8-9th August, 2008 School of Materials Science & Nanotechnology,Jadavpur University
SIMULATION OF THERMOMAGNETIC CONVECTION IN A RECTANGULAR CAVITY
Sumanta Banerjee1, Achintya Mukhopadhyay1, Swarnendu Sen1, Ranjan Ganguly2
1 Mechanical Engineering Department, Jadavpur University, Kolkata, 700032, India
2 Power Engineering Department, Jadavpur University, Kolkata, 700098, India
INTRODUCTION
Studies on thermal advection enhancement in a shallow rectangular cavity filled with an electrically-conducting fluid by applying a transverse external magnetic field [1] discusses the growing interest in problems of magnetic field-fluid interactions. This is primarily due to the numerous industrial processes in which these interactions assume prominence. For example, unavoidable convection movements during the manufacturing of crystals can be dampened with the help of a magnetic field [2].
The motivation for the present work is in addressing the problem of heat dissipation in electronic circuits and/or MEMS devices of high component densities [3]. With progressive miniaturization of circuits, the increasing heat flux densities adversely affect the performance, life and reliability of the devices, in addition to augmenting signal noise [4]. The junction temperature of a semiconductor device should, therefore, be kept below the safe operating temperature specified by the manufacturer. Under a situation where the heat sinks are to be designed to dissipate a constant value of heat energy under all working conditions, the thermal load among the components should be optimally distributed to prevent the maximum temperature on any device from exceeding a certain prescribed value.
The global objective is, thus, to maximize heat transfer density or minimize hot-spot temperatures when the total heat generation rate, circuit-board layout and/or other constraints are specified [5]. In generic configurations (e.g. in sealed enclosures housing PCBs), this amount to, at least partly, in investigating the influence of the aspect ratio behind the evolution of the flow field inside the cavity. This is evident from the significant volume of analytical [6], numerical [7,8,9] and/or experimental work [10,11] in the domain of natural/magnetothermal convection studies in confined fluid-filled enclosures of different configurations, under a variety of thermal boundary conditions and/or fluid rheology. However, the area of thermomagnetic convection analysis in shallow rectangular cavities (with more than one wall DHS modeling power-dissipating components) remains particularly unexplored, although such a study is an essential pre-requisite to thermal analyses of micro-electronics and Nano-CMOS chips [4, 12].
The transport of momentum and energy in miniaturized devices is, in general, diffusion dominated and requires very long transport time scales [13].By using ferrofluids and manipulating the flow pattern by external magnetic fields, thermomagnetic convection proves to be a viable alternative to enhance convection in these devices. This is due to the fact that the flow fields established inside a cavity by the ferrofluid depends both on the field gradient as well as the local fluid susceptibility gradients set up by the presence of heat sources [14].
In the present paper, numerical investigation of steady-state, thermomagnetic heat transfer is carried out in a shallow rectangular enclosure, where the cavity-width is twice the height. The effect of gravity has been neglected. Two discrete heat sources are flush-mounted on the bottom wall, representing heat-generating electronic components. A line dipole placed below the bottom wall sets up a two-dimensional, non-uniform magnetic field. Under the constraint where the overall heat energy supplied by the heaters is a constant and the dipole is placed symmetrically halfway along the cavity length, the study analyzes the evolution of the flow fields (depicted through streamfunction, heatline and isotherm plots [15]) for two representative cases of unequal heater lengths and strengths. The results are contrasted against those obtained for a square cavity configuration, keeping all other factors identical. The comparative study shows that the aspect ratio plays a central role in thermal energy transport and temperature field evolution within the cavity.
1
MATHEMATICAL MODEL
Figure 1(a) illustrates the geometry and the boundary conditions of the chosen physical configuration. The top wall and the non-heated portions of the bottom wall are adiabatic. The sidewalls are maintained isothermally cold at TCT=, providing the heat sinks. The height and the breadth of the shallow cavity are Wand respectively. The cavity extends to infinity in the third dimension such that the resulting flow field is two-dimensional. The finite-sized heaters are of lengths and, and their flux strengths are respectively qW21L2L1′′ and2q′′. The dimensionless length 1ε of the left heater is kept fixed at a value of 0.2, while the normalized right heater length 2ε is varied. The centerlines of the heaters are symmetrically placed (about the vertical mid-plane) and are separated by length such that S0.1=WS. All boundaries satisfy the no-slip velocity conditions. The line dipole is placed adjacent to the bottom wall, halfway along the enclosure length at and at Wx=04W0y= below its inner surface.
