the blockage effect of the baffles. The baffles cause convection suppression and heat transfer reduction relative to the undivided enclosure at the same Rayleigh number. Kahveci [5] studied natural convection in an enclosure partitioned with a finite thickness partition and found that the average Nusselt number decreases with an increase of the distance between the hot wall and the partition, and the thickness of the partition has a little effect on natural convection. It is well known that unavoidable hydrodynamic movements can be damped with the help of a magnetic field. Investigations of heat transfer for melt flows under crystal growth conditions permit one to qualify the critical operating parameters of crystal growth, Hence there has been increased interest in the flows of electrically conducting fluids in cavities subjected to an external magnetic field. Study and thorough understanding of the momentum and heat transfer in such a process is important for the better control and quality of the manufactured products. The study of Oreper and Szekely [6] shows that the magnetic field suppresses the natural-convection currents and the magnetic field strength is one of the most important factors for crystal formation. Ozoe and Maruo [7] numerically investigated the

*Corrected. This paper first appeared in IEEE Trans. Magn., vol. 46, pp. 24892492, June 2010. Due to a production error, Figs. 1 and 2 were not correct Manuscript received November 03, 2009; accepted January 26, 2010. Date of current version August 20, 2010. Corresponding author: M. Pirmohammadi (e-mail: [email protected]). Digital Object Identifier 10.1109/TMAG.2010.2060379

natural convection of a low Prandtl number fluid in the presence of a magnetic field and obtained correlations for the Nusselt number in terms of Rayleigh, Prandtl, and Hartmann numbers. Garandet et al. [8] proposed an analytical solution to the governing equations of magneto hydrodynamics to be used to model the effect of a transverse magnetic field on natural convection in a two-dimensional enclosure. Rudraiah et al. [9] numerically investigated the effect of a transverse magnetic field on natural-convection flow inside a rectangular cavity with isothermal vertical walls and adiabatic horizontal walls and found that a circulating flow is formed with a relatively weak magnetic field and that convection is suppressed and the rate of convective heat transfer is decreased when the magnetic field strength increases. Al-Najem et al. [10] used the power law control volume approach to determine the flow and temperature fields under a transverse magnetic field in a tilted square enclosure with isothermal vertical walls and adiabatic horizontal walls at Prandtl number of 0.71 and showed that the suppression effect of the magnetic field on convection currents and heat transfer is more significant for low inclination angles and high Grashof numbers. Pirmohammadi et al. [11] studied the effect of a magnetic field on buoyancy-driven convection in a differentially heated square enclosure. They showed that the heat transfer mechanisms and the flow characteristics inside the enclosure depend strongly upon both the strength of the magnetic field as well as the Rayleigh number. It was concluded that the magnetic field considerably decreases the average Nusselt number. Recently Pirmohammadi and Ghassemi [12] investigated magneto-convection in an enclosure heated from the left wall and cooled from the right wall while the other walls were kept adiabatic and one adiabatic baffle attached to the left vertical wall. They found that the total heat transfer across isothermal walls reduces considerably because of blockage effect of baffle and braking effect of Lorentz force. The present study considers laminar natural convection in the presence of a longitudinal magnetic field in an inclined partitioned enclosure. The enclosure is filled with an electrically conducting fluid whose Prandtl number is 0.054. The object of the

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where, Pr, Ra, and Ha are the Prandtl, Rayleigh, and Hartmann numbers and are defined as follows: (6) where, g is the gravitational acceleration, is the kinematic visis the magcosity, is the coefficient of thermal expansion, nitude of magnetic field and is the electrical conductivity. is proportional to the strength of the magnetic field and is a measure of the relative importance of the electromagnetic to the visin (3) is achieved by simcous forces. It is noted that using constant magnetic plifying the Lorentz force term field [11]. In order to compare total heat transfer rate, Nusselt number is used. The local and average Nusselt numbers are defined as follows: Fig. 1. Geometry and coordinates of enclosure configuration with magnetic effect.

