HEAT TRANSFER IN ELECTROCONDUCTIVE FLUIDS. A HIGHLY SIMPLIFIED MARK & CELL SIMULATION

Transcription

1 HEAT TRANSFER IN ELECTROCONDUCTIVE FLUIDS. A HIGHLY SIMPLIFIED MARK & CELL SIMULATION HIDEO KAWAHARA 1, ALEXANDRU MIHAIL MOREGA 2, MIHAELA MOREGA 3 Key words: Electroconductive fluids, Convection heat transfer, Nuerical siulation, Highly siplified ark and cell (HSMAC). Convection of electrically conducting fluids such as liquid etals in the presence of agnetic field is a ajor area of investigation, e.g., crystal growth processes. The usage of a agnetic field is known to inhibit the theral flow in the liquidus phase, and it is utilized in the design phase of casting systes. This study is concerned with the heat transfer and flow in electro-conductive elts in solidification systes. A atheatical odel of the underlying physical processes is established, and nuerical siulation results obtained by a hoe-ade code, based on the finite differences HSMAC schee, are reported: second order central finite differences for the viscous ters, third order upwind (UTOPIA) finite differences for the convective ters, and first order forward finite differences for the tie derivative. 1. INTRODUCTION In electroagnetic aterial processing, e.g., crystal growth, solidification, laser elting, casting and vapor deposition, etc. [1-4] heat transfer plays a key role. In any circustances, therally driven flows in the elt are seen as enace for the processes, therefore, the knowledge and the control of convection heat transfer are iportant in obtaining high quality aterials and in optiizing the fabrication technologies. Although extensive experiental studies are carried on to understand the role of convection heat transfer upon icrostructures, the progress reported in the theory of natural and forced convection systes revolves ainly in the area of nonlinear dynaics, e.g. ultiple solutions, periodic oscillations, chaos. On the other hand, there is a wide interest in developing nuerical odels aied at understanding the solidification phenoena, as well as developing design tools 1 Oshia National College of Maritie Technology, Departent of Shipping of Technology, Yaaguchi, Japan, E-ail: kawahara@oshia-k.ac.jp 2 Faculty of Electrical Engineering, University POLITEHNICA of Bucharest, Splaiul Independenţei nr. 313, sector 6, Bucharest, , Roania, E-aiL: a@ie.pub.ro 3 Faculty of Electrical Engineering, University POLITEHNICA of Bucharest, Splaiul Independeţei nr. 313, sector 6, Bucharest, , Roania, E-ail: ihaela@ie.pub.ro Rev. Rou. Sci. Techn. Électrotechn. et Énerg., 54, 1, p , Bucarest, 2009

2 68 Hideo Kawahara, Alexandru Mihail Morega, Mihaela Morega 2 usable in the optiization of fabrication technologies [1]. During crystal anufacturing, unwanted convective flows ay be suppressed in liquid etals and other electrically conducting fluids by applying external agnetic fields. This technique is also successfully used in solidification processes to weaken the buoyancy driven flows and to odify the rate of solidification. Extensive studies were therefore conducted to exaine the effect of agnetic field on the flow structure and stability of liquid etals [4-6]. Convection heat transfer with phase change solidification pose strongly nonlinear, coupled atheatical probles, characterized by abrupt changes in the teperature field that is particularly difficult to odel nuerically [7,8]. 2. MAGNETIC FIELD CONTROL IN CONVECTION HEAT TRANSFER SOLIDIFICATION 2.1. THE MATHEMATICAL MODEL Figure 1a shows a sketch of the physical doain for the proble under investigation. We focus on the transition fro liquidus to solidus phase for a elted copper alloy, which is an electro-conductive, low Prandtl nuber viscous fluid. Fro the hydrodynaic point of view, the walls of the enclosure correspond to no-slip boundary conditions. a. Physical syste. b. Matheatical odel; boundary conditions. Fig. 1 Coputational doain cylindrical coordinates.

3 3 Heat transfer in electroconductive fluids by a HSMAC schee 69 The top wall and botto wall are isotheral (hot), at different teperature, and the outer wall is isotheral (cold). Two buffer rings separate the heater plates and the chiller. The fluid is pereated by an axial, unifor agnetic field B = { 0, Boz}. The working fluid is assued to be Newtonian, Boussinesq, with constant properties, and its therally driven flow is assued lainar, incopressible [2]. Under these conditions, the therally expansive fluid is subject to theral gradients that generate a theral flow. When an external agnetic field is superposed, the fluid is also siege of an induced electrical field [3]. In electroconductive fluids (e.g., elted etals), a non-negligible current field J =σ[ v B] is also present, giving rise to electro-dynaic, Lorentz body force; v = { v r,v z } is the velocity field, and σis the electric conductivity of the fluid. This fundaental proble of theral flow agnetic field interaction provides insights in the echaniss that ay be used to control the syste. The atheatical odel consists of the following non-diensional equations: Moentu balance (Navier-Stokes) Mass balance Energy balance v τ + ( v ) v = p + Pr 2 v + Ra Pr θe z, (1) v = 0, (2) θ τ + ( v )θ= 2 θ, (3) in the doain 0 r 1, 0 z A (here, A = cell height / cell radius = 2, is the cell aspect ratio). Here v, θ, p, are non-diensional velocity, teperature, and pressure fields. The scales for tie, velocity, and pressure are L 2 ν, U 0 = α L and ρu 2 0, respectively. T is the diensional teperature, the non-diensional teperature is θ= ( T Telt ) ( Th Tc ), T elt is the elting teperature, T h and T c are the hot and cold wall teperatures, respectively, and the agnetic field is scaled by B 0 (reference). The non-diensional groups in eq. (1) are: Rayleigh nuber Ra = gβ( T ) 3 h Tc L / να; Hartann nuber Ha = B 0z L σ /ρν ; Prandtl nuber, Pr =ν/α. Other quantities are: g gravitational acceleration (in z-direction); β theral expansion coefficient [7]; ν kineatic viscosity; α theral diffusivity [7]; ρ ass density. The following initial and boundary conditions close the odel