Fig. 1(a): Schematic of physical system
Fig. 1(b): Field configuration for symmetric dipole placement
The magnetic field inside the ferrofluid medium can be expressed in the form [14]: B
()
−+=ϕϕϕχμerermBrmˆcosˆsin1220. (1)
The r−ϕ coordinate system (with respective unit vectors e and) is depicted in Fig. 1(a). The centerline of the coil is considered as the virtual line-origin of the dipole. Figure 1(b) depicts the imposed normalized magnetic field configuration and the corresponding lines of force. The contours are circular in nature, with centers at the virtual origin of the dipole. rˆϕeˆ
The assumed magnetic field and H (which inside a magnetic medium is ()[]mBHχμ+=10) conform to Maxwell’s equation in static form [16]. Here, 0μ is the permeability (= 4π × 10-7 N/A2)
The governing steady state, two-dimensional continuity, momentum, and energy equations are: 0=∂∂+∂∂yvxu, (2)
2
()xyxCHyTHxTHxHTTyuxuxpyuvxuu
∂∂+∂∂−∂∂−−
∂∂+∂∂+∂∂−=∂∂+∂∂∗ρβχμβρχμνρρρ200200222221, (3) ()yyxCHyTHxTHyHTTyvxvypyvvxvu
∂∂+∂∂−∂∂−−
∂∂+∂∂+∂∂−=∂∂+∂∂∗ρβχμβρχμνρρρ200200222221, (4)
and
∂∂+∂∂=∂∂+∂∂2222yTxTyTvxTuα. (5)
Here, is the effective pressure, defined as ∗p()2200Hppχμ−=∗, where is the absolute pressure. The last term in the momentu equations (Eq. (3) and Eq. (4)) contains the Kelvin body force (KBF) per unit volume p
m ()BMf∇⋅=that a magnetic fluid experiences in a spatially non-uniform magnetic field [17].The magnetic Rayleigh number, denoted by, refers to the dimensionless group mRa)2WNuRH(20TmρναβρΔ=LHmax,θmax,θ0χμ[17] and is referenced with respect to the flux strength of the left heater. The heater Nusselt numbers (,) and their maximum non-dimensional temperatures ((,)) are defined as in [14]. LHavg,RHavgNu,
The working fluid chosen is a representative ferrofluid continuum considered at a reference temperature of 300K, and having the following properties: density ρ = 1180 kg/m3, specific heat C= 4180 J/kg-K, kinematic viscosity pν= 5.93×10-6 m2/s, Pr = 49.6, reference magnetic susceptibility = 0.1 and compressibility coefficient 0,mχρβ = 5.6 × 10-4 /K [17].
NUMERICAL PROCEDURE
The coupled mass, momentum and energy equations are solved by a finite volume method using SIMPLER algorithm developed by Patankar [18]. The set of algebraic equations are solved sequentially by TDMA (Tri-Diagonal Matrix Algorithm). The power-law differencing scheme by Patankar is used for the formulation of the convection-diffusion terms in the equations. The pertinent variables ()TpV,, and the fluid properties (thermophysical and magnetic) are described using a staggered grid arrangement. Solution is obtained by progressive minimization of the mass residual. The computation is terminated when the root mean square value of the residuals get below10. The non-uniform grid, which is required to resolve the sharp gradients near the walls and close to the location of the magnetic dipole, is based on a sinusoidal mesh size distribution. The code is validated by comparing the simulations for buoyancy-driven convection with the benchmark results of de Vahl Davis [19] for thermogravitational convection, and with the results of Ganguly et al. [17] for thermomagnetic convection. 11−
RESULTS AND DISCUSSIONS
A numerical study already carried out by the present group shows that pertinent geometrical parameters (the length and/or strength ratio of heaters) can be suitably adjusted to achieve certain desired operating conditions [Error! Bookmark not defined.]. The study shows that, at higher values of , the fraction of heat energy dissipated through a cold sidewall is roughly proportional to the fraction of the total heat energy pumped in by the heat source placed adjacent to the wall (when the dipole is centrally placed). For the same overall energy input, the present study takes up the chosen mRa
3
system (Fig. 1(a)) to investigate the influence of aspect ratio on heat load division (between the sidewalls) under steady-state advection conditions.