(7) study is to obtain numerical solutions for the velocity and temperature fields inside the enclosure and to determine the effects of the magnetic field strength on the transport phenomena. II. BASIC EQUATIONS Steady, laminar, natural-convection flow in the presence of a magnetic field in a square enclosure of length H was considered. Dimensional coordinates with the x-axis measuring along the bottom wall and y-axis being normal to it along the left wall are used. The geometry and the coordinate system are schematically shown in Fig. 1. is applied longitudinally. The Magnetic field of strength top and bottom walls are insulated and the fluid is isothermally heated and cooled by the left and right walls at uniform temperand , respectively. atures of Dimensionless variables in the analysis are defined as

(1) where, u and v is the velocity components, p is the pressure, T is the temperature, is the thermal diffusivity, is the density. According to the above dimensionless variables, the governing equations in this study are based on the conservation laws of mass, linear momentum; energy and magnetic induction are given in dimensionless form as (2)

(3)

(4) (5)

The Nusselt number is the ratio of entire heat transfer to heat transfer by heat conduction. Boundary conditions are (8a) (8b) On the partitions (9)

III. NUMERICAL PROCEDURES The governing equations associated with the boundary conditions were solved numerically using the control-volume based finite volume method. The hybrid-scheme, which is a combination of the central difference scheme and the upwind scheme, was used to discretize the convection terms. A staggered grid system, in which the velocity components are stored midway between the scalar storage locations, was used. In order to couple the velocity field and pressure in the momentum equations, the well known SIMPLER-algorithm was adopted. Grid dependency is investigated for the standard case. The solution of the fully coupled discretized equations is obtained iteratively using the TDMA method. The criterions, needed to check the truth and validity of results obtained from this study have been looked over and it has been seen that the results provided these criterions. First, sweep number has been changed to observe how it affects the convergence since solution is closely related to sweep number. In the second step, values of spot and residual have been checked for physical variables such as temperature, pressure and velocity. Convergence has been considered as being achieved when summation of residuals became less than , which is the case for most of the dependent variables. The simulations have been achieved with cell numbers ranging

PIRMOHAMMADI et al.: EFFECT OF INCLINATION ANGLE ON MAGNETO-CONVECTION INSIDE A TILTED ENCLOSURE

Fig. 2. Streamlines (a) and isotherms (b) for different inclination angles, Ha = 100, H = 0:25

from 31 31 to 61 61 for and constant properties. Coarser grid distributions are not able to give accurate results in agreement with data from the literature. IV. RESULTS AND DISCUSSION The accuracy of results is verified with that of Rudraiah and is presented in Pirmohammadi et al. [11]. In this study there are two mechanisms for suppressing the convection heat transfer, the one is applying magnetic field and the other is considering two partitions on the adiabatic walls. In the first, magnetic field induces Lorentz force in opposite the

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Fig. 3. Variation of the mean Nusselt number with inclination angle for different Hartmann number and nondimensional partition height: (a) H = 0:166, (b) H = 0:25, (c) H = 0:33.

buoyancy force and in the second, the partition blocks fluid flow and causes heat transfer between two isothermal walls reduces because of reduction in advection. Fig. 2 presents the streamline and isotherm plots for Rayleigh number , Hartmann number 100 and dimensionless partition , one primary height 0.25. In Fig. 2(a) it is shown that at vortex is formed between two partitions and as inclination angle increases there is a tendency to appear secondary vortex in enclosure, so that at two secondary vortices are observed.