4 70 Hideo Kawahara, Alexandru Mihail Morega, Mihaela Morega 4 Initial conditions τ=0: Boundary conditions: v = 0 θ=0 for 0 r 1, 0 z 2 (4) : v = 0, θ = Horizontal botto wall ( z = 0) Horizontal top wall ( z = A) Vertical inner wall ( r = 0) : v = 0, θ r = 0. : v = 0, θ = Vertical outer wall ( r =1) : v = 0, θ = 0.5. The heat transfer rate, at the botto wall, is given by the average Nusselt nuber 1 Nu = Nudr z= 0, Nu = θ. (5) z z = THE NUMERICAL MODEL A HSMAC SCHEME To solve the atheatical odel we developed a hoe-ade code based on the HSMAC ethod [8,9] that utilizes the finite differences approxiation: second order central finite differences for the viscous ters, third order up-wind finite differences (Uniforly Third Order Polynoial Interpolation Algorith, UTOPIA) [10] for the convective ters, and first order forward finite differences for the tie derivative. Accuracy tests indicate that nodes, staggered, unifor esh provide grid-independent solutions. We investigated theral flows with Pr = 0.082, Ra = , and Ha = The non-diensional tie step is τ = Details of the nuerical algorith are given next. The discrete for of eq. (1) is ( v r ) i, j τ n ( v r ) i, j = ( p ) i, j r ( p ) i+1, j + ( DSFR) n i, j, (6) ( v z ) i, j τ n ( v z ) i, j = ( p ) i, j z ( p ) i+1, j + ( DSFZ) n i, j, (7) n θ +1 n i, j θ i, j τ = ( DSEN) n i, j. (8) Here n is the tie step and is the iteration index. DSFR, and DSEN stand for the convection and viscous ters. The grid nodes have the indices (i,j) in r and z directions, respectively.

5 5 Heat transfer in electroconductive fluids by a HSMAC schee 71 The ass balance (2) provides an equation for pressure. Introducing the discrete derivative D i, j = ( v r) i, j ( v r ) i 1, j r ( + v z) i, j ( v z ) i, j 1 the grid pressure p i, j ay be calculated by Newton ethod +1 () p = i, j () p i, j The velocity is given by v +1 r, ( i, j) v +1 z, ( i, j) D i, j Di, j p i, j = n v +1 r, ( i, j) + = n v +1 z, ( i, j) + The calculation procedure is then: 1 v r, ( i, j) p ( i, j) v z, ( i, j) p ( i, j) z = () p i, j 1. Use eqs. (5), (6) to copute the velocities +1 p = p n., (9) () i, j n +δ p +1. (10) n δ +1, (11) p r, ( i, j) n δ +1. (12) p z, ( i, j) +1 ( v ), +1 v { r i, j ( ) z i, j } for 2. Repeat the calculation until +1 D ε is satisfied, eq. (8)-(11) [in this study ε =10 3 ]; report v r v z p. ( ), ( ), 3. Calculate θ using eq. (7). 4. Repeat the cycle (1)-(3), until the final tie. 3. RESULTS AND DISCUSSION We analyzed the effect of a agnetic field on the theral flow by nuerical siulation in the absence (Ha = 0), and in the presence of a agnetic field (Ha < 40). The fluid properties used in the nuerical siulations are listed in Table 1. Figure 2 shows the dynaics of the flow by Nu group, Eq. (4), at the botto wall. The initial, transient regie (τ = ) is discarded, and we focus on the oscillatory regies that follow. Depending on Ha value, the frequency content and the aplitude of Nu is different.

6 72 Hideo Kawahara, Alexandru Mihail Morega, Mihaela Morega 6 Table 1 Heat transfer and flow properties of the odel copper alloy [11] Quantity Liquidus Solidus ρ [kg/ 3 ] C p [J/ol K] k [W/ K] η [N s/ 2 ] Higher frequencies that appear ay indicate either nuerical noise or incipient turbulence. For Ha < 20 the doinant frequency of oscillation is low, and Ha independent: e.g., f = 629 for Ha = 0 ~20. Further studies are needed to find the breaking point ore accurately. Fig. 2 Nu nuber at the botto wall, for different Ha (non-diensional tie). Figures 4, 5 evidence the periodic nature of the flow and the heat transfer for Ha = 0, at different tie oents arked in Fig. 3. Fig. 3 The average Nu nuber at the botto wall for Ha = 0.