Temperature non-uniformity inside the cavity due to the presence of heat sources produces non-uniformmχ values within the cavity. The KBF experienced by a ferrofluid element increases with a rise in temperature. The resultant force is primarily directed along()HH⋅∇, or towards the virtual origin of the dipole [17]. Due to the combined effects of non-uniform temperature and magnetic fields, the resulting steady-state thermomagnetic convection depends on such factors as the selection of the heater sizes, magnitudes of their flux strengths, and dipole placements.
In this paper, the following representative case (leading to asymmetry in wall heat transfer) is taken up for discussion:
• For symmetric placement of the dipole (external field distribution as in Fig. 1b) and heaters of the same flux strength (=1.0), the heater length ratiorq′′rε is taken as 0.5.
For central placement of the line dipole, the base situation corresponds to =rq′′rε=1.0 and . This all-symmetrical situation serves as the global standard for the present study, with reference to which any representative asymmetric situation is considered. 5107×≈mRa
Figures 2a.1, 2b.1 and 2c.1 depict the streamline, heatline and isotherm plots for rε=0.5 in the convection-dominated regime of . This value of ensures that the total heat input for the present case is the same as the base case of [14]. 5103.9×≈mRamRa5107×≈mRa
As the heaters are of different sizes, they pump in unequal amounts of energy. The resulting convection rolls are, therefore, not symmetric. The dominant roll is formed over the left heat source (Fig. 2a.1), as the latter contributes to %67≈ of the total energy input. A smaller roll sits atop the right heater (the weaker source). The major portion of heat (%64≈) is dissipated through the cold wall adjacent to the left source (see Fig. 2b.1), with the larger vortex affecting bulk convection. The larger number of heatlines terminating on the left wall is a visual evidence of the same. The overall advection of heat through the right wall is comparatively poor (%36≈). The contours-layout depict the coupled action of the stronger left vortex and the weaker right vortex, serving to dissipate a portion of the thermal energy pumped in by the left heater through the right wall (Fig. 2b.1). This explains why the heat advection through the right wall is %3≈ higher than the contribution of the right heater in the overall energy input. The warping of the corresponding isotherm contours (Fig. 2c.1) roughly corresponds to the predominance of the clockwise circulation. The conductive heat flux is higher at the left wall than at the right, as evident by the denser packing of isotherms at the left wall. The isotherm contours are packed more densely over the left heat source, which is stronger of the two.
The distribution of heat load among the sidewalls of the present physical system (rectangular cavity with ) is seen to be markedly different from an analogous situation studied for a square cavity () [14]. Figures 2a.2, 2b.2 and 2c.2 depict the streamline, heatline and isotherm plots for the square cavity. The flow-field inside the latter evolves in such a way that the heat dissipation through the right wall remains at 0.2=AR.1=AR0%33≈ for rε=0.5 for all regimes of heat transfer.
Table 1: Comparison of heat transfer parameters between 0.2=AR and 0.1=AR
AR
LHavgNu,
RHavgNu,
LWf
RWf
LHmax,θ
RHmax,θ
2.0
7.28
8.74
0.64
0.36
0.155
0.121
1.0
14.96
18.14
0.67
0.33
7.53E-02
5.81E-02
Table 1 clearly brings out the influence of the aspect ratio in heat energy distribution among the sinks in a convection-dominant regime. The energy dissipated through available heat sinks, for a given boundary condition and/or external field setup, need to be estimated to determine the safe operational
4
conditions for a device. The functional dependence of flow-field evolution on the aspect ratio warrants further study.
Fig. 2a.1
Fig. 2a.2
Fig. 2b.1
Fig. 2b.2
Fig. 2c.1
Fig. 2c.2
CONCLUSIONS
Heat transfer characterization is done in a convection-dominated regime for two values of the cavity aspect ratio, for a representative asymmetry situation of rq′′=1.0 and rε=0.5. The total heat input to the cavity is maintained constant. The flow-field evolutions are visualized through streamline, heatline and isotherm plots, and the influence of cavity geometry in transport of thermal energy borne out through the heatline plots. The values of pertinent heat transfer parameters tabulated in Table 1 quantitatively bears out the influence of aspect ratio in advection in rectangular enclosures.