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As shown in Fig. 2(b) at and the temperature gradient near the hot and cold walls is higher than the other anthe boundary layer near the isothermal walls gles. At heat transfer between hot is vanished. This means at and cold walls is minimum. Fig. 3 depicts the variation of the mean Nusselt number with inclination angle for different Hartmann numbers and nondimensional partition height. Generally and increases the mean Nusselt number we can see as decreases and this means that the total heat transfer between two isothermal walls is reduced. Also the variation of average Nusselt number with inclination angle in low Hartmann number is more considerable compare to high Hartmann number. The effects of Hartmann number on average Nusselt number for is shown in Fig. 3(a). It is seen that for the cases and , maximum Nusselt number occurs at and at is higher than the case . When the average Nusssselt number is maximum . Fig. 3(b) shows the variation of the mean Nusat selt number with inclination angle and Hartmann number for . It can be observed that at mean Nusselt number is maximum for the cases and . The variation of the mean Nusselt number with inclination angle is presented in for different Hartmann numbers and Fig. 3(c). It is observed that the Nusselt number increases up to and then decreases as increases. about V. CONCLUSION • The heat transfer mechanisms and the flow characteristics inside the enclosure depend strongly upon both the strength of the magnetic field and partition height. • The variation of average Nusselt number with inclination angle in low Hartmann number is more considerable compared to high Hartmann number.

• High Hartmann number reduces the advection effect and therefore the conduction heat transfer becomes the dominant mode of heat transfer. REFERENCES [1] X. Shi and J. M. Khodadadi, “Laminar natural convection heat transfer in a differentially heated square cavity due to a thin fin on the hot wall,” J. Heat Transfer, vol. 125, pp. 624–634, 2003. [2] S. H. Tasnim and M. R. Collins, “Numerical analysis of heat transfer in a square cavity with a baffle on the hot wall,” Int. Comm. Heat Mass Transfer, vol. 31, pp. 639–650, 2004. [3] V. Mariani and I. Moura Belo, “Numerical studies of natural convection in a square cavity,” Thermal Eng., vol. 5, pp. 68–72, 2006. [4] M. Ghassemi, M. Pirmohammadi, and G. A. Sheikhzadeh, “A numerical study of natural convection in a tilted cavity with two baffles attached to its vertical walls,” WSEAS Trans. Fluid Mech., vol. 2, pp. 61–68, July 2007. [5] K. Kahveci, “Natural convection in a partitioned vertical enclosure heated with a uniform heat flux,” Numer. Heat Transfer, vol. 51, no. 10, pt. A, pp. 979–1002, 2007. [6] G. M. Oreper and J. Szekely, “The effect of an externally imposed magnetic field on buoyancy driven flow in a rectangular cavity,” J. Cryst. Growth, vol. 64, pp. 505–515, 1983. [7] H. Ozoe and M. Maruo, “Magnetic and gravitational natural convection of melted silicon-two dimensional numerical computations for the rate of heat transfer,” JSME, vol. 30, pp. 774–784, 1987. [8] J. P. Garandet, J. P. Alboussiere, and T. Moreau, “Buoyancy driven convection in a rectangular cavity with a transverse magnetic field,” Int. J. Heat Mass Transfer, vol. 35, pp. 741–748, 1992. [9] N. Rudraiah, R. M. Barron, M. Venkatachalappa, and C. K. Subbaraya, “Effect of a magnetic field on free convection in a rectangular cavity,” Int. J. Eng. Sci., vol. 33, pp. 1075–1084, 1995. [10] N. M. Al-Najem, K. M. Khanafer, and M. M. El-Refaee, “Numerical study of laminar natural convection in tilted cavity with transverse magnetic field,” Int. J. Numer. Meth. Heat Fluid Flow, vol. 8, pp. 651–672, 1998. [11] M. Pirmohammadi, M. Ghassemi, and G. A. Sheikhzadeh, “Effect of a magnetic field on buoyancy-driven convection in differentially heated square cavity,” IEEE Trans. Magn., vol. 45, no. 1, pp. 407–411, Jan. 2009. [12] M. Pirmohammadi and M. Ghassemi, “Numerical study of magnetoconvection in a partitioned enclosure,” IEEE Trans. Magn., vol. 45, pp. 2671–2674, 2009.