7 7 Heat transfer in electroconductive fluids by a HSMAC schee 73 A higher, doinant frequency (f = 2857) eerges about Ha = 20, where a change in the global dynaics is observable (e.g., see Fig. 7, where Ha = 40). This period-ultiplication is accopanied by sooth decay and then increase of Nu aplitude with increasing Ha. a b c d e f g h Fig. 4 Flow coplexity and periodicity seen through strealines, Ha = 0. Isothers show regions of stable (lower part), and unstable theral stratification (upper part). a b c d e f g h Fig. 5 Isothers show regions of stable (botto) and unstable theral stratification (top), Ha = 0.

8 74 Hideo Kawahara, Alexandru Mihail Morega, Mihaela Morega 8 For the selected aspect ratio of the cell (A = 2), and the transfer conditions, the theral flow is confined in the upper part of the cell (Fig. 4). On the other hand, the agnetic field in this study is axial therefore it acts upon (daps) the horizontal coponent of the flow. Figures 7, 8 display the flow and heat transfer processes for Ha = 40, at different tie oents arked in Fig. 6. Fig. 6 The average Nu nuber at the botto wall for Ha = 40. High aplitude oscillations are now observed, and the theral stratification in the botto region of the cell is steeper. In fact, the flow is confined to the upper half of the cell. a b c d e f g h Fig. 7 Flow coplexity and periodicity seen through strealines, Ha = 40.

9 9 Heat transfer in electroconductive fluids by a HSMAC schee 75 a b c d e f g h Fig. 8 Isothers show regions of stable (botto) and unstable theral stratification (top), Ha = 40. Depending on the aspect ration of the cell (here, A = 2), its daping effect ay vary. The vertical flow develops on a longer distance as copared to the horizontal flow. Since the agnetic field acts only on the horizontal coponent, its daping effect ay be less effective. Further studies are needed to fully clarify the underlying echaniss of this process of transition through period-ultiplication. 4. CONCLUSIONS In this study we investigated the heat transfer and flow field in the phase change solidification casting processes. The presence of a agnetic field results in Lorenz body forces, which ay be used to odify the flow. The theral flow is oscillatory. For Ha < 20 the doinant oscillation frequency is low, Ha independent: e.g., f = 629 (Ha = 0 ~20). A higher frequency (f = 2857) eerges about Ha = 20, and a change in the global dynaics is observable. This period-ultiplication is accopanied by a sooth decay and then an increase of Nu aplitude with increasing Ha. The aspect ratio of the cell and the heat transfer conditions and properties of the syste are factors that ay be utilized to influence the elt theral flow this study akes the object of future research. Received on July 14, 2008

10 76 Hideo Kawahara, Alexandru Mihail Morega, Mihaela Morega 10 REFERENCES 1. A. M Morega, T. Nishiura, Cobined heat and ass double-diffusive natural convection, ICHMT Int. Syposiu Transient convective heat and ass transfer in single and two-phase flows, TRCON-03, August 2003, key lecture, Cese, Turkey. 2. T. Nishiura, A. M. Morega, Oscillatory natural convection by cobined heat and ass transfer, Trends Heat Mass & Moentu Transf., 6, pp , T. Nishiura, M. Wakaatsu, A. M. Morega, Oscillatory double-diffusive convection in a rectangular enclosure with cobined horizontal teperature and concentration gradients, Int. J. Heat Mass Transfer, 41, pp , A. Moraru, A. Panaitescu, D. Mocanu, A. M. Morega, Ileana Panaitescu, Antoaneta Soci, Coputational odels for phenoena in aluinu electrolysis cells, Rev. Rou. Sci. Techn. Électrotechn. et Énerg., 47, 1, pp. 3-34, T. Tagawa, H. Ozoe, Enhanceent of heat transfer rate by application of a static agnetic field during natural convection of liquid etal in a cube, J. Heat Transfer, 119, pp , A. M. Morega, Magnetic field influence on the convective heat transfer in the solidification process, Rev. Rou. Sci. Techn. Électrotechn. et Énerg., 33, 1, Part I, pp , and Part II, 2, pp , A. Bejan, Convection heat transfer, Wiley, NY, R. Peyret, T. D. Taylor, Coputational ethods for fluid flow, Springer Series in Coputational Physics, Springer Verlag, NY, V. R Voller, C. Prakash, A Fixed Grid Nuerical Modeling Methodology for Convection- Diffusion Mushy Region Phase Change Probles, Int. J. Heat Mass Transfer, 30, pp , T. Tagawa, H. Ozoe, Enhanceent of heat transfer rate by application of a static agnetic field during natural convection of liquid etal in a cube, J. Heat Transfer, 119, pp , J. Fjellstedt, Outokupu Copper, R&D, in [9].