5
6
ACKNOWLEDGEMENTS
This work was partially supported by the Centre for Nano Science and Technology, Jadavpur University under the UGC Scheme of University with Potential for Excellence. We acknowledge Council of Scientific and Industrial Research (CSIR), Government of India for the financial support provided to the first author.
REFERENCES
1 S. Alchaar, P. Vasseur, and E. Bilgen, Natural convection heat transfer in a rectangular enclosure with a transverse magnetic field, Journal of Heat Transfer, vol.117, pp. 668-673, 1995.
2 C. Vives, and C. Perry, Effects of magnetically dampened convection during the controlled solidification of metals and alloys, Int. J. Heat Mass Transfer, vol.30, pp. 479-496, 1987
3 S. P. Gurrum, S. K. Suman, Y. K. Joshi, and A. G. Fedorov, Thermal Issues in next-generation integrated circuits, IEEE Transactions, vol. 4, pp. 709-714, 2004
4. R. Remsburg, Thermal design of electronic equipment, chap.1, CRC Press LLC, Boca Raton, Florida, 2001
5. A. K. Da Silva, S. Lorente, and A. Bejan, Optimal distribution of discrete heat sources on a wall with natural convection, Int. J. Heat and Mass Transfer, vol.47, pp.203-214, 2004
6 M. Prud’homme, and H. Bougherara, Weakly non linear convection in a shallow horizontal cavity under uniform cross-fluxed heat fluxes, Int. J. Heat and Mass Transfer, vol. 48, pp. 2278-2289, 2005
7 O. Polat, and E. Bilgen, Natural convection and conduction heat transfer in open shallow cavities with bounding walls, Heat Mass Transfer, vol. 41, pp. 931-939, 2005
8 N. B. Cheikh, B. B. Beya, and T. Lili, Aspect ratio effect on natural convection flow in a cavity submitted to a periodical temperature boundary, Journal of Heat Transfer, vol.129, pp. 1060-1068, 2007
9 D. Zablockis, V. Frishfelds, and E. Blums, Numerical investigation of thermomagnetic convection in a heated cylinder under the magnetic field of a solenoid, J. Phys.: Condens. Matter, vol. 20 (2008) 204134 (5pp)
10 E. Blums, A. Mezulis, and G. Kronkalns, Magnetoconvective heat transfer from a cylinder from a cylinder under the influence of a nonuniform magnetic field, J. Phys.: Condens. Matter, vol. 20 (2008) 204128 (5pp)
11 D. S. R. Ramambason, and P. Vasseur, Influence of a magnetic field on natural convection in a shallow porous enclosure saturated with a binary fluid, Acta Mechanica, vol. 191, pp. 21-35, 2007
12. F. P. Incropera, Convection heat transfer in electronic equipment cooling, J. Heat Transfer, vol.110, pp.1097-1111, 1988
13 H. J. Kim, and A. Beskok, Quantification of chaotic strength and mixing in a microfluidic system, J. Micromech. Microeng. vol. 17, pp. 2197-2210, 2007
14 S. Banerjee, A. Mukhopadhyay, S. Sen, and R. Ganguly, Optimizing thermomagnetic convection for electronics cooling, Numerical Heat Transfer, Part A, vol. 53, pp.1231- 1255, 2008
15. S. Kimura, and A. Bejan, The “heatline” visualization of convective heat transfer, J. Heat Transfer, vol.105, pp.916-919, 1983.
16 R. E. Rosensweig, Ferrohydrodynamics, pp.1-4, Dover Publications Inc., New York, 1997
17 R. Ganguly, S. Sen, and I.K. Puri, Thermomagnetic convection in a square enclosure using a line dipole, Physics of Fluids, vol.16, pp.2228-2236, 2004.
18. S. V. Patankar, Numerical Heat Transfer and Fluid Flow, Hemisphere, New York, 1980.
19. G. de Vahl Davis, Natural Convection of air in a square cavity: A benchmark numerical solution, Int. J. Num. Methods Fluids, vol.3, pp.249-264, 1983.
Custom Search
