[雙語翻譯]外文翻譯--鋁電解槽熱磁耦合問題數(shù)值模擬（英文）

1、Numerical simulation of thermal problems coupled with magnetohydrodynamic effects in aluminium cell qY. Safa *, M. Flueck, J. RappazInstitute of Analysis and Scientific Computing, E ´ cole Polytechnique Fe ´de

2、´rale de Lausanne, Station 8, 1015 Lausanne, SwitzerlandReceived 27 December 2006; received in revised form 4 February 2008; accepted 8 February 2008 Available online 29 February 2008AbstractA system of partial diff

3、erential equations describing the thermal behavior of aluminium cell coupled with magnetohy- drodynamic effects is numerically solved. The thermal model is considered as a two-phases Stefan problem which consists of a no

4、n-linear convection–diffusion heat equation with Joule effect as a source. The magnetohydrodynamic fields are gov- erned by Navier–Stokes and by static Maxwell equations. A pseudo-evolutionary scheme (Chernoff) is used t

5、o obtain the stationary solution giving the temperature and the frozen layer profile for the simulation of the ledges in the cell. A numer- ical approximation using a finite element method is formulated to obtain the flu

6、id velocity, electrical potential, magnetic induction and temperature. An iterative algorithm and 3-D numerical results are presented. ? 2008 Elsevier Inc. All rights reserved.Keywords: Aluminium electrolysis; Chernoff s

7、cheme; Heat equation; Magnetohydrodynamics; Ledge; Solidification1. IntroductionA phase changing problem motivated by the modelling of thermal problem coupled with magnetohydro- dynamic effects in a reduction cell is stu

8、died. In a smelting cell operating with Hall–He ´roult process, the metal part is produced by the electrolysis of aluminium oxide dissolved in a bath based on molten cryolite [1]. Var- ious phenomena take place in s

9、uch a cell for which a transverse section is schematically pictured in Fig. 1. Running from the anodes through liquid aluminium and collector bars, the steady electric current spreads in the electrolytic bath. The import

10、ant magnetic field generated by the currents carried to the alignment of cells, coupled with the currents running through the cells themselves gives rise to a field of Laplace forces which maintains a motion within these

11、 two conducting liquids. A magnetohydrodynamic interaction takes place in the cell. In the other hand a heating source is produced by the Joule effect due to the electric resistivity of the bath.0307-904X/$ - see front m

12、atter ? 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2008.02.011q Sponsors: Alcan-Pechiney Company and Swiss National Science Foundation; Grant No. 200020-101391. * Corresponding author. Tel.: +41 22 379 23

13、66; fax: +41 22 379 22 05. E-mail addresses: yasser.safa@epfl.ch, yasser.safa@obs.unige.ch (Y. Safa).Available online at www.sciencedirect.comApplied Mathematical Modelling 33 (2009) 1479–1492www.elsevier.com/locate/apmT

14、he unknown physical fields with which we shall deal are listed as follows: Hydrodynamic fields:? u: velocity field in Xi; i ¼ 1; 2; (u ¼ 0 in solid ledges), ? p: pressure.Electromagnetic fields:? b: magnetic in

15、duction field, ? e: electric field, ? j: electric current density.Thermal fields:? H: enthalpy, ? h: temperature.The material properties are defined as? q: mass density, ? rb and r: electrical conductivity in and, respec

16、tively, outside the bath, ? g: viscosity of the fluids, ? l0: magnetic permeability of the void, ? k: thermal conductivity, ? Cp: specific heat, ? ‘: latent heat.2.2. Physical assumptionsThe model leans on the following

17、basic hypotheses:1. The fluids are immiscible, incompressible and Newtonian. 2. In each domain Xi, i = 1, 2, the fluids are governed by the stationary Navier–Stokes equations. 3. The electromagnetic fields satisfy the st

18、ationary Maxwell’s equations, Ohm’s law is moreover supposed to be valid in all the cell K. 4. The electrical current density outside the cell is given (current in the collector bars). 5. The electrical conductivity r is

19、 function of temperature h in the fluids and electrodes parts. 6. The viscosity g, the density q and the specific heat Cp are temperature independent. 7. The volumes of the domains X1 and X2 have given values (mass conse

20、rvation). 8. The only heat source is produced by the Joule effect due to the current crossing the cell. 9. Effects of chemical reactions [7], Marangoni effect [8,9], surface tension as well as the presence of gas flow ar

21、e neglected.2.3. The hydrodynamic problemIn this part we consider the temperature field h and the electromagnetic fields j and b as known. We choose to represent the unknown interface between aluminium and bath by a para

22、metrization of the form Cð? hÞ ¼ ½ðx; y; zÞ : z ¼ ? hðx; yÞ; ðx; yÞ 2 D?, where D is usually a rectangle corresponding to the parametrization of aluminium–cathode in

23、terface. We denote the dependence of X1; X2 and C with respect to ? h by usingXi ¼ Xið? hÞ; i ¼ 1; 2; C ¼ Cð? hÞ:Y. Safa et al. / Applied Mathematical Modelling 33 (2009) 1479–1492 1481